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

The Problem with Multiple Representations

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I propose we add a representation to the holy trinity of graphs, equations, and tables: “backwards blue graphs.” Have a look.

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Expert mathematicians and math teachers instantly see the uselessness of the backwards blue graph representation. It offers us no extra insight into or power over the data. But my suspicion is that many students feel that way about all the representations. They’re all the backwards blue graphs.

Students will dutifully and even capably create tables, equations, and graphs but do they understand the advantages that each one affords us? Or do they just understand that their grades depend on capably creating each representation?

At Desmos, we created Playing Catch-Up to put students in a place to experience the power of equations over other representations. Namely, equations offer us precision.

So we show students a scenario in which Julio Jones get a head start over Rich Eisen, but runs at half speed.

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We ask students to extend a graph to determine when Rich will catch Julio.

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We ask students to extend a table to answer the same question.

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Finally we offer them these equations.

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Our intent with this three-screen instructional sequence is to put teachers in a place to have a conversation with students about one advantage equations have over the other representations. They offer us more precision and confidence in our answer.

Without that conversation, graphs, tables, and equations all may as well be the backwards blue graphs.

Your Turn

The power of equations is precision. We put students in a place to experience that power by asking them to make predictions using the imprecise representations first.

In what ways are graphs uniquely powerful? Tables? How will you put students in a place to experience those powers?

BTW

Principles to Actions is great here:

Students should be able to approach a problem from several points of view and be encouraged to switch among representations until they are able to understand the situation and proceed along a path that will lead them to a solution. This implies that students view representations as tools that they can use to help them solve problems, rather than as an end in themselves.

Featured Comment

Sue Hellman:

I think an analogy here are the 3 ‘representations’ of location/directions provided by Google: a map, written directions, and street view. They all provide similar or at least related information but each offers advantages depending on the purpose and background knowledge of the user.

Katie Waddle:

I’ve noticed that kids who get the click that they all are connected understand stuff down the road a lot better, so I build in explicit teaching around seeing those connections (where is the y-intercept in the table? how can I see the slope in the equation?). It would be neat if kids could color code and write on things in this exercise, but computers are not good at letting you add stuff like that.

[Pseudocontext Saturday] Runner

This Week’s Installment

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Poll

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

Pseudocontext Saturday #5

  • Solving systems of equations (58%, 276 Votes)
  • Finding least common multiples (42%, 197 Votes)

Total Voters: 473

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: 0

Pseudocontext Submissions

Michelle Pavlovsky

This is may be the worst math problem I’ve seen in my life.

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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

Well well well … score one for Team Commenters.

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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.

Were you sleep-running the entire time? Why can’t you remember where you ran to and from?

This is Pseudocontext Saturday, so rather than overstep my jurisdiction I’ll let someone else critique the scaffolds in problem #32.

Featured Comment

Hallie:

Dan went for a run. Every 13th stride he sneezes. Every 17th stride he blinks. Every 5th stride a shiver runs down his spine thinking about his homework he has neglected to do. When will he shiver, blink and sneeze at the same time? (Ignore that it is impossible to sneeze with your eyes open.)

What I’m Working on the Day After the Election

Every morning, the four members of the teaching team at Desmos post a note to their Slack channel listing all the tasks they’re working on that day. We use the hashtag #workingon for easy reference. This is what I posted this morning.

I don’t know how any of you voted and I won’t make assumptions. (It’s clear that a lot of people who represented themselves one way to pollsters voted another way, and that likely holds true for our company as well.) But you may have voted like I did yesterday, leaving you bereft today, and struggling to locate some kind of purpose for your work, struggling to participate in the solution to a problem that has many names. If that’s you, then this what I’m telling myself about our work this morning.

If the name of that problem is economic anxiety, if President-elect Trump was propelled to power by people whom globalization, open borders, and free trade have left behind, I encourage us to locate political and social solutions to their problems, definitely, but also to help those people (and their children, particularly) learn better math better. Capitalists continue to automate routine manual jobs, leaving behind more and more non-routine cognitive jobs. Non-routine math tasks are difficult to design, difficult to teach, difficult to learn, and increasingly essential to full economic participation. We can help design them and we can give teachers tools to make them easier to teach.

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If the name of that problem is bigotry, then we should help teachers facilitate constructive arguments, cultivate empathy, and emphasize patience. One dimension of bigotry is impatience, a sense that “I know everything there is to know about a person based on his or her most easily observed characteristics.” The traditions of many math classes – completing short problems resulting in simple answers that are easily verified in the back of the textbook – only exacerbate this problem. Christopher Danielson’s “Which One Doesn’t Belong,” by contrast, invites students to realize that all of those objects don’t belong for one reason or another, that we can negotiate those reasons productively, and that we can understand the world through the eyes of another.

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Obviously we have lots of work to do in our neighborhoods, our churches, our social networks, our local and state governments, and in ourselves, work that is probably larger than anything we’ll do at Desmos today. But if yesterday’s election left you wondering what work you can do at Desmos to help solve a problem with many names, this is what I’m #workingon.

BTW. I’m watching Twitter for examples of math teachers helping their students understand where they live today. I’ll continue to update this post throughout the day.

Matt Enlow:

When you learn mathematics, you also learn a lot of other things. Here are three of those things.

John Golden:

We did Elizabeth Statmore’s talking points for Math Mindsets Chapter 7 (tracking), then for the election, then we looked at Megan Schmidt’s Social Justice Math slides.

wwntd offers her classes some words of consolation.

Dianna Hazelton asks her class:

What does the word empathy mean? How do you show empathy?

[Pseudocontext Saturday] Photos

This Week’s Installment

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Poll

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

Pseudocontext Saturday #4

  • Reasoning with proportions (63%, 523 Votes)
  • Calculating exponential growth (19%, 154 Votes)
  • Simplifying expressions (18%, 151 Votes)

Total Voters: 828

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: 3
Team Commenters: 0

Pseudocontext Submissions

Michelle Pavlovsky

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Paul Hartzer

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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

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The judges rule that this problem satisfies both criteria for pseudocontext:

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

I invite any commenter to rationalize the constraint that exactly 15 photos must be purchased and we don’t know which of them will be small or large. More often (always?) people begin with the photos they want, or perhaps they work from a total budget. “I can only buy 15 photos and the number of large photos I purchase can vary from zero to fifteen,” said no one ever.

Given that question, the assigned method isn’t a method most human beings would use to find it.

If most human beings were going to find out the cost of five large photos and ten small photos, they’d multiply each kind of photo by its price. Variables aren’t a useful tool.

So the textbook has made the world serve the math when math should serve the world. If the world doesn’t need math’s service, then math should be gracious enough to step out of the way.

Featured Comments

Jonathan Claydon:

I guessed correctly. The first and third choices made too much sense. Always step up to the plate thinking curveball.

Rachel:

The problem here is that the customer has no use for a general equation, but the store owner might–she’s got to deal with people who call in with all kinds of crazy orders and questions. Still, it’s unlikely the store owner would write an equation for just small and large pictures. It’s much more likely that she’d come up with a pricing scenario for unusual picture sizes.