# Category Archives: BMTN

## Claim-Evidence-Reasoning in Geometry

Last year I used the process of Claim-Evidence-Reasoning, or CER, to teach statistics. I wrote about it a lot. I mean a lot. Seriously. More than I’ve written about anything else. (two more posts here & here.) But that was about teaching statistics. This term I have a geometry class and as my students were struggling with a proof, I had an “aha!” moment. Why not use the claim-evidence-reasoning process?

We were trying to prove why the sum of the interior angles of a triangle equals 180 degrees. Here’s what I gave them to start with.

That’s right. I just gave them a triangle. The traditional way to approach this proof is to draw a line through point C so that it’s parallel to side AB. And I did include that on the back, in case kids got stuck. But here’s the interesting thing – faced with just the triangle and a background of transformational geometry, they began rotating this triangle to tessellate a line of three triangles. It looked like this:

Then I asked kids to think about this approach, talk about it with each other, and then write a proof for why the sum of the interior angles of a triangle equals 180 degrees.

I collected what everyone had written and, like before, transcribed it for students to analyze for evidence and reasoning. Then we reviewed as a class to try to sort out the evidence (highlighted in cyan) from the reasoning (highlighted in orange). From there, I was able to give them another try with some direction from our conversation about evidence and reasoning. This was more successful. I pulled 4 examples to share with the class – I could have easily shared twice that number, which was more than half of the papers that I received from the class of 22. Here’s one of the 4 examples:

Unfortunately, what you can’t seen in this scan is how the individual statements are numbered. Thinking doesn’t always come in deductive order. Sometimes you just have to write down what you know and why you know it. Then you can go back and organize it. It’s like making a rough draft of the proof.

Today, as we were reviewing these exemplars, I asked my students how often they wrote rough drafts for their humanities essays (all the time) or how often they wrote rough drafts of their science CER (claim-evidence-reasoning) papers (all the time). So, it shouldn’t be surprising that a rough draft might be in order for a geometry proof.

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## Systems Without Mobiles

In this fourth entry about using structure to teach algebra, I’d like to focus on how we moved our kids from the picture-based systems to using traditional symbols. We focused on systems of two equations because even though they could solve systems involving more than two equations using emojis and mobiles, moving to traditional symbols could be more intense.

We began with systems that could have been represented by mobiles.

Many students even drew their own mobiles using x’s and y’s. Others went right to the equation 2x + 5y = 4x + 2y and came up with 3y = 2x. Again, they used this relationship as a direct substitution. Some substituted into the top equation to get 3y + 5y = 48, while others transformed the bottom equation into 6y + 2y = 48. Once they were able to solve for y, they were able to solve for x.

But we didn’t just want systems that immediately transformed into mobiles, so we also gave them ones like this:

We were quite curious about what they would do with this kind of system. Our instinct and experience would be to solve the bottom equation for y, but that’s not what the kids did. They added 3 to both sides of the bottom equation so that both equations were equal to 22. This left them with the equation 2x + 3y = 3x + y + 3, or 2y = x + 3. There’s no direct substitution here, though, so kids needed to reason further. Some used y = 0.5x + 1.5 while others said that x = 2y – 3, which led them to 2x = 4y – 6. Again, they did this on their own, without any direct teaching from us.

We also gave them systems like this one, which seemed pretty obvious to us:

Just about every student was able to come up with the equation 3x – 2 = 4x + 1, but lots of kids weren’t sure what to do next because the equation didn’t contain x & y. Those students needed a bit of prompting until they realized that they could use that equation to solve for x.

It doesn’t look all that different from the others. Most students added 24 to the bottom equation and proceeded as above. But I had one student who decided to multiply the bottom equation by -5. What?! When I asked him why he decided to multiply, he said that he wanted to make the bottom equation equal 20 and multiplying by -5 would do that. Fair enough.

The entire journey, from emoji’s to mobiles to traditional symbols took us to a completely different substitution method for solving systems. Had we not been open to following our students’ lead, we never would have learned these ideas that were completely intuitive to them. Remember, these weren’t “honors” kids, but they were willing to try, to think, and to take risks. And we gave them the space and time to play with the ideas.

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## Mobile Algebra

Following the introductory use of structure and emoji math to introduce systems, my teaching partner and I continued with mobiles as suggested by the authors of “An Emoji is Worth a Thousand Variables.” EDC has this great website, SolveMe Mobiles, that has 200 mobile puzzles like this:

Each shape in this mobile has a value (or weight) and the total value (or weight) in this mobile is 60 (units). Go ahead and solve the mobile.

This mobile represents a system of four unknowns. Using traditional algebra symbols it might look like this:

A couple of those equations have just one variable, so it may not be quite as intimidating to look at the traditional symbols. On the other hand, the mobile shapes are just so accessible to everyone!

We needed to move our students away from systems that had one variable defined for them, though, and the SolveMe site, as great as it is, always includes some kind of hint. So we started to make up our own mobiles.

As first, students used a lot of educated guessing to solve the mobiles. Then there was a breakthrough.

Take a closer look at the left-hand mobile.

Students realized that they could “cross off” the same shapes on equal branches and the mobile would stay balanced. In the example above, you can “cross off” two triangles and one square. Whatever remains is equivalent, though it no longer totals 36. Therefore, two triangles equals one square. Using that relationship, some students then substituted two triangles for the one square in the left branch. Then they had a branch of 6 triangles with a total of 18. So, each triangle is worth 3. Other students used the same relationship to substitute one square for the two triangles in the right branch, resulting in a branch of 3 squares with a total of 18. So, each square is worth 3.

We were floored. We had never discussed the idea of substitution, but here it was, naturally arising from students reasoning about the structure in the mobile.

Looking closer at the center mobile, students used the same “cross out” method to find the relationship that 2 triangles equals 3 squares. If we’d been teaching the substitution method in a more traditional way, kids would have been pushed to figure out how much 1 triangle (or 1 square) was worth before making the substitution step. We knew substitution was happening here, but we didn’t invent this approach so we just followed closely to see where our students took us. Since 2 triangles equals 3 squares, some kids substituted 3 squares for the two triangles on the right branch of the mobile. Others made two substitutions of 6 squares for the 4 triangles on the left branch. Either way the result was a branch of 7 squares that totaled 14. It seemed quite natural to them.

What would you do with this one?

Next up: Moving to traditional symbols. The final (?) post of this saga.

Filed under #CCSS, BMTN, problem solving, teaching

## Emoji math

The October 2017 issue of Mathematics Teacher included the article “An Emoji is Worth a Thousand Variables,” by Tony McCaffrey and Percival G. Matthews. (Note that you need to be an NCTM member to access the article without purchasing it.) The authors introduced their students to systems by using sets of equations comprised of emojis, similar to those puzzles that are found on Facebook. Lots of people, including those who say they hate math or they aren’t good at math or, “I’m not a math person” will do puzzles like these. They get lots of likes, answers posted in the comments, and shares –  probably because this doesn’t look much like math. Take a moment and solve the puzzle.

What if, instead of the emoji puzzle, I had posted this puzzle:

The two are actually equivalent, but the abstract nature of the second representation is enough to make our students who see themselves as not math people shut down.

My fall term teaching partner and I used the emoji approach with our students. We hadn’t discovered the website yet, so we gave our students this short worksheet. Many of our students had struggled with math prior to coming to Baxter Academy. They were in self-contained or pull-out special education settings or in pre-algebra classes in middle school. A few probably had something closer to algebra 1, but they certainly hadn’t solved systems of three or more equations. They had no problems understanding what the emoji puzzles were asking of them. They weren’t put off by the number of equations or the number of icons. They were able to explain their solution process clearly and with great detail.

In a follow-up exercise, asking students to make connections between the emoji representation and the more traditional representation, nearly 80% of my students saw and could articulate the direct connection between the icons and the variables.

If the emoji system is more engaging and more accessible, then why don’t we use more of them to introduce systems of equations? Is it because emoji systems seem to “dumb down” the mathematics? The authors of the article make the case against that view:

“The algebra represented in [the emoji system] is not dumbed down at all. Notice that the puzzle presents a linear system in three variables … First-year algebra students are generally not exposed to three-variable systems; indeed, when McCaffrey checked all the first-year algebra texts in his school’s faculty library, none included systems of three variables. Although McCaffrey’s students had never seen three-variable systems before this class, most found the puzzle intuitive enough to solve.”

It is important to transition from emojis to more formal algebra, but it’s not important to start with the most abstract representation – the one that leaves too many of our students behind.

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## Using Structure to Solve Equations

Last November at the ATMNE 2017 fall conference, I attended a session where Gail Burrill highlighted some TI-Nspire documents that helped kids build concepts about expressions and equations. Sure, they’re targeted at middle level, but you use the tools that your students need, right, not where you wish they were?

Anyway, the one that really caught my eye was called “Using Structure to Solve Equations.” It was the perfect activity at the perfect time because my students were kind of struggling with solving (what I thought) were some fairly simple equations. Their struggle was about trying to remember the steps they knew someone had showed them rather than trying to reason their way through the equation. Just before learning about this activity, I had found myself using these strategies. For example, to help a student solve an equation like 3x + 8 = 44, I found myself covering up the 3x and asking the student, “What plus 8 equals 44?” When they came up with 36, I would follow up with, “So 3x must equal 36. What times 3 equals 36?” This approach helped several students – they stopped trying to remember and began to reason.

There were too many students who were still trying to remember, though. Their solutions to equations involving parentheses like 6(x – 4) – 7 = 5 or 8(x+ 9) + 2(x + 9) = 150 included classic distributive property mistakes. The “Using Structure …” activity was just like my covering up part of the equation and asking the question. But each student could work on their own equation and at their own pace. The equations were randomly generated so each kid had something different to work with. Working through the “Using Structure … ” activity helped kids to stop and think for a moment rather than employing a rote procedure.

In a follow-up to the activity, I asked kids to explain why 8(x+ 9) + 2(x + 9) = 150 could be thought of as 10(x + 9) = 150. My favorite response was because “8 bunches of (x + 9) and 2 bunches of (x + 9) makes 10 bunches of (x + 9).” From there, they saw it as trivial to say that x = 6. These were students who had previously struggled with solving equations like 3x + 8 = 44 because they couldn’t remember the procedure. After “Using Structure …” they didn’t have to remember, they just had to think!

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## Welcome to the Math Kitchen

A couple of weeks ago, at an ATOMIM Dine & Discuss launching our Becoming the Math Teacher You Wish You’d Had book study, Tracy Zager shared the following quote:

The front and back of mathematics aren’t physical locations like dining room and kitchen. They’re its public and private aspects. The front is open to outsiders; the back is restricted to insiders. The front is mathematics in finished form—lectures, textbooks, journals. The back is mathematics among working mathematicians, told in offices or at café tables.

Front mathematics is formal, precise, ordered, and abstract. It’s broken into definitions, theorems, and remarks. Every question either is answered or is labeled: ‘open question.’ At the beginning of each chapter, a goal is stated. At the end of the chapter, it’s attained.

Mathematics in back is fragmentary, informal, intuitive, tentative. We try this or that. We say, ‘maybe,’ or ‘it looks like.’

-Reuben Hersh, Professor Emeritus, Department of Math and Statistics, UNM

Tracy explained at the meeting that restaurants and theaters have a “front” where everything is presented perfectly to the public and a “back” where the chaos happens. This is the metaphor that Hersh is using. Too often our students are only exposed to the “front” of mathematics and none of the “back.”

I recently shared this at a BMTN meeting when a colleague coined the term “math kitchen.” And then she said, “Put on your apron – it’s going to get messy in here.” It made me think about how often my students want to have their math papers be perfect. Every mistake must be erased. Nothing can look messy. Am I alone here?

Another colleague said that she used to make all of her students do math in pen. That way they had to cross out mistakes. They couldn’t erase them. I think this is a brilliant idea.

Too often I hear my students say things like, “I remember doing something like this” or “I’m trying to remember what my teacher told me” or, God forbid, “I never learned this before.” What are we doing to our students that makes them think that they should have memorized or learned before what we are trying to teach them now?

So, in the spirit of exposing students to the “back of math” I say, “Welcome to the math kitchen. Grab a pen and put on your apron. It’s going to get messy in here.”

Filed under Baxter, BMTN, teaching

## My new PDSA

It’s the beginning of another year with the Better Math Teaching Network, so it’s time to figure out a new change idea. Okay, to be honest, the beginning of the new BMTN year happened back in July, but I was struggling with a new change idea. At least I’m teaching a bunch of algebra this year.

The aim of the Better Math Teaching Network (BMTN) is to increase the number of students who connect, justify, and solve with depth in algebra. Here’s how we’ve defined what that means:

• Connect. Making connections among mathematical algorithms, concepts, and application to real-world contexts, where appropriate.
• Justify. Communicating using mathematical thinking as well as critiquing the reasoning of others.
• Solve. Making sense of and find solutions to challenging math problems beyond the rote application of algorithms.

Last year, I was solidly in the justify category and it was really fun. Even though I’m not teaching that class again this year, I have lots of ideas about how to infuse my classes with the concepts of “claim, evidence, reasoning.” In fact, I think that will be part of my term 2 PDSA since I’ll be teaching Intro to Logic again.

Back to this term. This term I am teaching 9th graders. Not only are they new to our school, they are coming from so many different backgrounds. These students did not grow up together. So, part of the purpose of the class is to help them to get to know each other. Another purpose of the class is to introduce them to a math class (possibly) unlike any other that they’ve experienced. This is a math class where the teachers don’t tell the students exactly what to do so they can “practice it” 50 more times on “exercises” where only the numbers have changed. You see, that’s not deeply engaging with math, or algebra in this case.

Given that I would be teaching this class in term 1, I had to figure out what I could focus on that would make this experience better for my students. Last year, a bunch of people in the BMTN attempted the Connect strand and found it to be really difficult. I was thinking to myself, I like the Justify strand – I’m really comfortable there. Solve wouldn’t be too bad, either. But the more I thought about it, the more I thought that I really needed to get into that Connect strand. So that’s where I am.

Ultimately, I landed on connecting to the concept of slope. It’s a huge concept, with so many connections. But when you ask kids about slope, they typically say something like “rise over run” or they’ll quote a formula or they’ll say “y = mx + b.” It’s not their fault that they don’t have a deep understanding of slope. It’s ours.

So, my term 1 PDSA is about giving my students opportunities to see slope in different contexts. I wonder if I’ll broaden their thinking … stay tuned.

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## Claim, Evidence, Reasoning: Final results

In this final post about the Claim, Evidence, Reasoning approach to teaching statistics, I will share some student results. Fundamental questions with the PDSA approach to reflecting on and improving practice are:

• Will students engage?
• Will students learn what I am attempting to teach?
• Will students produce quality work?

Nearly all of the students in these two classes had prior experiences with statistics which allowed me the freedom to find a new approach. That said, there were definitely times when it became clear that some content instruction was needed, especially when we got into correlation and linear regression. But instead of trying to front-load all of the content, I waited until the need arose. For example, in looking at what students wrote about theit became clear that some instruction about regression lines and correlation coefficients was needed.

### Will students engage?

They didn’t at first – in that disastrous failure only 10% completed the first assignment. But I certainly learned from that experience, regrouped and restructured my approach. And then they engaged. My data show that 100% of my students engaged with the class, process, and content at some point and that 90% engaged consistently by the end of the term.

### Will students learn what I am attempting to teach?

I was attempting to teach my students how to apply the claim, evidence, reasoning process that they had previously learned in humanities and science to statistics. Reviewing work against the rubric helped to build an understanding of what quality looks like. It also kept us focused on the goal of claim, evidence, reasoning. By then end of the class, 95% of students were able to review statements through this lens and identify whether or not they were on target.

### Will students produce quality work?

This is the big question, right? It’s great if they will engage – that’s the first step – but if they aren’t working to producing quality work then what have they actually learned? Here are some representative examples of student work.

Analyzing movie data This assignment followed the best actor/actress investigation.

Education vs unemployment  Vinyl vs digital album sales  Juvenile incarceration rates This was the final assignment of the univariate data unit. Students had their choice of data to analyze.

Analyzing cars This assignment followed the class data investigation and included the opportunity for students to revise their work following feedback.

Fast food nutrition  1919 World Series This was the final assignment of the bivariate data unit. Students had their choice of data to analyze.

I will leave the question of whether these examples represent quality work to you, the reader. I hope you will let me know what you think.

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## Claim, Evidence, Reasoning: About the data

In the last post, I shared the general process that I developed to teach statistics through the lens of Claim, Evidence, Reasoning. This process was tested and refined through several iterations. The data that I chose for these assignments & iterations was critical to student engagement and learning.

How do I know what kind of data is going to be interesting to students? Well, I ask them. I’ve been asking them for a lot of years. Every data set isn’t going to be interesting to every student, but overall, I have been able to identify and collect pretty good data sets.

In the spring term I used these data sets (and the associated class devised claims):

• Minutes between blast times for Old Faithful (Claim: The time between blasts will be 90 minutes plus or minus 20 minutes.)
• Ages of Best Actress and Best Actor Oscar winners (Claim: The ages of the Best Actress Oscar winners is typically less than the ages of the Best Actor Oscar winners.)
• Box office (opening weekend, domestic, worldwide), critics & audience ratings for “original” movies and their sequels (Claim: Original movies are better than sequels.)
• Juvenile detention/incarceration rates for various types of crimes by sex and race (Claim: African-American males are incarcerated at a higher rate than any other subgroup.)
• Education level and unemployment rates (Claim: People with a higher level of education have lower unemployment rates.)
• Class measurements such as height, arm span, kneeling height, forearm length, hand span, etc (Claim: Human body measurements are related in a predictable way.)
• Car data including curb weight, highway mpg, fuel type, and engine size (Claim: Highway mpg depends the most on fuel type.)
• Fast food burger nutrition including calories, fat, protein, carbohydrates, etc (Claim: Fast food burgers are unhealthy.)
• Baseball data from the 1919 Chicago White Sox (Claim: The evidence supports the decisions made about the accused players in the 1919 World Series.)

Even with all of these options, students added their own:

• Skateboarding data including ages and birthplaces of known skaters and number of skate parks in a state (Claim: Professional skateboarders are most likely to come from California.)
• Olympic male swimming data (Claim: Michael Phelps is the best Olympic swimmer of all time.)

## What’s important about all of these data sets?

They all provide multiple variables and opportunities for comparison. They offer students multiple ways to investigate the claims. They allow students to create different representations to support their reasoning. So, the lesson here is that the data sets used much be robust enough for students to really dig into.

Imagine what could happen if the course were integrated with science or social studies.

Next post: The results

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## Claim, Evidence, Reasoning: Starting fresh

I’m not an AP Stats teacher. I did that once. What’s important to me is that all students who graduate from high school have an opportunity to think and reason about real data in a deep and meaningful way. AP Stats is typically reserved for a few juniors or seniors. Maybe they don’t want to do AP Calc, or maybe they’ve already done it. Both of these reasons are unacceptable to me. Data literacy needs to have a higher profile – it needs to be more important than being able to simplify rational expressions. Our students need to be able to reason about data that’s presented to them in the press, or on social media, or by our elected officials. That’s my personal crusade.

Since last December I’ve been on this journey to improve my statistics teaching and the learning of my students. I shared my catastrophic failing first attempt and progress made with that group.  One of the beautiful things about our trimester schedule is that it allows me to immediately apply new learning to a new group – assuming that I am teaching a new section of the same course, I don’t have to wait a whole year to apply what I’ve learned. Luckily, this was the case this year. So, in late March I was able to begin anew, armed with what I learned during the previous term.

My spring term class was also a small group, but quite different from the winter class. This new class had more than 50% who struggled with writing. Since the focus of our work would be “claim, evidence, reasoning,” I would have to find alternative ways for these students to share their learning and their arguments. I wrote my new PDSA form and jumped in, hoping that I had learned enough from the winter term to be somewhat successful this time. (For information about PDSA cycles, see here and here.)

In general, I used this process to introduce concepts:

• Tell students about the data, usually on paper and verbally, and give them time to make predictions about what they expect from the data. Students do not have access to the data yet. Have some discussion about those predictions. Write them on the board (or some other medium). These predictions become claims to investigate.
• Give students access to the data & some graphical representations, usually on paper, and have them think about how the data might or might not support the claims that they made. Then ask them to discuss the data with a partner and determine whether or not the data support their claim.
• Ask them to write a statement about whether or not the data support the claim and why. The why is important – it’s the evidence and the reasoning piece of the “claim, evidence, reasoning” approach.
• Collect students’ statements, collate them into one document, then have students assess the statements according to the rubric. The focus here is on formulating an argument, not on calculating statistics or representing data. That comes later.

I completed this cycle twice, with two sets of data: minutes between blast times for Old Faithful and ages of winners of Best Actor and Best Actress Oscars.

These are the scaffolds that I provided for the first couple of steps for the Oscar data: predictions & analysis. Remember, the objective at this point is on making an argument, not calculating statistics or creating representations. Taking that piece out of the mix allowed students to focus on finding evidence and formulating reasoning for the claim that we had produced as a class. The next step is to collectively look at the statements that the students produced and assess where they fall on the rubric. This was the second time that we reviewed student work against the rubric. All of this introduction was treated as formative, so although the assignment (and whether or not it was completed) went into the grade book, no grade was attached.

The process for practicing was similar, but included less scaffolding and did not include the step of reviewing student statements. It generally went like this:

• Tell students about the data, usually on paper and verbally, and give them time to make predictions about what they expect from the data. Students do not have access to the data yet. These predictions become claims to investigate.
• Give students access to the data, generally in digital form, and a template to help them organize their thinking.
• Have students calculate statistics and create representations to provide evidence to support or refute their claims.
• Have students paste their representations into the template and write a statement or paragraph explaining the evidence (this is the reasoning step).

I did this cycle twice for our unit on univariate data: once using data about movies and their sequels and again using a variety of data from which students could choose. By the 4th cycle this is what the assignment directions and template looked like. This was the end of unit assignment for the spring term.

At the beginning of this post I mentioned that more than 50% of this particular class had been identified as having difficulties with writing. So, what did I do? I pushed them to write something – at least one statement (or, in some cases, two) – and then offered to let them talk through their evidence and reasoning with me. I knew that there was good reasoning happening, and I wasn’t assessing their writing anyway. So, why not make the necessary accommodations?

Next post: The importance of data choices.

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