Tag Archives: NCTMP2A

#NCTMLive and #T3Learns webinar: Implement tasks that promote reasoning and problem solving, and use and connect mathematical representations.

On Wednesday, May 2, 2018, Jill Gough (@jgough) and I co-facilitated the second webinar in a four-part series on the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All.

Implement tasks that promote reasoning and problem solving,
and Use and connect mathematical representations.

Effective teaching of mathematics facilitates discourse among learners to build shared understanding of mathematical ideas by analyzing and comparing approaches and arguments.

  • How might we implement and facilitate tasks that promote productive discussions to strengthen the teaching and learning of mathematics in all our teaching settings – teaching students and teaching teachers?
  • What types of tasks encourage mathematical flexibility to show what we know in more than one way?

Our slide deck:

View this document on Scribd

Our agenda:

7:00 Jill/Jennifer’s Opening remarks

  • Share your name and grade level(s) or course(s).
  • Norm setting and Purpose
7:05 Number Talk: 81 x 25

  • Your natural way and Illustrate
  • Decompose into two or more addends (show it)
  • Show your work so a reader understands without asking questions
  • Share work via Twitter using #NCTMLive or
7:10 #LL2LU Use and connect mathematical representations

  • Self-assess where you are
  • Self-assessment effect size

Think back to a lesson you taught or observed in the past month. At what level did you or the teacher show evidence of using mathematical representations?

7:15 Task:  (x+1)^2 does/doesn’t equal x^2+1
7:25 Taking Action (DEI quote)
7:30 #LL2LU Implement Tasks That Promote Reasoning and Problem Solving
7:35 Graham Fletcher’s Open Middle Finding Equivalent Ratios
7:45 Illustrative Mathematics: Jim and Jesse’s Money
7:55 Close and preview next in the series

Some reflections from the chat window:

I learned to pay attention to multiple representations that my students will create when they are allowed the chance to think on their own. I learned to ask myself how am I fostering this environment with my questioning.

I learned to pay attention to the diversity of representations that different students bring to the classroom and to wait to everyone have time to think

I learned to pay attention (more) to illustrating work instead of focusing so much on algebraic reasoning in my approach to teaching Algebra I. I learned to ask myself how could I model multiple representations to my students.

I learned to pay attention to multiple representations because students all think and see things differently.

I learned to make sure to give a pause for students to make the connections between different ways of representing a problem, rather than just accepting the first right answer and moving on.

I learned to pay attention to the ways that I present information and concepts to children… I need to include more visual representations when I working with algebraic reasoning activities.

Cross-posted on Experiments in Learning by Doing


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A New Function

One of the NCTM Principles to Actions mathematics Teaching Practices is support productive struggle in learning mathematics. In the executive summary, we read “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

In calculus, we started the semester with a unit on Transcendental Functions. On the first day, students figured out everything they could about F(x).

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What is F’(x)?

What is F’’(x)?

What is F(1)?

Where is F(x) increasing, decreasing?

Where is F(x) concave up, concave down?

What is the domain for F(x)? the range?

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Then they sketched a graph of F(x) from what they figured out, and determined that F(x)=ln(x), and F’(ln(x))=1/x.

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(I found the suggestion for students coming up with F(x)=ln(x) by thinking through these questions somewhere else. But I don’t remember where, and I can’t find it anymore.)


So the next day, I asked them to differentiate y=log(2x).

I had not given them any “formula” for differentiating logarithmic functions. They had only figured out that the derivative of ln(x) was 1/x.

I sent the question to them as a Quick Poll to watch their progress.

I watched for a long time.

I saw and I heard productive struggle.

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And eventually, their struggle turned into success.

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We can cover so many more examples when we don’t give students time to grapple with mathematical ideas and relationships. But how effective are the examples without the productive struggle?

Ultimately, are my students better off having struggled to think through change of base to get to the derivative of log(2x) using what they already know about the derivative of ln(x)? Or would they have been better off with me giving them the textbook way to calculate the derivative of logb(x)?

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I’m hoping for the former, as the journey continues …


Posted by on February 9, 2015 in Calculus


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Productive Struggle: The Law of Sines

NCTM’s Principles to Actions suggests eight Mathematics Teaching Practices for teachers. One of them is to support productive struggle in learning mathematics. The executive summary states: “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

What does productive struggle look like? What does it sound like?

I saw a glimpse of what productive struggle looks like yesterday. I get to share a room with a teacher (who happens to be a former student of mine), and so I listen with one ear when I’m in the room working at my desk during her Precalculus class. The lesson was on the Law of Sines, but Trisha didn’t tell the students from the beginning that was the learning goal. Instead, the students focused on the math practice make sense of problems and persevere in solving them.

She presented a situation. And the students made assumptions and asked questions.

One I remember hearing was “I guess we can’t just use a measuring tape?”

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Then she asked them to solve the problem.

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And so they did. These students didn’t balk at the task. They all worked. They didn’t even talk very much at first … you could hear them thinking in the silence that encompassed the room. That’s when I looked over and realized that I was seeing productive struggle in action. Productive struggle isn’t always quiet, but it definitely started that way for these students. Eventually, students listened to Ain’t No (River Wide) Enough while they worked.

When solving the non-right triangle without knowing the Law of Sines, the students used another Math Practice – look for and make use of structure – to draw auxiliary lines. Some drew an altitude for the given triangle to decompose it into two right triangles. Some composed the given triangle into a right triangle.

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Trisha collected evidence of what students could do using a Quick Poll.

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So if we are given one side length and two angle measures of a triangle, is there a faster way to get to the other side?

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More productive struggle … the numbers are now gone, students are reasoning abstractly to make a generalization.

And they did.

And they derived the Law of Sines in the meantime.

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How often do we give our students a chance to engage in productive struggle? In how many classrooms is the Law of Sines just given to students to use, devoid of giving students the opportunity to “grapple with mathematical ideas and relationships”?

When I discussed what I saw with Trisha, she noted that last year, only a few of the students in her class successfully solved the triangle prior to learning about the Law of Sines. This year, all of them tried and most of them succeeded. These are the students with whom we started CCSS Geometry year before last. These are the students who have been learning high school math with a focus on the Math Practices. These are students who are becoming the mathematically proficient students that we want them to be. Because we are letting them. As the journey continues, we are learning to leave the front of the classroom behind so that we can support productive struggle in learning mathematics.


Posted by on November 6, 2014 in Geometry, Trigonometry


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