#CMCMath Opening Keynote Address – Practice Problems

This is the keynote address I gave at CMC North this weekend with my co-presenters Shira Helft, Juana de Anda, and Fawn Nguyen.

The premise:

For a long time I worried I had chosen the wrong career. Other careers seemed like they had so much in their favor – better pay, less homework, more flexibility on the timing of bathroom breaks, etc. If you followed this blog ten years ago, you witnessed that worry.

Then a conversation with some of my close friends convinced me why I – and we – never have to envy any other career:

We have the best questions.

At least for me, no other job has more interesting questions than the job of helping students learn and love to learn mathematics.

A career in teaching means freedom from boredom.

To illustrate that, I interviewed three teachers at different stages in their careers – a teacher in her first decade, her second decade, and her third decade of teaching. I asked them, “What questions are you wondering right now?” Then we each took ten minutes to share our four questions.

But our talks weren’t disconnected. An important thread connected each of them, and I elaborated on that connection at the end of the talk.

Chapters

Please pitch in. Tell us all in the comments:

What question motivates you this year? What question wakes you up in the morning and energizes you throughout your day?

Featured Comments

Kathy:

The question that drives me is “How can I present this in a fashion that will be so interesting that they will not only want to learn it, but they will remember it next week, next month, and next year?”

Andrew Stadel:

Whether with my family (most important), the teachers I support, or students I work with:
How am I being present?

[Pseudocontext Saturdays] Rock Climber

This Week’s Installment

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Poll

What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #7

  • Identifying quadrilaterals (60%, 227 Votes)
  • Counting to 100 by 10's (40%, 149 Votes)

Total Voters: 376

Loading ... Loading ...

(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Team Me: 4
Team Commenters: 2

Pseudocontext Submissions

Jennifer Pazirandeh:

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Jon Orr:

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Michelle Pavlovsky:

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Rules

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)

Answer

I lose again. (But aren’t we all winners on Pseudocontext Saturdays? No? Just you. Okay.)

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The judges rule that this violates the first rule of pseudocontext:

Given a context, the assigned question isn’t a question most human beings would ask about it.

I think we can neutralize this pseudocontext by simply deleting the context. Delete the rock wall and we delete the lie that rock climbers are concerned with quadrilaterals while simultaneously preserving a task with a lot of admirable qualities.

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Then ask:

Which quadrilaterals can you locate in this grid? Can you find a trapezoid? How do you know it’s a trapezoid? Show a neighbor.

For whatever it’s worth, if there were some way to help Livia climb the wall by communicating with her through quadrilaterals, I’d re-evaluate this entire post.

[via John Golden]

“All the time.”

David Cox:

Yesterday, a student gave me step-by-step directions to solve a Rubik’s Cube. I finished it, but had no idea what I was doing. At times, I just watched what he did and copied his moves without even looking at the cube in my hands.

When we were finished, I exclaimed, “I did it!”, received a high-five from the student and some even applauded. For a moment, I felt like I had accomplished something. That feeling didn’t last long. I asked the class how often they experience what I just did.

They said, “All the time.”

Featured Comment

Lauren Beitel:

Is there an argument to be made that sometimes the conceptual understanding comes from repeating a procedure, then reflecting on it? Discovering/noticing patterns through repetition?

Great question. I wrote a comment in response.

[Pseudocontext Saturdays] Smoke Jumper

This Week’s Installment

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Poll

What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturdays #6

  • Calculating mean, median, mode. (60%, 224 Votes)
  • Calculating angles of elevation (40%, 152 Votes)

Total Voters: 376

Loading ... Loading ...

(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Bad trend here. I do not like it.

Team Me: 4
Team Commenters: 1

Pseudocontext Submissions

Curmudgeon

pseudo-func1

Cathy Yenca

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And no fewer than three people – Bodil Isaksen, Jocelyn Dagenais, and David Petro – sent me the following problem, created by a French teacher.

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And I don’t know. The jist of the problem is that two soccer players are arguing about the perfection of one of their dabs. They consult a universal dabbing rulebook which says that in a perfect dab those triangles above must be right triangles. And it’s all pretty winking, right? It can’t be pseudocontext if it isn’t actually trying to be context in the first place, right? The judges give it a pass.

Rules

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)

Answer

The commenters win a second straight week.

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The judges rule that this problem satisfies the first criterion for pseudocontext:

Given a context, the assigned question isn’t a question most human beings would ask about it.

A question that might neutralize the pseudocontext is: “Can all of these smoke jumpers ride in the same plane together? How would you arrange them so the plane is properly balanced?”

Instead, the task here is to find mean, median, mode, standard deviation, first quartile, third quartile, the interquartile range, the maximum, the minimum, the variance, etc, etc.

Do you get my point? Yes, all of those operations could be performed on those numbers. We often assign all of the math that could be done in a context without asking ourselves, what math must be done in the context? What math does the context demand?”

Engagement in Math: Three Places to Start

Mark Chubb, today on Twitter:

If a teacher sees students as disengaged and not liking math, what would be one good thing to watch, one good thing to read, one good thing to try?

Watch: Beyond Relevance & Real World.
Read: Why Don’t Students Like School?
Try: Estimation180.

Andrea Davis, later today in the comments:

Will you please give me the top three pieces of advice you have for the teachers of our youngest learners? We are K-6 and want to start now.

One, ask informal, relational questions (questioning, estimating, arguing, defining, etc.) as often as formal, operational questions (solving, calculating, simplifying).

Two, pose problems that have gaps in them – look up headless problems, tailless problems, and numberless problems, for three examples – and ask students to help you fill in those gaps. The most interesting problems are co-developed by teachers and students, not merely assigned in completed form by the teacher.

Three, before any explanation, create conditions that prepare students to learn from that explanation. These for example.

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What are your suggestions for Andrea and Mark?

Featured Comments

Tim Teaches Math:

Play.

Let’s try to describe a big number using a small amount of syllables (Berry’s Paradox). For example, 777777 takes 20 syllables, but saying “777 times 1001” takes 15. For a number like “741” which is seven syllables, “Nine cubed plus twelve” is much better. More complicated expressions test our perception of order of operations. Have students come up with a scoring system to rank abbreviations.

Sarah Giek:

Read: Mathematical Mindsets
Watch: Five Principles of Extraordinary Math Teaching.
Try: Number Talk Images