Dan Meyer

Total 1588 Posts
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

[Desmos Design] Algebra Is Power, Not Punishment

This is the first of several posts where I’ll use the activities Desmos created last quarter to illustrate our design principles.

One of those principles is:

Create an intellectual need for new mathematical skills.

Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they so often mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the fruit of that work.

In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.

Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.

Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.

We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work numerically, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”

And then we use their expression to power ten pool borders.

And ten lawnmowers.

Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression once.

In each case, students are more likely to see algebra as power than punishment.

You Can’t Break Math

We were solving linear equations in Ms. Warburton’s eighth grade class last week and I learned (or re-learned, or learned at greater depth) a couple of truths about mathematics.

As I approached B and R, I misread them as disengaged. In fact, they were thinking really, really hard about this beast:

B suggested they multiply by two as a “fraction buster.”

(One small pleasure of guest teaching is trying to identify and decode the vernacular of each new class. I heard “fraction buster” more than once.)

R asked, “But do we multiply this by two or the whole thing by two?”

If you’ve taught math for a single day, you know the choice here.

You can tell them, “You multiply the whole thing by two.” That’d be helpful by the definition of “helpful” that includes “completing as many math problems as correctly and quickly as possible.” But in terms of classroom management, I’ll be doing myself no favors, having trained B and R to call me over whenever they have any similar questions. More importantly, I’ll have done their relationship with mathematics no favors either, having trained them to think of math as something that can’t be made sense of without an adult around.

Variables like x and y behave just like numbers like -2 and 3.” I said. I wrote this down and said, “Try out both of your ideas on this version and see what happens.”

After some quick arithmetic, they experienced a moment of clarity.

In the next class, students were helping me solve 2x – 14 = 4 – 2x at the board. M told me to add 2x to both sides. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand.

So at the board I was asking myself, “Why are we adding 2x to both sides? What if we added a different thing?”

Then I asked the students, “What would happen if we added 5x to both sides? What would break?”

Nothing. We decided that nothing would break if we added 5x to both sides. It wouldn’t be as helpful as adding 2x, but math isn’t fragile. You can’t break math.

BTW

  • I haven’t found a way to generate these kinds of insights about math without surrounding myself with people learning math for the first time.
  • One of my most enduring shortcomings as a teacher is how much I plan and revise those plans, even if the lesson I have on file will suffice. I’ll chase a scintilla of an improvement for hours, which was never sustainable. I spent most of the previous day prepping this Desmos activity. We used 10% of it.
  • Language from the day that I’m still pondering: “We cancel the 2x’s because we want to get x by itself.” I’m trying to decide if those italicized expressions contribute to a student’s understanding of large ideas about mathematics or of small ideas about solving a particular kind of equation.


Featured Comments

Here is an awesome sequence of comments in which people savage the term “cancel,” then temper themselves a bit, and then realize that their replacement terms are similarly limited:

Corey Andreasen:

I have a huge problem with ‘cancel.’ It’s mathematical slang, and I’m OK with its use among people who really understand the mathematics. But among learners it obscures the mathematics and leads to things like “cancelling terms” in rational expressions.

Melissa Lechleiter:

I think the word “cancel” is misused in math when teaching students. We are not canceling anything we are making ones and zeros.

Susan:

I never say cancel. I’ve worked hard to eliminate it from any teaching I do. Same with cross multiply, never say it. Instead I say “add to make zero” or “divide by or multiply by the reciprocal of to make one”.

Sam Shah:

We use “cancel” to mean too many things and so they use the term anytime they want to get rid of something or slash something out. The basis for my concern: when I ask kids “why can you do that?” they often can’t explain.

Jeremy Hansuvadha:

However, when it comes to squaring a square root, what is most accurate to say? I don’t correct the kids in that case, and I tell them that cancel means the same thing as “undo”.

Paul Hartzer:

If the point is to be rigorous, “apply the inverse” is more rigorous and technical than “cancel”.

More miscellaneous wisdom on language in mathematics:

David Butler:

In the topic of “get x by itself”, I’ve started saying “We want to say what x is. What would that look like?” They usually say “It would look like x = something”. Then they’ve chosen what the final equation ought to look like for themselves.

I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

Joel Penne:

Even in my Advanced Algebra 2 classes I have started using the phrases “legal” and “useful”. In the original post adding 5x to both sides was definitely “legal” just not as “useful”.

Laura Hawkins:

One of my refrains is that an algebraic step is “correct, but not useful.” Inspired by my dance teacher, I also talk about how a particular procedure is lovely, just not in today’s choreography (which is geared towards solving for an unknown/simplifying a trig expression/ finding intercepts, etc).

[Pseudocontext Saturdays] Exoskeleton

This Week’s Installment

Poll

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

Pseudocontext Saturday #11

  • Calculating angles of rotation (57%, 249 Votes)
  • Graphing rational functions (28%, 121 Votes)
  • Proving trigonometric identities (15%, 67 Votes)

Total Voters: 437

Loading ... Loading ...

(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: 6
Team Commenters: 4

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.)

Answer

Mega-Grandma!

Later the text goes on to ask students to construct and graph the formula for the average cost of producing the Mega-Grandma, which turns out to be a rational function given the constraints.

But the text might have been better served by asking students to solve for generic widgets, or tennis balls, or something a little less gonzo-bananas than the Mega-Grandma exoskeleton, which is all I’m going to remember from this unit in the textbook.

BTW. Thanks to Jasper LaFortune for the submission.

Here Are Ten New Desmos Activities

The Desmos quarter just ended and this was a huge one for the team of teachers I support.

First, we made substantial upgrades to our entire activity pool. Second, we released ten new activities in the same amount of time it took us to release one activity two years ago. This is all due to major improvements to our technology and our pedagogy.

Technologically, our engineers created a powerful scripting language that hums beneath our activities, enabling us to set up more meaningful interactions between teachers, students, and mathematics.

Pedagogically, my teaching team has spent the last year refining our digital mathematics pedagogy through daily conversations, lesson pitches, lesson critiques, summary blog posts, occasional lunch chats with guests like the Khan Academy research team, and frequent consultation with our Desmos Fellows.

The result: we cut an activity pool that once comprised 300 pretty good activities down to 127 great ones, and we gave each one of those 127 a serious upgrade, making sure they took advantage of our best technology and pedagogy. Then we added ten more.

I don’t think I’ve learned as much or worked as hard in a three-month span since grad school, and I owe a debt of gratitude to my team – Shelley Carranza, Christopher Danielson, and Michael Fenton – for committing the same energy. Also, it goes without saying that none of our activity ideas would have been possible without support from our engineers and designers.

In future posts, I’ll excerpt those lessons to illustrate our digital pedagogy. For today, I’ll just introduce the activities themselves.

Picture Perfect

Hang loads of pictures precisely and quickly using arithmetic sequences.

Game, Set, Flat

Your shipment of tennis balls has been contaminated. Use exponential functions to find the bad ones.

Graphing Stories

Graphing Stories comes to Desmos just in time for its tenth birthday.

Pool Border Problem

One of the oldest and best problems for exploring algebraic equivalence. We wouldn’t have touched it if we didn’t think we had something to add.

Laser Challenge

Use your intuition for angle measure to bounce lasers off mirrors and through targets.

Lawnmower Math

Use Algebra and the properties of circles to help you mow ten lawns automatically and quickly.

Land the Plane

Use linear equations to land airplanes safely and precisely.

Circle Patterns

Practice circle equations by completing artistic patterns.

Constructing Polynomials

Develop your understanding of the behavior of polynomial graphs by creating them piece by piece, factor by factor.

What’s My Transformation?

This is my favorite introduction to the concept of a transformation. “Actually, there’s really only one parabola in the world – we just move it around to make new ones.”

We are still testing these activities. They are complete, but not complete complete, if you know what I mean. You won’t find all of them in our search index yet. We welcome your feedback.

[Pseudocontext Saturdays] Tornado!

This Week’s Installment

Poll

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

Pseudocontext Saturday #10

  • Calculating probabilities of independent events (69%, 238 Votes)
  • Interpreting bar graphs (20%, 70 Votes)
  • Calculating area of parallelograms (11%, 38 Votes)

Total Voters: 346

Loading ... Loading ...

(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: 4

Pseudocontext Submissions

William Carey has offered two additional genres of pseudocontext that are worth your attention:

First:

One motif in pseudocontextual questions seems to be treating as a variable things that, you know, don’t vary.

Second:

The car question follows a fascinating pattern that shows up in lots of physicsy work: it begs the question. Physicists like to measure things. Sometimes measuring something directly is tricky (or impossible), so we measure other things, and then calculate the thing we actually want.

Questions like that have as their givens the thing we can’t measure and ask us to calculate the thing that we can measure. It’s absolutely backwards.

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.)

Answer

The commenters bit down hard on the lure this time, folks. The correct answer – “calculating area of parallelograms” – was selected least.

Delicious pseudocontext, right? The judges all suffered massive strokes when they saw this problem so I couldn’t get their official ruling, but I don’t think it matters. This context fails the “Come on, really?” test for pseudocontext.

“This unpredictable force of nature is threatening a precisely-bounded parallelogram? Come on, really?”

How could we neutralize the pseudocontext? I would be thrilled to see a task that invited students to select and approximate important regions with various quadrilaterals, but let’s not lie about where our tools are useful.