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Blanton & Kaput modify the Christmas carol The Twelve Days of Christmas for an algebraic reasoning task befitting the season:

How many gifts did your true love receive on each day? If the song was titled “The Twenty-Five Days of Christmas,” how many gifts would your true love receive on the twenty-fifth day? How many total gifts did she or he receive on the first two days? The first three days? The first four days? How many gifts did she or he receive on all twelve days?

“The X Days of Christmas.” I like it.

Malcolm Swan, on how to begin a lesson:

Every lesson should begin by getting [students] to articulate something about what they already understand or know about something or their initial ideas. So you try and uncover where they’re starting from and make that explicit. And then when they start working on an activity, you try to confront them with things that really make them stop.

And it might be that you can do this by sitting kids together if they’ve got opposing points of views. So you get conflict between students as well as within. So you get the conflict which comes within, when you say, “I believe this, but I get that and they don’t agree.” Or you get conflict between students when they just have fundamental disagreements, when there’s a really nice mathematical argument going on. And they really do want to know and have it resolved. And the teacher’s role is to try to build a bit of tension, if you like, to try and get them to reason their way through it.

And I find the more students reason and engage like that then they can get quite emotional. But when they get through it, they remember the stuff really well. So it’s worth it.

Christopher Danielson and Megan Schmidt have both written recently and compellingly about the trouble students have when taught that math is a series of correct “steps.”

Danielson, doing his best Howard Beale:

THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.

Let me add to the conversation the category of “steps that are correct but useless.” These are great. They come from a conversation I had, like, fifteen minutes ago with a teacher named Leah Temes here at NWMC 2014.

Leah teaches Algebra II. We were talking about solving systems of equations. It’s really easy to teach the solution to a system like this as a series of correct, useful steps:

2x + 3y = 10
5x – 3y = 4

  1. Add the second equation to the first one.
  2. Solve for x.
  3. Substitute x in either equation to solve for y.
  4. Check that pair in the other equation for full credit.

Leah said she was tired of seeing her students mimic those correct steps without understanding why they worked. So instead of showing her students steps that were useful and correct, she asked them if she was allowed to add the following two equations:

2x + 3y = 10
5 = 5

To get:

2x + 3y + 5 = 10 + 5

Is everything still correct? Yes.

Was that useful? No.

This experience awakened her students to a category of steps in addition to the correct and useful ones they’re supposed to memorize and the incorrect and useless ones they’re supposed to avoid – correct and useless steps.

Alerting your students to that category of steps may make math seem less intimidating and more interesting. Math isn’t any longer a matter of staying on the right side of a line between the incorrect and correct steps. There’s another region out there, one that’s a bit less tame, a place for explorers, a place where the worst thing that can happen is you did something right but it just wasn’t useful. That category of steps also requires justification – “how do you know this is correct?” – which can help bend the student away from memorization and back towards understanding.

BTW. All of this implies a fourth category of steps – incorrect but useful. Can anybody give an example?

Featured Comments

Cathy Yenca:

I do a similar thing when solving equations in one variable by asking students if I can add 1,000,000, let’s say, to each side of an equation… or if I can subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.

David Petro:

I have long been a fan of deliberately sabotaging a solution to something that I might be doing on the board so that somewhere down the road things become obviously wrong. This is so students can start to develop strategies for what to do when this happens.

Many will tell you that it’s important for students to make mistakes (in fact, that they learn the most when they do). But that sometimes runs counter to what they see in class. That is, a teacher demonstrating flawless execution of mathematics. Even some of our best students often won’t even attempt a problem unless they are sure they will get it correct. If they are ever going to become comfortable with making mistakes as part of the normal process then we have to include managing those mistakes as part of our day to day in class.

Moana Evans:

[It’s] incorrect but useful to estimate things like area problems, in order to find out a ballpark figure and check if you’ve done the math right.


In the September 2014 edition of Mathematics Teacher, reader Thomas Bannon reports that his research group has found that the applications of algebra haven’t changed much throughout history.


Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent 13 years in his dotage; how old is he?


A man bought a horse and carriage for $500, paying three times as much for the carriage as for the horse. How much did each cost?


The Panama Canal will be 46 miles long. Of this distance the lower land parts on the Atlantic and Pacific sides will together be 9 times the length of the Culebra Cut, or hill part. How many miles long will the Culebra Cut be? Prove answer.


Shandra’s age is four more than three times Sherita’s age. Write an equation for Shandra’s age. Solve if Sherita is 3 years old.

I’m grateful for Bannon’s research but his conclusion is, in my opinion, overly sunny:

Looking through these century-old mathematics book can be a lot of fun. Challenging students to find and solve what they consider the most interesting problem can be a great contest or project.

My alternate reading here is that the primary application of school algebra throughout history has been to solve contrived questions. Instead of challenging students to answer the most interesting of those contrived questions, we should ask questions that aren’t contrived and that actually do justice to the power of algebra. Or skip the whole algebra thing altogether.


If you told me there existed a book of arithmetic problems that didn’t include any numbers, I’d wonder which progressive post-CCSS author wrote it. Imagine my surprise to find Problems Without Figures, a book of 360 such problems, published in 1909.

For example, imagine the interesting possible responses to #39:

What would be a convenient way to find the combined weight of what you eat and drink at a meal?

That’s great question development. Now here’s an alternative where we rush students along to the answer:

Sam weighs 185.3 pounds after lunch. He weighed 184.2 before lunch. What was the weight of his lunch?

So much less interesting! As the author explains in the powerful foreword:

Adding, subtracting, multiplying and dividing do not train the power to reason, but deciding in a given set of conditions which of these operations to use and why, is the feature of arithmetic which requires reasoning.

Add the numbers back into the problem later. Two minutes later, I don’t care. But subtracting them for just two minutes allows for that many more interesting answers to that many more interesting questions.

[via @lucyefreitas]

This is a series about “developing the question” in math class.

Bryan Anderson and Joel Patterson simply subtracted elements from printed tasks, added them back in later, and watched their classrooms become more interesting places for students.

Anderson took a task from the Shell Centre and delayed all the calculation questions, making room for a lot of informal dialog first.


Patterson took a Discovering Geometry task and removed the part where the textbook specified that the solution space ran from zero to eight.


“It turns out that by shortening the question,” Joel Patterson said, “I opened the question up, and the kids surprised themselves and me!”

I believe EDC calls these “tail-less problems.” I call it being less helpful.

BTW. These are great task designers here. I spent the coldest winter of my life at the Shell Centre because I wanted to learn their craft. Discovering Geometry was written by friend-of-the-blog Michael Serra. This only demonstrates how unforgiving the print medium is to interesting math tasks, like asking Picasso to paint with a toilet plunger. You have to add everything at once.

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