Friday, December 11, 2015

Finding Time to Differentiate

Differentiating 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:  How to differentiate and when to differentiate.

Let's start with the when, 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 how also, because differentiation strategies are not universally  effective, but first we should make sure we can carve out time during the lesson to differentiate.

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.

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.

Collecting information.

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 in math."  What part of math?  "This student really excels in math."  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.

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.

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 information I don't mean paragraphs or even sentences.  I just mean information.  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?"

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.

Carefully managing instructional time.

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.

  • 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-- mental math, and it is not really a time to make sure everybody has the correct answer and understands everything.  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 explain 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..
  • 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.
  • 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.   
  • 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 my explanation, which is stated in my own words, expressed only as 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 anti-differentiation way.  For now, I suggest not going too far down that road for K-6 instruction.
Infusing differentiation into your lesson.

Now is when we will begin to merge the how in with the when.  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.

  • 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.    
  • 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.  
  • 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.  
  • Offer extensions/ extra practice activities as optional homework or classwork.  I am not a fan of assigning extra work 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.  
  • 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.
  • 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. 
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.


Friday, October 16, 2015

Flexible Language and Everyday Math

In 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.
  • Number sentences and number models are terms Everyday math uses to represent expressions, equations and inequalities.  It is especially important to familiarize your students with the word "equation" alongside the Everyday Math terms, as equation does not show up regularly in Everyday Math units until grade 4.  There will be many times when students come across the term equation, 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:  "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."  Or another example would be:  "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."
  • Number Stories  are word problems.  "Number story" sounds a lot more pleasant to deal with than "word problem," 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:  "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."  
  • Turn-around rule is the term Everyday Math uses to refer to the commutative property 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: "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." 
  • Names is the term Everyday Math uses to refer to different representations 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."
  • Use your reading and writing terminology 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! 
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.


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:




Thursday, September 17, 2015

Videos!

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.

Added December 2015:

For those interested in an alternative reporting program to ConnectED for trimester 2:
Assessment Spreadsheet Instructional Video

Added October 2015:

10/27: Scoring CHALLENGE items on the cover sheet

10/26: Intro to the new NEW cover sheets (Unit 2 and beyond?)

Change Quick Entry/ Evaluation to 4-Point Scale with Exceeds

This is a GREAT 11 minute video that is HIGHLY RELEVANT to how we are approaching problem solving:    Dan Meyer's 2010 Video about Problem Solving

Added September 2015:

9/25: How to use the new cover sheets

Adding Student Content to ConnectED

How to Enter Data for BOYA

How to Enter Data for Unit Assessments

Whoa, 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.





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.

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 entertain 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 create imaginary problem solving scenarios to occupy and activate our brains because it is fun 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 interesting and engaging problems.  Sometimes they will solve problems independently, but more often your students will be working in groups and pairs to solve problems.

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.

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 being told how 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 expecting to be told how to solve a problem before being given the task to solve a problem.  They are used to being given the strategies to solve problems, rather than being given time and collaborators to help discover strategies and find their own solutions.

For this transition to happen, students need opportunities to practice solving problems in teams and with partners.  It is important to not 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 have to solve the problem.  Check in with them frequently to make sure they don't check out 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.

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.

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.

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?"

It is a transition year for us, with new, updated units, and new total alignment to Common Core standards, so give yourself and your students time to practice and get comfortable with the changes.
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.  

Tuesday, May 12, 2015

Problematic Behaviors, Classroom Management, and Teaching Math

When 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.

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.

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.

(You can learn more about Responsive Classroom here:
https://www.responsiveclassroom.org/sites/default/files/pdf_files/rc_brochure.pdf)

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.

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.

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 most 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.

The four RC domains are engaging academics, positive community, effective management, and developmental awareness.

In addition to the domains, RC also has seven guiding principles, and they are also completely relevant to math instruction at all levels.  Let's look:

1. The social and emotional curriculum is as important as the academic curriculum.

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 not 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.

2. How children learn is as important as what they learn

When Mr. White shared this guiding principle, I simply wrote in my notes, "SMPs!"  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 Goals for Mathematical Practice, essentially showing you where these different learning experiences lie within and throughout the curriculum.   So yes! How children learn math is 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.

3. Cognitive growth occurs through social interaction

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.

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  

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.

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.

5. Knowing the children we teach-- individually, culturally and developmentally-- is as important as knowing the content we teach.

This comes from formative assessment, 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.

6. Knowing the families of the children we teach is as important as knowing the children we teach.

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 weird.  There is greater emphasis on understanding 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 you about their son or daughter, and y'all have an awful lot to talk about.

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 is a young mathematician, and that we are doing everything we can to help them understand, enjoy and excel at mathematics.

7. How we, the adults at school, work together is as important as our individual competence.  Lasting change begins with the adult community.

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.




Teacher Talk vs Turn-and-Talk: Some Ideas for Increasing Mathematical Discourse

In a book published by Marilyn Burns' Math Solutions, called Classroom Discussions in Math*, the authors outline four steps toward productive math talk in the classroom:

1.  Helping individual students clarify and share their own thoughts

2.  Helping students orient to the thinking of others

3.  Helping students deepen their own reasoning

4.  Helping students engage with the reasoning of others

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 gather, clarify and share their own thoughts.

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:

https://www.teachingchannel.org/videos/multiplication-division-in-the-core

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.

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.

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.

A more traditional style of teaching math, at least ideally, went something like this:

First:  The teacher talks the class through a new math procedure in front of the class.

Then:  The teacher gives the class an example to work through together, with guidance from the teacher.

Last:  The students work on a series of problems, similar to the above work, independently.

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.

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 how something is done 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.

There are two main dangers that linger when this happens.

The first danger 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.

The second danger 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.

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.

Here is a neat way to think about the learning experience as it pertains to mathematics;

Think of learning math and problem solving as a social experience.

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+.  Como Estas? Muy Bien, gracias!  Spanish is easy!  You took Spanish II, and it was harder, but you learned a lot!  Spanish III?  Oh my, accents, novellas, dictations, I sure am getting good at Spanish!  Spanish IV made you a master!  I must be fluent now! I feel so confident!  You took a spring break vacation to Cancun and it was: What are they saying? How do I respond?? Ayudame!  Any foreign language teacher of professor will tell you, you don't become fluent until you immerse yourself in the language.

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 with each other and construct their own solutions together.  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.

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.

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.

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:


  • 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.
  • 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.
  • Practice specific protocols for communicating in groups, and have students do it every day.  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 Can you clarify that for me? and I think you made a mistake.  Can I show you?)
  • 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: How could I get them to work this out without me showing them how to do it?
  • Assign roles to group members, such as presenter, note-taker, ambassador, fact-checker, editor, time-keeper, equalizer (making sure everybody is contributing), etc.  
  • 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! 
  • 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.  
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!

*The book mentioned at the top, Classroom Discussions in Math, was published in 2013 by Math Solutions of Sausalito, CA, and was written by Nancy Anderson, Catherine O'Connor, and Suzanne Chapin.

Recently published (Fall 2015) articles on Mathematical Discourse:

Orchestrating Mathematical Discourse to Enhance Student Learning

Establishing a Culture of Collaborative Learning

Tips for Launching an Inquiry Based Classroom

Learning Is Loud







Thursday, April 2, 2015

Some 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.

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:


  • 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 not 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 we use the word rectangle, even if it is somewhat close to being a square.
  • Students and adults get three-dimensional and two-dimensional figures confused all the time.  Like the rectangle-square issue, there is also a square-cube issue.  A cube is not 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. 
  • 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 commutative property..."
  • 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. 
  • 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!  
  • 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.

How Much Should I Trust the Spiral? Everyday Math, Engagement, and Best Practices

Recently, 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.

We refer to these changes we are seeing in our math instruction as an "instructional shift" toward greater understanding, 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 using multiple strategies from David Ginsburg via EdWeek:


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.

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.

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.

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.

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.

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.

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.

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 how 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.

Teachers of Everyday Math are often asked to "trust the spiral," but I think the better way to look at it is to understand the spiral, to know that it is there.  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.

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 you 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.




Tuesday, March 17, 2015

'Explain Your Thinking' Revisited

Now 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.

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.

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.  

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.

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.



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."


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?"


Halfway through 2nd grade, students are "encouraged to use each other's strategies," and explain how more than one strategy can work.  See above.


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.

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).

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 Standards for Mathematical Practice and their accompanying Everyday Math Goals for Mathematical Practice. These practice standards are key to the instructional shifts we speak of:

SMP1: Make sense of problems and persevere in solving them

GMP1.1 Make sense of your problem
GMP1.2 Reflect on your thinking as you solve your problem
GMP1.3 Keep trying when your problem is hard
GMP1.4 Check whether your answer makes sense
GMP1.5 Solve problems in more than one way
GMP1.6 Compare the strategies you and others use

SMP2: Reason abstractly and quantitatively

GMP2.1 Create mathematical representations using numbers, words, pictures, symbols, gestures, tables, graphs, and concrete objects
GMP2.2 Make sense of the representations you and others use
GMP2.3 Make connections between representations

SMP3: Construct viable arguments and critique the reasoning of others

GMP3.1 Make mathematical conjectures and arguments
GMP3.2 Make sense of others’ mathematical thinking

SMP4: Model with mathematics

GMP4.1 Model real-world situations using graphs, drawings, tables, symbols, numbers, diagrams, and other representations
GMP4.2 Use mathematical models to solve problems and answer questions

SMP5: Use appropriate tools strategically

GMP5.1 Choose appropriate tools
GMP5.2 Use tools effectively and make sense of your results

SMP6: Attend to precision

GMP6.1 Explain your mathematical thinking clearly and precisely
GMP6.2 Use an appropriate level of precision for your problem
GMP6.3 Use clear labels, units, and mathematical language
GMP6.4 Think about accuracy and efficiency when you count, measure, and calculate

SMP7: Look for and make use of structure

GMP7.1 Look for mathematical structures such as categories, patterns and properties
GMP7.2 Use structures to solve problems and answer questions

SMP8: Look for and express regularity in repeated reasoning

GMP8.1 Create and justify rules, shortcuts, and generalizations

Tuesday, February 10, 2015

Smarter 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!

Here are some important things to consider when looking ahead to these tests.

  • 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.   
  • 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...
  • The assessment can teach us a lot!  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.  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.  
  • 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.
  • 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.
  • 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.  Oh no, we thought to ourselves, my students don't know operations with fractions!  We'll have to go over that tomorrow!  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.

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).

Smarter Balanced Practice Tests

Grade 3:

The word "unknown" appears many, many times.

The word "equation" appears many, many times (as opposed to "number sentence").

The word "expression" appears also.

The relationship between multiplication and division is emphasized heavily

There are a number of questions where students are asked to place fractions on a number line

Filling in fraction bars to model equivalent fractions

Area and perimeter of multi-sided right-angled figures (for example, a rectangle with a inverted corner)

There was one question about telling time on a traditional clock

"mass" in grams is mentioned in a word problem; students are expected to understand what "mass" is.

One question asked students to make a rhombus that is also a rectangle.

Parentheses appeared in a number of problems, with order of operations used.



Grade 4:

Many basic operations problems are written as word problems... There are many, many of these word problems.

Understanding that angles are properties of rectangles, rhombi, and parallelograms will be helpful

Understanding the relationship between measures of length, such as millimeters and centimeters will be important

Students will benefit greatly from being able to model division with remainders, in other words, they should be able to tell you a story that involves division and remainders.

Understanding the relationship between numerators and denominators is important, i.e., what it means when one is bigger than the other.

Students should be familiar with solving problems that involve fractions as amounts of things

Understanding how to represent decimals as fractions with 10, 100, and 1,000 as denominators is important

Students should know what an equation is (never once did the assessment use the term "number sentence").

Mixed numbers are all over the place; comparing them, adding them, representing them pictorially..

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)

Visual representations/ models of fractions and mixed numbers are frequent



Grade 5:

Exponents!  Especially with a base number of 10..

Word problems with mixed numbers... adding them, subtracting them, multiplying them with like and unlike denominators

Word problems with measurement of mass in kilograms, and using the word "mass"

Construction and deconstruction of mixed numbers and improper fractions

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.

Gallons and cups!

Comparing decimals to the hundred-thousandth place

Area models used frequently for multiplication and division

Plotting points on the coordinate plane!