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Here is what I mean. Ask a student to:

Give an algebraic function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Maybe the student can determine the function. At some point, an advanced algebra student should determine the function. But what do I learn from a student who can’t determine the function? What does a blank graph tell me?


The student might understand what roots, intercepts, and asymptotes are. She might understand every part of the task except how to form the function algebraically. I won’t know because I’m asking a very formal task.

This is why a lot of secondary math teachers ask a less formal question first. They ask for a sketch.

Sketch a function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.


Think of what I know about the student that I didn’t know before. Think of the feedback that’s available to me now that wasn’t before.

Desmos just added sketching into its Activity Builder. That was the result of months of collaboration between our design, engineering, and teaching teams. That was also the result of our conviction that informal mathematical understanding is underrepresented in math classes and massively underrepresented in computer-based mathematics classes. We want to help students express their mathematical ideas and get feedback on those ideas, especially the ones that are informal and under development. That’s why we built sketch before multiple choice, for example. I’m stating this commitment publicly, hoping that one or more of you will help us live up to it.

Mike Caulfield:

But the biggest advantage of a tutor is not that they personalize the task, it’s that they personalize the explanation. They look into the eyes of the other person and try to understand what material the student has locked in their head that could be leveraged into new understandings. When they see a spark of insight, they head further down that path. When they don’t, they try new routes.

EdSurge misreads Mike pretty drastically, I think:

What if technology can offer explanations based on a student’s experience or interest, such as indie rock music?

Mike is summarizing what great face-to-face tutors do. They figure out what the student already knows, then throw hooks into that knowledge using metaphors and analogies and questions. That’s a personalized tutor.

But in 2016 computers are completely inept at that kind of personalization. Worse than your average high school junior tutoring on the side for gas money. Way worse than your average high school teacher. I don’t think this is a controversial observation. In a follow-up post, Michael Feldstein writes, “For now and the foreseeable future, no robot tutor in the sky is going to be able to take Mike’s place in those conversations.”

So it’s interesting to see how quickly EdSurge pivots to a different definition of personalization, one that’s much more accommodating of the limits of computers. EdSurge’s version of personalization asks the student to choose her favorite noun (eg. “indie rock music”) and watch as the computer incorporates that noun into the same explanation every other student receives. Find and replace. In 2016 computers are great at find and replace.

This is just a PSA to say: technofriendlies, I see you moving the goalposts! At the very least, let’s keep them at “high school junior-level tutor.”

BTW. I don’t think find-and-replacing “indie rock music” will improve what a student knows, but maybe it will affect her interest in knowing it. I’ve hassled edtech over that premise before. In my head, I always call that find-and-replacing approach the “poochification” of education, but I never know if that reference will land for anybody who isn’t inside my head.

Blue Point Rule

What is the rule that turns the red point into the blue point?


My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally and also informally.

In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and most formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).

It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for machines to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.

But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be good at math.

The informal expressions aren’t just interesting work but they also make the formal expressions easier to learn. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.


  • Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
  • Before I ask for a graph, I ask for a sketch.
  • Before I ask for a proof, I ask for a conjecture.
  • David Wees: Before I ask for conjectures, I ask for noticings.
  • Before I ask for a calculation, I ask for an estimate.
  • Before I ask for a solution, I ask students to guess and check.
  • Bridget Dunbar: Before I ask for algebra, I ask for arithmetic.
  • Jamie Duncan: Before I ask for formal definitions, I ask for informal descriptions.
  • Abe Hughes: Before I ask for explanations, I ask for observations.
  • Maria Reverso: Before I ask for standard algorithms, I ask for student-generated algorithms.
  • Maria Reverso: Before I ask for standard units, I ask for non-standard units.
  • Kent Haines: Before I ask for definitions, I ask for characteristics.
  • Andrew Knauft: Before I ask for answers in print, I ask for answers in gesture.
  • Avery Pickford: Before I ask for complete mathematical propositions, I ask for incomplete propositions.
  • Dan Finkel: Before I ask for the general rule, I ask for a specific instance of the rule.
  • Dan Finkel: Before I ask for the literal, I ask for an analogy.
  • Kristin Gray: Before I ask for quadrants, I ask for directional language.

At this point, I could use your help in three ways:

  • Offer more shades between informal and formal for the blue dot task. (I offered three.)
  • Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [your turn]. That work has begun on Twitter.
  • Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.

BTW. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.

Featured Comment

Allison Krasnow points us to Steve Phelp’s Guess My Rule activities.

David Wees reminds us that the van Hiele’s covered some of this ground already.

Our first approach in preparing a new lesson is often to ask, “Where does this skill apply in the world of work or in the world outside the classroom?” There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So instead, I try first to ask myself, “Why did we invent this skill? How does this skill resolve the limits of older skills? If this skill is aspirin, then what is the headache and how do I create it?”

Two examples from my recent past.

Combining Like Terms

Why did we invent the skill of combining like terms in an expression? Why not leave the terms uncombined? Maybe the terms are fine! Why disturb the terms?

One reason to combine like terms is that it’s easier to perform operations on the terms when they’re combined. So let’s put students in a place to experience that use:

Evaluate for x = -5:

3x + 5 + 2x2 – 7 + 8x – 5x2 – 11x + 4 – 5x + 3x2 + 4 + 3x – 6 + 2x + x2

Put it on an opener. The expression simplifies to x2, giving students an enormous incentive to learn to combine like terms before evaluating.

[I’m grateful to Annie Forest for bringing the example to mind. She also adds a context, if that’s what you’re into.]


When students first learn to graph points, the parentheses are the first convention they throw out the window. And it’s hard to blame them. If I told you to graph the point 2, 5, would you need the parentheses to know the point I’m talking about? No.

So why did mathematicians invent parentheses? What purpose do they serve, assuming that purpose isn’t “tormenting middle school students thousands of years in the future.”

It turns out that, while it’s very easy to graph a single point with or without parentheses, graphing lots of points becomes very difficult without the parentheses. So let’s put students in a place to experience that need:

Graph the coordinates:

-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3

You can’t even easily tell if there are an even number of numbers!

[My thanks to various workshop participants for helping me understand this.]


The need for combining like terms is Harel’s need for computation and the need for parentheses is Harel’s need for communication. I can’t recommend his paper enough in which he outlines five needs for all of mathematics.

My point isn’t that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn’t mean the mathematics was invented arbitrarily or for no reason or for malicious reasons. There was a need.

Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.


We explored these ideas in a summer series.

Henry Pollak, in his essay, “What Is Mathematical Modeling?

Probably 40 years ago, I was an invited guest at a national summer conference whose purpose was to grade the AP Examinations in Calculus. When I arrived, I found myself in the middle of a debate occasioned by the need to evaluate a particular student’s solution of a problem. The problem was to find the volume of a particular solid which was inside a unit three-dimensional cube. The student had set up the relevant integrals correctly, but had made a computational error at the end and came up with an answer in the millions. (He multiplied instead of dividing by some power of 10.) The two sides of the debate had very different ideas about how to allocate the ten possible points. Side 1 argued, “He set everything up correctly, he knew what he was doing, he made a silly numerical error, let’s take off a point.” Side 2 argued, “He must have been sound asleep! How can a solid inside a unit cube have a volume in the millions?! It shows no judgment at all. Let’s give him a point.”

What a fantastic dilemma.

Pollak argues that the student’s error would merit a larger deduction in an applied context than in a pure context. In a real-world context, being wrong by a factor of one million means cities drown, atoms obliterate each other, and species go extinct. In a pure math context, that same error is a more trivial matter of miscomputation.

The trouble is that, to the math teachers in the room, a unit cube is a real-world object. They can hold a one-centimeter unit cube in their hands and, more importantly, they can hold it in their minds.

The AP graders aren’t arguing about grading. They’re trying to decide what is real.

What a fantastic dilemma.

Featured Comment

Shannon Alvarez:

Whenever I wanted to give students the most amount of partial credit, my coop teacher would ask me the poignant question, “What exactly are you assessing?” I found this was a great question to continue asking myself. So, in the example you gave, are you assessing students’ ability to perform mathematical functions correctly or are you assessing their ability to connect those math functions to the real world?

Benjamin Dickman alerts us that Pollak’s piece is online, free, along with a number of his modeling tasks.

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