Becoming the Math Teacher Book Study, Part 3

I hope you're enjoying this book study as much as I am! There's so much to reflect on that I'm having a hard time narrowing it down. If you missed the first 2 parts, you can catch up with Chapters 1-3 HERE and Chapters 4-5 HERE.
Mathematicians Rise to a Challenge
Since so many teachers are required to use their district-mandated math curriculum, former high school math teacher Dan Meyer gave a TED Talk called Math Class Needs a Makeover. If you have not seen it yet, you must watch it HERE.  He compares the difference between reading a word problem in a textbook and demonstrating the same problem in real life. Meyer argues that "students lack initiative, perseverance and retention because they spend most of their time plugging in formulas without understanding the mathematics." (p. 114) Our goal should be for students to learn and understand math by working on problems, NOT just to get the write answers to textbook problems. Amen!

Author Tracy Zager claims that problems in textbooks often have too many parts, often missing the main math objective. Or the problems spoon feed students and give them too much information so that no thinking is necessary. I've noticed this when students blindly multiply numbers in a word problem because the chapter is about multiplication or the other practice problems were all multiplication. They need to develop an understanding of what the problems are asking, building on what they already know, and then figure out how to solve problems themselves. (p. 118)

Low Thresholds, High Ceilings, and Open Middles
I love the following "math problem as room" analogy by Seyour Papert (1980). The best math problems have low threshold (entry points accessible by all), high ceilings (allow for differentiation and deeper investigation), yet open middles (multiple solution paths to the same answer).
Check out this fantastic website, Open Middle . It's a free resource organized by strand and grade level. There are hundreds of "Open Middle" tasks posted!

Productive Struggle versus Destructive Struggle
Another section for thought in Chapter 6 is the difference between productive struggle vs. destructive struggle. Destructive struggle leads to frustration and students feeling inadequate. Productive struggle is when students are challenged and learning. We want our kids to think and try and understand. In our attempts to "teach" and help students, teachers (including me for sure!) have tried to rescue our struggling students, depriving them of the opportunities to fully make sense of the math. (NCTM 2014, 48) Zager states two scenarios: 1) breaking the problem down into smaller steps for students (guilty!) and 2) teaching students tricks, rules, algorithms, or procedures so they can get an answer (guilty!) (p. 129) I had to sit and think about this for a while: if the teacher breaks down the problem and "teaches" what to do for each step, then the teacher is the one who has done the problem solving, thinking, and learning. The student simply implements the steps, whether he understands them or not. Hmmm. . . I thought I was being helpful. Likewise, teaching rules does not help students to understand the problem. (especially if you don't know when to apply which rule!)

When reading, I have students use symbols on post-its to take notes: points they're confused about, important parts they want to discuss, etc. However, I had not thought about teaching students to use this same strategy! Margin symbols are simple icons kids draw in the margin of their work to let the teacher know their progress. These are the six symbols Zager used:
Did you notice the growth mindset language: to be successful, assuming all students WILL be successful? What a great strategy! This is teaching students to be responsible for their learning and to learn how to say what they need. (Remember the importance of precision in Chapter 5? Review that HERE) Using symbols also allows students to move on, with the intent to revisit the problem. I love this! I'm going to introduce it to my class this year!

Growing up, I memorized formulas and rules, yet I never understood the WHY these rules worked and therefore, I didn't know when to apply them. When I became a teacher, I was determined to understand math so I could help my students understand math. A website to help us teachers develop our own mathematical content knowledge and improve our teaching is Ask Dr. Math , a free question and answer service, sponsored by NCTM (National Council of Teachers of Mathematics).  Although it is no longer taking new questions about math, there is a terrific searchable archive available by grade level and math topic, as well as summaries of Frequently Asked Questions (the Dr. Math FAQ).  Try it; type in a math question to see if it is in their database. It's so interesting!

Mathematicians Ask Questions
Developing Questions
Chapter 7 focuses on the importance of questions, as opposed to the typical math classroom that focuses on finding answers. Giving students the opportunity to ask questions without the pressure of finding answers begins to encourage risk taking, inquisitiveness, and playfulness. A great website Zager recommends is 101questions ( There are hundreds of thought provoking photos and short video clips, such as this San Francisco house below.
What questions come to mind when you look at this house? Have students call out their questions as they are recorded on the board/chart. This is a wonderful way to set the expectation that students should approach math with the same inquisitiveness and curiosity. I have often used a similar strategy (a la CGI) of showing a picture while students discuss what they observe and wonder, as a set up to the particular problem we are focusing for the day. This gets them excited about the problem, as well as gives context. Some follow up questions Zager recommends:

  • What question do you think most people ask?
  • What question would you really want an answer to?
  • Did anyone else's question inspire you to think of a new question or add on to theirs?
  • What's a question you heard today that was surprising/interesting/thought provoking to you?
  • Did you ask any questions we could figure out using mathematics, if we wanted to?
As a veteran teacher, I was a little hesitant to open up discussions to seemingly random, off topic "kid questions".  So I was relieved to read "As much as I love 101questions, we do need to find answers in math class... how can we maintain the inquisitiveness of 101questions while teaching problem solving?" (p. 141) Phew! Exactly what I was thinking!  In comes a strategy by Max Ray-Riek called Notice and Wonder:

Give students an image or a scenario without a question. Here's an example from a fourth grade textbook:
Julia was buying DVDs of her old favorite TV series. She bought eight DVDs at the store and she bought seven online.
After recording student noticings (observations) and wonderings (questions), either reveal a question you'd like them to solve (the problem) or have students come up with a question by asking, "If this picture/scenario were the beginning of a math problem, what could the math problem be?" Ray-Riek described the benefit of Notice and Wonder: "Leaving off the question and just sharing an initial story/scenario/image increases participation from struggling students because there are no right or wrong answers to 'What do you notice?' and 'What are you wondering?'... Through wondering, students see that math problems come from their own thinking." (p. 144) What a terrific way to get students to think and apply their math knowledge! This helps to eliminate kids immediately grabbing numbers and trying to "do things to them".  It's also interesting to see if students come up with the same question the textbook intended!

To bring the focus back to math and helping students determine what is or isn't a mathematical question,  Ray-Riek suggests asking:

  • Which of these noticings have to do with math?
  • Which of these wonderings could we use math to help us answer/prove?
  • Which of these did you use math to think of?
  • Which of these could we use math to explore more? 
I'm excited for school to start so I can teach my new students to use math margin symbols and to notice and wonder! Have you tried remaking your textbook word problems to encourage kids to slow down and think before solving? I'd love to hear your experiences!

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