I have always been bothered by the concept of “constructivism.“ which I understood as the idea that students construct their own understanding of math concepts, and a teacher’s carefully designed explanation can’t do that for them  The problem with this is that while I believe it is true, it isn’t very helpful.  How do they do that?  How do I help them to do that, and help them to get better at it?  

When I first tried to implement the idea 20+ years ago, I fell into a trap of thinking I just had to stop giving students the answers and they would start coming up with answers themselves.  Sometimes people would say “students should discover things for themselves.” Any of you with me at the bottom of that pit know what a failure that is.  And there is a much more subtle trap that I am still wrestling with - the idea that the less I say in math class the better, that I should always avoid just telling a student something.  I suspect that we all wrestle with this every day, we know that we “shouldn’t” tell students what to do, but then there are those students who are struggling a bunch, so we give them a hint, or a suggestion, or an example, to help them get back in the game.  But somehow that feels like cheating, or less than ideal.  

There is often a vast gap between what most people imagine math to be (the process of remembering how to do something the teacher taught you, like adding fractions, and then getting the answer) and the math experience we are trying to build in our classrooms.  The difference shows up in both the tasks we are asking the students to do, which is currently pretty well-defined and easy to observe, and the ways we hope students will engage with those tasks, which is less well-defined.  In trying to better define that second piece I realized that the process I want my students to go through on a daily basis (almost minute by minute) is to build on what they know to figure out something they don’t yet know or understand.  It is not so much about constructing understanding (which will happen, I assume, although I still don’t know how) as it is about asking students to construct their answers from the skills and knowledge they possess.  Not “get” the answer through mental gifts or divine intervention, but through remembering, reasoning, modeling, etc., construct the answer.

Defining constructivism as “constructing answers” rather than “constructing understanding” is a subtle difference but it helps me a lot.  “Constructing understanding” is almost a tautology – of course people construct their own understanding.  It may be an insight, but it doesn’t really guide me in how to do things differently.  “Constructing answers” reminds me that I am teaching not just individual skills, but also the process of selecting and sequencing those skills appropriately for a context.  Students need opportunities to practice that before I tell them what to do.  There is a criticism that problem-solving requires too much of novices, and should be the domain of experts.  But I think it is a matter of scale.  Why not include an appropriate, novice level of decision making in the work we give our students? Why not ask them what they notice or what they can figure out before we share our expert insights?  

“Constructing answers” gives me insight about when it may be beneficial to step in or not step in, the pieces that may be missing for a given student or class, and the kinds of information I might share, questions to ask, etc.  It is less about how much student talk vs. how much teacher talk, and more about who is talking when.  I want my students to talk first, so I can see how they are constructing their answers.  That way I will know what I have to say when it is my turn.

Ever used a model in your math class? Maybe you got out some manipulatives to work with a concept, or you used hangers or tape diagrams or algebra tiles to solve equations.  How did you use them - were they for modeling or for demoing?
There is an important difference between the teacher using a model to demonstrate and the students using a model to aid their thinking and develop understanding.  When a teacher puts a model up in front of the students, demonstrates a problem or more and explains the connection between the model for the students, that is a demo.  Even if the teacher asked "What do you notice? What do you wonder? What do you think?" and had the students explain some of the connections, that's a demo.  A demo has limited value, because the students don't have an opportunity to develop deeper understanding.
We have all had the experience when someone explains something to us with the utmost clarity, pictures, diagrams, tables, whatever... and we still don't quite get it until we have had time to think it through ourselves.  In math class, we want to give students that opportunity to develop their understanding of the concepts.  In order to do that, they need time and they need something that they can think about.  The purpose of a model is to make the relationships and connections visible so the students can literally see what we want them to understand, and think about it for themselves.  With a hanger diagram, for instance, the concepts of balance, doing the same thing to both sides, and simplifying both sides with opposite operations are all inherent in the diagram, and students can reason with those concepts independently with relative ease.  Working with algebraic symbols before those concepts are in place is a purely mechanical exercise and students rely on the teacher to do all the thinking for them. 

Giving the students a quick demo with a model is not utilizing its full potential.  It may clarify things for some students, but for many, it is likely to build few lasting connections or even be more confusing.  Here are some things to remember when using a model:
1)  Make sure it is a model that the students can manipulate and reason about on their own.  Start with simple examples that illustrate the key relationships, then gradually make the problems more complex.  If you can pose a problem and the students can solve it on their own or mostly on their own, it is working.  If the students are frequently asking how to proceed or why a step comes next, there is something wrong with the model or the problems you are asking them to do.
2)  Give the students time to work with the model, use it to solve problems independently, and notice things on their own.
3)  Once the concepts seem to be well-developed with the model, you can begin to use the mathematical symbols to represent them.  Begin by pairing model representations with symbols and asking the students to provide one or the other.  Can they draw a model for these symbols?  Can they write the symbols that represent this model.  Continue to cross-reference the symbols as long as the students need it. 

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At CMC-South, I attended a workshop by David Chamberlain on the distributive property.  At the end he posed the question: is this a trick?  These days that question might be really asking: Is this something I can teach my students responsibly or is this intellectual poison?  “Tricks” are recipes that students follow without understanding, and they can even shut down further thinking because the solution is so readily accessible that the process becomes automatic.  Calling something a trick is pejorative, and implies that it doesn’t belong in a classroom where real learning is going on… but that is an overly simplistic point of view.   I prefer the label mnemonic to distinguish “tricks” from models.  Both are useful tools, each with a time and place, and the labels help us recognize and communicate the role each plays in our teaching, which in turn makes it easier to think about when to use them.

So instead let’s ask, is this a model or a mnemonic?

Models have a number of important characteristics.  The ones we use in math often make an abstract idea visible.  They also are usually something that students can manipulate and experiment with.  These two characteristics make the third possible:  students can do their own problem-posing and -solving. This is the defining characteristic of a model in math class.  A model allows students to think for themselves and make their own discoveries.  So the model must allow students to explore uncharted territory, without teacher guidance, and uncover new ideas.

An example of a model is a balance or hanger diagram to represent an algebraic equation.  Many of us use these in algebra classes to help students discover or justify for themselves the steps involved in solving an algebraic equation.  Using balance diagrams makes it clear for students why we “do the same thing to both sides” and when it is more direct to add or subtract first, or when multiplying or dividing first might be better.

A mnemonic is a tool for recall.  PEMDAS, cross product, keep-change-change (for dividing fractions) are all mnemonics, helping students remember the order of operations or the sequence of steps to work with fractions.  “Do the same thing to both sides” as a stand alone phrase is a mnemonic for an important part of the solving process. This is not wrong.  We assume that understanding something means we will remember it, and vice versa - if we don’t remember something, we must not have understood it.  But it turns out this is not the case (Google Robert Bjork and memory or listen to some of Craig Barton’s math podcasts if you want to learn more).  Understanding and recall are more loosely connected, and each needs to be cultivated and strengthened.  Students build recall strength by recalling, and if they don’t practice remembering, the accessibility (but not necessarily the knowledge) fades.

So what about the wrist tool that David described - is it a mnemonic or a model?
For me, it seems to be more of a mnemonic for remembering that when multiplying two sums, there are four products that need to be calculated and then summed.  Using wrists doesn’t provide any justification for that, and there is nothing inherent in the model that tells students what to do next, the same way that there is no way to tell if you should add denominators together or not when you see a fraction addition problem on a page.  You have to get that information somewhere else.

That said, the line between a model and a mnemonic is not black or white.  Using wrists makes the process more visible, and one can see what is happening better than on paper.  The closest it came to being a model for me was when we factored polynomials.  Then I could see why shifting from (x+3)(x+4) to (x+2)(x+5) would leave the first and middle terms the same, but change the last term.  And sometimes mnemonics communicate some of the underlying concepts.  Models themselves can also serve as mnemonics, helping students remember what steps to take.  But in general we can ask does the tool mostly help students think or does it mostly help them remember?  If we asked them to give an explanation, would they be explaining their reasoning or listing their steps? During the session, when we described what we had done, I felt it was mostly the latter.

Why is the distinction important?  We want to cultivate both understanding and memory in our students.  Neither one alone is sufficient for success in mathematics.  David referred to area models and, as I recall, we talked about using the wrist model after using some other context to explore the concepts behind the distributive property.  So we need to be aware of when we are asking the students to think and discuss and justify their reasoning, and when we are asking them to remember and perhaps share their process.  We want to make sure we are giving proper time to both.

David quoted one of his students saying “Why did we have to go through all that other stuff [the area model I assume]?  Why didn’t you teach us this first?”  Robert Bjork also has researched desirable difficulties, and the fact that learning happens when things feel hard - once it feels easy and comfortable, we aren’t learning any more.  But managing frustration (for both students and parents) in math class is a real issue, and mnemonics are an important tool for that.  We just have to be careful that ease does not surpass thinking and reasoning as the main goal for our math classes.  Making intentional choices about when to use models and when to use mnemonics will help.

I am not an expert on cognitive load theory.  I first heard of it on the most amazing math podcast I know: Mr Barton Maths Podcast  run by Craig Barton.  The specific episode I am referring to is this one with Greg Ashman, where they discuss something called cognitive load theory.  Basically, the concept is a familiar one.  Remember the idea that your brain can hold only about 7 unconnected things in short term memory?  So we have 7 digits in phone numbers, you forget a grocery list for more than a handful of items, etc.  But the key word is unconnected.  If you can group them into a meaningful chunk, you can remember more.  Cognitive load applies this to completing tasks.  Short term memory actually is less when you have to do something with the information - the steps take up short term memory too.  This is an explanation why some students may get lost solving multistep equations, or doing long procedures like long division, solving simultaneous equations, or writing a linear equation from two points. 

Even more challenging is applying those procedures to solving problems.  Imagine a student who is still shaky on the skills for multiplying and dividing fractions having to solve a problem that involves choosing which of the two operations is appropriate.  The cognitive load there is overwhelming compared to the student who has automatized (chunked!) those skills, and can focus solely on the method selection embedded in the problem. (Method selection is the ability to decide what arithmetic is going to apply to the problem at hand.  I first heard the term in this podcast with Mark McCourt.  Problem sets with problems from previous lessons mixed in give students a chance to practice method selection.)

This doesn't really change the student landscape of my classroom, but somehow this totally transforms the way I look at them.  I have always been uncomfortable with providing exemplars (worked examples for reference) during tests to students who were struggling with procedures, because then what am I testing?  It should have been obvious before, but it suddenly is now that I am testing two things: procedural fluency and the bigger concepts that we applying the fluency to.  So I need to separate those and assess and provide feedback on both.  Perhaps a small quiz on one day, offering students the choice to solve procedural problems outright (fraction division, solve an equation), or with steps missing, or in an example - problem pair.  On another day, with exemplars available, assess method selection and conceptual knowledge.  Also, in my room I plan to post exemplars for students to use on the days when we are doing more conceptual work and those skills are required, and just be more thoughtful overall about when and how I can most effectively provide those supports.

Also how I teach those skills.  I see three strands now when I used to see one.  Take fraction multiplication. 

First there is understanding the operation itself... what does 1/2 x 3/5 mean?  Using manipulatives and models, making sure that students build an understanding of what multiplying by a fraction means concretely. That may be a task without end, but I can establish some particular goals. 

Second, there are the arithmetical mechanics of how you get the answer efficiently.  It has been a long evolving idea of mine that these first two strands appear closely related, but are perhaps only distantly related, and I still am unsure exactly how.  The goal of this second strand clearly is chunking.  But what leads to lasting chunking?  Providing meaning is one way:  if the last four digits of someone's phone number are 1812, I can just remember the War of 1812.  But it doesn't seem that the conceptual understanding of how dividing fractions works provides meaning for remembering the procedure.  You might think that it would, but maybe the cognitive load of re-imagining the procedure and re-establishing the link between the procedure and the symbol manipulation does not provide a stable hook for memorization.  We can prove the Pythagorean Theorem, but retracing the path from geometry to proof to formula is not how experts remember and apply it.  Math gurus/curriculum often say that when students learn conceptually, it helps them to remember the procedures longer.  I am just not convinced.  It seems that whenever the conceptual understanding leading to a procedure is a chain of reasoning two or more steps long, a significant number of students will have forgotten the procedure within days, if not hours.  I think part of it is their limited understanding or experience with what it means to justify something by reasoning.  Proof is a concept we are still establishing.

So how to best support this second strand?  Of course, an obvious answer is rehearsal.  But seeing clearly that this problem of chunking the mechanics is its own issue, I can ask also if there are other ways to chunk?  I have been exposed to other ideas that I have discarded and mostly forgotten, ie mind maps, that I might go back and take another look at.  Example-problem pairs are something that Craig Barton frequently brings up with his guests, and I plan to take a deeper look into those.  With just the knowledge I have so far, however, they make a lot of sense, and seem to combine some of the features we have seen so far that seem effective.  Students see a problem either presented or demonstrated, and then do a similar problem alongside.  There are lots of scaffolding variations: a complete example, an example with blanks present, examples with successively fewer steps included, etc.  I can see how you can continually adjust the complexity of the examples and ebb and flow the scaffolding from full to minimal, full to minimal each time a level of complexity is added.  The procedures for things like solving equations are not entirely rote, there are still decisions that can be made, so the gradual addition of complexity might be beneficial.  With example-problem pairs, we combine sense-making and pattern-seeking with rehearsal, and the research apparently shows this is effective.

The third strand is method selection, being engaged in problem-solving where the particular skill is needed but the student has to recognize that without clues or prompting from the teacher or the materials.  I have not thought as much about this, but I recognize that providing these kinds of problems for students are an important part of being able to effectively use mathematics outside the context of a structured lesson.

Cognitive load theory provides an actionable explanation for some of the things I have observed and struggled with in math class.  It has given me a number of ideas of new things to experiment with (providing exemplars, structuring practice with example-problem pairs, thinking more about effective ways to differentiate students and instruction or integrate instruction better based on who has chunked the prerequisite skills and who has not). I will also be looking into what kinds of evidence there is for the theory or against.  But for now, it has changed the way I am thinking about my classroom significantly.

One of the parts of teaching I love the most is trying to figure out what is holding a student back.  Recently I was working with a student who could not write out the algebraic steps for solving an equation.  We are finishing Math 7 Unit 6, and while his mental math is excellent, and he can solve tape diagrams, hanger diagrams and word problems consistently, his skills don’t easily transfer to abstract processes. 
 
The other day, we were working after school with the equation 4x - 7 = 41.  I started with reminding him of the “why” behind the algebra work.  “We have to add 7 back on both sides, get the simpler equation 4x = 48, divide by 4 to find 1x,” etc.  Dean nodded and said he didn’t have any questions.  His turn. 
 
As he worked to solve 8x – 11 = 53, the work spattered across the white board.  He got the answer, and 8x = 64 was swimming in the middle, but Dean clearly was not seeing it the way I was.  I asked him to compare the two problems as they appeared on the board: the one I had written out and the one he had written out.  What was the same?  What was different?  While he was able to point out some details like specific numbers or the answer, he was not seeing the structure of two sides in balance or the sequence of steps. 
 
We went on to another problem, this time 7(x – 5) = 49.  Again Dean got the answer quickly by inspection.  I thought I would highlight each step of his thinking process and record each one algebraically as we went along.  “Great,” I said.  “What did you start with?” “12” he said.  What?  “You start with 12, then subtract 5, then multiply by 7 and get 49.” I realized that Dean was focused on the numbers, not the process.  When we were working together he would often include the answer when he was trying to write an equation.  Once he had found the value for x, x disappeared.  And once he has done the work, he has a hard time thinking backward to review the process.
 
That was all we had time for, but I had another opportunity to sit with him during class the next day.  As we worked I saw more clearly what I had seen the day before: Dean was moving through the process so quickly that he wasn’t even aware of it, so of course he couldn’t record it.  We had been using tape diagrams yesterday at his suggestion.  I thought hanger diagrams might make it easier to look back over the process once we had finished.  So I made a hanger diagram for 3x + 5 = 20 and asked him what x was.  He quickly gave me the answer and I asked him how he got it.  “20 – 5 is 15 and ÷ 3 = 5,” he said.
 
“Okay,” I said.  “Let’s take this one step at a time and see what it looks like.” As he described each step, he could more easily see on the hanger diagram the idea of doing the same thing to both sides.  He also understood that each step produced a new, simpler hanger relationship.  We turned to another problem, 3x + 11 = 47. 
 
He solved it, described the process and we wrote adjustments on the hanger diagram.  “Can you write an equation for this hanger?” He could.  “Okay.  Now, when you write out the algebra for the problem, it’s like running a race in slow motion.  You have to slow your thinking down and show each step.  The equation is like the starting line before the race.  What happened next?”  As we worked through the problem, he was able to show each step and the work that led up to it.  With two more examples he was able to notate the solutions algebraically, and though I intervened on a few steps (we both were having fun with SSLLOOWW MOOOTION), he seemed to have much more autonomy and understanding.  I think we have a good foundation to move forward with, and I think that Dean has found access to a new level of abstraction.
 
All the while, I was thinking about the value of modeling and having multiple models available.  Hanger diagrams worked for Dean.  Tape diagrams work for other students.  In fact, for Dean it was possibly his exposure to both that helped prepare him for a new level of understanding.  When we anchor the work in models where students can see or feel the concepts, and keep stepping back until we find the level where they can think for themselves, then we can use that thinking as a foundation to extend their understanding to new insights.  For Dean, I am hoping that algebraic notation is going to be his next new modeling tool.  Time for some clotheslines, I think…

- The pattern on the left ranges from positive infinity to negative infinity. The pattern on the right ranges from positive infinity to positive infinitesimals (infinitely small pieces).  That's cool how this joins the two forms of infinity, and how one side includes negatives and the other side excludes them.
- The two patterns arrive at the identity for their respective operations together.  0, the identity for addition/subtraction is matched with 1, the identity for multiplication/division.  In a way, they are equated in parallel universes.  I love that. 

There is a story that I tell my students about the Garden of Addition and the Garden of Multiplication that help them to see how addition/subtraction is really one operation, likewise multiplication/division, and the role of 0 and 1 as identities, and the unique (and frightening) role 0 plays in the Garden of Multiplication.  You could probably make up your own, but I will post it here in a little bit.

It took a while for me to start to incorporate the Cool-Downs that end each lesson in the Open Up Resources Math 7 curriculum.  I have always been used to starting class with a pattern from Fawn Ngyuen's site, Visual Patterns, an Estimation180 activity, or maybe a discussion using Clothesline Math.  I found that I wasn't getting through the entire lesson, so the Cool-Downs seemed like an easy thing to skip.  Recently I decided that I was going to begin class with the Open Up lesson and use the supplemental activities at the end.  Now I have time to get to the Cool-Downs and I have found they are a great tool for assessing what the students got from the lesson, what I might want to revisit, and which individual students might be left struggling with the lesson concepts for that day.

After I hand out the Cool-Downs (I always remind students that they can also use them to ask me questions or send me a note if they like) the students complete the problem and return them to one of 4 baskets labelled:  4 - I could explain this to others; 3 - I feel confident with this concept; 2 - I have a question or two; 1 - I need help with this concept (thanks to Morgan Stipe @mrsstipemath ).  After flipping through a few days worth and making piles, I realized this was valuable feedback I would like to track.  Here is what I do:

I record the basket that the student placed their answer in with tally marks.  I find that makes it easier to see at a glance how many 4's or 2's I have than if I had written the numeral.  Then I put a check or an "x" depending on whether a student got the answer right or not.  Sometimes I will write "se" for a small error, like a computation mistake. 
A number of things jump out at me:
- Scanning down a column, I can see which questions I need to revisit with the class, or who I might want to check in with individually.
- Scanning across a row, I can see how a student is doing overall in the unit.
- Comparing tally marks with checks or x's, I can see which students are over- or under-estimating their abilities. 

I hope you give this a try.  It has been a relatively easy way to gather some really useful information about how my classes are progressing through the unit. 

If you have any other recording ideas you use with your students, please add them to the comments below.

Here are the reformatted materials for Unit 3: teacher presentations, student task statements, and practice.  Again, I have reformatted the student task statements as "student_summaries", with the summary moved to the top, as a resource for parents and students.  When I needed the tables or graphs, I made "student_worksheets."  I reformatted the teacher presentations to gather stray sentences that cross pages or make charts or tables and instructions are on the same page.  This time I have also included materials from Grade 6 Unit 1 that covered the area of parallelograms and triangles, since my students did not come from a classroom that used the IM materials in grade 6. 

Depending on when you view this, the materials may not be complete, but I will be adding to them as I move through the unit.

Unit 3 Teacher Presentation PDFs
Unit 3 Homework and Classwork Docs
Unit 3 Student Summary Docs
Unit 3 Student Summary and Classwork PDFs
Unit 3 Homework PDFs

It is easy for students to get fixated on the "how" of mathematics.  As much as we try to emphasize thinking and discussion and as often as we praise clear explanations and celebrate insights, there is still a lot of pressure from parents and siblings and tutors and society at large to focus on getting the answer.  And let's be honest, for many students (especially the middle schoolers I see each day), any schoolwork is time taken away from the truly important stuff.  So just tell them what they need to do so they can be done and move on.

In trying to get them to value both the process of exploring and the connections that come from it, I realized that they might not actually see the difference between rote procedures and conceptual understanding the way I do.  After all, from a student's point of view, if she knows how to do a problem, then she understands it.  So this year I put a poster up in my room that says Little Steps / Big Ideas.  When we started using fraction operations at the beginning of the year, and we re-examined the equivalence of multiplying by 1/3 and dividing by 3, or why we can multiply just the numerator in a problem like 3 · 4/5, we were able to talk about the Big Ideas behind the Little Steps and the difference between the two.

Area formulas are a perfect opportunity to highlight the difference between the memorized rules and the geometric principles behind them.  Once students appreciate the difference, then they can see the value of assessment items that look for evidence of understanding the Big Ideas.  Someone once told me the students know what teachers value by what they test and grade.  Is that Big Idea going to be on the test?  If you give tests, it should be.

I like to remind them of the long history of development in mathematics; that the procedures we have today weren't always obvious to even full-time mathematicians, and took a long time to evolve.  Anytime your students' explorations culminate in a procedure or a "shortcut"  that makes calculations quicker and easier, you can bring into relief the algorithm that makes the mathematics handy, and the big ideas that justify it and connect it to  the rest of the field of mathematics.