What can I say? It’s been a rough 18 months. I hope to get back to recording my thoughts and trials and experiences of teaching soon. It’s not like I’m waiting for that perfect moment. I know that will never come. It’s more that I haven’t been very inspired to write about teaching recently.
Another graduating class. This makes three.
Last year, I wrote about the Class of 2017 and their plans. Seems only right that I should write about the Class of 2018. They, too, are a remarkable group of individuals who care about the world they live in – the world they will soon be leading. So, without further ado, here are the schools that look forward to welcoming our graduates.
- Acadia University
- Bard College
- Bates College
- Bennington College
- Clark University
- Earlham College
- Hampshire College
- Hobart and William Smith Colleges
- Landmark College
- Lesley University
- Maine College of Art
- Maine Maritime Academy
- Marlboro College
- Mount Holyoke College
- The New School
- Saint Anselm College
- Saint Lawrence University
- Salem State University
- Smith College
- Syracuse University
- Southern Maine Community College
- University of Maine
- University of Maine, Farmington
- University of Southern Maine
- Wentworth Institute of Technology
- Worcester Polytechnic University
- Yale University
Among the group who stated majors, there are 22 scientists, 10 engineers, 10 working with arts or design, 7 looking toward the liberal arts, 2 business, 2 working toward skilled trades, 4 entering the work force – including one with his own landscaping/snowplowing company – several are undecided and 7 are taking a gap year.
This group is also full of the pioneering spirit that brought our original group of students to Baxter. Like NASA’s “next nine,” they are the “next class.” And the world #bettermakeroom.
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.
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.
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.
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.
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.
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!