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.

]]>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.

And then there was this one:

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.

]]>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.

]]>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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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 3*x* + 8 = 44, I found myself covering up the 3*x* and asking the student, “What plus 8 equals 44?” When they came up with 36, I would follow up with, “So 3*x* 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 3*x* + 8 = 44 because they couldn’t *remember* the procedure. After “Using Structure …” they didn’t have to *remember*, they just had to *think*!

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.”

]]>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:

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

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

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.

So, let me clarify. Today was the first day with my new advisory. I’ll admit I was nervous. I couldn’t quite figure it out. I’m typically nervous meeting new people, but I know what that feels like. This felt different. And then I realized that “my people” weren’t going to be there today. They’ve all graduated and gone of to wherever it is they’ve gone off to. So that was part of it. Then my husband/colleague said to me, “And they’ve always been there.” Wow. That was it. That thing I was feeling. I was going off to school to meet 16 new Baxter students and the people I had come to really depend on weren’t going to be there. So this was my transition day. Graduation didn’t make me sad. Today made me a little sad.

And then I met them. We played my silly, stupid name game which, even though some of them hated it today, I know they will appreciate why I made them do it at some point in the future. We spent an hour together, just us with a couple of Baxter Ambassadors (returning students who know the ropes), getting to know each other, getting the rundown of today’s schedule. Then we met up with four other advisory groups at the park and did some fun team-building activities, led by other Baxter Ambassadors and fabulous colleagues. The afternoon held a couple of workshops about Baxter, in mixed advisory groups, and a “Genius Session” about a cool thing that other faculty wanted to offer.

This week is just for the 9th graders. There will be a total of six workshops, two Genius Sessions, a Scavenger Hunt, building a float, and a little bit of testing. I like that we are giving time to develop them as a group – an advisory group, a workshop team, and the Class of 2021. They come to us from all over southern Maine. In this advisory group I have students coming to Portland from as far away as Bridgton, Alfred, and Auburn and as close as Portland, Westbrook, and Scarborough. It’s worth the time to help them get to know each other. They leave their hometown friends behind to come to Baxter. That’s kind of a big transition. And each one has their own reason for coming to us.

Every year we get to iterate the start of school. Every year it gets better. I am grateful to work in a school that learns by doing and reflects on how to improve next time.

]]>**Full disclosure**: I’ve been using TI-Nspire technology for over a decade. I’m not new to this game. I have also used TI-8* family of graphing calculators as part of my teaching since 1990. I still have a TI-84, but I prefer the teaching power of TI-Nspire. I am also a T^{3} Regional Instructor. I became an instructor because I am passionate about how TI-Nspire can be used to help students learn math better. Even if I were not an instructor, I would give the same recommendations and say the same great things about TI-Nspire.

Back to the why.

**Coding**: I’ve created an “hour of code” lesson for 9th grade orientation. Since they won’t have their laptops yet, we’ll be using TI-Nspire handhelds. Sure, some students may already have experience programming, but the materials allow me to easily differentiate. And the Innovator Hub provides an additional challenge for those who need it. Best of all, coding with TI-Basic is pretty straight forward for teaching programming structures.

**CAS**: Using the CAS capability as a learning tool helps students to see structure in the mathematics. Sometime it causes us to ask interesting questions about whether an unexpected result is equivalent to the result we expected to see.

**Modeling**: I can import a picture and superimpose the graph of a function. I can drag the graph to conform to the shape I’m trying to model (as long as that function is appropriate for the shape). I can add a point on the graph and identify its coordinates. My students can have some really interesting conversation about what all the numbers mean.

**Beyond algebra & graphs**: The statistics and geometry applications are unparalleled in a single device. I’ve taught statistics with Fathom and geometry with Geometer’s Sketchpad. The TI-Nspire apps remind me of these two powerful programs. Last year I taught a lot of statistics classes. TI-Nspire was really valuable when it came to representing and analyzing the data. And the geometry app is dynamic, too.

**Operates like a computer**: The operating system is file and menu driven, so it’s easy to think about TI-Nspire documents like computer files. All of the same keyboard shortcuts apply, too, which kids really love. There’s even a touch pad that operates like a mouse. When I first introduce the handheld to my students, I point out the important buttons: menu, esc, tab, ctrl. These can get you out of anything you’ve messed up by mistake. You can keep undoing until you get back to what you want. Just keep hitting ctrl-z.

**Beyond the handheld**: We only have a few classroom sets of the handhelds, but we have enough software licenses for all of our students’ laptops. That’s where we use TI-Nspire most often. It’s great because the handheld and computer versions are functionally identical and the computer version offers a lot more screen real estate. Sometimes that comes in handy when you’re comparing a lot of variables.

**Advanced options**: Coding with Lua can extend document design/creation options for more experienced programming students. Using science probes for data collection helps to integrate the two disciplines. The TI-Nspire Navigator is a powerful tool for formative assessment and student feedback.

**TI Support**: There is a vast library of activities available for free on the TI website. These are curated, organized by topic, and searchable. Each activity includes a student directions sheet in Word format so that any teacher can modify it for their students or context. If I ever have a problem, TI Cares is right there to help. I’ve never had an issue renewing a software license (because my school laptop was reimaged over the summer) or getting help working through a network issue getting the Navigator up and running. People, right there in Dallas, ready to answer my questions and help me out. I really appreciate that they listen and aren’t just running through a script.

So, why do I love my TI-Nspire? Because it’s powerful, flexible, and backed by a company with over 30 years in education: one that listens to teachers and continues to improve.

]]>- 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 the class data it became clear that some instruction about regression lines and correlation coefficients was needed.

Now, to answer those questions.

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.

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.

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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