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.”

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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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Year 5, Day 1

Today was the first day of school. Okay, that’s not entirely true. We had a whole week, last week, of teacher workshops – and they were all great. Even the insurance guy was hilarious – and that takes some skill, right?

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.

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Why I love my TI-Nspire

As the new school year approaches and I contemplate how I might structure my classes for the coming year, I remember how grateful I am that there are so many resources out there to support me and my students. One resource, though, rises above the rest: TI-Nspire. Why?

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 T3 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.

hoop shot

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.

mms

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.

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

Now, to answer those questions.

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.)
  • Sales of vinyl records and digital album downloads (Claim: Sales of vinyl records will soon overtake digital album downloads.)
  • 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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