Hey history teacher-friends!

Lately, I’ve been interested in the math teaching opportunities that arise when we delete and then progressively reveal details of a task. Digital media offers us that luxury while paper denies it.

I saw an opportunity to apply the same approach in history and geography. I took the license plates from all fifty United States and removed explicit references to the state or its outline. Then Evan Weinberg turned it into an online quiz.

Feel free to send your students to that quiz, or to use the images themselves [full, deleted, animated] in any way you want. If you’re feeling obliging, stop by and let us know how it went in the comments.

]]>I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend *I’ve still got it* in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

**When Should You Bet Your Coffee?**

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers *such* a ticklish application of the Intermediate Value Theorem:

If the bowl only has *one* other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had *one million* cards in it, you’d want to hold onto your card. So *somewhere* in between one and one million, there is a number of cards where your decision *switches*. How do you figure out that number? (PS. I realize the IVT doesn’t hold for *discrete* functions like this one. Definitions offer us a lot of insight when we *stretch* them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

Here’s my work and the 3-Act Task for download.

**How Many Bottles of Coca-Cola Are in That Pool?**

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

“12 feet across by 30 inches high,” he responded.

Even though the *frame* of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

Here’s my work and the 3-Act Task for download.

**How Do You Solve Zukei Puzzles?**

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, *justification*. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus – every side congruent – but also to be able to *argue* from that definition.

And the challenge that tickled my brain *most* was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.

**Commenter-friends:**

- What are your professional resolutions for 2017?
- What recreational mathematics are you working on lately?
- If any of you enterprising programmers want to make a Zukei puzzle solver, I’d
*love*to see it.

**2017 Jan 2**. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

**2017 Jan 2**. Shaun Carter is another contender.

First, Wilson’s hypothesis:

Wilson fears that students who depend on technology [calculators, specifically –

dm] will fail to understand the importance of mathematical algorithms.

Next, Wilson’s experiment:

Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didn’t. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.

I trust my readers will supply the answer key in the comments.

**BTW**. I’m not saying there *isn’t* evidence that calculator use will inhibit a student’s understanding of mathematical algorithms, or that no such evidence will ever be found. I’m just saying this study *isn’t* that evidence.

**Featured Tweet**

Got one!

@ddmeyer I think you just found me a new example for chapter 4 (experimental design)...

— David Griswold (@DavidGriswoldHH) December 23, 2016

**Featured Comment**

]]>The most clarifying thing that I can recall being told about testing in mathematics came from a friend in that business: you’ll find a positive correlation between student performance on almost any two math tests. So don’t get too excited when it happens, and beware of using evidence of correlation on two tests as evidence for much.

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you *knew* this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to *solve* systems and struggle to *remain awake* while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem – and other problems *like* this problem?

I asked that question on Twitter and I received responses from, roughly speaking, two camps.

One group recommended we **change the adjectives and nouns**. That we make the problem more *real* or more *relevant* by changing the *objects* in the problem. For example, instead of analyzing an *animated* movie, we could first survey our classes for the movie genres they like most and use *those* in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who *aren’t* already adept at that work.

Another strategy is to ignore the adjectives and nouns and **change the verbs**, to change the *work* students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is *hard*, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges $4 for child tickets and $12 for adult tickets.

- What’s a large amount of money you could make?
- What’s a small amount of money you could make?
- Okay, your no-good kid brother is working the cash register. He told you he made:

- $2,550 on Friday
- $2,126 on Saturday
- $1,968 on Sunday
He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about *relationships* between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by *one set* of those quantities. Other times, *lots* of sets solve those relationships and other times those relationships are so constrained that they’re *never* solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he *wasn’t* lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to *visualize* their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization *quickly*, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student *learn that work* is lots and lots of scaffolding and our only hope for helping her *remain awake* through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

**BTW**. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.

**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*: 5

*Team Commenters*: 3

**Pseudocontext Submissions**

**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.)

This was a nail-biter between Team Commenters and Team Me this week, with Team Commenters narrowly tipping the scales in their favor.

The judges rule that this satisfies the second rule of pseudocontext:

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

Reasonable people might wonder about the dimensions of a water tank. The judges rule that most human beings would use a tape or a stick or any other kind of measuring device to answer it, not a cubic polynomial.

I can’t think of any way to neutralize this pseudocontext. The number of actual contexts for cubic polynomials with non-zero quadratic and linear terms is *vanishingly small*.

Here is an activity I would much prefer to use to teach the construction of polynomials. It doesn’t involve the real world but it does ask students to do real work.

**Featured Comment**

]]>One motif in pseudocontextual questions seems to be treating as a variable things that, you know,

don’t vary. I have a funny video playing in my mind of some surprised fish watching the volume of their tank become negative. But happily the volume of that tank is not varying, inasmuch as it’s sides are made of glass.

Also this week I received an email from May-Li Khoe, a researcher at Khan Academy, reflecting on her experience seeing Fawn Nguyen keynoting CMC-North. Both May-Li and Fawn are Asian-American.

I did not expect to be so affected by having Fawn speak during the keynote. Obviously the content of her presentation made an impression on me, but reflecting back later, I realized that I have never seen anyone remotely resembling myself as a keynote speaker, at any conference, ever.

We want all students to see themselves as people who can *do mathematics*, regardless of their race, ethnicity, gender, or any other variable. The power of mathematical thinking is good for *everybody*, and nobody should feel like their identity excludes them from that power.

The project of extending that access will require a diverse corps of teachers, which will require that a diverse corps of teachers sees teaching as a career full of advancement possibilities. Which means, among other efforts, that we need a more diverse corps of teachers speaking in front of large rooms of teachers.

So if you’re organizing a conference, I’m asking you to consider inviting any of the names below to give a talk before you consider inviting another tall, white dude. I’ll personally vouch for all of their abilities to deliver outstanding talks to large rooms of people. I have included Twitter contact information for each of them, along with websites and sample talks. I’m also happy to connect you with any of them personally. Let me know.

**Maria Anderson**. Applying research to instruction. [Twitter, Web, Sample]**Harold Asturias**. Teaching mathematics & academic language to emerging bilingual students. [Twitter, Sample]**Deborah Ball**. Teacher development; mathematical knowledge for teaching. [Twitter, Web, Sample]**Robert Berry**. Formative assessment; equitable experiences for all math students; #blackkidsdomath. [Twitter, Sample]**Jo Boaler**. Cultivating a growth mindset in mathematics. [Twitter, Web, Sample]**Marilyn Burns**. Helping students make sense of math. [Twitter, Web, Sample]**Ed Campos, Jr**. Technology integration. [Twitter, Web]**Peg Cagle**. Creating engaging mathematical experiences. [Twitter, Sample]**Shelley Carranza**. Technology integration. [Twitter]**Rafranz Davis**. Technology integration; creating equitable experiences for all math students. [Twitter, Web, Sample]**Juli Dixon**. Teaching students with special needs. [Twitter, Web, Sample]**Annie Fetter**. Mathematical thinking and problem solving. [Twitter, Sample]**Kristin Gray**. Creating engaging mathematical experiences. [Twitter, Web, Sample]**Rochelle Gutierrez**. Creating equitable experiences for all math students (and their teachers). [Twitter, Sample]**Shira Helft**. Instructional routines that promote discourse and sensemaking. [Twitter, Sample]**Ilana Horn**. Cultivating a student’s mathematical identity. [Twitter, Web, Sample]**Elham Kazemi**. Understanding a student’s mathematical thinking. [Twitter, Sample]**Jennie Magiera**. Technology integration. [Twitter, Sample]**Danny Martin**. Creating equitable experiences for all math students. [Sample]**David Masunaga**. Mathematical inquiry, particularly in geometry.**Fawn Nguyen**. Mathematical thinking and problem solving. [Twitter, Web, Sample]**Cathy O’Neil**. The powerful and sometimes pernicious effect of algebraic models in the world. [Twitter, Web, Sample]**Carl Oliver**. Integrating social justice and mathematics education. [Twitter, Web]**Megan Schmidt**. Integrating social justice and mathematics education. [Twitter, Web]**Marian Small**. Creating engaging and productive mathematical experiences. [Twitter, Web, Sample]**Joi Spencer**. Integrating social justice and mathematics education. [Twitter, Sample]**Lee Stiff**. Technology integration; creating equitable experiences for all math students. [Sample]**John Staley**. Teaching mathematics for social justice. [Twitter, Sample]**Greg Tang**. Creating engaging and productive mathematical experiences for elementary students. [Twitter, Web, Sample]**Megan Taylor**. Creating engaging and productive mathematical experiences. [Twitter, Sample]**Kaneka Turner**. Cultivating a student’s mathematical identity. [Twitter, Sample]**Sara Vanderwerf**. Creating equitable experiences for all math students. [Twitter, Web]**Jose Vilson**. Creating equitable experiences for all math students. [Twitter, Web, Sample]**Audrey Watters**. Analyzing technological trends and their effect on education and society. [Twitter, Web, Sample]**Anna Weltman**. Integrating creativity, art, and mathematics. [Twitter, Web, Sample]**Talithia Williams**. Statistics; diversity in higher education. [Twitter, Sample]**Jennifer Wilson**. Helping students make sense of mathematics; #slowmath. [Twitter, Web, Sample]**Cathy Yenca**. Technology integration. [Twitter, Web, Sample]**Tracy Zager**. Literally anything – have her read the tax code. (Also once her book comes out, your probability of getting her for your conference decreases asymptotically to zero. Buy now.) [Twitter, Web, Sample]

Add someone deserving or promising in the comments. Attach the same information you see above.

[Photos by Cathy Yenca and Kristin Hartloff.]

**2016 Dec 14**. The commenters have already caught a bunch of my *really* embarrassing omissions. Thanks for picking up my slack, everybody.

**2016 Dec 16**. In response to this critique from TODOS, I’d like to clarify that, yes, this list is incomplete, and my hope was that it would be made more complete in the comments. Additionally, my process in constructing the list is inherently biased towards a) speakers who have already given addresses to large rooms, which likely reflects the institutional biases of organizations who rent large rooms, b) speakers I have already seen, many of whom probably don’t challenge my privilege in ways I’d find uncomfortable, c) speakers who address secondary educators on themes of technology and curriculum design, themes reflective of my own disciplinary interests, d) speakers whom I could remember, which reflects my own lousy memory.

In spite of all those biases, I decided it was better for this list to exist than to not exist. I’m interested in hearing from TODOS (or anybody else) how this project could have done a better job advancing the interests of students and teachers of color.

**Featured Comment**

]]>I was in graduate school before I had my first Persian teacher (if you exclude my education in Iran). It was an amazing experience, and I did every ounce of work possible in that class.

Reader William Carey via email:

Last year I realized that Pre-Calculus is really a class about moving from the particular to the general. We take particular skills and ideas students are comfortable with — like solving a quadratic equation — and generalize them to as many mathematical objects as we can — solving all polynomial equations. As we worked our way through polynomials, we wanted to move from reasoning about particular quadratic equations like

y = xto reasoning about all quadratic equations:^{2}+ 2x + 1y = ax. For homework, the students had to graph about twenty quadratics with varying^{2}+ bx + ca,b, andc.Then we got together to discuss the results in class. They remembered that

acontrols the “fatness” or “narrowness” of the parabola and sometimes flips it upside down. They remembered thatcmoves the parabola up and down. They weren’t totally sure whatbdid. A few students adamantly maintained that it moved the parabola left and right (with supporting examples). After about fifteen minutes of back and forth, we decided to go to Desmos and just animateb.

Shock and disbelief: the vertex traces out what looks like a parabola as. Furious math and argument ensue. Ten minutes later, a student has what seems to be the parabola the vertex traces graphed in Desmos. Is it the right parabola? Why? Can we prove that? (We could and did!)bchanges

**Previously**: WTF Math Problems.

**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**

I’m kicking the number of options back up to three. Two options simply doesn’t give y’all the challenge I know you need.

*Team Me*: 4

*Team Commenters*: 3

**Pseudocontext Submissions**

*John Gibson*

I don’t know if this is pseudocontext, but I *for sure* don’t know under what circumstances anyone would wonder about resultant momentum. In my head right now it’s like wondering about the middle names of the people who manufactured that car. It feels like trivia! I’m not saying it *is* trivia, but I *am* wondering if someone can put me in a position where knowing how to calculate resultant momentum would feel like power rather than punishment.

**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 took this one right on the nose. The pseudocontext was in the last place they looked.

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.

Moreover, I just don’t see any congruent triangles *in the picture*. None. I know I’ll see some if you widen the camera’s angle, but there aren’t any in the frame *right now*, which makes this a uniquely poor context.

The only way I can think to neutralize this pseudocontext:

Show students four spaghetti bridges. They have to decide which ones are fragile and which ones are strong. Understanding congruency somehow (waves hands) makes them more accurate in their decision-making.

**Featured Comment**

Dick Fuller:

]]>I like physics. And math. One without the other is school.

**Tracy Zager** illustrates a key feature of some of my favorite math tasks: their constraints are simple, but they create paths for complex thinking and ever more interesting questions:

I think my name is worth $239. Beat me? Haven’t figured out my $100 strategy yet.

**Lisa Bejarano** is a recipient of our nation’s highest honor for math teachers, so when she admits “I have no idea what I am doing” and starts sketching out a blueprint for great classrooms, I tune in:

Now, beginning with the first day of school, I intentionally work at building a unique relationship with each student. I make sure to find reasons to genuinely value each of them. This starts with weekly “How is it going?” type questions on their warm up sheets and continues by using their mistakes on “Find the flub Friday” and through feedback quizzes. I also share a lot of myself with them. When we understand each other, my classes are more productive. I still make plans, but I allow flexibility to meet my students where they are.

**David Cox** describes “a difficult thing for students to believe”:

Once students begin to believe that the way they see something is the currency, then our job is to simply help them refine their communication so their audience can understand them. Only then does the syntax of mathematics matter.

“Help me understand you.”

“Help me see what you see.”

**Kevin Hall** thoughtfully deconstructs his attempts to teach linear function for meaning, and includes this gem:

Once you introduce the slope formula, slope becomes that formula. It barely even matters if today’s lesson created a nice footpath in students’ brains between “slope” and the change in one quantity per unit of change in another. Once that formula comes out, your measly footpath is no competition for the 8-lane highway that’s opened up between “slope” and (y

_{2}–y_{1})/(x_{2}-x_{1}).

**Featured Tweet**

This math-name game is also a great way to learn names, build community, and honor names from diverse cultures @ddmeyer #equity #edchat https://t.co/uA5Glj8mPJ

— Jeremy Koselak (@koselak37) December 8, 2016

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?