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*

- Introduction.
- Shira Helft’s question.
- Juana de Anda’s question.
- Fawn Nguyen’s question.
- My question.
- Conclusion.

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**

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?”

]]>Whether with my family (most important), the teachers I support, or students I work with:

How am I being present?

**Poll**

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

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(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:*

*Jon Orr:*

*Michelle Pavlovsky:*

**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.)

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

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.

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]

]]>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**

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.

]]>**Poll**

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

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(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*

*Cathy Yenca*

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

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.)

The commenters win a second straight week.

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*?”

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**

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.

]]>

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**

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.

]]>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.

**Poll**

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

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.(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.

**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.)

Well well well … score one for Team Commenters.

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**

]]>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.)

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

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

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?

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

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.**Current Scoreboard**

*Team Me*: 3

*Team Commenters*: 0

**Pseudocontext Submissions**

*Michelle Pavlovsky*

*Paul Hartzer*

**Rules**

*isn’t* pseudocontext, collect a personal point.

(See the rationale for this exercise.)

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**

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

]]>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.

Tricia Poulin makes some awesome moves in her #bottleflipping lesson, including this one:

Okay, so now the kicker: will this ratio be maintained no matter the size of the bottle?

Graham Fletcher offers us video of kindergarten students interacting in a 3-Act modeling task:

It’s always great to engage the youngins’ in 3-Act Tasks. I’ve heard colleagues say, “I don’t have time to do these types of lessons.” I hope this helps show that we don’t have time to not have the time.

Wendy Menard offers her own spin on the Money Duck, one of my favorite examples of expected value in the wild:

The students designed their own “Money Animals”, complete with a price, distribution, and an expected value. This was all done on one sheet; the design, price and distribution were visible to all, while the calculations were on the back. After everyone had finished, we had our Money Animal Bonanza.

Sarah Carter hosts the Mini-Metric Olympics, a series of data collection & analysis events with names like “Left-Handed Sponge Squeeze” and “Paper Plate Discus”:

]]>After the measurements were all taken, we calculated our error for each event. One student insisted that she would do better if we calculated percent error instead, so we did that too to check and see if she was right. In the future, I think I would add a “percent error” column to the score tracking sheet.