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.

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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…]]>

Most of you have probably done a lesson on non-positive exponents by looking at a pattern similar to this one. (Thanks to Desmos for the table). I typically start with showing the students something like this. I ask them about the pattern of values on the right, focusing on the way the values change going up (most of the time students see that first) and the way the values change going down. I also point out that there is a pattern on the left and we talk about that one. We take some time to distinguish between the two patterns - one involving addition and subtraction, and the other involving multiplication and division. And then I ask whether we could continue the patterns, and see what the students say. There are lots of directions to go from there, depending on what your students say, but it is always an interesting discussion, and fun to follow up with another exploration deriving the power rules for multiplication and division and tying that back to what you uncovered here. The deeper symmetry that I appreciate in this is how the exponent function maps addition onto multiplication, and there are two features that I realized for the first time recently. |

- 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.

]]>- 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.

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.

]]>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

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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.

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- how to write the general equation for a table and what it meant

- understanding the units attached to the constant of proportionality

- seeing the direction of the relationship in a table... the independent and dependent variables

So in the lessons and the practice, I made some changes.

Lesson 4:

- I asked more explicitly about the units of the constant of proportionality

- I included a multiplication sign in the equation so that they would understand m = 3 · h as "the number of miles is equal to three times the number of hours" instead of "1 mile equals 3 hours."

- I stressed the direction of the relation, using phrasing like "relating hours to miles" instead of "relating hours and miles."

Lesson 5:

- I broke this into two parts and reordered some of the problems. In part 1, I added more basic problems and scaffolded the problems to ask for two tables instead of one, emphasizing the two different ways to look at the same relation: relating miles to hours and hours to miles. We started with tables instead of equations.

- In part 1, I also used whole number/unit fraction constants of proportionality.

- In part 2, we moved into more challenging constants of proportionality, still beginning with tables. In the final activity, I asked the students to reason about equations.

- Also, as a warm up I made a note to ask the students to fill in tables using equations to again give them experience with the multiplication in the equation. They often want to interpret the equation as a ration.

- I also took a day before part 2 and after part 2 for a lab on tongue twisters as an application of proportional relationships.

Lesson 6:

- This year the students were not really using the equations to solve for the unknown quantities... they were really using tables. The math is the same, but they weren't introduced to the usefulness of equations. I rewrote the lesson to emphasize the use of equations, and I will need to rewrite the practice sheet next year.

Here are links to the teacher presentation PDFs (I use Openboard... let me know if you would like the Openboard files) and classwork and practice sheet docs and PDFs. I have reformatted the student sheets two ways. I usually just have students work off the screen from my teacher presentations. If there are tables or graphs for them to use, I make a student worksheet. I have also reformatted the students sheets so that parents can see what we are doing in class. I moved the summary section to the top, and tightened up the formatting to make them shorter page-wise. I call those Student Summaries.

Unit 2 Teacher Presentation PDFs

Unit 2 Homework and Classwork Docs

Unit 2 Student Summary Docs

Unit 2 Homework PDFs

Unit 2 Student Summary and Classwork PDFs

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As an example, my students are in the middle of Unit 2 in the Illustrative Mathematics/Open Resources Grade 7 curriculum. We have been doing a lot of problems where the students multiply a whole number times a fraction, and they have been doing pretty well with it. Some use mental math, and some want to write out the algorithm and cancel (which drives me nuts, but it works for them). To push all of the students to think a little more deeply, I thought about asking an OpenMiddle-style question:

The first round did not push the students much out of their comfort zone. I got repetitions of problems we had been doing already: 2/3 · 5 = 10/3 (I didn’t specify whole numbers in the boxes. But later on I was glad I didn’t). There were other students who answered with whole numbers, but they were pretty comfortable with those already.

I wonder what they will do with this:

With a little bit of constraint, the answers revealed some additional insights on their part. I got things like (I apologize for not taking the time to put all the fractions in a better format):

- 1/1 · 6 = 6 , and general rumbling there are an infinite number of those.
- And lots of ½ · 12 = 6, 1/3 · 18 = 6, ¼ · 24 = 6, and again more seeming consensus that we could keep going with those.
- One student offered 1/(1/2) · 3 = 6. That was pretty cool. I wish I could remember how he got there… I think it was continuing a pattern he saw, because he asked if that was possible. I don’t think he did the division, but I wonder…?

I know exactly what to do for the next round.

I wonder what they will do with this:

I got a lot of related problems: 2/4 · 12 = 6; 2/8 · 24 = 6; 2/16 · 48 = 6… which were cool, but somehow the thinking didn’t feel deep to me. I realized that I was hoping for a general insight that the two missing numbers had to form a quotient of three. That wasn’t what I had started out looking for, but as the conversation continued over three days, that was something I noticed was missing. The students naturally went to a doubling pattern and just ran that out infinitely, without looking for a broader, more general pattern.

**What did I notice? What do I wonder?**

First of all, I notice that the order of the steps was reversed. I started out wondering, running an experiment, and then noticing. Of course, it is not really reversed… notice/wonder/notice/wonder is really a cycle of inquiry; I guess I was just spending time on the other side of the circle. But it reminds me… as I have begun to I ask my students to notice and wonder more often in class, I need to continue the cycle around, and help them to formulate actions inspired by “I wonder” for another round of “I notice.”

While writing this, I noticed that my understanding of the math behind this problem and the levels of understanding that my students might achieve evolved and changed tremendously.

I noticed that I can’t tell you exactly who understood what, or who progressed. But I did see energetic, engaged conversation, and there were a variety of strategies shared. I don’t think I would spend more time assessing individual students, but I think it would be worth pushing their understanding further with more of these questions, and also worth the time to shift to a different modality working the same concepts, like a number talk looking at:

2/3 · 9 =

2/5 · 15 =

2/7 · 21 =

2/7 · 5 =

2/14 · 10 =

I noticed that I had just been taking answers from the students, and I never stopped to ask how any of the solutions were related. In using resources like Visual Patterns or Which One Doesn’t Belong? or OpenMiddle questions, you develop a patter that can help highlight the deeper connections. And it takes many, many uses of those formats to develop those ideas. I will definitely be adding “How are the solutions related?” or “Are there any patterns you see in the different solutions?” or something like that to my OpenMiddle patter.

As I mentioned before, sometimes I wonder what concept I am actually working to develop and sometimes that changes. In this case, I wonder if the more general idea is important enough to spend more time on. I am not sure. But I don’t think that’s a problem. I am taking a skill or concept that the students are familiar with, and making some new connections, stretching it a little bit. I added time and day notations so that you realize this is just a warm up, and we do lots of other things in a class period. What gets learned will play out differently for different students. And I think that is perfect. What they all will get is more time practicing mathematical thinking. And that is my number one goal every day.

]]>First of all, I notice that the order of the steps was reversed. I started out wondering, running an experiment, and then noticing. Of course, it is not really reversed… notice/wonder/notice/wonder is really a cycle of inquiry; I guess I was just spending time on the other side of the circle. But it reminds me… as I have begun to I ask my students to notice and wonder more often in class, I need to continue the cycle around, and help them to formulate actions inspired by “I wonder” for another round of “I notice.”

While writing this, I noticed that my understanding of the math behind this problem and the levels of understanding that my students might achieve evolved and changed tremendously.

I noticed that I can’t tell you exactly who understood what, or who progressed. But I did see energetic, engaged conversation, and there were a variety of strategies shared. I don’t think I would spend more time assessing individual students, but I think it would be worth pushing their understanding further with more of these questions, and also worth the time to shift to a different modality working the same concepts, like a number talk looking at:

2/3 · 9 =

2/5 · 15 =

2/7 · 21 =

2/7 · 5 =

2/14 · 10 =

I noticed that I had just been taking answers from the students, and I never stopped to ask how any of the solutions were related. In using resources like Visual Patterns or Which One Doesn’t Belong? or OpenMiddle questions, you develop a patter that can help highlight the deeper connections. And it takes many, many uses of those formats to develop those ideas. I will definitely be adding “How are the solutions related?” or “Are there any patterns you see in the different solutions?” or something like that to my OpenMiddle patter.

As I mentioned before, sometimes I wonder what concept I am actually working to develop and sometimes that changes. In this case, I wonder if the more general idea is important enough to spend more time on. I am not sure. But I don’t think that’s a problem. I am taking a skill or concept that the students are familiar with, and making some new connections, stretching it a little bit. I added time and day notations so that you realize this is just a warm up, and we do lots of other things in a class period. What gets learned will play out differently for different students. And I think that is perfect. What they all will get is more time practicing mathematical thinking. And that is my number one goal every day.

2.5.1 Lesson Plan

2.5.1 Lesson (Problems)

2.5.1 Practice (Word)

2.5.1 Practice (PDF)

2.5.2 Lesson Plan

2.5.2 Lesson (Problems)

2.5.2 Practice (Word)

2.5.2 Practice (PDF)

Looking ahead, I think I will need to adjust Lesson 6 as well. My students did not go through the Grade 6 materials, so some of the skills are new to them.

]]>I began by asking who knew a good tongue twister. Most of my students did, and we took a few minutes to share some. Then I explained:

“I am going to give each pair of you an unfamiliar tongue twister. One of you is going to repeat the tongue twister as many times as you can while the other one counts and watches the clock. You will time your partner for 10 seconds, 20 seconds, 30, 40, and 50 seconds, and record the number of times they say the tongue twister for each time in table. What I want to know is: do you think that table will show a proportional relationship?”

We had a good conversation about that. I was surprised that most of my students said no. They thought saying a tongue twister was just too unpredictable. You can challenge them to explain how they will know if it is or it isn’t, and see how much of the previous lessons they have understood and can apply on their own.

Then they got to work. We have just finished collecting the data. In conversations with pairs we looked at tables that seemed to not be proportional at first, but then seemed more proportional as we noticed other patterns. At one table we realized the number of times the student said the tongue twister was always 1 off from 1/5 of the seconds. At another it was always just above or just below. When we reconvene after our annual trip off-campus, I will be sharing all the tables with students, asking which tables have a constant of proportionality or something close, writing equations for those tables, and talking with them about the roles of approximation and modeling in applying math to science. I think this will be a great way to lock in some of the key concepts for this unit, and make them memorable.]]>