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.

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

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

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.

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

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

Finally we offer them these equations.

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.

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.

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.

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.

I am sure about discussion yet with my freshman so I am starting with this. #MTBoS #EduColor pic.twitter.com/uurIr92CCk

— Anne Schwartz (@sophgermain) November 9, 2016

Free-write this morning. #Election2016 #education pic.twitter.com/zlEs9PFL8A

— Sahar Khatri (@KhatriMath) November 9, 2016

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?