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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.

September Remainders

Awesome Internetting from the last month.

New Blog Subscriptions

  • Tracy Zager has been one of my favorite math voices on Twitter this school year and she’s now blogging. She’s also recently announced a fight with breast cancer and has requested that we “Please help me remember that I have thinking and ideas to share, and am involved in a world bigger than this right now.”
  • Annie Fetter’s work at the Math Forum has always been impressive and it’s a total oversight I hadn’t realized she writes a blog until now.
  • Tim McCaffrey and I share a lot of the same enthusiasms. He helps districts run lesson studies around three-act tasks and just started blogging about it.
  • Matt Bury had positively invaluable commentary during last month’s adaptive learning discussions.
  • Dan Burf, a/k/a Quadrant Dan, is a new teacher who has been using my old, old lessons, which is kind of fun to watch.
  • Amy Roediger, whose writing on Classkick was extremely useful.
  • Julie Wright is full of promise.
  • Just Mathness is full of promise.

New Twitter Follows

Multimedia Math

I make an open offer to my workshop participants to help them with their video editing. A couple of newcomers to multimedia modeling came up with these two tasks:

Great Tweets

Max Goldstein:

Proofs are social documents not compiled code.

Press Clippings

  • The Ontario Ministry of Education filmed an interview series with me and other math education-types in Toronto.
  • An interview with a teen writer from The Santa Fe New Mexican.
  • An interview with AFEMO, a Francophone group of math educators.

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.

Mo Jebara, the founder of Mathspace, has responded to my concerns about adaptive math software in general and his in particular. Feel free to read his entire comment. I believe he has articulated several misconceptions about math education and about feedback that are prevalent in his field. I’ll excerpt those misconceptions and respond below.

Computer & Mouse v. Paper & Pencil


Just like learning Math requires persistence and struggle, so too is learning a new interface.

I think Mathspace has made a poor business decision to blame their user (the daughter of an earlier commenter) for misunderstanding their user interface. Business isn’t my business, though. I’ll note instead that adaptive math software here again requires students to learn a new language (computers) before they find out if they’re able to speak the language they’re trying to learn (math).

For example, here is a tutorial screen from software developed by Kenneth Tilton, a frequent commenter here who has requested feedback on his designs:


Writing that same expression with paper and pencil instead is more intuitive by an order of magnitude. Paper and pencil is an interface that is omnipresent and easily learned, one that costs a bare fraction of the computer Mathspace’s interface requires, one that never needs to be plugged into a wall.

None of this means we should reject adaptive math software, especially not Mathspace, the interface of which allows handwriting. But these user interface issues pile high in the “cost” column, which means the software cannot skimp on the benefits.

Misunderstanding the Status Quo


Does a teacher have time to sit side by side with 30 students in a classroom for every math question they attempt?


But teachers can’t watch while every student completes 10,000 lines of Math on their way to failing Algebra.


I talk to teachers every single day and they are crying out for [instant feedback software].

Existing classroom practice has its own cost and benefit columns and Jebara makes the case that classroom costs are exorbitant.

Without adaptive feedback software, to hear Jebara tell it, students are wandering in the dark from problem to problem, completely uncertain if they’re doing anything right. Teachers are beleaguered and unsure how they’ll manage to review every student’s work on every assigned problem. Thirty different students will reveal thirty unique misconceptions for each one of thirty problems. That’s 27,000 unique responses teachers have to make in a 45 minute period. That’s ten responses per second! No wonder all these teachers are crying.

This is all Dickens-level bleak and misunderstands, I believe, the possible sources of feedback in a classroom.

There is the textbook’s answer key, of course. Some teachers make regular practice of posting all the answers in advance of an exercise set, also, so students have a sense that they’re heading in the right direction and focus on process not product.

Commenter Matt Bury also notes that a student’s classmates are a useful source of feedback. Since I recommended Classkick last week, several readers have tried it out in their classes. Amy Roediger writes about the feature that allows students to help other students:

… the best part was how my students embraced collaborating with each other. As the problems got progressively more challenging, they became more and more willing to pitch in and help each other.

All of these forms of feedback exist within their own webs of costs and benefits too, but the idea that without adaptive math software the teacher is the only source of feedback just isn’t accurate.

Immediate v. Delayed Feedback

Most companies in this space make the same set of assumptions:

  1. Any feedback is better than no feedback.
  2. Immediate feedback is better than delayed feedback.

Tilton has written here, “Feedback a day later is not feedback. Feedback is immediate.”

In fact, Kluger & DeNisi found in their meta-analysis of feedback interventions that feedback reduced performance in more than one third of studies. What evidence do we have that adaptive math software vendors offer students the right kind of feedback?

The immediate kind of feedback isn’t without complication either. With immediate feedback, we may find students trying answer after answer, looking for the red x change to a green check mark, learning little more than systematic guessing.

Immediate feedback risks underdeveloping a student’s own answer-checking capabilities also. If I get 37 as my answer to 14 + 22, immediate feedback doesn’t give me any time to reflect on my knowledge that the sum of two even numbers is always even and make the correction myself. Along those lines, Cope and Simmons found that restricting feedback in a Logo-style environment led to better discussions and higher-level problem-solving strategies.

What Computers Do To Interesting Exercises


Can you imagine a teacher trying to provide feedback on 30 hand-drawn probability trees on their iPad in Classkick?


Can you imagine a teacher trying to provide feedback on 30 responses for a Geometric reasoning problem letting students know where they haven’t shown enough of a proof?

I can’t imagine it, but not because that’s too much grading. I can’t imagine assigning those problems because I don’t think they’re worth a class’ limited time and I don’t think they do justice to the interesting concepts they represent.

Bluntly, they’re boring. They’re boring, but that isn’t because the team at Mathspace is unimaginative or hates fun or anything. They’re boring because a) computers have a difficult time assessing interesting problems, and b) interesting problems are expensive to create.

Please don’t think I mean “interesting” week-long project-based units or something. (The costs there are enormous also.) I mean interesting exercises:

Pick any candy that has multiple colors. Now pick two candies from its bag. Create a probability tree for the candies you see in front of you. Now trade your tree with five students. Guess what candy their tree represents and then compute their probabilities.

The students are working five exercises there. But you won’t find that exercise or exercises like it on Mathspace or any other adaptive math platform for a very long time because a) they’re very hard to assess algorithmically and b) they’re more expensive to create than the kind of problem Jebara has shown us above.

I’m thinking Classkick’s student-sharing feature could be very helpful here, though.



So why don’t we try and automate the parts that can be automated and build great tools like Classkick to deal with the parts that can’t be automated?

My answer is pretty boring:

Because the costs outweigh the benefits.

In 2014, the benefits of that automation (students can find out instantly if they’re right or wrong) are dwarfed by the costs (see above).

That said, I can envision a future in which I use Mathspace, or some other adaptive math software. Better technology will resolve some of the problems I have outlined here. Judicious teacher use will resolve others. Math practice is important.

My concerns are with the 2014 implementations of the idea of adaptive math software and not with the idea itself. So I’m glad that Jebara and his team are tinkering at the edges of what’s possible with those ideas and willing, also, to debate them with this community of math educators.

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Mercy – all of them. Just read the thread if you want to be smarter.

Can Sports Save Math?

A Sports Illustrated editor emailed me last week:

I’d like to write a column re: how sports could be an effective tool to teach probability/fractions/ even behavioral economics to kids. Wonder if you have thoughts here….

My response, which will hopefully serve to illustrate my last post:

I tend to side with Daniel Willingham, a cognitive psychologist who wrote in his book Why Students Don’t Like School, “Trying to make the material relevant to students’ interests doesn’t work.” That’s because, with math, there are contexts like sports or shopping but then there’s the work students do in those contexts. The boredom of the work often overwhelms the interest of the context.

To give you an example, I could have my students take the NBA’s efficiency formula and calculate it for their five favorite players. But calculating – putting numbers into a formula and then working out the arithmetic – is boring work. Important but boring. The interesting work is in coming up with the formula, in asking ourselves, “If you had to take all the available stats out there, what would your formula use? Points? Steals? Turnovers? Playing time? Shoe size? How will you assemble those in a formula?” Realizing you need to subtract turnovers from points instead of adding them is the interesting work. Actually doing the subtraction isn’t all that interesting.

So using sports as a context for math could surely increase student interest in math but only if the work they’re doing in that context is interesting also.

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Marcia Weinhold:

After my AP stats exam, I had my students come up with their own project to program into their TI-83 calculators. The only one I remember is the student who did what you suggest — some kind of sports formula for ranking. I remember it because he was so into it, and his classmates got into it, too, but I hardly knew what they were talking about.

He had good enough explanations for everything he put into the formula, and he ranked some well known players by his formula and everyone agreed with it. But it was building the formula that hooked him, and then he had his calculator crank out the numbers.

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