Tuesday, April 25, 2017

Fraction representations should be accurate.. which is not always easy

I 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.

Before I get into some examples, let me just say that the main message of this post is this:  When modeling fractions, always do so with accuracy and precision, 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.

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:


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.

Now observe the image below.


Here is the same model, but drawn as a square 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 algorithm for calculating only that is entirely devoid of conceptual representation of quantity.  We do not want students confusing rectangular area models with lattice.  

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.

***

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:


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 almost accurate, and after all, it is divided into three parts, which is the important part, right?  Not so much.

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 much of the whole they actually represent.  Partitioning a whole into thirds means three EQUAL parts, and their models must represent that. 

***

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.

Thursday, March 9, 2017

More Meaningful Math Visual Displays...

Allow me to share with you a few more excellent visual displays I have encountered in my recent travels.

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.

What IS great information to display in math?  How do I make sure my students utilize visual displays for math?

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:

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.



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. 




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!!  ;^)



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.



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.  



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.



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."

Grades 3-5: Talkin' Fraction Immersion Blues

It'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.

Take this problem, for example, from a fifth grade, Unit 5 math box:

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 multiplying fractions, but... is this a multiplication problem?  Let's have a look at each part.

Gwen is 7/8 of the way through a race.  It is important students recognize that means Gwen is almost finished with her race.  It's an essential piece of information, and it needs to be understood.

Next, we learn that he family was cheering for her when she was 2/3 of the way through the race.  It is important for students to know what that means as well.  At 2/3 of the race, is Gwen almost finished?  Is it as close to the finish as 7/8?

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 prove that the operation is multiplication.  They won't be able to, because it is a subtraction problem.

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 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).

So the number model is: 7/8 - 2/3 = X.

The key here is the meaningful conversations, the discourse 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.

Fractions are really hard for children to grasp, especially for those who have not become fluent and automatic with multiplication, and/or have not mastered 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.

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 explain 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.

Allowing maximum time for students to collaborate and communicate their thinking around fractions is essential this time of year.

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.

Wednesday, February 1, 2017

Subtraction and Multiplication: Fact Fluency and Automaticity in the Early Grades

It 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.

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.

Since high school advanced math courses are electives, students have either mastered the prerequisite skills to take the course, or they have almost 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.

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.  Research 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?

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?

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 rational number? What's the difference between a ratio 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 divide fractions?!

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 use 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.

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.

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.

3rd grade.  Multiplication strategies and introduction to fractions are a big focus at this point.  This is where we expect students to become both fluent in multiplication (able to understand how to multiply, what multiplying is, and to choose from multiple strategies to solve multiplication problems) and to achieve automaticity in multiplication facts (the ability to know 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 fluent and automatic 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.

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, How many are left? How many more? How many less? What is the difference? Take away, subtract, minus, and how many in all NOW?  Subtraction presents the first real algebraic thinking students encounter, as the difference represents an unknown addend in an addition problem.  Simply memorizing subtraction facts is not enough for students to truly master and become both fluent and 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.

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 and automatic in subtraction and multiplication will 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.

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.

Monday, January 30, 2017

Keeping Up with Pacing

We'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 nearly on pace (this is good!), few were actually on 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 some 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.


Here's why pacing is important.
  • While it is definitely not effective to cruise through lessons at breakneck speed, skipping over important practice activities, game opportunities, or other important parts of the lessons, we do 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 do the best we can to assure that everyone has had an equal opportunity to learn grade level mathematics.

Here are some tips to help you keep from falling behind.
  • 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. 
  • 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 often is your students will get used to the routine and learn to work more efficiently.
  • Be as brief as possible in the warm-up and the math message.  Remember, the mental math in every warm-up is a quick 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 Focus portion of the lesson.  Progress through the Math Message swiftly so students can spend more time working on the rest of the Focus and Practice parts of the lesson.

Here are some things you can do to catch up if you are falling behind the pacing guide.
  • 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.

Please don't hesitate to contact me regarding any pacing concerns you may have.