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…

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. 

I realized the other day that I sometimes pose problems with a completely different mindset than I used to have.  Instead of formulating a sequence of questions that will guide the students in a certain direction, I sometimes start the class with a problem just out of curiosity: I wonder what my students will make of this?  What do they know that I don't yet know they know?  It is a wonderful frame of mind to be in.
 
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:

Round 1 (10 minutes)
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.

 Next Day: Round 2 (10 minutes)
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.
 
Next Day: Round 3 (10 minutes)
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.

I am teaching three sections of Grade 7 this year, and when we hit the Water Cooler problem in lesson 5 during my first period class, I knew I had to hit the emergency stop.  The students were not ready to analyze equations at all, let alone with decimal equivalents to fractions.  We shifted to something else, and I quickly tried to regroup for the following classes.  I felt that the students needed more experience with writing equations before they analyzed the meaning of the parts, and they needed a less steep approach.  I split the activity into two days, the first with whole numbers and unit fractions, and the second using the more complicated fraction relationships.  I also built in more scaffolding in the way of careful discussion of the parts of each equation throughout, and pushed the equation analysis to the end of day 2.  Here are the the lessons, my adapted lesson plans, and practice for Lessons 2.5.1 and 2.5.2:
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.

This is a fun activity that engages and challenges the students.  I heard of a similar activity during a visit to Nueva School in San Mateo.  By introducing an “approximately” proportional relationship, they have to think carefully about what it means to be proportional.  Depending on the depth and time you can bring to the class discussion, the students can apply most of the Standards for Mathematical Practice during this activity.
 
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.

I love language almost as much as I love numbers, so I tend to get excited about vocabulary.  We talk about Latin roots in my class, and why we use the terms we do in math class.  I have always stressed “constant” and “variable,” but I really got excited when I saw how contrasting the variety of possible scale factors in a table with the constancy of the “constant of proportionality” highlighted the concept of a mathematical constant.  By talking about proportional tables and asking questions like “What’s the same throughout the table and what keeps changing?” my students have already heard me say the word “constant” (with something I could point to) more times than all of last year.  When we start to write equations, it will be the word “variable,” and I expect that the idea of a variable will make perfect sense in the context of the proportional relationships and tables.  And by spending as much time as we are going to on proportional y = ax equations before we move on to y = ax + b, students will have a deeper understanding of the subtler differences between constants, constant terms, and coefficients.

In my prealgebra classes I have used my own curriculum for a number of years, but it seemed that I was always re-writing or re-sequencing.  This year I decided to try out the Illustrative Mathematics curriculum for Grade 7, and let someone else make the decisions for me.  So far I think I put myself in the right hands.  These are good problem-based units, they stretch the students in ways that I realize I have been shying away from, and they develop multiple threads of skills at the same time.  Sometimes that sequencing is obvious, sometimes more subtle.
 
The first decision I made for myself was in fact a mistake: I was concerned about time for some of my own topics and activities that I wanted to keep, and I thought I could do without Unit 1.  Scale drawings and Proportional Relationships seemed like they would be covering similar ground, and scale drawings weren’t as critical a topic as proportions.  But the two units together develop the two kinds of relationships you find in proportional relations:

Scale factors are the relationships between pairs of ratios and the constant of proportionality is the relationship across each equivalent ratio … which stays constant for each row of the table.  Unit 1 develops the idea of scale factors and Unit 2 applies scale factors to tables and adds the constant of proportionality.  So they really build nicely on each other.
 
Fraction skills are also a concern of mine (more on that later) and Unit 1 is an important part of that sequence as well.  In Unit 1, the students multiply by unit fractions and are reminded of the equivalence of dividing by 4 and multiplying by ¼.  They also find 3/5 of a number or 2.5 of a number, and are reminded of reciprocals.  In both Unit 1 and 2 IM ventures into fractions gently, introducing a skill with whole numbers, then using numbers that the students can make sense of mentally or with relatively easy calculations.  I have tried to do that myself in the past, but it is nice to have the details thought through for me, and so far it has worked well for my students. 
 
I have been able to make things work this year with a little extra time and attention to scale factors, but I will be using Unit 1 next year.