tag:blogger.com,1999:blog-84978674946925260702017-08-03T07:32:26.272-07:00Turning Math Hysteria Into Math EuphoriaObservations, Reflections and Ideas for K-8 Math InstructionJames Galenoreply@blogger.comBlogger34125tag:blogger.com,1999:blog-8497867494692526070.post-42597987981928670022017-04-25T08:25:00.002-07:002017-04-25T08:25:08.569-07:00Fraction representations should be accurate.. which is not always easyI was in a couple classrooms lately where I stumbled upon some fraction representation issues that could be problematic, if we are not careful. The folks at Everyday Math have put a lot of research into the models they use to help students understand fractions, but there are some potential problems that could exacerbate students' confusion around fractions.<br /><br />Before I get into some examples, let me just say that the main message of this post is this: When modeling fractions, <i>always do so with accuracy and precision</i>, and make sure your students do the same. That might mean asking students to erase their work and start again, or it might mean supplying students with multiple blank pages so they can practice a few times before they get it right.<br /><br />OK, here are a couple of examples. The first involves using area models of rectangles to demonstrate multiplication of fractions and mixed numbers. Have a look at the image below:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-aBqRrB7gfWw/WP9dEdVmQEI/AAAAAAAAAak/teoO_-NuW7cPifRwd1aMjb2pq0ahm0bBACLcB/s1600/Screen%2BShot%2B2017-04-25%2Bat%2B10.27.46%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="https://2.bp.blogspot.com/-aBqRrB7gfWw/WP9dEdVmQEI/AAAAAAAAAak/teoO_-NuW7cPifRwd1aMjb2pq0ahm0bBACLcB/s320/Screen%2BShot%2B2017-04-25%2Bat%2B10.27.46%2BAM.png" width="320" /></a></div><br />Notice that the model has been designed to show that the portion of the rectangle representing 1 is larger than the portion of the rectangle representing 3/4. Likewise, the portion of the rectangle representing 2 is larger than the portion representing 1/3. They are not exactly proportionally accurate, but they show the fraction as smaller than the whole. It is a geometric model, and should serve as a visual representation of multiplying fractions and not as merely another algorithm for multiplying.<br /><br />Now observe the image below.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-09rEWzbnXA0/WP9fne-7rSI/AAAAAAAAAa8/6r2umCBN8RETymbW8b5Gh5vB8wrkJEV6gCLcB/s1600/Screen%2BShot%2B2017-04-25%2Bat%2B10.38.43%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-09rEWzbnXA0/WP9fne-7rSI/AAAAAAAAAa8/6r2umCBN8RETymbW8b5Gh5vB8wrkJEV6gCLcB/s200/Screen%2BShot%2B2017-04-25%2Bat%2B10.38.43%2BAM.png" width="191" /></a></div><br />Here is the same model, but drawn as a <i>square</i> and partitioned as though all parts are the same size. With this version, I have reduced the model to an abstract representation of the multiplication task. And worse, it looks like lattice, which is an <i>algorithm</i> for calculating only that is entirely devoid of conceptual representation of quantity. We do not want students confusing rectangular area models with lattice<i>. </i><br /><br />It is important as a teacher when using rectangular area models for representation, to do your best to partition the rectangle so that the smaller quantities are represented with smaller rectangular partitions, and the larger quantities are represented with larger rectangular partitions. It may be difficult for students to follow this protocol when they depict their own rectangular area models, but if their teacher does, it will help them make better use and sense of the models.<br /><br /><div style="text-align: center;">***</div><br />For the next example, let's look at fraction strips. The Common Core emphasizes fractions on a number line a great deal, and Everyday Math gives students a lot of opportunities to work with fractions on a number line. But many will benefit from making their own fraction strips to visualize fractional partitions of a whole before they start placing fractional representations on a number line. Accuracy and precision are very important in both scenarios, otherwise the actual quantities that fractions represent can be lost on some students. Unfortunately, it is super easy to add to students' confusion when we have them work with fraction strips. So for our students to gain the most benefit from this crucial tool for understanding, we need them to make accurate fraction strips. Absolute perfection to the millimeter is not the goal, but accurate visual representation is. Observe the image below:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-QhI3WuO1lJ4/WP9lMv3yUwI/AAAAAAAAAbY/iqLqIm59HpkXZQg_Rvfyl1BTNV0vVGOTQCLcB/s1600/Screen%2BShot%2B2017-04-25%2Bat%2B10.56.01%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="303" src="https://4.bp.blogspot.com/-QhI3WuO1lJ4/WP9lMv3yUwI/AAAAAAAAAbY/iqLqIm59HpkXZQg_Rvfyl1BTNV0vVGOTQCLcB/s320/Screen%2BShot%2B2017-04-25%2Bat%2B10.56.01%2BAM.png" width="320" /></a></div><br />The top and bottom fraction strips represent visually accurate representations of 1/3 and 1/4, respectively, but the middle strip is a little bit off. As a sample of student work, we might let the middle one slide, as it appears to be <i>almost</i> accurate, and after all, it is divided into three parts, which is the important part, right? Not so much.<br /><br />Look at the first partition in the middle strip, labeled 1/3. Notice how close it is in length to the representation of 1/4 just below it. For students who are not fully grasping the concept of fractional partitioning, the middle strip reinforces the misconception that any whole partitioned into three parts is divided into "thirds," and two parts are always halves, and four parts are always fourths, no matter how <i>much</i> of the whole they actually represent. Partitioning a whole into thirds means <i>three EQUAL parts</i>, and their models <i>must</i> represent that. <br /><br /><div style="text-align: center;">***</div><br />The more confident they are with fractions, the more competent they will be, and vice-versa. The more we can prevent those misconceptions from settling into their brains for the long-haul the better, and fractions so often represent, along with long division, one of the first truly intimidating concepts students encounter in their mathematical journeys. But it doesn't have to be that way (!) if we do our best to keep accuracy and precision at the forefront of our mathematical modeling.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-6251739876488562462017-03-09T12:43:00.000-08:002017-03-09T12:45:05.926-08:00More Meaningful Math Visual Displays...Allow me to share with you a few more excellent visual displays I have encountered in my recent travels. <br /><br />Math displays can be tricky. Sometimes there is too much information, and it gets ignored. Other times it's great information, but difficult to read or see.<br /><br />What IS great information to display in math? How do I make sure my students utilize visual displays for math?<br /><br />Think about strategies that your students are frequently practicing, and consider constructing a strategy wall for your classroom. Here are a few nice ones that have caught my eye:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-N5ZlAbFZSRg/WMG5QPum_5I/AAAAAAAAAX4/-zexK02t1o0RpgkC4Fz2XH3-VpGPkQ10ACLcB/s1600/Grade%2B3%2BStrategy%2BWall.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-N5ZlAbFZSRg/WMG5QPum_5I/AAAAAAAAAX4/-zexK02t1o0RpgkC4Fz2XH3-VpGPkQ10ACLcB/s400/Grade%2B3%2BStrategy%2BWall.JPG" width="300" /></a></div><div class="separator" style="clear: both; text-align: center;">I like this 3rd grade strategy wall (above) because it is designed as a work in progress (note the added strategies below) and it is displayed LOW to the ground, and big enough to read, so it can function as a teaching tool. It is also on display near a table where small groups assemble to do work with the teacher. Multiplication fact strategies... so, so important.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-5ETI2vm9Bmc/WMG53lN77iI/AAAAAAAAAX8/hnCN_xavSlkenp-DyUi0hZx86yBn1FVAgCLcB/s1600/Grade%2B3%2BStrategy%2BWall2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://4.bp.blogspot.com/-5ETI2vm9Bmc/WMG53lN77iI/AAAAAAAAAX8/hnCN_xavSlkenp-DyUi0hZx86yBn1FVAgCLcB/s400/Grade%2B3%2BStrategy%2BWall2.JPG" width="400" /></a></div><div style="text-align: center;">Here's another 3rd grade strategy wall that's completely different. It is displayed high, but it is clear enough to see from student seats and is also at the front of the room, making it easy to refer to as an instructional tool. Very clear and easy to read. </div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-O8fHrsbuw38/WMG6ob-xSEI/AAAAAAAAAYI/hE2Zh07R8II-KlIdc7CVyWcDoL4lPiQOACLcB/s1600/Grade%2B4%2BWord%2BWall.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://3.bp.blogspot.com/-O8fHrsbuw38/WMG6ob-xSEI/AAAAAAAAAYI/hE2Zh07R8II-KlIdc7CVyWcDoL4lPiQOACLcB/s400/Grade%2B4%2BWord%2BWall.JPG" width="400" /></a></div><div style="text-align: center;">Look at all the great stuff on this 4th grade word wall! It is big and colorful and takes up a whole bulletin board... Who devotes an ENTIRE bulletin board to math?? A GREAT TEACHER, that's who!! ;^)</div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-w-IrMHOj2II/WMG7M6Sah5I/AAAAAAAAAYQ/4_NAmQVlyPQvevhtKh1ORdq_TvWLiGxmACLcB/s1600/Grade%2B1%2Bdisplay.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-w-IrMHOj2II/WMG7M6Sah5I/AAAAAAAAAYQ/4_NAmQVlyPQvevhtKh1ORdq_TvWLiGxmACLcB/s400/Grade%2B1%2Bdisplay.JPG" width="300" /></a></div><div style="text-align: center;">This one might be my favorite of all. Can I make my explanation stronger? YES! I love the one star, two star, three star system. I love the base ten notation included at the top. This is displayed in a FIRST GRADE classroom. It's never too early to develop great persuasive writing and mathematical thinking habits.</div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Qh5m67jMIRc/WMG78tb5HMI/AAAAAAAAAYc/p4YHR03A3R8OUnJLz1_hxKGHur85JN4CwCLcB/s1600/Grade%2B1%2BDisplay2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://2.bp.blogspot.com/-Qh5m67jMIRc/WMG78tb5HMI/AAAAAAAAAYc/p4YHR03A3R8OUnJLz1_hxKGHur85JN4CwCLcB/s400/Grade%2B1%2BDisplay2.JPG" width="400" /></a></div><div style="text-align: center;">Here's another one from a first grade classroom. Number sentences have symbols, numbers, and an answer! Such a simple, crucial message, and students in this class can't ignore it. </div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/--Y3iYM4jO0k/WMG8UJIvg0I/AAAAAAAAAYk/jKW4Lpo_wzghT2luXJ6jzDzNboolfygnQCLcB/s1600/Grade%2B6%2BDisplay.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://2.bp.blogspot.com/--Y3iYM4jO0k/WMG8UJIvg0I/AAAAAAAAAYk/jKW4Lpo_wzghT2luXJ6jzDzNboolfygnQCLcB/s400/Grade%2B6%2BDisplay.JPG" width="400" /></a></div><div style="text-align: center;">This is from a sixth grade classroom. I love the message here, and the white boards placed exclusively for the purpose of posting learning targets. It makes it a lot easier to get into the habit of posting learning targets when they have their very own white boards.</div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-_DlLb_q9PWU/WMG84Y94XUI/AAAAAAAAAYs/sHur4ZOUe9IxsXzSwVR2EKNj67o93dHkACLcB/s1600/Grade%2B6%2Bdisplay2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://2.bp.blogspot.com/-_DlLb_q9PWU/WMG84Y94XUI/AAAAAAAAAYs/sHur4ZOUe9IxsXzSwVR2EKNj67o93dHkACLcB/s400/Grade%2B6%2Bdisplay2.JPG" width="400" /></a></div><div style="text-align: center;">Here's another display from a 6th grade classroom. Each component is laminated and kept on file so a unit's essential question can be on display for any period of time, and quickly replaced with ease. Essential questions are great things to display, since they remind students of the overall mission. I asked a student in this class to tell me what his class had been working on, and he referred to this display. "Ratios," he said, "and how they compare and stuff."</div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-18739184310290817492017-03-09T11:59:00.000-08:002017-03-09T11:59:57.315-08:00Grades 3-5: Talkin' Fraction Immersion BluesIt's springtime in Everyday Math Land, and fractions are all over the place. In our lessons, we're talking about understanding fractions, comparing fractions, fractions on a number line, area models for multiplying fractions, simplifying fractions, even subtracting and dividing fractions. And when the topic is fractions, there is SO much to talk about.<br /><br />Take this problem, for example, from a fifth grade, Unit 5 math box:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-xz_ixzPWucE/WMFr0K4VpjI/AAAAAAAAAXY/voj0ZvMRdJcuXXstfuqCWcrdTT0L9QB7gCLcB/s1600/Screen%2BShot%2B2017-03-09%2Bat%2B9.46.25%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="332" src="https://1.bp.blogspot.com/-xz_ixzPWucE/WMFr0K4VpjI/AAAAAAAAAXY/voj0ZvMRdJcuXXstfuqCWcrdTT0L9QB7gCLcB/s640/Screen%2BShot%2B2017-03-09%2Bat%2B9.46.25%2BAM.png" width="640" /></a></div>I totally love this problem, because there is SO MUCH TO TALK ABOUT here! There is no doubt in my mind fifth grade students struggled with this one. For one thing, it falls, right smack dab in the middle of a lesson on <i>multiplying </i>fractions, but... is this a multiplication problem? Let's have a look at each part.<br /><br /><i style="font-weight: bold;">Gwen is 7/8 of the way through a race.</i> It is important students recognize that means Gwen is <i>almost finished</i> with her race. It's an essential piece of information, and it needs to be understood.<br /><br />Next, we learn that <i style="font-weight: bold;">he family was cheering for her when she was 2/3 of the way through the race.</i> It is important for students to know what that means as well. At 2/3 of the race, is Gwen <i>almost finished</i>? Is it <i>as</i> close to the finish as 7/8? <br /><br />Each of the above two pieces of information could be discussed in a turn-and-talk. It's likely many students will assume this is a multiplication problem (because of the theme of the lesson), so they need to be given the opportunity to justify that, to <i>prove</i> that the operation is multiplication. They won't be able to, because <b><i>it is a subtraction problem</i></b>.<br /><br />Through meaningful discourse, students should be able to explore the information given to realize what is being asked. First we are given one value, 7/8 of the whole (of the race), and then we are given another value, 2/3 of the whole (of the race) and we are asked to identify <i style="font-weight: bold;">how much of the whole (of the race) is in the difference (the distance between 2/3 of the race and 7/8 of the race)</i>.<br /><br />So the number model is: <i style="font-weight: bold;">7/8 - 2/3 = X</i>.<br /><br />The key here is the meaningful conversations, the <i style="font-weight: bold;">discourse</i> we want our young friends to experience so we can allow them to form their understanding of the problem. Encourage debate with problems like these; see if students can justify or prove why they chose the operation they chose.<br /><br />Fractions are really hard for children to grasp, especially for those who have not become fluent and automatic with multiplication, and/or have not <i>mastered</i> the concept of division. It is essential they have these conversations to build their conceptual understanding of what fractions are, and what happens when we add them, subtract them, multiply them and divide them.<br /><br />We need to do what we can to avoid those fraction immersion blues that come in March and April. It affects students when they are struggling with understanding fractions, and then things just progress too fast. It seems like one minute they were identifying fractions on a number line, and the next they were dividing mixed numbers. The fraction immersion blues affects teachers when they find themselves having to <i>explain</i> fractions to frustrated learners a lot, and having to watch their students suffer through the pain of doing math they don't understand for weeks at a time. <br /><br />Allowing maximum time for students to collaborate and communicate their thinking around fractions is essential this time of year. <br /><br />Make room for more collaborative work and meaningful math talk by jigsawing journal activities and spot-checking their journal work and math boxes as they work on them. And let me know if there are ways I can help.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-91081392458902886122017-02-01T12:35:00.000-08:002017-02-01T12:35:33.207-08:00Subtraction and Multiplication: Fact Fluency and Automaticity in the Early GradesIt has recently been brought to my attention in more ways than one how important fact fluency and automaticity are in the early to mid elementary grades. Subtraction and multiplication, in particular, are operations we need to pay extra attention to. <br /><br />One way to understand this is to start at the end and work backward, so let's do that. Let's start at advanced high school mathematics.<br /><br />Since high school advanced math courses are electives, students have either mastered the prerequisite skills to take the course, or they have <i>almost</i> mastered the prerequisites and convinced the powers that be they can handle the rigor and rise to the occasion. Some struggle in advanced math , but some for whatever reason, don't struggle as much. It could be they have developed a love for complex problem solving, or it could be they have had stellar mathematics instruction through the years, or it could be something else. Many students, however, don't get the chance to fail or succeed in advanced math courses, because they don't take them.<br /><br />We know that a significant number of students in high school don't take calculus or pre-calculus before graduation. Some high schools don't even offer calculus. <a href="http://kut.org/post/high-school-math-courses-predict-whether-students-go-college-study-finds">Research</a> says students who take advanced math in high school are more likely to enter into higher education, so why is it that so many do not qualify for or opt out of advanced math in high school?<br /><br />Moving one step further back, we look at regular high school math, that is, algebra and geometry. Students who don't take advanced math, or fail to earn prerequisite credit for advanced math, either struggled with algebra and/or geometry, or they did not have a positive enough experience to want to take the step to advanced math. Why is that?<br /><br />High school teachers report that many 9th graders show up for their algebra class without a solid understanding of fractions and division. When it comes time to operate with fractions, they get confused and start sweating. They wonder: Do I invert and multiply? HOW do I invert and multiply? WHY and WHEN do I invert and multiply? Which one is the numerator? When is it that denominators have to be common, and how do I get them that way? What is a <i>rational number?</i> What's the difference between a <i>ratio</i> and a fraction? I'm not even sure I know how to do long division, so how on earth am I supposed to be able to <i>divide fractions</i>?!<br /><br />These questions indicate a lack of confidence in understanding and operating with fractions. So let's take the next step back and dive into middle school math, which tackles operating with rational numbers and integers (part of what is often called "pre-algebra") as well as basic linear algebra concepts. The basis of linear algebra on a coordinate plane, dependent and independent variables and relationships, "rise over run"and slope, and y-intercept all require students to not only understand fractions, but to <i>use</i> them and multiply with them in equations. A linear equation in slope-intercept form is a two-step equation that involves multiplication and division to solve. Students will struggle with the basics of linear algebra in 7th and 8th grade if they do not have a functional understanding of fractions and mastery of basic and long division. <br /><br />The next step back brings us to fifth and sixth grade and application of multiplication and division, as well as fractions, into solving problems relating to money and probability. Without an understanding of the relationship fractions have to decimals, students will likely struggle with everything from finding unit rates to determine likelihood of an outcome in simple probability. Adding, subtracting, multiplying and dividing fractions and mixed numbers becomes a tedious task of memorizing steps rather than a process of estimating, applying understanding, and choosing an efficient strategy.<br /><br />Back still more to 4th grade. Here the relationship between division and fractions is front and center, unless it isn't. For students who fail to grasp that relationship, conceptual understanding of fractions, (beyond unit fractions and equal parts, is likely to be weak. Students are still developing their fluency in division of whole numbers, exploring division algorithms, and understanding remainders. Students who do not make the connection between remainders and fractions will struggle to understand and operate with mixed numbers.<br /><br />3rd grade. Multiplication strategies and introduction to fractions are a big focus at this point. This is where we expect students to become both <i>fluent</i> in multiplication (able to understand <i>how</i> to multiply, what multiplying is, and to choose from multiple strategies to solve multiplication problems) and to achieve <i>automaticity</i> in multiplication facts (the ability to <i>know</i> in an instant that a multiplication fact is accurate, having already understood multiplication as repeated addition, and used a variety of models like arrays and areas of rectangles to visualize and deconstruct multiplication problems). If students are not both <i>fluent</i> and <i>automatic</i> with their multiplication facts, division will likely not make a lot of sense, and bigger multiplication problems will be a more arduous and time consuming cognitive process.<br /><br />Alas, 2nd grade, where students are mastering subtraction strategies and gaining understanding of place value. Subtracting is far less intuitive than addition, and presents a linguistic challenge. Students used to finding out "how many are in all" with addition are now expected to understand, <i>How many are left? How many more? How many less? What is the difference? Take away, subtract, minus, </i>and <i>how many in all NOW?</i> Subtraction presents the first real algebraic thinking students encounter, as the <i>difference</i> represents an unknown addend in an addition problem. Simply memorizing subtraction facts is not enough for students to truly master and become both fluent <i>and</i> automatic with subtraction. It is essential students understand exactly what it is they are doing when they subtract. Without a complete understanding and automaticity of this operation, students will likely struggle with every algorithm for division.<br /><br />We could go further back... The building blocks of subtraction and place value are assembled starting in kindergarten and first grade, but I singled out subtraction in 2nd grade and multiplication in 3rd grade because these are the first real significant hurdles that many students have trouble clearing. Students who do not become fluent <i>and</i> automatic in subtraction <i>and</i> multiplication <i>will </i>struggle when they attempt to take on division and fractions. The algorithms they use will not be as efficient as intended, and division and fractions will become a roadblock to understanding, to engagement, to finding joy in mathematics and mathematical problem solving. <br /><br />To conclude, we need to plan our units in ways that emphasize opportunities to explore subtraction and multiplication concepts in the early grades, and division and fractions in the upper grades, and follow that up with consistent practice for fluency and automaticity. It's not easy, but I want to be a resource for making that happen, and utilizing all that EM has to offer, in any way I can.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-32855463755196406952017-01-30T07:32:00.000-08:002017-01-30T07:35:12.528-08:00Keeping Up with PacingWe've surpassed the halfway point in the school year, and I want to give everyone a chance to take stock of where you are in the pacing guide (The pacing guides are all accessible in your grade level math Google folder... If you would like me to reshare the folder at any time, let me know). A couple weeks ago, I traveled to just about every classroom over the course of two days and took inventory of where everyone was. While most classrooms were <i>nearly</i> on pace (this is good!), few were actually <i>on</i> pace with the pacing guide. It is ok if your students are not on the very lesson the pacing guide suggests for any given date, and there is <i>some</i> flexibility built into the calendar to account for falling behind. After all, there are snow days, assemblies, power outages and other interesting disruptions to your pacing this time of year. But as we progress toward spring, we'll have MEA testing and other end-of-the-year activities to contend with, and this all happens to coincide with some of the most dense math lessons in the curriculum.<br /><br /><br /><b>Here's why pacing is important.</b><br /><ul><li>While it is definitely <i>not</i> effective to cruise through lessons at breakneck speed, skipping over important practice activities, game opportunities, or other important parts of the lessons, we <i>do</i> want students at every grade level to finish the year with a consistent background in mathematics. It might be that one grade level in one building finishes a unit ahead of the same grade level in another building, and that creates an inequitable scenario. Students from one class are moving ahead to the next grade level less prepared than students from another class. We can't expect everyone to be at the same place cognitively; we will always have a diversity of learners to work with. But we can <i>do the best we can</i> to assure that everyone has had an equal opportunity to learn grade level mathematics.</li></ul><br /><b>Here are some tips to help you keep from falling behind.</b><br /><ul><li>Use flex/game days to differentiate and re-teach. This is the single most effective, in my humble opinion, way to keep lessons on schedule. Rather than taking more than one day to teach a lesson that is meant to be taught in one day, reserve an activity for flex day, or select students who are struggling to participate in a re-teach or extra practice or a readiness activity. Flexible grouping with other grade level team members to allow for differentiating on flex days allows opportunities to fill in holes in instruction or understanding that opened up during the week, and it also offers opportunities to reach some of your learners who might benefit from more challenging extensions. </li></ul><div><ul><li>Jig-saw some of their journal activities. Most lessons have a couple journal activities, and if you try to get every student to do every item in every activity, it is less likely they will get to the games (which are really, really important) or even math boxes. Instead, try assigning parts of the activity to pairs or small groups, and see if they can come to a consensus in their group and report out. This could save a lot of time, and can be done on a regular basis. The benefit to doing this kind of thing <i>often </i>is your students will get used to the routine and learn to work more efficiently.</li></ul></div><div><ul><li>Be as brief as possible in the warm-up and the math message. Remember, the mental math in every warm-up is a <i>quick </i>activity designed to transition into mathematical thinking time, and also as a gauge for you to see how your students are grasping various concepts. It is not a time to make sure every student has a chance to share strategies, or a time to make sure every student gets it right and understands. It is a time to see who gets it and who doesn't, take note of it, and move on to the next part of the lesson... which is the Math Message. Similarly, the Math Message is not intended to take a long time. It is an activity that can be done quickly to preview the kind of work and thinking students will be doing in the <i>Focus</i> portion of the lesson. Progress through the Math Message swiftly so students can spend more time working on the rest of the <i>Focus</i> and <i>Practice</i> parts of the lesson.</li></ul></div><div><br /><b>Here are some things you can do to catch up if you are falling behind the pacing guide.</b></div><div><ul><li>Contact me. I can help! We can work together to come up with a plan. Is there a schedule problem? Do we need to adjust your pacing guide? The goal is to move forward with instruction that sacrifices little if any of the essential grade level content between now and the end of the school year.</li></ul><div><br /></div><div>Please don't hesitate to contact me regarding any pacing concerns you may have. </div><div><br /></div></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-44376345068100430162016-12-09T08:42:00.001-08:002016-12-09T08:42:05.114-08:00Let Them Make MistakesOne of the pitfalls we find ourselves falling into as math teachers (I use the <i>first</i> <i>person plural</i> here because I have done this and I see others doing it too) is we facilitate our instruction with a goal that everyone gets it right.<br /><br />It is easy to forget that the Everyday Math lessons we teach are structured in a way that allows and even <i>encourages</i> children to make mistakes. In any lesson on any given day, we should expect students to make mistakes while they work, and especially when they think aloud.<br /><br />Let's look at this in terms of the parts of a lesson.<br /><br />Warm-up.<br /><br />For the Warm-Up portion of Everyday Math lessons, Mental Math and Fluency is a check-in opportunity. This is not a time to make sure all students get it; it is a time to check for mastery. That's why there are three levels of difficulty for the mental math segment of every lesson. If most students appear to struggle with the first level, don't go onto the next. Mental math is a chance for you to have a snapshot of your students' levels of math fluency before you get into the bulk of the lesson. What you see in mental math might impact how you facilitate the rest of your lesson. Mental Math is not the time to clear up misconceptions, or to keep providing more examples until everyone gets it right. "What do you see?" when flashing a Quick Look card, or "How do you see that?" are great questions to ask that will elicit formative information. But asking <i>every</i> student to share a strategy, or making sure <i>every</i> student gets it right will take too long and disengage students.<br /><br />Mental Math.<br /><br />Mental math is your students' opportunity to get their feet wet, so let them. They might make mistakes. Give them opportunities to make those mistakes, and then let them talk to each other about their mistakes. The Teacher's Lesson Guide gives you a little suggested script to follow up after the Math Message. Notice it never says, <i>Keep quizzing children until everyone gets it right.</i> Instead, it usually offers a differentiation strategy with suggestions for scaffolding. Your scaffolds should not be "hints," but rather sentence structures to help them grasp a strategy, or visual aides to help them understand how to use a tool. We still want to give them opportunities to figure out the problem for themselves (and to occasionally make mistakes, even with scaffolds).<br /><br />Math Journal work.<br /><br />They can make mistakes here too! But now their mistakes are visible on paper. The journal work is an excellent opportunity for students to work together, check each other's work, and compare strategies. Two partners have different answers? Wonderful! Have them see if they can come to a consensus. The "growth mindset" in math <i>requires</i> that students make mistakes, and do it fairly often. Every time a student truly discovers the root of his or her mistake, that student has gained significant mathematical understanding. <br /><br />Practice.<br /><br />Mistakes are <i>still</i> encouraged here, while playing a game or working on math boxes. Here, mistakes are likely smaller and quickly resolved, but they still can and do happen. This is where we want students to be able to catch and correct their own mistakes, either on their own or with the help of a partner. If there are a lot of mistakes at this point of the lesson, that informs you that some extra practice or re-teaching may be necessary. <br /><br />You may find that persistently <i>allowing</i> students to make mistakes actually <i>saves</i> you instruction time, because you are not so busy going from child to child making sure every student has gotten every part of the problem right, or demonstrated every strategy correctly. Sometimes, that can postpone the entire class from proceeding to the next part of the lesson. Instead, use those mistakes as learning opportunities, turn-and-talk topics, group consensus opportunities, and formative evaluation of their understanding for future instructional decisions. <br /><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-76175026720243901262016-12-01T08:11:00.001-08:002016-12-01T11:21:26.774-08:00Whole Class Instruction: Know When To Say WhenMuch of our instructional time in math lessons is taken up with small group and partner work, which allows students to explore, experiment, debate and take risks with their mathematical thinking. Whole group instruction is minimal in our Everyday Math lessons, but when it happens, there are some things to keep in mind.<br /><br />Recently in a fourth grade classroom and I saw a masterful decision take place on the part of the teacher. Students had been working on a task, and it came time to share out. A student volunteered to share his strategy for dividing three 8" pizzas among two friends evenly.<br /><br />"Well.. basically what I did was.. I started by taking the three pizzas, and then I.. well, they're each 8 inches in diameter but I don't think that really matters for this problem, but I thought I would mention that, and then I, um.. I cut the pizzas in half because then I would have a number of parts of pizza that I could distribute evenly for everybody, and then I had four pizzas instead of two, or rather four half pizzas... Wait... I think I made a mistake..."<br /><br />The teacher then intervened. Instead of asking the student to re-think his strategy, or to start from the beginning, or asking if there were any other students who could help this student with his thinking, or asking if there were any other students who had a <i>different</i> strategy, the teacher instead put a halt to the share-out and asked the rest of the class to turn and talk with a partner to share strategies. After a couple minutes, students had another opportunity to share, and lo and behold:<br /><br />"We both cut the three pizzas in half so there were six halves, which is also divisible by two, so there would be three halves for each person."<br /><br />The decision to intervene in the initial student's share-out was masterful because of the <i>timing</i> of the intervention. She realized the student had not fully thought out his conclusion, or was not yet confident enough in his strategy to present it to the class. Sure enough, by the time the student had said, <i>"Well.. basically what I did was.."</i> he had lost most of his peers' attention. Eyes started turning away, pencils were being picked up, body language across the room said, almost universally, "We're not paying attention anymore." <br /><br />The student's hesitation alone had lost the confidence in his peers, and many had decided it was not worth their time to listen. The class had been well trained not to be distracting when distracted themselves, so there was little noise or even whispering going on, but it was clear very few, if anyone, was listening intently to the student share out his temporarily confused thought process. <br /><br />The teacher had masterfully gauged student engagement and transitioned to an interactive partner review of the problem. We all know what would likely have come next if she had allowed the student to continue, or if she had called on another student to follow up. A visible or audible distraction from somewhere in the room, perhaps a spoken redirection or two from the teacher, and a prolonged share-out with even fewer attentive students. Instead, students turned to a partner and rehearsed their thinking out loud together, interactively. When it came time for a volunteer to share out, the result was clear, precise and succinct.<br /><br />We always want to elicit mathematical thinking from our students, and we have a tendency to want to hear multiple students share a variety strategies all in a quick whole group share-out, so that little lightbulbs might turn on around the room as new understandings are shared and discovered by all. Unfortunately, it is a challenge for elementary and middle school aged children, and even adults sometimes, to articulate mathematical thinking on the spot. And listening to someone think out loud is not often that engaging, especially for those of us who are not primarily auditory learners. If what we hear doesn't make instant sense, we shut down. <br /><br />So a couple of important lessons can be learned from this that pertain to whole-group instruction. <br /><br />Firstly, student attention is short-lived in a whole group setting unless the topic is purely engaging. Even though a teacher might not be <i>lecturing</i>, sitting and listening while student after student attempts to articulate a strategy or defend an answer to a problem will not yield a great deal of new understanding. Moreover, it will often lead to distracted students and sometimes misbehavior. Masterful teaching moves I have seen include expertly timed turn-and-talks, whole class physical transition to a new location (such as moving to the floor from their desks or vice versa), and transitioning to the next stage of a lesson. If students are not comprehending, they are not going to invest their energy in listening. If they are bored, they are going to struggle to keep focused. Something has to change. No matter how animated and excited <i>we</i> get in the front of the room, it still often feels like a losing battle:<br /><br /><i>"Oh! Great thinking! Did everybody hear what Timmy just said? Timmy can you say that again so everybody can hear? Ok, everybody, listen to Timmy's awesome strategy.. Suzie, you too.. and Billy, that means you.. Ok Timmy, go ahead.. but wait until everybody is quiet..." </i><br /><br />Ugh. Not fun for you, not fun for them. Sometimes a student just hits the nail on the head and you want everyone to absorb the learning opportunity, but if it took too long to elicit that sweet morsel of demonstrated knowledge, beware. We might expend less energy typing it up and inserting it into 23 fortune cookies than we would trying to get every student to devote his or her complete attention to it.<br /><br />Another lesson to be learned about whole group instruction involves giving directions. In problem solving tasks, directions are often complex and require considerable effort on the part of students (and sometimes teachers) to understand. It becomes very important in those cases for the <i>teacher</i> to be the one to read or deliver instructions slowly and carefully, with all the right intended emphases. We often want to give students opportunities to read aloud and take ownership, but instructions to problem solving tasks are not appropriate for those opportunities. Even some of the best student readers still need work on their delivery and might not execute the right emphasis. Some read very fast, others read very quietly, others read with a monotone, still others insert careful but perhaps awkward space between words. All of these can make it difficult for their peers to understand, and thus result in students abandoning the effort of paying attention. <i>"Two.. brothers.. go.. to.. lunch.. and.. share.. three.. pizzas.. equally.. how.. much.. pizza.. does.. each.. brother.. get.." </i>By the fifth word, chances are wandering eyes and body language will begin to present visible evidence that students are not eagerly anticipating the task. A confident adult reader is important when giving instructions and sharing important details for mathematics problem solving tasks.<br /><br />Whole class instruction <i>can</i> be engaging, and it <i>can</i> be effective. It is important to be keenly aware of student engagement and to be practice teaching moves that maintain engagement or re-engage students. Then your energy can be more devoted to <i>listening</i> to what students are saying, gauging student understanding, and confidently moving forward with facilitation of learning. Knowing <i>when to say when </i>will keep whole class instruction limited, and leave more time for partner and group work, as well as those all-important games and hands-on activities. James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-73664193843342820182016-10-13T12:42:00.001-07:002016-10-13T12:42:04.739-07:00Assessing for UnderstandingSo by now the school year is well underway and you are beginning to learn a lot about the mathematicians in your classes. As we gather evidence that will eventually serve to provide a grade on the trimester report card, I've had a few questions regarding how to collect that evidence, and how we are scoring assessments this year.<br /><br /><span style="color: #cc0000;">What do I do if the student got it wrong, but I know they can get it right?</span><br />You'll see in the email response below that this will be a common scenario. It's important to remember that we are always formatively assessing our students. "Formative assessment" is an ongoing process that occurs constantly as you teach your lessons, observe students at work and the discourse they share, and review completed assignments. <i>Always be looking for evidence of understanding</i>. If a student makes an error in their journal, or even on an assessment, you can always give them more opportunities to show they really get it. If it is a careless error, the student may simply need to see the error he or she made, and be given the opportunity to fix it. If it is an error that is a result of a lack of understanding, the student may need re-teaching. It might come back in a future lesson (see the spiral!), or you might be able to provide opportunities to revisit the concept and re-assess for understanding. <br /><span style="color: #cc0000;"><br /></span><span style="color: #cc0000;">The items on the test are either correct or incorrect and cannot be given partial credit.</span> This is true. Even if there are six answers to #7 (as in, there is a 7a,7b,7c,7d,7e, and a 7f), they ALL need to be correct in order for the item to be marked correct. Why is this? Here's how I answered this in a recent email:<br /><br /><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">If you think a student juuuust missed getting them all [the parts of the question] correct with a careless error, give him or her the chance to look over their work and make the fix.. If he or she can find and fix the error, then adjust the score to a 3. The other option is to find more evidence elsewhere in the unit or upcoming unit to determine if the student is achieving proficiency or developing. You can use items on the cumulative assessments for this, the ACIs, alternative test items, or other examples. (The spiral in the TLG might help you locate where the specific standard is assessed again in the book.)</span><br /><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;">The more I have learned about scoring assessments, the more I lean toward this kind of scoring. If the item has 4 parts, EM has essentially designed it that way to assess proficiency. Students are not proficient until they can get all those right. And if they don't but you feel they really should have gotten all the answers right, you can give students chances to provide you with that evidence.</div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><span style="color: black; font-family: Times; font-size: small;">Later, we got into discussing specifics. The items in question were a 5th grade order of operations and grouping problem and a volume-of-a-rectangular-prism problem. Should students be required to evaluate all for expressions correctly, and then answer the volume problem with the correct units of measurement (cubic centimeters)? </span></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><br /></div><div style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><div style="font-size: 12.8px;">For #3, the standard is 5.OA.1, "evaluate expressions that contain grouping symbols." There are four expressions to evaluate, but <i>a </i>and<i>b</i> go together and <i>c</i> and <i>d </i>go together as they use the same numbers with parentheses in different places. So if a student got <i>a</i> wrong but <i>b</i> right, that likely demonstrates a misunderstanding of what parentheses mean, as the answer to <i>b</i> would be the same even without the grouping symbols. The four separate parts to the item help to ensure that the student understands the standard. 5.OA.1 is listed on page 8 (the spiral page) of the Teacher's Lesson Guide as a mastery expectation, so with this assessment item we want to make sure the student gets it.</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">With units of measurement pertaining to volume of 3D figures, I have mixed feelings! Yes, it is important that they learn <i>how</i> to compute volume, but we also want them to <i>understand</i> how and why lxwxh=V. Otherwise, they are just memorizing a formula which is susceptible to becoming confused with area and/or perimeter as they progress down the geometry/ measurement content strands... If the l,w, and h measurements are in cm, then the answer must be in cubic cm. NOW... That's tough for 5th graders. So the approach I like is to take those answers that are correct with the incorrect or missing unit of measurement, and bring them back to the student. "Did you forget something?" ...or, "Is your answer just '40,' or is it '40 <i>somethings...?</i>" Give the student a chance to demonstrate the unit of measurement. If the student shrugs his or her shoulders and says "I don't get it, my answer is 40.." then they do not grasp how or why they are using that formula. The CCSS language is "recognize volume as an attribute of solid figures and <i>understand concepts of volume measurement</i>." But again, this is hard for 5th grade, so that's why I suggest giving students every opportunity to grasp the concept... You can always re-teach, and give students another chance to demonstrate proficiency. </div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">Also, there is certainly some subjectivity to scoring these items. You know your students more and more as the weeks go on, and you will know who needs re-teaching and who doesn't. The one thing I recommend as you make these judgments, though, is to keep looking for evidence of <i>understanding </i>when you score assessments. And the more opportunities they have to explore and discover these concepts during the units (to problem solve collaboratively <i>and</i> independently by trial and error and with manipulatives), the more they will achieve understanding.</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="color: black; font-family: Times; font-size: small;">Lastly, we discussed what can be done in the event we want to find additional evidence to show a student is achieving the standard when the initial assessment item did not: </span></div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;">Sometimes there might be a domain at the end of a trimester that has minimal assessment items represented... You can certainly base a student's domain grade on more than the two or three assessment items they attempted pertaining to that domain. Use the items in the cumulative assessment if you like, or the ACIs, or alternative assessments to find more evidence. I'm always happy to help with this too (and even adjust cover pages if necessary).</div><div style="font-size: 12.8px;"><br /></div><div style="font-size: 12.8px;"><span style="color: black; font-family: Times; font-size: small;">More thoughts on all this? Let me know!</span></div></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-89584048038643311992016-05-04T08:14:00.001-07:002016-05-04T08:14:44.548-07:00Don't Panic About PacingAs we quickly shift into the home stretch of the school year, there is considerable anxiety regarding pacing. Will my class finish all the lessons and units before the end of the year?<br /><br />Despite the new Everyday Math units being somewhat more manageable in terms of how many lessons are taught and how much goes into each lesson, finishing out the year having covered every lesson is clearly a challenge for many. This time of year, the lessons are getting more complex and dense, and with all the field trips, fire drills, assemblies and celebrations scheduled, it is easy to fall behind the pacing guides.<br /><br />Here are a few important things to keep in mind that might help with pacing the rest of this year and next year too. Before I get to the bullets, though, remember that for most of us, this year is the very first year of a program that is 80% new, so we are still in our learning phase. <br /><br /><br /><ul><li>The unit assessments are long, and are taking multiple days to complete in some cases. If you feel your students need a little more time than one lesson period to finish the unit assessment, that is understandable. But think of the unit assessment as a <i>formative</i> tool in addition to a summative one. <span style="color: #cc0000;">If the assessment is taking a long time, it may be best to prioritize items and cut the assessment short</span>. Learning is most important, and if students are not able to complete every part of an assessment, that is ok. Remember there are extra practice and readiness activities in the upcoming lessons you can rely on to provide a little extra support for students who might not have been able to demonstrate mastery in certain domains of prior units. Moving on to the next unit may be more important for your students than every student finishing the assessment. For 2016/2017, we are looking at abbreviating the unit assessments some in order to focus more on assessing <i>content </i>standards, so the assessment process will hopefully become more efficient in time. </li></ul><ul><li>Math boxes are an important part of every lesson, but they are intended to take up <span style="color: #cc0000;">no more than 15 minutes </span>of time. If students are taking longer than 15 minutes to finish math boxes, they are either struggling with the material, which is important formative information for you to know, or they are distracted. If students are struggling to finish the math boxes, it is ok to modify the assignment, to take one or two of them off their plate. If students are distracted, that is a behavior/disciplinary issue, and should be addressed as such. </li></ul><ul><li>Don't be afraid to <span style="color: #cc0000;">use a timer</span> when you teach Everyday Math lessons. When I teach, the danger lies in the warm-up. There always seem to be so many teachable moments! If you look in the Teacher's Lesson Guide, most <span style="color: #cc0000;">Warm Ups (Mental Math and Fluency) are designed to take only 5 minutes.</span> <i>These are also formative assessments. </i>The first part of the lesson is <i>not</i> intended for discussion or correcting students' thinking. That's what the Math Message is for. The Warm Up is a quick gauge of where your students are at before you start the lesson, and an opportunity for students to get their brains into a math mindset. No real <i>teaching</i> is happening. Have them write their answers on their slates and move on. It is ok, to tell them the answer, and answer a question or two, but beyond that, the Warm Up is not a time for justifying answers, clearing up major misconceptions, or having teachable moments. It is hard, I know! But for the lesson to be most effective, save the great mathematical discussions and teachable moments for the Math Message and the rest of the Focus portion of the lesson. </li></ul><ul><li>Remember that the math you teach right now is foundational prerequisite knowledge for the math they will be learning later. When tough decisions need to be made regarding pacing, <span style="color: #cc0000;">work with your team</span> to see what strategies your colleagues might be employing, and contact your friendly Math Strategist to help you map out your next lessons and units. You are working hard to give your students the most meaningful math experience possible to prepare them for future learning and problem solving, so you don't want to misdirect those efforts. Together with your colleagues, you can carve a sensible path forward.</li></ul><ul><li>Next year <span style="color: #cc0000;">your hard work will pay off</span>; not only will your students go to their next year's teachers with a richer mathematical background, but your students will come to you as more capable learners as well. And you will have a year (or two, in the case of grades K-2) of teaching this new program under your belt. So have faith that good things are happening, and they are only getting better next year.</li></ul><div>This has been (and still is!) a great first year with all grades (K-6) teaching Everyday Math 4. I am very much looking forward to next year when we can focus even more on deeper understanding of concepts and mathematical thinking. If your pacing hasn't been perfect this year, it will get better next year. We'll all be working hard to make that happen. While it is not necessarily reasonable to expect everyone to be teaching the same lesson on the very same day, it is helpful for teams to be able to collaborate when grade level classrooms are within a few days of each other in the unit. This way, Open Response lessons can be taught simultaneously. Wait! This should be another bullet...</div><div><ul><li><span style="color: #cc0000;">Teach Open Response lessons on the same day as your grade level colleagues! </span> You can teach these lessons out of order if you are not exactly on the same page as your teammates in order to coordinate the days when you will teach Open Response lessons. The benefit to this is grand. You can teach day 1 of the lesson, and then <i>bring student work with you to review with your colleagues between day 1 and day 2</i>. Then you collaborate to determine what day 2 of your O.R. lesson will look like. This helps you to see a wider range of student work, which will not only help you see trends of confusion and misconceptions, but it will also help you determine what to look for when you are scoring Open Response assessments after odd numbered units. </li></ul><div>I hope this has been a helpful bit of information regarding pacing. Please contact me if you would like to discuss any issues with pacing as they arise. </div></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-23692978675892043272016-04-04T07:13:00.001-07:002016-04-04T07:13:17.870-07:00Spring Classroom Reflections: How Great Visuals HelpFor the last couple of months, I have been taking note of the excellent visuals I see on classroom walls that help students with their mathematical routines.<br /><br />I've always had mixed feelings about math visuals.. In middle school, when the phenomenon of math anxiety really begins to set in for some, the last thing I ever wanted to do was scare my students every time they came in my classroom with equations and formulae all over the walls. I'm a math teacher, someone who genuinely sees beauty in mathematics, and a room full of numbers sounds kind of nauseating even to me. But over the years I have grown to appreciate carefully planned visuals that truly help students with learning. I've learned that what those visuals are and how they are displayed is just as important as <i>how they are utilized</i>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-zJJ_EtZxOyA/VwJwxUARJOI/AAAAAAAAANk/Fw6psFmhQ58N6ev0CVr5kXFq3hqP7i1JQ/s1600/IMG_0201%2B%25281%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://2.bp.blogspot.com/-zJJ_EtZxOyA/VwJwxUARJOI/AAAAAAAAANk/Fw6psFmhQ58N6ev0CVr5kXFq3hqP7i1JQ/s320/IMG_0201%2B%25281%2529.JPG" width="320" /></a></div><br /><br />For example, some of the posters that I see in first and second grade classrooms display strategies for subtraction. They are clearly visible and attractive, but most importantly, I <i>see teachers referring to them</i>. A poster on the wall is just a poster on the wall until it is demonstrated and modelled as a <i>tool</i> that can be utilized by students at any time.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-GYjmeTLkmSU/VwJw-AvHONI/AAAAAAAAANo/2zvK-rV96rIPhPa5Bv3MM8G9rbQYrH_hQ/s1600/IMG_0185%2B%25281%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://1.bp.blogspot.com/-GYjmeTLkmSU/VwJw-AvHONI/AAAAAAAAANo/2zvK-rV96rIPhPa5Bv3MM8G9rbQYrH_hQ/s320/IMG_0185%2B%25281%2529.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tViNMmvf5pM/VwJxAZqhEiI/AAAAAAAAANs/ORjTQWNU1HYHvHhdPSFIMEVe-bknG1b3g/s1600/IMG_0186%2B%25281%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://1.bp.blogspot.com/-tViNMmvf5pM/VwJxAZqhEiI/AAAAAAAAANs/ORjTQWNU1HYHvHhdPSFIMEVe-bknG1b3g/s320/IMG_0186%2B%25281%2529.JPG" width="320" /></a></div><br /><br />Word walls are another thing that can be useful if displayed and utilized intentionally. Otherwise, it is just a group of words on the wall that might as well be in a foreign language. In the Everyday Math lessons, unit vocabulary is listed in the <i>Mathematical Background</i> section of the Teacher's Lesson Guide, located just before the first lesson of each unit. Building a word wall and evolving it with each unit is a great practice, but actually <i>referring to it</i> and encouraging students to refer to it is best. Wall space is precious in most classrooms I visit, and one wouldn't want that precious space to go un-utilized. <br /><br />Many of the excellent visuals I see when I visit classrooms are interdisciplinary. Most notably, posters and wall art that displays think-stems and sentence starters for writing exercises are perfectly appropriate for students who need help expressing their mathematical thinking on paper. This is one of our students' biggest challenges, and having those visual aids on the wall can be extremely helpful to them. Developing confidence in writing is a big deal, and I have a feeling it is going to be a major source of collaboration in RSU 5 among the strategists in both literacy and math in the future.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-6Lj5M78lHh8/VwJxlG-BQWI/AAAAAAAAAN0/uJJISCXzvOU24hQlLBbQ40PQce4_NLQvA/s1600/IMG_0188.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://4.bp.blogspot.com/-6Lj5M78lHh8/VwJxlG-BQWI/AAAAAAAAAN0/uJJISCXzvOU24hQlLBbQ40PQce4_NLQvA/s320/IMG_0188.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-GiZ0yxSerDw/VwJxmzMBttI/AAAAAAAAAN8/Z_oiws3DqBkF8Ys6HEz1YMBfwqS3EsduQ/s1600/IMG_0191.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://1.bp.blogspot.com/-GiZ0yxSerDw/VwJxmzMBttI/AAAAAAAAAN8/Z_oiws3DqBkF8Ys6HEz1YMBfwqS3EsduQ/s320/IMG_0191.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Ya7RGrT_f-Y/VwJxpkLc6kI/AAAAAAAAAOA/byyaHvNSnRc3AdPQT8oQvslEKgyaYwRRg/s1600/IMG_0192%2B%25281%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://2.bp.blogspot.com/-Ya7RGrT_f-Y/VwJxpkLc6kI/AAAAAAAAAOA/byyaHvNSnRc3AdPQT8oQvslEKgyaYwRRg/s320/IMG_0192%2B%25281%2529.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-kvwHmdTR4dg/VwJxu6FcKmI/AAAAAAAAAOE/u5LqbYzMerEAs11kI8OwqVTpi1olgvF4Q/s1600/IMG_0195.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-kvwHmdTR4dg/VwJxu6FcKmI/AAAAAAAAAOE/u5LqbYzMerEAs11kI8OwqVTpi1olgvF4Q/s320/IMG_0195.JPG" width="239" /></a></div><br /><br />Both science and social studies displays can include mathematical components as well. I have seen planetary displays (miles and kilometers), timelines (years, positive and negative numbers), and thermometers (also positive and negative numbers) displayed in classrooms<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/--xG9suaiSSg/VwJyxIb4CQI/AAAAAAAAAOM/jMId2dV9XJctj6GHm0er90SaYdqdeZKmQ/s1600/Screen%2BShot%2B2016-04-04%2Bat%2B9.56.30%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/--xG9suaiSSg/VwJyxIb4CQI/AAAAAAAAAOM/jMId2dV9XJctj6GHm0er90SaYdqdeZKmQ/s320/Screen%2BShot%2B2016-04-04%2Bat%2B9.56.30%2BAM.png" width="299" /></a></div><br /><br />Most of the representations of classroom visuals I chose to include came from RSU 5 classrooms, and most of them are hand-made. There are some nice math posters you can buy online, but you never know if it will pertain to exactly what you want on your wall. Nice teacher-made wall art takes time and effort, but it can be used year after year, and can even be laminated. Also, Every EM4 teacher kit came with a series of posters that display the eight <i>Standards for Mathematical Practice</i>, along with EM4's aligned <i>Goals for Mathematical Practice. </i>These can be displayed all at once or rotated in and out to focus on specific practices. The key, though, as with any display, is to refer to them often and model their use. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-3dw_B7sfZIM/VwJ1C41vKOI/AAAAAAAAAOk/qxjXjVyFywMdEytP2wAm3cs3Az87H-wHA/s1600/IMG_0202.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-3dw_B7sfZIM/VwJ1C41vKOI/AAAAAAAAAOk/qxjXjVyFywMdEytP2wAm3cs3Az87H-wHA/s320/IMG_0202.JPG" width="239" /></a></div><br /><br />It is great to see such carefully thought-out and smartly displayed mathematical wall displays in so many places. I will keep taking more pictures and posting them as they are all ideas worth sharing. James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-63887450594096459472016-02-03T07:00:00.000-08:002016-02-03T07:00:01.573-08:00Subtraction Strategies in grades 1 and 2, and how they impact learning in the upper elementary grades.Subtraction is the first nearly universal struggle math learners stumble upon in the early elementary years, before division, fractions, and eventually algebra. It doesn't work like addition, it is not commutative (i.e. the turn-around rule does not work in subtraction), the answer is called the "difference" (confusing word in math), and we use about ten different words to represent it (minus, take away, subtract, minuend/subtrahend, how many less, decreased by, how many left over, how many fewer, how many more, what is the difference, etc!). <br /><br />By the time students reach third grade, they have been introduced to a variety of different subtraction strategies (see grade 2, unit 3 for a list), but as the authors of the Everyday Math units tell us, we should not expect every student to be able to master every subtraction strategy. Students are introduced to subtraction in kindergarten, and practice only basic subtraction in first grade. By the end of second grade, they should have tried all the strategies introduced, and they likely will have latched onto a couple. In third grade, we really want them to be considering strategies to use beyond counting on their fingers. <br /><br />Third grade involves a lot of subtraction practice in units 2 and 3, where your students should be "building their understanding of place value to develop methods for subtracting 2- and 3-digit numbers." <i>(<-- from the Mathematical Background: Content section in the grade 3 TLG)</i> Here, expand-and-trade takes front and center as a major algorithm for students to practice and use. This is an important piece of their understanding for future computation and problem solving, and really tests their understanding of place value. <br /><br />It is not intended to be an algorithm they latch onto for life; expand-and-trade is like training wheels for the more traditional vertical subtraction most adults are familiar with. Parents may see their child struggle with expand-and-trade, and not have the faintest idea how to help them. "That's not how I learned it," they may say to themselves, or express to you in a frustrated note attached to a homework assignment. Unfortunately, the temptation is to just go ahead and teach the traditional vertical subtraction method instead, because it makes more sense to the <i>adult</i>. But do what you can to prevent this. Expand-and-trade is designed to slow down the steps and deconstruct the numbers so students can see precisely when they are subtracting ones, tens and hundreds. When they eventually transition to more efficient methods, they are well-versed in what each step of subtraction represents, and are better equipped to catch their own errors when they happen and know when their answers are in the ballpark or not. In short, they become more confident with their subtraction. <br /><br />It is the confidence that we are aiming at, because that confidence carries over to the rest of their math learning. If they can add and subtract with confidence, then learning how to multiply and divide will not produce as much anxiety. When they can multiply and divide with confidence, fractions are less confusing and intimidating. <br /><br />Subtracting numbers is one of the most important things your students will learn how to do while in first, second and third grade. The more opportunities your students have to work on subtraction in the early years, the better they will do in the later years. And your fourth, fifth and sixth grade teaching colleagues will really notice a difference.<br /><br />Here is a link to the VLC with some resources related to subtraction:<br /><br /><a href="http://vlc.uchicago.edu/resources/search?utf8=%E2%9C%93&tag_ids%5B%5D=642&tag_ids%5B%5D=&tag_ids%5B%5D=&tag_ids%5B%5D=&tag_ids%5B%5D=&tag_ids%5B%5D=&tag_ids%5B%5D=&tag_ids%5B%5D=&editions%5B%5D=4&keyword=&commit=Search">Subtraction Resources on the VLC</a><br /><br />Lastly, don't forget to access the <b style="font-style: italic;">Grade Level Resources</b> option toward the bottom of the grey menu when you log onto ConnectED. There are lots of tutorial videos and visual aides and other resources in this section that can help you provide subtraction practice opportunities for your students. Contact me if you need help accessing these tools.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-3878590594922076502016-02-02T06:57:00.000-08:002016-02-02T06:57:00.210-08:00What's Happening in Classrooms: Fantastic Turn-and -TalksRecently I have visited a lot of math lessons where turn-and-talks are frequent and intentional, and I want us to <b>celebrate and encourage that practice</b>.<br /><br />Teaching math with intentional student discourse involves taking risks, but they are risks worth taking.. These are some of these risks and ways to overcome them:<br /><br /><b>It might be disruptive and might not work. </b> When you ask your students to talk to each other during a lesson, you run the risk of creating a noisy, disruptive environment. For this reason, we need to have students practice a specific protocol for turn-and-talks. Make sure they are using indoor voices. It is also helpful to have explicit instructions for what to discuss, and make sure they only last a minute or so. I encourage everyone to have students practice math turn-and-talks, because math is something students don't always feel comfortable talking about. You could say, for example:<br /><br /><i>OK, friends, let's practice a turn-and-talk. Here's what I would like you to discuss. 13 plus 12. What is the answer, and what are two different ways to show or prove your answer is correct. Ready? OK, turn and talk.</i><br /><i><br /></i>You might wish to add extra protocols, like making sure each partner has a chance to talk. <br /><i><br /></i><i>OK, one, two, three, eyes on me. (Wait for all to respond stop talking).. Now I would like you to turn-and-talk again, just to make sure both partners had a chance to speak. Turn and talk again now please.</i><br /><i><br /></i>Practicing and re-teaching protocols for math talk is really important during the middle of the year to maintain the most productive and comfortable learning environment.<br /><br />I used to wonder if <b>too many turn-and-talks might mean too many transitions and disruptions to learning. </b>Could Turn-and-talks could get old? <b> </b>This is actually not something to worry about. You can have students turn and talk repeatedly in one mini-lesson, three or four times in just a few minutes. As long as you encourage them to use the protocols, if there are things to talk about, give them the opportunity to do so. The more they do this, the more comfortable they will become communicating their thinking with one another, with you, to their class, and on paper. The benefits of turn-and-talks are especially great in math classes and every class should be utilizing them (in my humble opinion..). It is a good idea to insert a little space or discussion between turn-and-talks, but the actually risk of doing them <i>too often</i> is almost zero. <br /><br /><b>Not enough turn-and-talks is the greatest risk of all...</b> When your students are not communicating with each other about their thinking in math on a regular basis, they are not getting opportunities to reflect on their work, to reflect on their own thinking, and to hear the thinking of their peers. Critiquing and analyzing the mathematical thinking of others is an important mathematical practice, and turn-and-talks are a critical way to make that happen.<br /><br />It's important to diversify students' conversational experiences in math. Turn-and-talks, small group work, class discussions, peer conferences, partner work-- They all provide excellent opportunities for students to reflect on their own thinking and learn from their peers. But Turn-and-talks are special, in that they offer students a chance to communicate directly with a partner about an idea that is right there in the moment. They only take a few seconds, and they keep your students engaged in your lesson. If you would like support from a colleague in using turn-and-talks in math lessons, chances are good you have someone right next door or across the hall who can help you... Or you can ask your friendly math strategist to come model turn-and-talks in math. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-iXoonFlODww/VrDCOvZUkPI/AAAAAAAAALs/EHG5mSbLdiI/s1600/Screen%2BShot%2B2016-02-02%2Bat%2B9.49.45%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-iXoonFlODww/VrDCOvZUkPI/AAAAAAAAALs/EHG5mSbLdiI/s1600/Screen%2BShot%2B2016-02-02%2Bat%2B9.49.45%2BAM.png" /></a></div><br /><b><br /></b><b><br /></b>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-86429439132935118352016-01-19T09:03:00.003-08:002016-01-19T09:07:24.243-08:00EM4 and How We Teacher Common Core MathWe are doing the right thing. That is the most important message I can give you at this time. When we teach for greater understanding of mathematical concepts, and coach our young people into becoming better problem solvers, we are doing the best thing we can for them. So the questions we need to be asking ourselves when we are faced with <i>how</i> to teach a lesson, a unit, or a concept, is: Will this help my students become better problem solvers? This is one of the reasons we went with Everyday Math 4-- The emphasis is on problem solving and deeper understanding of mathematical concepts.<br /><br />Common Core math is constantly under scrutiny in social media and even in the news, mostly because teaching for deeper understanding is not how previous generations were taught, as a rule. We were primarily given instructions to follow, rather than a puzzle to work at, take risks with, and eventually solve, perhaps with some help from a neighbor or two. Puzzles aren't always easy to understand, and they often involve making mistakes before a solution is found. For some puzzles, it helps to look at them from multiple perspectives, which is why we often have our students problem solve with partners and small groups. Other puzzles are best approached independently (have you ever tried to solve one of those nine-square puzzles with birds or reptiles or another such scene on them, with a group? Unless one member is a very dominant type, it is almost impossible!*).<br /><br />Let us remember that Common Core is merely a set of standards, and not entirely different than standards we've had in the past. The major difference is within the eight Standards for Mathematical Practice and the content standards, there is greater emphasis on deeper understanding and problem solving. This is not understood by some parents and other vocal critics, who see what looks like strange and unusual homework assignments coming home (and others, with questionable authenticity, posted on Facebook) they have trouble helping their children with, let alone understanding themselves. While the Common Core math standards are the same in 42 states, how they are implemented is not always the same. Different districts in these 42 states have different math programs, and within those districts there is a varying degree of fidelity to those supposedly Common Core aligned math programs. As someone who has studied the Common Core math standards since before they were released, while still in draft form, my opinion is that EM4 has done an impressive job aligning to the standards and capturing the emphasis on understanding and problem solving. As with any mass-published resource, we each will find the occasional weirdly worded problem, or curiously designed activity, but the mathematical content and instructional shifts that are the focus of EM4 are its major strengths, and teachers in RSU5 are doing a fabulous job in the inaugural years of using these resources.<br /><br />I want to share a NY Times article that was published recently which attempts to address some concerns about Common Core math, just to help celebrate and acknowledge what we are doing. I think you will read this short article and say to yourself, "I'm doing that," or, "that's already happening in my classroom." So congratulations, you are doing the right thing. I am confident that as the next couple years progress, we will see more and more great things coming out of the work our students produce, because I see the difference one year makes every day in the classrooms I visit. Imagine what will happen when we all <i>really</i> get good at this. <br /><br />Thank you for taking risks in your work, for allowing opportunities for your students to problem solve together, for shifting your instruction, and for putting so much love and hard work into your math lessons.<br /><br />Here's the link:<br /><br /><a href="https://www.washingtonpost.com/news/answer-sheet/wp/2016/01/15/are-the-common-core-math-standards-being-misinterpreted/">NY Times Common Core Math piece</a><br /><br /><br />* There's a story that goes with this puzzle thing... I worked at the Spurwink School, among other intense alternative education environments, prior to getting my teaching certificate, and one day I pulled out one of those nine-piece square puzzles for a boy that was dealing with a lot of emotional issues and had recently gotten himself in some serious trouble at school (that would eventually involve charges being brought against him). I brought in the puzzle because my wife and I had been working on it at home periodically, and it took almost two weeks for us to stumble upon the solution. Intuition led me to believe this particular child might really focus on such a thing, and he was obsessed with songbirds. This puzzle had songbirds all over it. The first thing he did was correctly name each bird species, and then he went to town on the puzzle, quietly mumbling to himself as he rapidly swung cards around in different positions. He solved it in less than 60 seconds. Now one could come to a number of conclusions from such a phenomenon, one being that my wife and I are perhaps not all that sharp... but I prefer to use this as an example of not just a special kid (I have never seen anyone else solve the puzzle so fast), but of one of those situations where other perspectives actually <i>complicate</i> the problem solving process. When my wife sat beside me at the table, she was looking at the puzzle from a different angle. Every time she moved a piece in or out of place, it completely messed with what I was seeing, and I had to switch gears in my thinking. When I would adjust the puzzle, my wife would let out a displeased sigh. For the child I gave the puzzle to, it was just a matter of rapid-fire trial and error, in total quietude, without disruption, and bam. Puzzle solved. It didn't end up being a very lengthy distraction for him, but he did get to feel good about himself for a few minutes.<br /><br />It just goes to show that while problem solving is a process most often benefitted from collaborative work environments, we should continue to still provide opportunities for our students to do some work independently. Sometimes that is where their strengths manifest themselves.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-53959301610009356032015-12-11T09:43:00.000-08:002015-12-11T09:43:23.595-08:00Finding Time to DifferentiateDifferentiating instruction continues to be the single most challenging task when teaching math lessons. Let's consider two important factors when planning and executing differentiated instruction in the math classroom: <i>How</i> to differentiate and <i>when</i> to differentiate.<br /><br />Let's start with the <i>when</i>, since one of the most common concerns about differentiating instruction is the limited time that exists during a lesson to make it happen. We'll get to the <i>how</i> also, because differentiation strategies are not universally effective, but first we should make sure we can carve out time during the lesson to differentiate.<br /><br />If we try to take time to work one-on-one with every struggling student in the class, it is likely time will run out and the lesson will not proceed as planned. Just for the record, differentiated instruction is not individualized instruction, and it should not require anyone to be a one-on-one tutor and a classroom teacher simultaneously. <br /><br />Differentiating really means two things. You are collecting information and adjusting your instruction based on that information. Both of those things are physical activities that take up time and need to be incorporated into a lesson in order to differentiate. <br /><br /><b>Collecting information.</b><br /><b><br /></b>Chances are, you have students who struggle, and students who outpace the majority of the class with ease. Both groups need your attention, or else the strugglers will fall continuously behind and the other group will get bored. The thing about math is it is many subjects in one. It is hard to justify the statement, "This student really struggles <i>in math</i>." What part of math? "This student really excels <i>in math</i>." Oh? I want to know what aspects of math are this student's true strengths. It is actully quite rare that a student struggles and/or excels in all aspects of mathematics. <br /><br />I had an identified gifted 8th grader one year who could have performed well in a high school algebra II class, but when it came to geometry and geometric proofs, he fell apart. He really struggled! It took him bursting into tears out in the hallway with me one day, after I called him out on refusing to participate in a small group activity, to admit to me, "I don't do well with shapes!!" It was such a shocker, because this student exceeded expectations in almost every academic field. Yet here he was, sobbing out in the hall over internal and external angles of polygons. I had to really be a cheerleader for this student and put in extra effort for him, because I felt I had failed him. I had not collected enough information to acknowledge his self-perceived weaknesses in geometry. And here I was, taking up valuable class time to extract that little tidbit of valuable information out in the hallway.<br /><br />Here's one thing I wish I had done with him that you can do with your students. Keep a class roster handy and take notes as you observe your students working. Notice I said "roster" and not "log book." A roster is just a list of students with some room to record information on to the right of each student's name. And by <i>information</i> I don't mean paragraphs or even sentences. I just mean <i>information.</i> For example, as students embark on independent or small group work, grab your roster and wander around, and when you see a student struggling with something you think they should have mastered by now, put an "S" by their name (or, say, a "B" for beginning). For students that are on task and are demonstrating intended understanding, just put a visible dot, or a check mark. For students that seem to be outpacing their peers, or are demonstrating exceptional understanding, put an "E" by their name (for "extending"). You can use as many different letters or symbols as you want, but if you keep your list handy, and do this quick 2-minute "walk-around" every day, by the end of the week you will have some pretty valuable information about who in your class might need some differentiated instruction. Be sure not to interrupt your two minutes-- or you might not have time to visit every student. If students are asking for your help, you can always say, "I'll be with you in just a few minutes, but in the meantime, have you asked a neighbor for help?"<br /><br />Once you have that information, perhaps combined with other evidence, such as further observations and student work, you can begin to address the needs of the students who indicate they are consistently either outpacing or are outpaced by their peers. <br /><br /><b>Carefully managing instructional time.</b><br /><b><br /></b>It is so easy to let that hour fly by. I'm just going to bullet some common pitfalls, both from personal experience and from observation, that tend to eat up time when teaching, and inhibit differentiation.<br /><br /><ul><li>Mini-lessons that are not mini enough. That's right-- most Everyday Math lessons begin with not one, but two short introductions. Mental Math and the Math Message are both meant to be short exercises to help students prepare for the primary content of the lesson. It is easy, during either Mental Math or the Math Message, to find one's self going off on a tangent and drawing out these short segments into major segments. Mental math is exactly what it says it is-- <i>mental</i> math, and it is not really a time to make sure everybody has the correct answer and understands <i>everything</i>. Take note of who has it and who doesn't, but then move on. And the math message, while an opportunity to introduce a new topic, is not necessarily a time to <i>explain</i> the topic to the whole class. They will have time to ask each other questions and gain better understanding during the Focus of the lesson. Mini-lessons are not time for unexpected teachable moments, however tempting they may be. Which brings us to the next bullet..</li><li>Teachable moments that could wait. Sometimes a student asks a question or shares a delightful "ah-ha" moment, and you just want to harness that thirst for knowledge and divulge into an epic and glorious discussion about the true meaning of mathematics. Sometimes, it is just inevitable, and such teachable moments can yield profound learning opportunities.. But they can also be big distractions, and can cause a rather awkward break in the intended rhythm of a mathematics lesson. Be careful not to open the door to an untimely distraction.</li><li>Too many students taking turns demonstrating in front of the class. It feels great to have a student demonstrate for the entire class how he or she solved a problem, and that student certainly benefits from the experience. The problem is, it is very time-consuming. Sometimes, it can even be counter-effective, because the student might present in a confusing way. But one thing that almost guarantees you will lose teaching and/or learning time is when multiple students take turns presenting to the class. Keep in mind that when one student is presenting, everyone else is expected to listen and maintain attention for the duration-- which can be a major challenge for some students. If two or three students present, then the time "sitting and getting" is doubled or tripled, and chances students will "check out" multiplies also. After the first or second student has presented, the rest of the class has most likely lost interest. It is only interactive for the few students who presented. In most cases, it might be best to have a student present, ask a few others to summarize what was said, and move on to the next part of the lesson. </li><li>General "sit-and-get" style teacher talk. Some of my middle school students used to get frustrated with me when I set them to work collaboratively to solve problems. "Why can't you just explain it to us?" they would ask. "Because that would be boring! Where is your sense of adventure?" I would say in response. I was telling the truth. I am long-winded, and when I explain stuff, it takes a while. Also, students hear only <i>my</i> explanation, which is stated in <i>my</i> own words, expressed only as <i>I</i> understand it. I learned this from experience; I am a boring and less effective teacher when I put my energy into explaining. And then I have wasted a good portion of my lesson only connecting with the few of my students who understand my way of thinking. The "sit-and-get" style of teaching is really the <i>anti-differentiation way</i>. For now, I suggest not going too far down that road for K-6 instruction.</li></ul><div><b>Infusing differentiation into your lesson.</b></div><div><b><br /></b></div><div>Now is when we will begin to merge the <i>how</i> in with the <i>when</i>. Really, the best answer to "when to differentiate" is... Always. Just remember that when you are collecting information (formative evaluation/ assessment), you are beginning the process of differentiating. And you are always collecting information, by simply listening to your students when they share their thinking with you, and observing them as they work together. Once you have that information, the next step is to address the learning needs of your students accordingly. Let's go back to the bullets to explore ways of doing this.</div><div><br /></div><div><ul><li>Meet with a group of struggling or accelerated learners. During the mental math, the Focus, or the practice portions of your lesson, you can assemble a group to work with you while others work in their own groups or pairs. Practicing this with your students is important, because your entire class will need to cooperate in order for you to be able to devote attention to a group of high-needs students. It can be just a five minute activity, or it can be for a longer period of time, but the value in working with three to five students can be significant. Such groups do not need to be entirely arranged by ability, either. Maybe two of four students in your group are struggling, and two are not. That way, the strugglers are working alongside peers that can help guide them. Also, it helps to rotate a variety of students in and out of these groups, so individuals do not alienated. Mix it up and try doing it just two or three times a week, to allow students to also work with their peers in pairs and in groups without the teacher always immediately present. </li><li>Substitute portions of the lesson with enrichment or readiness activities. Everyday Math provides some decent activities with the vast majority of its lessons (Open Response lessons do not have these, as the OR lessons are designed to be collaborative problem solving experiences for all students), that might be appropriate for your students. These enrichment and readiness activities are found on the page opposite the beginning of each lesson in your Teacher's Lesson Guide. There is also an extra practice activity. </li><li>Extend or scaffold a math box or journal assignment. In a small group or independently, certain problems can be extended or scaffolded based on a student's demonstrated ability. Asking a student or group of students to "try this," while working might be just what that student or group of students needed to stay engaged or to grasp a concept. </li><li>Offer extensions/ extra practice activities as optional homework or classwork. I am not a fan of assigning <i>extra work</i> to students who struggle or excel, but sometimes when given an opportunity to take home an extra practice assignment or an extra challenge, either in place of a regular home link/ homework assignment or on a night when no homework is assigned, students will jump at the opportunity. </li><li>Explain your thinking! Whether a student has solved a problem early or is struggling to find a way to solve a problem, having him or her take turns sharing their thinking with a partner might be a helpful exercise. Both struggling students and consistently high-performing students can benefit from this. Struggling students can benefit from both working through their own thoughts and also listening to a partner explain his or her thinking. Students who solved the problem swiftly with ease might benefit by being forced to slow down their thinking and deconstruct their strategy in a way they are not used to, and also by listening to a partner share his or her own thinking.</li><li>Try a 'help table' during independent work time. Leave a table vacant with three or four seats open, and a space for you (preferably where you can see most of your students). Let them know that anyone who is really having trouble can come work with you at the 'help table.' If the table fills up and a line forms, it may be time for some re-teaching or another activity to students understand relevant concepts. </li></ul></div>What are some successful differentiation techniques you would like to share? I would love to add to the bullets above. Submit a comment or send me an email. <br /><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-35384008940582832262015-10-16T12:51:00.000-07:002015-11-04T10:42:58.349-08:00Flexible Language and Everyday MathIn an effort to be developmentally appropriate, the Everyday Math units we teach include some unique terminology within the huge bank of mathematical vocabulary we deliver to our students. The vast majority of Everyday Math vocabulary terms are universally acknowledged in mathematics, and are emphasized in the Common Core math standards, but some have been substituted by the authors. Just as we teach a variety of algorithms for adding , subtracting, multiplying and dividing, we also should include standard mathematical terminology alongside Everyday Math terminology, or we risk students running into roadblocks of understanding down the road.<br /><ul><li><b>Number sentences</b> and <b>number models </b>are terms Everyday math uses to represent <i style="font-weight: bold;">expressions, equations </i>and<i style="font-weight: bold;"> inequalities</i>. It is especially important to familiarize your students with the word <i>"equation"</i> alongside the Everyday Math terms, as <i>equation</i> does not show up regularly in Everyday Math units until grade 4. There will be many times when students come across the term <i>equation</i>, including on standardized assessments, and we want them to be prepared and familiar with it. Does that mean you need to test your students on the meaning of "equation" and "inequality" ? No it does not. But taking regular opportunities to use these words when you teach is important. For example: <i> "Janet just gave us a nice number model, also known as an equation, to show us a way of writing six plus five equals eleven. Equations are number models that include an equal sign in them." </i>Or another example would be: <i>"Four times three is a number model for that array, because it shows four rows of three, or for groups of three. 'Four times three' is also known as an 'expression,' or an 'operation,' because there is no equal sign, just two factors and a multiplication sign."</i></li></ul><div><ul><li><b>Number Stories </b>are word problems. "Number story" sounds a lot more pleasant to deal with than "<b><i>word problem</i></b>," but we want students to know these two terms are the same. So it is encouraged to introduce your students to the term, "word problem,"and use it from time to time alongside the term "number story." For example: <i>"The directions are for you to use the data in the graph to write your own number story for your partner to solve. Number stories are also called "word problems," as they are like regular math problems that use words as well as numbers to help you understand them." </i></li></ul></div><div><ul><li><b>Turn-around rule</b> is the term Everyday Math uses to refer to <b><i>the commutative property</i></b> of addition and multiplication. It has a lot of syllables, but it is a specific rule of mathematics we cannot allow our students to ignore. Use it in your instruction when you can. For example: <i>"Yes, Janet, four times two equals two times four is an example of the turn-around rule, also known as...???? Who can tell me the official mathematical term for the turn-around rule? Let's all say it together... The Commutative Property! Well done. The commutative property of multiplication means you can multiply two factors and as long as you don't change the factors, you will always get the same answer, no matter what order those two factors are in." </i></li></ul></div><div><ul><li><b>Names</b> is the term Everyday Math uses to refer to different <i style="font-weight: bold;">representations</i> of the same number. Directions in the student journal may ask students to "create at least four different names for the number 8," and a student might write the word "eight," draw a four by two array, draw eight tally marks, and write the expression/ number model "4 + 4." It is important for students to know that this use of the word "names" refers to "representations" of the given quantity. In programs other than Everyday Math, it is unlikely they will ever see representations of quantities referred to as "names."</li></ul><div><ul><li>Use your <b>reading and writing terminology</b> in your math lessons. Susan Dee came to me recently and asked if fifth grade teachers could use the term "think stems" and "thinking prompts" when Everyday Math uses "sentence frames." These are clauses that help students find ways to introduce an idea or a strategy in their problem solving. Perfect! It is really important that our students carry over their reading and writing strategies into their problem solving in math. Thank you, Susan Dee and Matt Halpern for thinking of our young mathematicians! </li></ul></div><div>Are there other Everyday Mathematics terms you have encountered that should be addressed? Are there other words and phrases you use in other classes that are completely relevant to teaching math?Please let me know and I will add them to this list. Thank you for taking the time to consider this topic.. It really is important that we paint a complete picture of the language of mathematics for our students.</div></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-pwmmN5_0ySE/ViFU_SUJQnI/AAAAAAAAAJ4/CYottSOXWqg/s1600/Screen%2BShot%2B2015-10-16%2Bat%2B3.49.39%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="228" src="http://2.bp.blogspot.com/-pwmmN5_0ySE/ViFU_SUJQnI/AAAAAAAAAJ4/CYottSOXWqg/s320/Screen%2BShot%2B2015-10-16%2Bat%2B3.49.39%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;">Addendum: Here are links to a couple articles related to the language we use in mathematics. When is it appropriate to retire certain terms, phrases, and mathematical shortcuts? The first link is directly related to elementary mathematics, but it also may be worth your time to check out the second article about middle school. Not only is there some overlap, but it is also good to know what is in store in your students' next stages of mathematics learning after they leave you. Here are the articles:</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue1/tcm2014-08-18a_pdf/">Elementary Expiration Dates in Math</a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/2015/Vol21/Issue4/mtms2015-11-208a/">Expiration Dates in Middle School Math</a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div><br /></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-1322726200375014282015-09-17T06:45:00.002-07:002015-12-11T09:15:08.682-08:00Videos!Below are videos that may be helpful to you.. I recommend opening these videos on a separate tab or browser so you can pause the video when you need to and follow directions. I will add to these throughout the fall and the rest of the school year, so keep checking back. If there is a video you would like me to make for the benefit of yourself and others, please let me know and I will try to make that video.<br /><br />Added December 2015:<br /><br />For those interested in an alternative reporting program to ConnectED for trimester 2:<br /><a href="https://youtu.be/9AO_3J_-B4w">Assessment Spreadsheet Instructional Video</a><br /><br />Added October 2015:<br /><br />10/27: <a href="https://youtu.be/HN7LlMuUsEY">Scoring CHALLENGE items on the cover sheet</a><br /><br />10/26: <a href="https://youtu.be/M-Hw03QICVw">Intro to the new NEW cover sheets (Unit 2 and beyond?)</a><br /><br /><a href="https://youtu.be/ol26H2vjo28">Change Quick Entry/ Evaluation to 4-Point Scale with Exceeds</a><br /><br />This is a GREAT 11 minute video that is HIGHLY RELEVANT to how we are approaching problem solving: <a href="http://ww2.kqed.org/mindshift/2015/10/19/how-can-we-teach-math-to-encourage-patient-problem-solving/">Dan Meyer's 2010 Video about Problem Solving</a><br /><br />Added September 2015:<br /><br />9/25: <a href="https://youtu.be/KAHsHn4q8m4">How to use the new cover sheets</a><br /><br /><a href="https://youtu.be/RSX9xUcTYlk">Adding Student Content to ConnectED</a><br /><br /><a href="https://youtu.be/Sb56Ki_osbM">How to Enter Data for BOYA</a><br /><br /><a href="https://youtu.be/CDjmOwr3ZpM">How to Enter Data for Unit Assessments</a><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-61212257706229845992015-09-17T06:45:00.001-07:002015-09-17T06:45:15.075-07:00Whoa, These New Lessons Are Different...So here we are, a couple weeks into the new school year and by now math lessons are well under way. In RSU 5, teachers teaching grades kindergarten through sixth grade are now all officially in the same boat; we are teaching a brand new curriculum, and using a brand new online data reporting system. For this first entry of the new year, I will address beginning-of-the-year teaching strategies as they relate to the instructional shifts inherent in the new Everyday Math 4 units. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-XlwTnL-ZOEc/VfgXBxe6pzI/AAAAAAAAAIE/YA0i0AbyXmw/s1600/Photo%2Bon%2B9-8-15%2Bat%2B11.03%2BAM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://4.bp.blogspot.com/-XlwTnL-ZOEc/VfgXBxe6pzI/AAAAAAAAAIE/YA0i0AbyXmw/s320/Photo%2Bon%2B9-8-15%2Bat%2B11.03%2BAM.jpg" width="320" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-5QETVVKWpOc/VfgXMV9jwiI/AAAAAAAAAIM/z14Toa48G5U/s1600/Photo%2Bon%2B9-8-15%2Bat%2B11.02%2BAM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://3.bp.blogspot.com/-5QETVVKWpOc/VfgXMV9jwiI/AAAAAAAAAIM/z14Toa48G5U/s320/Photo%2Bon%2B9-8-15%2Bat%2B11.02%2BAM.jpg" width="320" /></a></div><br /><br />I could go on and on about the things I personally like about the instructional shifts in the new Everyday Math program, but I want to focus here on the challenges that come with these shifts. One big reason to celebrate what is happening with the new EM4 units (and the similar shifts that are happening in other popular math programs) is this focus on understanding, collaborating with peers, and communicating students' mathematical thinking can and should lead to an important outcome often missing from the K-6 math experience: Joy. This is not to say that before this edition of the program there was no joy in learning math, but with the greater focus on understanding and communicating mathematical concepts, students gain greater access to the art of solving problems.<br /><br />Whether it is through landscaping in the back yard, measuring ingredients of a recipe, or solving a labor dispute, real problem solving is part of our human nature, and we often solve problems with help from our peers. Playing games that involve strategy is something that we do to <i>entertain</i> ourselves when there aren't fun problems to solve in our immediate vicinity. Whether it is football, Monopoly, Risk, Pac-Man (Sorry, I grew up in the '70s and '80s), KenKen, Sudoku, or the latest edition of World of Warcraft, we <i>create</i> imaginary problem solving scenarios to occupy and activate our brains because it is <i>fun </i>to solve problems. If we are lucky, we will grow up and get paid to solve the kinds of problems we like to solve. Rarely will we be asked to solve those problems all by ourselves, but we need to be able to perform the foundational operations, with confidence, by ourselves. The new EM4 lessons, via the Common Core Standards for Mathematical Content and Practice, aim to provide opportunities for students to gain confidence in their foundational understanding of operations and algorithms, so they have the tools they need to solve <i>interesting and engaging</i> problems. Sometimes they will solve problems independently, but more often your students will be working in groups and pairs to solve problems.<br /><br />So what is the best way to prepare students to work collaboratively and to communicate their mathematical thinking (both to each other, and on paper)? They may not have done this as much in the past, and now they are suddenly expected to do it A LOT this year.<br /><br />Your students may not be used to spending so much time working collaboratively to solve problems, and they may not be comfortable or confident communicating their mathematical thinking. Especially at first, some students might miss <i>being told how</i> to solve problems and they may wonder why that is not happening so much anymore. Some may even have conditioned themselves into (sadistically?) enjoying the old fashioned skill-and-drill number work, but transitioning away from the more traditional "I do, we do, you do, then you do it over and over again" style of teaching procedural math leaves some students <i>expecting</i> to be told how to solve a problem before being given the task to solve a problem. They are used to being <i>given</i> the strategies to solve problems, rather than being given time and collaborators to help <i>discover</i> strategies and <i>find their own </i>solutions.<br /><br />For this transition to happen, students need opportunities to practice solving problems in teams and with partners. It is important to <i>not</i> just dive into collaborative work with long periods of unstructured time for groups to brainstorm... Structure their time in segments at first with specific tasks. Give them a few minutes to determine exactly what it is they need to do, and a few more minutes to determine all the tools and information they <i>have</i> to solve the problem. Check in with them frequently to make sure they don't <i>check out</i> and that they are actively seeking to answer questions to overcome roadblocks that might be holding them back. Open Response lessons will allow for this practice, and even give a second day to review their work and make revisions.<br /><br />Sometimes a specific protocol can help. In an earlier blog entry, having specific assigned roles when working in groups was discussed. One person can be given the role of ambassador, with special permission to check in with other groups to see what their strategies are. Another can be in charge of taking notes, recording all ideas and outlining them or diagramming them. Another group member can be in charge of making and/ or checking all calculations. Every student should be actively participating.<br /><br />Students will be expected to independently explain their mathematical thinking very often this year! This will be in assignments and on assessments, but it will also occur regularly when they work with a partner or a group. Students will have to share their thinking with peers, and evaluate each other's mathematical thinking. They start this in kindergarten, when their teachers help them decompose their solutions and quantitative grouping. Teachers might ask, "How did you know the blue pencil was longer than the pink pencil?" or, "You told me you made a triangle and a rectangle. Can you tell me how you know that one is the triangle?" In the first grade, teachers work throughout the year to help children write their thinking on paper and draw pictures to support and represent their thinking. By the time this year's first graders reach the second grade, they will have had two years of experience communicating their mathematical thinking.<br /><br />This year's fourth, fifth and sixth graders have not had as much experience with this kind of work, and it may be difficult for some of them. Often high performing students have trouble "explaining" why or how they solved a problem the way they did. "I just knew it," you will hear, or, "I did it in my head." Just like with 5 and 6 year olds, asking a lot of "how do you know" questions will help them deconstruct their strategies. "How do you know the area of that figure is 64 square centimeters? What is a 'square centimeter,' and how does it differ from a regular centimeter? How do you know which is the width and which is the length? Does that matter? Why? What does your drawing represent? Can you can prove your method is correct with another example?"<br /><br />It is a transition year for us, with new, updated units, <i>and</i> new total alignment to Common Core standards, so give yourself and your students time to practice and get comfortable with the changes.<br />Make sure expectations are clear. Students should be using indoor voices when solving problems and collaborating, and group/ partner work should be monitored with frequent check-ins, either on a group-by-group basis (circulating from table to table), or by checking in with all groups at once. This time of year, both teachers and students will need to practice, practice, practice. James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-18940414779660450382015-05-12T05:54:00.001-07:002015-05-12T05:54:25.909-07:00Problematic Behaviors, Classroom Management, and Teaching MathWhen math is taught as a social activity, students are communicating their thinking, learning from others, and gaining confidence in their ability to solve problems. In order to teach math as a social activity, we need to cultivate a learning environment that involves a safe setting for social interaction. Students need to know how to communicate at a low volume so others are not distracted, how to listen to other people's ideas, and how to respond in a respectful manner. We are actually encouraging our students to talk during class (!), but in a manner that is conducive to learning and problem solving.<br /><br />Recently I had the pleasure of sitting in on an hour-long professional development session presented by Paul White at Morse Street School. Mr. White is a Kindergarten teacher and Responsive Classroom trainer, and he gave us a nice overview of Responsive Classroom's four major domains and seven guiding principles. <br /><br />I had been thinking about the impact classroom management has on math instruction and found the Responsive Classroom (RC) model helped to illustrate the connections to the mathematics learning experience really nicely.<br /><br />(You can learn more about Responsive Classroom here:<br /><a href="https://www.responsiveclassroom.org/sites/default/files/pdf_files/rc_brochure.pdf">https://www.responsiveclassroom.org/sites/default/files/pdf_files/rc_brochure.pdf</a>)<br /><br />The first of four domains of Responsive Classroom (Engaging Academics) illustrates the cycle of engagement --> positive behaviors --> engagement --> positive behaviors that you may be familiar with. Excluding emotional outbursts (which can be due to any of a variety of factors), we know that when children are not engaged they are prone to boredom and that can lead to distracting behaviors. Immediately we are faced with a real challenge, because traditionally, math is not the subject every student immediately falls in love with. Engaging students in math takes practice and careful planning. <br /><br />Maintaining a positive community is another domain that is directly related to teaching math. Since anxieties around math begin at an early age, it is crucial that we create a safe climate in our classrooms where students feel comfortable making mistakes in front of their peers. That means setting guidelines for discourse and practicing those guidelines. <br /><br />Awareness of developmentally appropriate language and academics is also important in math, and is one of RC's domains. This is precisely what makes teaching elementary math as complex as teaching high school math, and it is the very part of teaching math that parents often have difficulty understanding. Perhaps you have heard: "Well, I learned that when you divide fractions, you flip and multiply. Why are they teaching with all these weird pictures and models?" Or, "I just don't understand this partial products stuff. It is so confusing!" If you are a parent, perhaps you have even felt these sentiments. We don't remember a lot about our early mathematics experiences, but even back when we were kids, chances are we used manipulatives and models before we learned algorithms. Those things helped us to understand the quantities we were working with before we simplified them into written equations and solved them with operations and algorithms. A great deal of research goes into determining what <i>most</i> children at varying grade levels are capable of and ripe for in terms of discovering and exploring mathematical ideas. As adults, we may remember learning the algorithms, but much of the meat of our mathematics learning happened before that. And for some of us that did not have as much modeling and practice prior to being taught algorithms, math can be more challenging, even as adults. To come full circle with the RC domain dedicated to developmental awareness and its connection to classroom management, consider that for those students who missed out on important early experiences with physical models, games involving quantities, pieces, parts, geometric shapes, and opportunities to discover, math is harder and less engaging. Thus, what we are trying to do with the emphasis on understanding, and experimenting with all these different algorithms, and encouraging our students to share their thinking and solve problems in more ways than one, all that effort is intended to make math a more meaningful and worthwhile experience for children. When the content is meaningful to the students, it becomes easier to teach and classroom management becomes less challenging.<br /><br />The four RC domains are <i>engaging academics, positive community, effective management, </i>and <i>developmental awareness.</i><br /><br />In addition to the domains, RC also has seven <i>guiding principles</i>, and they are also completely relevant to math instruction at all levels. Let's look:<br /><br /><i>1. The social and emotional curriculum is as important as the academic curriculum.</i><br /><i><br /></i>Above, I mentioned the importance of feeling safe in the classroom. Students must feel comfortable making mistakes, and learn how to critique others without ridiculing their effort. This is addressed in another post devoted to facilitating math discourse, but let's just summarize by saying that when a student does <i>not</i> feel safe, learning can be significantly stifled. Speaking personally, if I am in a room of people and I feel like I am the only one who does not understand what is being discussed, I am likely to shut down and not participate in the discussion. I don't want to burden my peers with my inadequate knowledge! It is very difficult for children, too, when they are put in that position. Keeping whole-class classic Q+A style discussions to a minimum and allowing students to work out their confusions in small groups is less stressful for them. Saying "I don't get this" to one or two peers is much less daunting than admitting it to the whole class. <br /><br /><i>2. How children learn is as important as what they learn</i><br /><br />When Mr. White shared this guiding principle, I simply wrote in my notes, "<i>SMPs!"</i> This is precisely why the Standards for Mathematical Practice were written into the Common Core. The SMPs were designed as a guideline for us to give students a wide variety of mathematical learning experiences. Post them in your room somewhere if it helps, because if you can manage to incorporate the SMPs into your instruction, it is one way of differentiating! And one of my very favorite things that Everyday Math has done in their newest edition is divide the Standards for Mathematical Practice into more specific <i>Goals for Mathematical Practice</i>, essentially showing you where these different learning experiences lie within and throughout the curriculum. So <i>yes!</i> How children learn math <i>is</i> as important as what they learn in math. And how does this impact classroom management? By giving your students all these different types of learning opportunities, you are helping to address their learning needs, preferences and styles, and also addressing their weaknesses. This means not only will students be performing more varied tasks in class and solving more interesting problems, but they will also be developing greater understanding which yields greater engagement.<br /><br /><i>3. Cognitive growth occurs through social interaction</i><br /><br />This guiding principle is heavily discussed in the post devoted to Mathematical Discourse, but I'd just like to throw out there that nowhere is this statement more true than in mathematics. Much of mathematics is a language, and languages are meant to be communicated. Students learn and grow by sharing their thinking with one another and having their conjectures confirmed or critiqued. The traditional 20-problem independent homework assignment is standard for engraving procedural sequences in one's brain, but there is not much use for it beyond that. It takes too long to go over 20 problems in class; students who have mastered the procedures are bored during that time; students who did not perform the procedures correctly must now revise the 20 questions (and re-wire their brains from the incorrect procedure sequence they practiced the night before). Today we see more deep problem solving being sent home, and fewer sets of problems to practice procedures with. With one or two problems to try at home, the work can be revisited in much shorter time in class, and mistakes are learned from without becoming so consequential. Moreover, time spent in class can be devoted to sharing strategies, exploring new concepts, and analyzing each others ideas and work. Cognitive growth in math definitely occurs through social interaction, as math is a social activity. <br /><br /><i>4. To be successful academically and socially, children need to learn a set of social and emotional skills that include cooperation, assertiveness, responsibility, empathy and self-control </i><br /><i><br /></i>In order for children to work on tasks together, whether it is in pairs or small groups, they need guidance, instruction, encouragement and practice with the communication skills required for cooperative problem solving. For example, listening is a skill some need more practice at than others, but it is essential for working together. Knowing how to listen to another's ideas, without interrupting, and how to ask questions about those ideas, re-state those ideas, and/or clarify those ideas is necessary before one person can endorse or criticize the ideas of another. Many adults have problems with this! In many ways, children are at an advantage. There is less history, less experience, a.k.a. "baggage," and that allows children to learn how to share different ideas and strategies with a common goal. In other words, children and adults often have a hard time sharing opposing ideas without getting impatient and competitive. But adults have a lifetime of being impatient, under-appreciated, self-conscious, shy, dominant, overconfident or overly anxious that can make this kind of collaboration difficult. For children, there is more room for learning how to collaborate effectively, even debate, with the understanding that everybody wins when a solution is reached.<br /><br />Communication skills for collaborative problem solving should be introduced at the start of every school year, practiced, and re-taught at various times throughout the year. <br /><br /><i>5. Knowing the children we teach-- individually, culturally and developmentally-- is as important as knowing the content we teach.</i><br /><i><br /></i>This comes from <i>formative assessment</i>, which includes the observations you make as a teacher, notes you take about every student, direct communication with students, and data from multiple summative and cumulative assessments. It also comes from communication with parents and other teachers, but that is covered in the next guiding principle. Knowing the children we teach through the various ways of formatively assessing informs our instruction. And in math, this means knowing more than who is strong and who is weak, who has test anxiety and who has trouble writing. Math includes a whole spectrum of different types of problem solving. You might learn that one of your students has a special love of algebraic patterns, while another has specific difficulty with angles and geometric properties. And we all are familiar with the dreaded "does not work well with others" label... You may discover that a student who previously owned that label actually works very well with others in certain types of problem solving scenarios. <br /><br /><i>6. Knowing the families of the children we teach is as important as knowing the children we teach.</i><br /><br />Communicating regularly with parents about math is more important now than ever, mostly because from their perspective, much of the math we are teaching their children just seems <i>weird.</i> There is greater emphasis on <i>understanding</i> concepts than ever before, and that often looks funny on paper. Whether it is a homework assignment or a unit assessment, parents have a lot of questions about what their children are doing in math class. Combine that with all the information parents can share with <i>you</i> about their son or daughter, and y'all have an awful lot to talk about. <br /><br />Another note about communicating with parents.. Year after year I encounter adults who insist that math was just never their cup of tea, and they expect their children to experience the same difficulties in or apathies toward math in school. This can be a real road block to learning, especially as the child gets older and nurtures the image of him or herself as mathematically hopeless. One cannot blame them for having this attitude; it is entirely likely these parents were not given enough opportunities to explore, play with, discover, and eventually appreciate math as children. But it is important for us to to what we can to share with them that their child <i>is</i> a young mathematician, and that we are doing everything we can to help them understand, enjoy and excel at mathematics. <br /><br /><i>7. How we, the adults at school, work together is as important as our individual competence. Lasting change begins with the adult community.</i><br /><i><br /></i>Just as math can (and should, in my humble opinion!) be considered a social activity, teaching math is in the same category. I am so lucky to be able to see so many great teaching styles and techniques as I make my way from school to school, but as far as exposing one teacher's great work to another, the best I can do is tell you about it or write to you about it. Talking to each other about teaching math and visiting each other's classrooms are the best ways to collaborate and learn from each other. Hopefully we can make more time to do just that in the future. The teaching that I witness is too good not to share. As a math strategist, I look forward to teaching with you, learning from you, and passing along the good teaching practices I witness far and wide. <br /><br /><br /><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-33192857867028016692015-05-12T05:54:00.000-07:002015-10-07T05:51:00.078-07:00Teacher Talk vs Turn-and-Talk: Some Ideas for Increasing Mathematical DiscourseIn a book published by Marilyn Burns' <i>Math Solutions</i>, called <u>Classroom Discussions in Math</u>*, the authors outline four steps toward productive math talk in the classroom:<br /><br />1. Helping individual students clarify and share their own thoughts<br /><br />2. Helping students orient to the thinking of others<br /><br />3. Helping students deepen their own reasoning<br /><br />4. Helping students engage with the reasoning of others<br /><br />Students often need to gather their own thoughts before even beginning to clarify them. In an earlier post, we explored the importance of providing think time for students before they engage in discourse (either with each other or when sharing out). Using that think time is an excellent way to help students <i>gather</i>, clarify and share their own thoughts.<br /><br />In the first couple minutes of the Teacher Channel video below, you can see a nice example of a sequence that encourages productive math talk. The teacher asks students to observe and think, then turn-and-talk, then share:<br /><br /><a href="https://www.teachingchannel.org/videos/multiplication-division-in-the-core">https://www.teachingchannel.org/videos/multiplication-division-in-the-core</a><br /><br />The student that is first asked to share what patterns he sees seems relaxed, confident and not intimidated. He was given an opportunity to look at the math, to think about it, and then to share his ideas with a peer and listen to his peer share his ideas (turn and talk), all before being asked to share anything with the whole group. Also, students are all seated on the floor, where they have become accustomed to communicating about math as a group.<br /><br />I have seen some terrific student discourse happening in classrooms I visit. It is late in the school year and students in many classrooms are developing stellar talents for communicating their mathematical thinking. Especially in the K-2 grades, where the new Everyday Math curriculum requires more collaborative thinking and sharing of mathematical thinking than ever before, students are getting used to sharing not just their solutions to problems, but also how they solved those problems. I have even seen students volunteering to restate others' strategies, sharing new strategies, and critiquing each other's strategies. I notice more and more students leaning over their neighbor's desk, walking him or her through a problem to help find an error. They are not just learning how to communicate their mathematical thinking, but they are becoming very good at it too.<br /><br />Once upon a time, in middle school and high school, collaborative learning opportunities used to be a rare thing in the math classroom. Desks were mostly in rows, and looking at your neighbor's strategies, or sharing your strategies with your neighbor, could have landed you in the principal's office for cheating. Teaching usually involved a lot of explaining and not much discovery. It was not all bad, at least for those students who already had discovered an appreciation for math and had confidence.<br /><br />A more traditional style of teaching math, at least ideally, went something like this: <br /><br />First: The teacher talks the class through a new math procedure in front of the class.<br /><br />Then: The teacher gives the class an example to work through together, with guidance from the teacher.<br /><br />Last: The students work on a series of problems, similar to the above work, independently.<br /><br />This approach is commonly referred to as the "I do, we do, you do" approach, and it is a logical way to teach someone how to complete a procedural task. In high school, I was taught how to graph a linear equation that way. <br /><br />Even as we experiment with different classroom techniques, employing a workshop model in the classroom, facilitating group and partner work, using centers, and incorporating movement into our lessons, the tendency to want to simply demonstrate <i>how something is done</i> remains dominant in most of us. "My students are clearly confused about area and perimeter. I know the distinctions between area and perimeter," we think to ourselves, "so I will just show them so they know too," and we launch into a mini-lecture on the topic, complete with nice diagrams and a laser pointer. <br /><br />There are two main dangers that linger when this happens. <br /><br /><i>The first danger</i> is that in order for even a very brief mini-lecture to be effective, you need complete student engagement. As soon as you have to re-direct just one student to face the front or keep his or her hands to his or her self, there is a disruption to your presentation of the topic. If it happens twice, it becomes a burden for those that are listening to remain focused. And this is assuming you began with 100% attentiveness. In the event your students are not all that interested in what you have to say from the start, your mini-lecture is doomed... The more behavioral disruptions the more you lose student engagement, the longer the mini-lecture becomes and the period of time you expect your students to remain focused on you.<br /><br /><i>The second danger</i> is just as significant as the first, but more simple: Your students may not connect with what you are saying. Any number of them might not understand your way of explaining the concept. Students learn math in a myriad of ways, and your explanation, algorithm or procedure may be lost on one or more of them. <br /><br />Since I know (from my own experience) how dominant this tendency to want to explain things to the class is, I am not condemning the practice, but I do offer a suggestion; monitor yourself very carefully when slipping into "explain mode." Know that if your explanation lasts more more than a minute, you have likely lost at least a couple of your students. Also, if it happens more than a couple times during a math class, your students may be missing out on more valuable learning and discovery time.<br /><br />Here is a neat way to think about the learning experience as it pertains to mathematics; <br /><br /><b><i> Think of learning math and problem solving as a social experience.</i></b><br /><br />It is like learning a language. If you have ever had the experience of trying to learn a foreign language, perhaps you can relate. You took Spanish I, and got an A+. <i>Como Estas? Muy Bien, gracias! Spanish is easy!</i> You took Spanish II, and it was harder, but you learned a lot! Spanish III? <i>Oh my, accents, novellas, dictations, I sure am getting good at Spanish! </i> Spanish IV made you a master! <i>I must be fluent now! I feel so confident! </i>You took a spring break vacation to Cancun and it was: <i>What are they saying? How do I respond?? Ayudame! </i>Any foreign language teacher of professor will tell you, you don't become <i>fluent</i> until you <i>immerse yourself</i> in the language. <br /><br />Luckily, we do not need to send our students to Mexico or Spain to immerse them in mathematics. We need to give them opportunities to problem-solve <i>with each other</i> and construct their own solutions <i>together</i>. I used to tell my middle school students scientists and mathematicians rarely make great discoveries in isolation. NASA does not usually send a solo astronaut into space to work on its most precious and complex projects. Problems are sometimes solved independently, but mathematicians and other specialists often work in teams to solve the world's most important problems.<br /><br />Solving problems in pairs and teams can allow students the opportunity to reach greater heights with their thinking, and celebrate greater victories. Even a classroom open response experience is better off spent tackled with a partner or two. A confident solution with a consensus should warrant a high-five! Eureka! Math is fun. <br /><br />As a social activity, math is about sharing, about experimenting, about collaborating, about persevering and about finding solutions. Every part of that problem solving process can be a stimulating, engaging, enjoyable experience.<br /><br />Here are some ways some teachers in RSU 5 are increasing that excellent mathematical discourse that is at the heart of a superior learning experience in the classroom:<br /><br /><br /><ul><li>Incorporate turn-and-talk opportunities into every delivery of whole-class instruction, so your students become accustomed to sharing their thinking with each other regularly.</li></ul><ul><li>Get into the habit of incorporating whole-class sharing and whole-class instruction in a part of the room where students can be seated on the floor, close together and close to you. Turn-and-talks are effective in this setting too.</li></ul><ul><li>Practice specific protocols for communicating in groups, and have students do it <i>every day.</i> For example, speaking at an indoor volume, always remain quiet when a group member is speaking, identify and use specific polite words when questioning or critiquing someone else's thinking (like <i>Can you clarify that for me? </i>and <i>I think you made a mistake. Can I show you?</i>)</li></ul><div></div><div><ul><li>Always be looking for opportunities for students to explore new ideas and new concepts, especially when you are tempted to explain stuff to them. Think: <i>How could I get them to work this out without me showing them how to do it?</i></li></ul></div><div><ul><li>Assign roles to group members, such as presenter, note-taker, ambassador, fact-checker, editor, time-keeper, equalizer (making sure everybody is contributing), etc. </li></ul></div><div><ul><li>Promote a strictly safe environment for sharing in class to promote confidence and reduce anxiety when sharing. Encourage everyone to take each other seriously so nobody is ever laughed at for giving a wrong answer or making a mistake. This includes the promoting of making "brave mistakes" as the essential element of achieving success! </li></ul></div><div><ul><li>Arrange desks in ways that allow students to easily communicate with each other. Tables are best for this, as they can sit across or next to or diagonal from each other and communicate, while having some distance from other groups. </li></ul><div>Do you have other ideas or tips for creating an environment for collaborative learning? Please let me know and I will add them to the above bullets!</div></div><br />*The book mentioned at the top, <u>Classroom Discussions in Math</u>, was published in 2013 by <i>Math Solutions</i> of Sausalito, CA, and was written by Nancy Anderson, Catherine O'Connor, and Suzanne Chapin. <br /><br />Recently published (Fall 2015) articles on Mathematical Discourse:<br /><br /><a href="http://smartblogs.com/education/2015/10/01/orchestrating-mathematical-discourse-to-enhance-student-learning/">Orchestrating Mathematical Discourse to Enhance Student Learning</a><br /><br /><a href="https://www.teachingchannel.org/blog/2015/09/24/collaborative-learning-gbt/">Establishing a Culture of Collaborative Learning</a><br /><br /><a href="http://ww2.kqed.org/mindshift/2015/09/21/10-tips-for-launching-an-inquiry-based-classroom/">Tips for Launching an Inquiry Based Classroom</a><br /><br /><a href="http://www.teachingquality.org/content/blogs/jessica-keigan/learning-loud">Learning Is Loud</a><br /><br /><br /><br /><br /><br /><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-77835719407963458262015-04-02T06:14:00.001-07:002015-04-02T06:14:44.273-07:00Some Things to Keep In Mind Regarding Elementary Geometry...Back in the old days, foundational geometry units were often reserved for the end of the year in many grade levels. Browsing through the preview materials of the new Everyday Math units for grades 3 through 6, I see geometry units and lessons occur at the beginning, middle and end of the year. I don't know why geometry wound up back-loaded into March, April, May and June, back in the day, but I remember it well. For me, geometry was fun and not difficult, whereas all other forms of math gave me greater challenges. For others, operations, algebra and statistics come easy but geometry poses problems.<br /><br />Here are some things we all should keep in mind when we are teaching the basics of geometry to prevent confusion and future geometric mishaps. Please let me know if you have others I should add to this list! These are in no particular order:<br /><br /><br /><ul><li>Many students have trouble remembering the relationship between a square and a rectangle. They often mistakenly use the terms interchangeably. It may take some time for your students to know that a square is a classification of rectangle, or a type of rectangle, and <i>not</i> the other way around. The more you model the correct terminology for them, the sooner they will clear up their misconceptions. A square is a closed figure with four ninety degree angles and four congruent sides. A rectangle only has the four 90 degree angles. When referring to rectangles, it is important that <i>we</i> use the word <i>rectangle</i>, even if it is somewhat close to being a square.</li></ul><div><ul><li>Students and adults get three-dimensional and two-dimensional figures confused <i>all the time</i>. Like the rectangle-square issue, there is also a square-cube issue. A cube is <i>not</i> a square, plain and simple. But nevertheless, it happens, we catch ourselves saying to a child, "Can you put those wooden blocks away when you are finished? Yes, the triangle ones and the square ones..." There is no such thing as a "square block." It is called a cube, even though cubes have squares on them. Using three-dimensional geometric terms appropriately is super important to encourage correct use of geometric terminology. Rectangular solid/prism, triangular prism, pyramid, cylinder, cone, cube and sphere, those are the words to use when describing, well, rectangular solids, triangular prisms, pyramids, cylinders, cones, cubes and spheres. </li></ul></div><ul><li>Don't be afraid to use other geometric terms, even if they sound too advanced. Words like congruent, vertex/vertices, figure, base, height, etc, should all be heard by students when discussing geometry. For example, you can say "the bottom side of the triangle..," just say "the bottom side, or the base, of the triangle..," so students become familiar with the terminology. The same is true for all categories of math, in fact. Everyday Math likes to use kid-friendly terminology, but the rest of the world does not often recognize EM-talk. When you say, "number model," or "number sentence," you can also say, "or an equation." When you refer to "turn-around facts," you can say, "also known as the <i>commutative property..."</i></li></ul><div><ul><li>Think carefully before making definitive statements about shapes, such as, "when you put two triangles together, you get a rectangle," or, "when you cut a rectangle in half, you get two triangles," as statements like these may be dependent upon certain conditions and could lead to significant misunderstandings. Sometimes, two triangles put together make a parallelogram, or even an irregular quadrangle, and sometimes when you cut a rectangle in half, you get two smaller rectangles. </li></ul></div><div><ul><li>Be careful when drawing/ modeling representations of shapes and angles for students. Your "right angle" might look right from where you are sitting off to the side of your whiteboard or easel, but from the vantage point of your students it might not look like a right angle at all. Sometimes I am horrified at what I have drawn or written for my students when I back away from the board and have a proper look! </li></ul></div><div><ul><li>When students are drawing polygons and circles, have them use rulers, compasses and stencils whenever possible. The more they are encouraged to represent such figures as accurately as possible, the less they will be inclined to think their distorted and/or deformed representations are acceptable. Also, these are important mathematical tools they should be comfortable using.</li></ul></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-81657734064451311122015-04-02T06:14:00.000-07:002015-04-02T06:14:36.003-07:00How Much Should I Trust the Spiral? Everyday Math, Engagement, and Best PracticesRecently, EdReports published a Consumer-Reports-like review of the major published K-8 mathematics curriculum materials available to schools right now. Everyday Math was not on the list of reviewed programs, because they are in the midst of updating to their latest edition (EM4), which we are currently using in grades K-2 and will be using also in grades 3-6 starting this coming fall. I did have a chance to look over some of the programs that were reviewed, and it left me feeling very good about the program we've got, and the changes in teaching practices that are occurring.<br /><br />We refer to these changes we are seeing in our math instruction as an "instructional shift" toward greater <i>understanding</i>, moving beyond teaching isolated procedures that are easily forgotten. By asking students to explore more than one strategy, critique others' thinking, and write about their own mathematical thinking, we are giving them greater opportunities to understand the math that they use in their problem solving. For a little more on "teaching for understanding," here is a recent blog post with an interesting example of looking at <i>using multiple strategies</i> from David Ginsburg via EdWeek:<br /><a href="http://4.bp.blogspot.com/-fQogal44si4/VRvjZAu6x-I/AAAAAAAAAE8/2rnWmedzLSE/s1600/Screen%2BShot%2B2015-04-01%2Bat%2B8.22.14%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="http://1.bp.blogspot.com/-ST0JNTz6Bn8/VRvjjad1kCI/AAAAAAAAAFE/l01wjbqVcv8/s1600/Screen%2BShot%2B2015-04-01%2Bat%2B8.22.36%2BAM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2F4.bp.blogspot.com%2F-fQogal44si4%2FVRvjZAu6x-I%2FAAAAAAAAAE8%2F2rnWmedzLSE%2Fs1600%2FScreen%252BShot%252B2015-04-01%252Bat%252B8.22.14%252BAM.png&container=blogger&gadget=a&rewriteMime=image%2F*" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-fQogal44si4/VRvjZAu6x-I/AAAAAAAAAE8/2rnWmedzLSE/s1600/Screen%2BShot%2B2015-04-01%2Bat%2B8.22.14%2BAM.png" height="385" width="640" /></a><img border="0" src="http://1.bp.blogspot.com/-ST0JNTz6Bn8/VRvjjad1kCI/AAAAAAAAAFE/l01wjbqVcv8/s1600/Screen%2BShot%2B2015-04-01%2Bat%2B8.22.36%2BAM.png" height="374" width="640" /><br /><br />Ginsburg's example of looking at two distinctly different ways to solve a problem is one way to "construct viable arguments and critique the reasoning of others," and it helps our students "make sense of problems and persevere in solving them." Almost all of the CCSS Standards for Mathematical Practices (SMP's), in fact, relate to this example. <br /><br />A successful mathematics program requires research, drafts, revisions and trials. There were years when I taught my own custom-designed lessons, and was it ever frustrating how many of my seemingly great lesson ideas flopped, and flopped hard! Sometimes what I thought was fun and engaging turned out to be embarrassing or boring to my students. I had a difficult time deciding how much homework to assign, how long to stick with a concept when my students are not demonstrating mastery, how to differentiate, and how often to incorporate group and partner work into my lessons. I was proud of the home-made lessons that had succeeded, but the many late nights of planning and organizing had taken its toll. <br /><br />When my district offered to give me a published math program to teach the next year, I was humbled at first, but eventually I welcomed the lessons, scope and sequence, pre-made assessments and professional development that came with it. It did not feel like a defeat! It felt like a revelation.<br /><br />I realized that while a curriculum designed by mathematicians at a big university and published by some large profit-seeking corporation may not be everything I, personally, want them to be, I have the expertise and experience as a teacher to deliver them effectively, and not feel as though I am sacrificing my instructional integrity and creativity. In addition, the lessons I designed, no matter how hard I tried to diversify them, tended to reflect my own teaching styles and preferred strategies. In other words, I didn't do a very good job of incorporating all of the SMP's into my instruction.<br /><br />So this brings us back to Everyday Math, the McGraw-Hill published program we use as our K-6 math curriculum. In its new incarnation, it is designed as a Common Core program, which is to say, all its lessons and units were designed to teach the Common Core State Standards for Mathematics, including the Standards for Mathematical Practice. The program spirals, which is to say it teaches concepts continually throughout the school year, and the kindergarten to sixth grade span of years. <br /><br />The spiral concept seems to have come slightly under attack in some of the discussions and readings I have come across. Older editions of Everyday Math have been criticized for not diving deeply enough into concepts before jumping ship and switching to a completely different concept, sometimes leaving students without a deep enough understanding of what was taught. By the time the concept "spirals back again," some students require total re-teaching, while others retain enough to take their understanding to a new level. <br /><br />In reality, almost every math program that spans more than one or two grade levels also spirals its content. The Common Core itself spirals its content. One major difference between the old Everyday Math and the new Everyday Math (EM4) is that the spiral now (more or less) parallels the spiral of the Common Core, teaching specific content within specific domains every year. In earlier professional development this year, we looked at how important place value is throughout different points of the K-8 span. It appears in the Common Core standards many times, and it shows up in the EM4 curriculum many times as well.<br /><br />Another major difference with EM4 (which we have addressed before and will continue to address again) is that it places a much greater emphasis on the teaching and learning practices outlined in the Common Core Standards for Mathematical Practice. In two years' time, third grade teachers will notice a big difference in their students' abilities when it comes to writing about their mathematical thinking. Students in first and second grade are already becoming far more versed at using diagrams and words to describe <i>how</i> they found the solutions to their problems. And kindergarten students are sharing more of their thinking out loud and are learning how to communicate their mathematical thinking as well.<br /><br />Teachers of Everyday Math are often asked to "trust the spiral," but I think the better way to look at it is to <i>understand</i> the spiral, to <i>know that it is there</i>. What our students are experiencing now is going to help them with their learning weeks, months and years from now. This is true both with regards to the content standards and the standards for mathematical practice. The place value lessons they get now, whether or not they completely master them 100%, will help them with their addition later in the year, and their ability to manipulate data next year, and their work with rational numbers two years after that. Likewise, the oral communicating they are doing in kindergarten math lessons will help them to express themselves on paper in first grade, which will help them become more confident with problem-solving tasks in third grade. In the end, we will see far more students demonstrating greater understanding of the math they are doing, and showing greater confidence in their solutions. Fewer students in sixth grade, when faced with the task: "Explain how you know your answer is answer is true," will respond with exasperation and fear.<br /><br />Finally, when it comes to teaching a math curriculum like Everyday Math, it is best to look at the curriculum as a presentation of math concepts (standards) put together by expert mathematicians that <i>you </i>will be collaborating with by teaching it! In fact, we are all collaborating together, for when we see something in Everyday Math we don't understand, or don't like, we seek each other out to make sense of the situation and come up with a solution. Your teaching expertise, along with the support of colleagues and myself, and the research, drafts, revisions and trials that went into the authorship of EM4, make up the collaboration needed to facilitate a unique, engaging and meaningful math learning experience for our K-6 students.<br /><br /><br /><br /><br />James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-3897344066122213282015-03-17T08:15:00.000-07:002015-03-20T06:46:54.892-07:00'Explain Your Thinking' RevisitedNow that we are well into the month of March, spring is rumored to be approaching, and the MEA's are here. With the implementation of the new Everyday Mathematics 4 program in grades K-2, there has been a significant instructional shift toward "self-explaining" in problem solving, and it might be helpful to revisit that at this time. It might also be helpful for teachers of grades 3 and up to know how it is going for their future students.<br /><br />On Friday the 13th of March, all K-2 teachers worked with Lisa Demick and me to review and assess how things are progressing this year, to unpack upcoming units, and to look ahead into next year. 'Explaining your thinking' has been a major theme with the new EM4 program, and it was a theme on our day of professional development too.<br /><br />There were some challenges at the start of the year. Students struggled with the self-explaining problems, often not knowing what to say and (especially) what to write. Teachers were faced with the difficulty of scoring the self-explaining problems on assessments, and finding ways to support students who were having difficulties. And of course parents were asking questions about the self-explaining problems and wondering why their children were suddenly "developing" when last year they were "achieving" or "exceeding." These challenges were discussed during Friday's professional development, and most if not all teachers shared these experiences between early in the year and now. <br /><br />The good news is teachers also reported seeing some real improvement in students' ability to explain their thinking, both orally and in writing. It was also noted that students are gaining confidence in their mathematical thinking! This was refreshing to hear, as it might be the most important intended outcome of this instructional emphasis being placed on self-explaining problems. The more opportunities we provide to share their thinking, to record their strategies, to collaborate and communicate with each other about their problem solving, the more confident our students become with their mathematical thinking.<br /><br />The theme of "explaining your thinking" and communicating mathematical thinking is emphasized in the Common Core Standards for Mathematical Practice. On Friday morning teachers reviewed the curriculum materials and recorded some of the specific expectations for each grade level at different points of the year. Here are some of the results.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-antXWCANWxw/VQg-AD1FXXI/AAAAAAAAAD4/Q3UxGP0fhk0/s1600/IMG_0079.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-antXWCANWxw/VQg-AD1FXXI/AAAAAAAAAD4/Q3UxGP0fhk0/s1600/IMG_0079.JPG" height="640" width="476" /></a></div><br /><br />The above image displays some of the ways Kindergarten students are given opportunities to share their mathematical thinking at about halfway through the year. One example is "students are asked to draw, describe and compare shapes and vertices." Another is "Students are asked to describe shape names and use positional words."<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-i-9aizMbxzw/VQg-jBP3KiI/AAAAAAAAAEA/UfAwXIPgFH8/s1600/IMG_0077.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-i-9aizMbxzw/VQg-jBP3KiI/AAAAAAAAAEA/UfAwXIPgFH8/s1600/IMG_0077.JPG" height="640" width="476" /></a></div><br />Here (above) are some examples of ways students are communicating their math thinking in early 1st grade. "Partners discuss their responses and self-evaluate," and students are asked, "Why is it important to name your unit of measurement?"<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-vnao3tspsRQ/VQhAKugqc1I/AAAAAAAAAEM/dL8BDAy1PbI/s1600/IMG_0073.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-vnao3tspsRQ/VQhAKugqc1I/AAAAAAAAAEM/dL8BDAy1PbI/s1600/IMG_0073.JPG" height="476" width="640" /></a></div><br />Halfway through 2nd grade, students are "encouraged to use each other's strategies," and explain how more than one strategy can work. See above.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-nn8Y0z62bHc/VQhAlYIWXRI/AAAAAAAAAEU/XDQWXMcCgp0/s1600/IMG_0071.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-nn8Y0z62bHc/VQhAlYIWXRI/AAAAAAAAAEU/XDQWXMcCgp0/s1600/IMG_0071.JPG" height="476" width="640" /></a></div><br />By the time students are in the 3rd grade, students continue to make sense of others' strategies and practice communicating their mathematical thinking with stems such as, "I noticed..., I wonder..., How did you..., and, Why did you..." Students are expected to communicate their thinking in writing more fluidly and are given many opportunities to practice.<br /><br />Self-explaining problems are the meat of the new Everyday Math program, and are hugely important in building understanding and confidence in mathematics. That is why it is emphasized so much in the Standards for Mathematical Thinking (SMP's). <br /><br />Everyday Math 4 has done us a big favor by taking the Common Core SMP's and breaking each SMP into several Goals for Mathematical Practice (GMP's). Below is a list of all 8 of the <b>Standards for Mathematical Practice</b> and their accompanying Everyday Math <b>Goals for Mathematical Practice</b>. These practice standards are key to the instructional shifts we speak of:<br /><br /><div class="p1"><span class="s1"><b>SMP1: Make sense of problems and persevere in solving them</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP1.1 Make sense of your problem</span></div><div class="p3"><span class="s1">GMP1.2 Reflect on your thinking as you solve your problem</span></div><div class="p3"><span class="s1">GMP1.3 Keep trying when your problem is hard</span></div><div class="p3"><span class="s1">GMP1.4 Check whether your answer makes sense</span></div><div class="p3"><span class="s1">GMP1.5 Solve problems in more than one way</span></div><div class="p3"><span class="s1">GMP1.6 Compare the strategies you and others use</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP2: Reason abstractly and quantitatively</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP2.1 Create mathematical representations using numbers, words, pictures, symbols, gestures, tables, graphs, and concrete objects</span></div><div class="p3"><span class="s1">GMP2.2 Make sense of the representations you and others use</span></div><div class="p3"><span class="s1">GMP2.3 Make connections between representations</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP3: Construct viable arguments and critique the reasoning of others</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP3.1 Make mathematical conjectures and arguments</span></div><div class="p3"><span class="s1">GMP3.2 Make sense of othersâ€™ mathematical thinking</span></div><div class="p4"><br /></div><div class="p1"><span class="s1"><b>SMP4: Model with mathematics</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP4.1 Model real-world situations using graphs, drawings, tables, symbols, numbers, diagrams, and other representations</span></div><div class="p3"><span class="s1">GMP4.2 Use mathematical models to solve problems and answer questions</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP5: Use appropriate tools strategically</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP5.1 Choose appropriate tools</span></div><div class="p3"><span class="s1">GMP5.2 Use tools effectively and make sense of your results</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP6: Attend to precision</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP6.1 Explain your mathematical thinking clearly and precisely</span></div><div class="p3"><span class="s1">GMP6.2 Use an appropriate level of precision for your problem</span></div><div class="p3"><span class="s1">GMP6.3 Use clear labels, units, and mathematical language</span></div><div class="p3"><span class="s1">GMP6.4 Think about accuracy and efficiency when you count, measure, and calculate</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP7: Look for and make use of structure</b></span></div><div class="p2"><span class="s1"><b></b></span><br /></div><div class="p3"><span class="s1">GMP7.1 Look for mathematical structures such as categories, patterns and properties</span></div><div class="p3"><span class="s1">GMP7.2 Use structures to solve problems and answer questions</span></div><div class="p2"><span class="s1"></span><br /></div><div class="p1"><span class="s1"><b>SMP8: Look for and express regularity in repeated reasoning</b></span></div><div class="p2"><br /></div><div class="p3"><span class="s1">GMP8.1 Create and justify rules, shortcuts, and generalizations</span></div>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-46738303541362502802015-02-10T06:49:00.000-08:002015-02-13T10:47:05.430-08:00Smarter Balanced is Coming, What Am I Going To Do?The new Maine Educational Assessments are coming, as designed by The Smarter Balanced Assessment Consortium, and they are creating a fair amount of anxiety. It is my hope to ease that anxiety with this post!<br /><br />Here are some important things to consider when looking ahead to these tests.<br /><br /><ul><li>This is the first time ANY classroom in Maine has officially taken these tests (minus the folks that did the trial tests last year), so EVERYONE will be in the same boat. </li></ul><ul><li>Let's remember, at the end of the day, it is just a test we are talking about here. One measure alone will not bring us down, or any school, or any class, or any teacher. It is a test to see what our students know, and we should milk it for everything it offers. Lisa reminds us also that this is a baseline year for this assessment, so there is really nowhere to go but up from here. And on that note...</li></ul><ul><li>The assessment can teach us a lot! <span style="color: blue;"><i>You can go online and take practice tests for grades 3, 4, and 5... I highly recommend this (link is below). Even if you teach K-2, taking one of these practice tests (they are about 26 questions long) will help to show you what kind of thinking we want to be encouraging in our classrooms.</i></span> Likewise, if you teach 3rd grade, taking the 5th grade assessment will give you an excellent idea of what you are currently preparing your students for. Is it a perfect test? No.. In fact I did see a few items I would change if it were up to me, but that has been the case with almost every math assessment I ever took, saw, or gave. </li></ul><ul><li>The test is hard, but really, tests should be hard. They are designed with adaptive software that, like the NWEA tests, selects new items based on how the student answered the previous question. Unlike the NWEA, the new MEA/ Smarter Balanced tests include more items that are more difficult to get right by chance. In other words, in order to get most of the questions right, you really have to know how to do the problem. For example, instead of having to choose which answer out of five options is correct (a 20% random probability), a student might have to choose more than one possible answer out of five. In many cases, students have to type their numerical answers rather than click on a choice.</li></ul><ul><li>Next year, and the year after, your students will do better and better on this assessment. In the case of the math assessment, the items are designed to allow students to demonstrate understanding of concepts they have learned during the year. The more we teach for greater understanding year after year, the better prepared our students will be for this assessment.</li></ul><ul><li>The old methods of "teaching to the test" really won't work in this case. Many of us used to cram the night before a major exam when we were in college, and many of us have crammed with our students during the weeks before a major assessment. <i>Oh no</i>, we thought to ourselves, <i>my students don't know operations with fractions! We'll have to go over that tomorrow! </i>The most reliable way to teach to these new tests is to teach for greater understanding, all year long. Give them opportunities to problem solve, to collaborate, to challenge each other's thinking, and to explore and play meaningful games under your supervision. The new Everyday Math units do a pretty good job of providing those opportunities in each lesson, so that will have a positive impact on our students' performances on assessments in time.</li></ul><br />So go ahead and try the practice tests. Don't let it be intimidating! Instead of worrying about how many students will struggle, think about what instructional adjustments can be made over time to increase understanding and confidence in your students in the years to come. Below the link are a few concepts I took note of while taking the practice assessments. I understand the temptation to want to abandon everything and focus on those concepts expected to appear on the new tests, but I strongly encourage all to think instead about ways to encourage understanding and make the learning experience as rich and as enjoyable as possible (for you and your students).<br /><br /><a href="https://login4.cloud1.tds.airast.org/student/V85/Pages/LoginShell.aspx?a=Student&c=SBAC_PT&logout=true">Smarter Balanced Practice Tests</a><br /><br />Grade 3:<br /><br />The word "unknown" appears many, many times. <br /><br />The word "equation" appears many, many times (as opposed to "number sentence").<br /><br />The word "expression" appears also.<br /><br />The relationship between multiplication and division is emphasized heavily<br /><br />There are a number of questions where students are asked to place fractions on a number line<br /><br />Filling in fraction bars to model equivalent fractions<br /><br />Area and perimeter of multi-sided right-angled figures (for example, a rectangle with a inverted corner)<br /><br />There was one question about telling time on a traditional clock<br /><br />"mass" in grams is mentioned in a word problem; students are expected to understand what "mass" is.<br /><br />One question asked students to make a rhombus that is also a rectangle.<br /><br />Parentheses appeared in a number of problems, with order of operations used.<br /><br /><br /><br />Grade 4:<br /><br />Many basic operations problems are written as word problems... There are many, many of these word problems.<br /><br />Understanding that angles are properties of rectangles, rhombi, and parallelograms will be helpful<br /><br />Understanding the relationship between measures of length, such as millimeters and centimeters will be important<br /><br />Students will benefit greatly from being able to <i>model</i> division with remainders, in other words, they should be able to tell you a story that involves division and remainders.<br /><br />Understanding the relationship between numerators and denominators is important, i.e., what it means when one is bigger than the other.<br /><br />Students should be familiar with solving problems that involve fractions as <i>amounts</i> of things<br /><br />Understanding how to represent decimals as fractions with 10, 100, and 1,000 as denominators is important<br /><br />Students should know what an <i>equation</i> is (never once did the assessment use the term "number sentence").<br /><br />Mixed numbers are all over the place; comparing them, adding them, representing them pictorially..<br /><br />I saw some emphasis on partial products for multiplication.. and lots of checks for understanding relationships between multiplication and division (such as A times B = C can also be expressed as C divided by B = A)<br /><br />Visual representations/ models of fractions and mixed numbers are frequent<br /><br /><br /><br />Grade 5:<br /><br />Exponents! Especially with a base number of 10..<br /><br />Word problems with mixed numbers... adding them, subtracting them, multiplying them with like and unlike denominators<br /><br />Word problems with measurement of mass in kilograms, and using the word "mass"<br /><br />Construction and deconstruction of mixed numbers and improper fractions<br /><br />Volume, volume, volume... It shows up in calculations and word problems. Not just being able to calculate the volume of a rectangular prism, but being able to understand the meaning of volume when it appears in a word problem.<br /><br />Gallons and cups! <br /><br />Comparing decimals to the hundred-thousandth place<br /><br />Area models used frequently for multiplication and division<br /><br />Plotting points on the coordinate plane! James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-77877419338377492502014-12-18T05:50:00.000-08:002014-12-18T05:50:59.327-08:00The Language Of Math, The Language in Math, and Math as a Foreign Language The math anxieties and math phobias so many people experience result from a lot of misconceptions about mathematics. Many people simply consider themselves not mathematically inclined, as if there is a dreaded gene responsible for being bad at math, or some sort of Mathematics Deficiency Syndrome (probably caused by drinking blue Kool-Aid as a kid, or eating those Pop Tarts with the red sprinkles). <br /><br />Sadly, many people have convinced themselves they are bad at math because when they were a child, an adult told them so. Sometimes it was a teacher that bore the bad news:<br /><br />"Some people just aren't math people, and based on your grade in my class, that some people is YOU." <br /><br />Other times it was a parent:<br /><br />"Don't worry, son/ daughter, you just happen to come from a long, proud line of good people who are very bad at math. Just remember that we love you anyway." <br /><br />What compounds the problem of anyone mistakenly believing he or she is hopeless at math is that it is commonly accepted and shrugged off, or even worn like a badge! It has become cool in some middle school and high school circles to be "mathematically challenged."<br /><br />There is no doubt that we all learn in different ways, and some of us excel in areas where others struggle. But math is such a large and vague category of something to be good at or bad at; while about 7% of the population are born with dyscalculia, the math equivalent of dyslexia, this does not mean such individuals cannot excel in problem solving, or geometry, or even advanced mathematics. Math is a great many subjects in one, and almost all barriers to success in each can be overcome.<br /><br />There are all sorts of learning disabilities out there, including dyslexias and dyscalculias, but we do not dismiss them as conditions we can do nothing about. When a child has a reading disability, we work diligently with the child to make sure he or she can read. Likewise, if a child has a learning disability related to numeracy, we work just as hard. These learning challenges affect a small percent of our students, yet far more adults and teenagers gleefully proclaim they are no good at math.<br /><br />What we are experiencing in our classrooms is not a great video game induced increase of Mathematics Deficiency Syndrome, or an increase in the number of children borne with the bad-math-gene. What is far more likely is that our students today are coming from a wider variety of math backgrounds. Important learning experiences may or may not have impacted their brains at that crucial stage of child development where kids have huge capacities for learning language and mathematics.<br /><br />That's right! Language <i>and</i> mathematics. Many parents remember when their children began soaking up words and phrases like a sponge, and those of us who are not parents get to see examples of this every day posted on facebook. "Watch this video of our Junior reciting the French alphabet backwards!" Brain research has determined we gain huge amounts of language knowledge between the ages of 3 and 9. And that also happens to be when children learn a lot of number sense and become fluent with their facts. That is, unless they don't. <br /><br />Just like a foreign language, math is a language that is best introduced at an early age, and not just with flashcards, but with actual verbal intercommunicating. We don't learn to speak a foreign language by memorizing long lists of vocabulary words alone, and children do not learn to become fluent in math by memorizing long lists of facts. We learn new languages by speaking them. Beginning Spanish students learn how to respond to "Como estas?" before they knew how to spell it, so they can gain an understanding of how the language works and what it sounds like. Likewise, we often teach the word "half" before we teach what a denominator and a numerator are, and how to even write fractions, so we can introduce the concept for our students to understand.<br /><br />A criticism of the earlier editions of Everyday Math and other broad math curriculum programs is that they tried to squeeze too many topics into one school year, and sometimes did not give children deep enough understandings of the concepts being introduced to them. Some students, especially those who for one reason or another may have missed out on important early childhood mathematical learning experiences, advanced through the grade levels with less than stellar understandings of basic operations and concepts. <br /><br />What we are seeing more and more of with modern mathematics instructional resources and curriculum materials is an effort to teach mathematics more for understanding, and less for procedural memorization. Even the Common Core features fewer content standards per grade level, with the expectation that math curricula will be constructed to allow for deeper exploration into math concepts. In addition to the content standards, the Common Core also includes Standards for Mathematical Practice, which are guidelines not for <i>what</i> math should be learned, but <i>how</i> that mathematics learning should occur. There is a tremendous effort to incorporate a wide variety of learning experiences into mathematics curricula nationwide.<br /><br />While this is seen by many as a big step toward increasing math proficiency, it also presents challenges. Students are being asked to problem solve, express their thinking on paper and share strategies with peers not only more often, but at a younger age (see previous post in this blog from November 20, Everyday Math, Cognitive Development, and "Explain.."). <br /><br />Learning mathematics relates to language learning, and today students are also being asked to use language more often in mathematics. This is hard for students who struggle with language learning, and students who might not be learning in their first language to begin with.<br /><br />In Lewiston, Maine, nearly 20% of all students are learning English as their second language. When I taught in Lewiston, I often found it difficult to determine whether my ELL (English Language Learner) students were struggling with the mathematics of a particular lesson or unit, or with the language itself. I quickly learned to slow my delivery in the classroom, and to check for understanding with greater frequency, but I still found it to be one of the most difficult challenges of my teaching career. I wanted to be able to speak in <i>their</i> first language. Progress in math can come more slowly for students who have difficulties with reading, and for students who have had fewer experiences playing with numbers and quantities in pre-school. Now consider a student who is just beginning to master the English language after being raised to speak one or more other languages.<br /><br />Jessica Sturges shared with me some important pieces for teachers to incorporate into their instruction when there is an ELL student on board (in fact, I would recommend these practices even if there are no ELL students in the class):<br /><br />#1: Speak slowly and clearly<br /><br />#2: Provide students with vocabulary lists that pertain to the current unit to take home and study before the unit is taught. <br /><br />Here is a sample Everyday Mathematics graphic organizer Ms. Sturges shared with me that shows how she presents unit vocabulary to some of her students:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-erm5_-D7f_A/VI8Ps-L994I/AAAAAAAAACQ/5HTaKdflGN4/s1600/photo.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-erm5_-D7f_A/VI8Ps-L994I/AAAAAAAAACQ/5HTaKdflGN4/s1600/photo.JPG" height="400" width="298" /></a></div><br /><br />In this example, there are some words listed in the student's first language, but that is not always an option, and it is not always necessary. A word bank used in this manner could be helpful to any student learning new concepts. <br /><br />Finally, an important thing to remember both with ELL students and other students who have trouble showing their thinking on paper, is that the language of mathematics can be <i>confusing</i>. Upper elementary, middle school, and even high school students can still be heard using words like "times" or "<i>timesed</i>" when they really mean <i>multiply</i> or <i>multiplied. </i>Early elementary is a time to encourage all students to use "add" and "plus" appropriately, as well as "subtract" and "minus." <i>Add</i> is the verb and <i>plus</i> is a conjunction. <i>Subtract</i> is a verb and <i>minus</i> is the conjunction ("take away" is often used in kindergarten and first grade in place of "minus," but by second grade, we should be encouraging the use of <i>minus</i> as the proper conjunction). And likewise, <i>multiply</i> is the verb and <i>times</i> is the conjunction. Division has some challenges as well... Remember <i>guzinta</i>? 6 <i>guzinta</i> 42 <i>how</i> many times?? Division all too often represents a great void of essential understanding. I think we'll save that for a separate post.<br /><br />It is hard for struggling students. Minus, subtract, take-away, what is the difference, how many more than, how many less than, how many are left... The list goes on. It is ok to use and encourage <i>all</i> of these representations of subtraction, but it is also important to check for understanding frequently.<br /><br />There have been entire books written on the language connection to mathematics, but for now let's just remember that as we present new ways of learning mathematics, we are trying to tap the natural hunger for learning that occurs in a child's developing brain, in order to build a greater foundation for understanding as they tackle more complex mathematics concepts in the future. This is why we use the word <i>fluency</i> in mathematics (more on that topic in a future post!). The more fluent we are in a language, the less it actually <i>hurts</i> our brains to use that language. The same is true of math. With greater "fact-fluency" and overall mathematical fluency, complex problem solving becomes less painful, and potentially enjoyable and engaging.<br /><br />For further reading on math and the brain, and the links between mathematics learning and language learning, here are a couple online resources:<br /><br /><a href="http://www.pbs.org/wgbh/misunderstoodminds/mathbasics.html">http://www.pbs.org/wgbh/misunderstoodminds/mathbasics.html</a><br /><br /><a href="http://www.mathematicalbrain.com/dysclink.html">http://www.mathematicalbrain.com/dysclink.html</a><br /><br /><a href="http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx">http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx</a><br /><br /><a href="http://www.doe.virginia.gov/instruction/esl/resources/strategies_teach_math.pdf">http://www.doe.virginia.gov/instruction/esl/resources/strategies_teach_math.pdf</a><br /><br />And here is a highly informative series of links from PBS' "Child Development Tracker," for several age levels relating to learning mathematics:<br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/five/mathematics.html">http://www.pbs.org/parents/childdevelopmenttracker/five/mathematics.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/six/mathematics.html">http://www.pbs.org/parents/childdevelopmenttracker/six/mathematics.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/seven/index.html">http://www.pbs.org/parents/childdevelopmenttracker/seven/index.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/eight/index.html">http://www.pbs.org/parents/childdevelopmenttracker/eight/index.html</a><br /><br />And relating to language learning:<br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/five/language.html">http://www.pbs.org/parents/childdevelopmenttracker/five/language.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/six/language%20.html">http://www.pbs.org/parents/childdevelopmenttracker/six/language%20.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/seven/language.html">http://www.pbs.org/parents/childdevelopmenttracker/seven/language.html</a><br /><br /><a href="http://www.pbs.org/parents/childdevelopmenttracker/eight/language%20.html">http://www.pbs.org/parents/childdevelopmenttracker/eight/language%20.html</a>James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0tag:blogger.com,1999:blog-8497867494692526070.post-31256538938081812182014-11-21T05:19:00.000-08:002014-11-21T05:19:19.712-08:00About Teaching Fractions... How Third Grade Math Links to Sixth Grade MathI had a conversation with an educator at Freeport Middle School recently about the great number of middle school students who still lack basic understanding of fractions. While I have a great appreciation for Everyday Math and all the research that goes into it, a common criticism of the program over the years is that it hasn't addressed many math concepts deeply enough. The spiral format allows students opportunities to experience and re-learn things they might have struggled with in the past, but often those learning experiences have been brief and far between. Having taught middle school math in multiple districts (some were Everyday Math districts and some were not), my students often displayed learning deficits when it came to fractions. And for the middle school teacher, that poses a far too common dilemma: Do I take extra time to teach them foundational mastery of fractions, or do I drag them through the unit with the limited understanding they currently possess?<br /><br />There is good news on two fronts! Firstly, Everyday Math is rolling out its latest edition of the program (Everyday Math 4), and from what we have seen in the new grade K-2 units we are piloting this year, there seems to be deeper exploration into key concepts. The new edition of Everyday Math 4 is aligned to Common Core, which brings us to good news item number two: The third grade Common Core standards emphasize teaching fractions not just pictorially, with fraction bars, rectangles, pizzas and divided quantities, but also representing fractions on a <i>number line</i>, so students can also become familiar with fractions as sequential numbers. <br /><br />Understanding fractions is crucial for success in middle school, but having a rich learning experience involving hands-on opportunities, part-to-whole relationships, parts-of-quantities relationships, and placing fractions on a number line will also have a positive impact on our students' understanding and mastery of multiplication and division.<br /><br />Below is a link to a short video that explains about how fractions are introduced in the Common Core math standards. The video gets a little wonky around 1:30, but it only lasts a few seconds, so you shouldn't (?) have to take any dramamine. It is worth watching, in my opinion. :^)<br /><br /><a href="http://www.edweek.org/ew/articles/2014/11/12/12cc-fractions.h34.html?cmp=ENL-EU-MOSTPOP">http://www.edweek.org/ew/articles/2014/11/12/12cc-fractions.h34.html?cmp=ENL-EU-MOSTPOP</a><br /><br />Please contact me if you are looking for more strategies relating to teaching fractions.James Galehttps://plus.google.com/112586452652974842740noreply@blogger.com0