Month: October 2013

Total 6 Posts

Great Classroom Action


Cathy Yenca gives Graphing Stories a go and the going gets tough (and interesting) when she runs into Christopher Danielson’s step-function:

The last video we tried today was Ponies in Frame. I heard the most awesome muttering as soon as the video began. “Oh! I get it. This one’s discrete.” [..] It wasn’t all lollipops and rainbows. A comment laced with negativity that resonated with Lauren and me was an outburst that “graphing used to be so easy, and this just made it hard.” How would you take a comment like that? What does that comment say about the student’s true level of understanding?

Jonathan Newman has his students analyze parametric motion by creating stop-motion videos.

Nicora Placa reminds us that the one of the best ways to assess a student’s understanding of direct proportions is to give her an indirect proportion and see if she treats it directly.

At a workshop last week, the following task caused a bit of confusion. “If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?” Several participants were convinced it was 144.

Megan Schmidt uses one of the Visual Pattern tasks and surprises us (me, at least) with all the different interesting equivalent ways there are to express the pattern algebraically:

They came up with the following pre-simplified expressions for the nth step:

2n(n+1) + 3
1 + (n2 +n2) + (n+1) + (n+1)
2[n(n+1)] + 3
2n(n+1) + 3
3 + [(n+1)n] + [(n+1)n]
3+2(n+1) + 2[(n+1)(n-1)]
2n2 + 2n + 3

For each of these, I had the student put the expression on the board. I then had different students explain the thinking of the student who came up with the expression and relate it to the pictured pattern. I saw a real improvement here from when I had them do this activity the first time last week. I had many more students volunteer to explain the thinking of their cohorts and much less hesitation to work out what the terms in the expressions represented.

Lifeless School Geometry & Questions That Require Proof

Christopher Danielson:

School geometry seems to me one of the most lifeless topics in all of mathematics.

Paul Lockhart [pdf]:

All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum.

Proof is part of the problem. There’s no mathematical practice with a greater difference between how mathematicians practice it and how it’s practiced in schools, between how exhilarating it can be and how inert it is in schools, than proof.

Here’s Christopher Danielson offering us a way forward:

… eventually we reach a question that sort of requires proof; it seems true, but is non-obvious, and it has arisen from the questions we have been asking about how properties relate to each other.

Then they prove.

Questions that require proof are hard to create, hard to package in a textbook, and probably impossible to crowdsource. You’re trying to nail that point where the seemingly-true hasn’t yet turned into the obviously-true and that spot varies by the class and the student.

For example, “Square matrices are always invertible” might strike that enticing balance for one student while for another its truth is too obvious-seeming to be worth the effort of a proof and for others it’s too foreign for them to have an opinion on its truth one way or the other.

This is tricky, right? And Danielson offers us a description but not a prescription. He describes the satisfying proof process in his classroom but he doesn’t prescribe how to make it happen in ours.

Here’s one possible prescription:

  • Ask students to produce something given some simple, loose constraints. Draw any rectangle you want and then draw the diagonals. Choose any three consecutive whole numbers and add them up. Draw a triangle with three side lengths that the class chooses. Add up two odd numbers.
  • Publicly display their productions and ask your students what they notice. The diagonals seem like they’re the same length. The sums are always multiples of three. Our triangles all look the same. Our sums are all even.
  • Ask students to tell you why that should be true given what we already know.
  • Ask students what other questions we can ask given our newly proven knowledge.

“You people want students to recreate 10,000 years of mathematical knowledge,” says the math reform-critic.

No one I respect thinks students should discover all of geometry deductively. But as Harel, et al, say in a paper that has fast become the most meaningful to my current work:

It is useful for individuals to experience intellectual perturbations that are similar to those that resulted in the discovery of new knowledge.

To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.

That’s more useful and more fun than the alternative:


The problem here isn’t just the coffin-like two-column stricture. The proof doesn’t arise from “a question that requires proof” but from a statement that has been assigned. That statement makes no attempt to nail the gray, truthy area Danielson describes. It informs you in advance of its truth. It’s obviously true! You just have to say why. Tell me anything more lifeless than that.

BTW: Ben Orlin is great here also.

Great Classroom Action


Nathaniel Highstein engineers a counterintuitive moment about graphing, one that subverts his students’ expectations and creates intellectual need for new knowledge:

I love this problem because the answer becomes totally clear when you make a time vs. elevation graph — and the answer violates nearly everyone’s expectations and leads to a surprise! Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect. When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

Option 1. Explain how to use place value.
Option 2. Explain how to use place value while first asserting its usefulness to humanity.
Option 3. Explain how to use place value while first putting students in a position to experience life without any kind of place value.

Anna Weltman took option three:

Three fingers is Na Na Na. But four fingers — now, that’s a lot of fingers. Na Na Na Na is quite a mouthful and it’s getting hard to tell the numbers apart. Here is where the cavemen bring in a new word. Na Na Na Na is Ba.

We continue counting. Ba Na, Ba Na Na (giggles), Ba Na Na Na — now what? The kids think until — Ba Ba, of course!

I remember Hung-Hsi Wu’s frustration with incomplete pattern problems. Paraphrasing him: “You can’t find the next term in the sequence ‘4, 10, 16, … ,’ because it could be anything.” He’s right, of course, but you can find a first term and Chris Hunter turns that fact into an icebreaker and a robust exercise in justification. He asks students to “Extend the pattern ‘Ann, Brad, Carol, … ,’ in as many ways as you can.”

Not mathy enough for you? Remember, not all teachers will have a positive attitude towards mathematics. This is a safe icebreaker. You can always follow it up with the mathier “Extend the pattern 5, 10, 15, … , in as many ways as you can.”

The first week of Exploring the Math Twitter Blogosphere asked teachers for their favorite tasks. Lots of people mentioned Four Fours. Megan Schmidt offers us an interesting cousin to that task and a useful description of what makes it effective for students.

The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids.

My guess: it’s the puzzle-like feel. (Extra credit: What makes a math task feel puzzling?)

The Digital Networked Textbook: Is It Any Different?

Let’s speculate that before this year’s cohort of first-year teachers retires from math education more than 50% of American classrooms will feature 1:1 technology. That’s a conservative prediction — both in the timeline and the percentage — and it’s more than enough to make me wonder what makes for good curricula in a 1:1 classroom. What are useful questions to ask?

Here’s the question I ask myself whenever I see new curricula crop up for digital networked devices like computer, laptops, tablets, and phones.

Is it any different?

That isn’t a rhetorical or abstract question. I mean it in two separate and specific ways.


If you print out each page of the digital networked curriculum, is it any different?


The answer here is “sort of.”

When I look at iBooks in the iBookstore from Pearson and McGraw-Hill or when I see HMH publish their Algebra Fuse curriculum in the App Store, I see lots of features and, yes, they require a digital medium. They have a) interactive slider-type demonstrations, b) slideshows that walk students through worked examples, c) stock video in the margins instead of stock photography, d) graded multiple-choice quizzes, e) videos of Edward Burger explaining math concepts and f) probably other items I’m forgetting. None of those features would survive the downgrade to paper.

So the question becomes, “Is it different enough?”

Are these offerings different enough to justify the enormous expense in hardware, software, and bandwidth? Do they take full advantage of their digital birthright?

I don’t think so.


“Is it any different?” here means “if you were hundreds of feet below the surface of the Earth, in a concrete bunker without any kind of Internet access, is the curriculum any different?”


Here, in September 2013, the answer is “no,” which is a shocking waste of very expensive, very powerful device.

Look at the apps you have on the home screen of your smartphone and ask yourself “how many of these are better because they have a large network of people using them?” Me, I have 12 apps on my homescreen and eight of them — Tweetbot, Messages, Instapaper, Instagram, Phone, Mail, Safari, Spotify — are so much better because of the crowd of people that use them with me. When I switch off my phone’s network connection, they get so much worse. Those are the apps I care most about also, the ones that enrich my life, the ones that justify the expense of a smartphone.

When you switch off the network connection, most curriculum stays exactly the same. It doesn’t suffer at all, which means it isn’t taking advantage of the network connection when it’s on.

More Different

Digital devices should allow you to:

  • Pose more interesting problems using more diverse media types and fewer words. (eg. three-act-style tasks).
  • Replace your textbooks’ corny illustrations of mathematical contexts with illustrations from their own lives. Students: find a trapezoid from your own life. Take a photo. Tap upload. Now it’s in your textbook.
  • Progressively disclose tasks over multiple screens so students don’t have to look at pages full of questions and information like this [pdf] and can instead start with a brief video and single sentence.

Networked devices should allow you to:

  • See all your friends’ illustrations from their own lives. The teacher should be able to see that gallery of trapezoids, promote certain illustrations, and offer comments on others that are visible to everybody.
  • Start lessons with integrated, formative polling. I’m talking about Riley Lark’s ActivePrompt software built right into the textbook.
  • Create student conversations. Use student data to find students who disagree with each other, pair them up, and have them work out their differences. All of that should happen without the teacher having to facilitate it because the device is smart.
  • Combine student data for better, more accurate modeling. (eg. Pennies, where each student collects a few data points which are then instantly collected into a much larger class data set.)

There are other possibilities, of course, some of which we’ll only start to realize as these tools are developed. But don’t just sit around and wait for an industry as reactive as textbook publishing to start making those tools for you. Publishers and their shareholders react to their market and that’s you. As long as they can still profit by repurposing existing print curriculum they will. It’s on you to tell your publishing reps that the curriculum they’re selling doesn’t do enough justice to the powerful, digital networked devices they’re putting them on. It isn’t different enough.

2013 Sep 27. And here’s LA Unified buying a billion dollars worth of iPads and then wasting the network that might make that investment worthwhile:

By Tuesday afternoon, L.A. Unified officials were weighing potential solutions. One would limit the tablets, when taken home, to curricular materials from the Pearson corporation, which are already installed. All other applications and Internet access would be turned off, according to a district “action plan.”

Featured Comment

Elizabeth Statmore:

This is always a problem in the early stages of a new technology. The “Technology Adoption Life Cycle” has proven itself over and over for the last 20 years to be the gold standard in analyzing tech markets.

The “innovators” adopt a technology because they need to be the first kids on their block to have whatever it is. The “early adopters” see strategic advantages and uses for it – and they are willing to put up with what they perceive as minor inconveniences like limited optimized uses in order to gain the advantages they seek.

That moment of “crossing the chasm” into the mainstream is that moment when a technology catches fire because vendors have figured out a way to reach beyond the techno-enthusiastic “early adopters” who have sustained their businesses to the techno-unimpressed “early majority” customers who are the major “show-me” skeptics. These skeptics form the first mass market for a technology, followed only later – and reluctantly – by a “late majority.”

Seems to me that we are still very much in an “early adopter” market in the race for digital textbooks. No one knows the “killer app” for digital curriculum is going to look like, but we do know it might bear some slight resemblance to the analog textbook. But this will not

As Steve Jobs always used to say, the “killer app” for the iPhone was making a phone call. But it was all the supporting infrastructure tht was built in (seamlessly integrated contacts, e-mail, texting, reminders, calendar, notes, & management of the technology) that transformed the act of making a phone call.

Pennies, Pearson, And The Mistakes You Never See Coming

I took machine-graded learning to task earlier this week for obscuring interesting student misconceptions. Kristen DiCerbo at Pearson’s Research and Innovation Network picked up my post and argued I was too pessimistic about machine-graded systems, posing this scenario:

Students in the class are sitting at individual computers working through a game that introduces basic algebra courses. Ms. Reynolds looks at the alert on her tablet and sees four students with the “letters misconception” sign. She taps “work sample” and the tablet brings up their work on a problem. She notes that all four seem to be thinking that there are rules for determining which number a letter stands for in an algebraic expression. She taps the four of them on the shoulder and brings them over to a small table while bringing up a discussion prompt. She proceeds to walk them through discussion of examples that lead them to conclude the value of the letters change across problems and are not determined by rules like “c = 3 because c is the third letter of the alphabet.”

My guess is we’re decades, not years, away from this kind of classroom. If it’s possible at all. Three items in this scenario seem implausible:

  • That four students in a classroom might assume “c = 3 because c is the third letter of the alphabet.” I taught Algebra for six years and never saw this conception of variables. (Okay, this isn’t a big deal.)
  • That a teacher has the mental bandwidth to manage a classroom of thirty students and keep an eye on her iPad’s Misconception Monitor. Not long ago I begged people on Twitter to tell me how they were using learning dashboards in the classroom. Everyone said they were too demanding. They used them at home for planning purposes. This isn’t because teachers are incapable but because the job demands too much attention.
  • That the machine grading is that good. The system DiCerbo proposes is scanning and analyzing handwritten student work in real-time, weighing them against a database of misconceptions, and pairing those up with a scripted discussion. Like I said: decades, if ever.

This also means you have to anticipate all the misconceptions in advance, which is tough under the best of circumstances. Take Pennies. Even though I’ve taught it several times, I still couldn’t anticipate all the interesting misconceptions.

The Desmos crew and I had students using smaller circles full of pennies to predict how many pennies fit in a 22-inch circle.


But I can see now we messed that up. We sent students straight from filling circles with pennies to plotting them and fitting a graph. We closed off some very interesting right and wrong ways to think about those circles of pennies.

Some examples from reader Karlene Steelman via e-mail:

They tried finding a pattern with the smaller circles that were given, they added up the 1 inch circle 22 times, they combine the 6, 5, 4, 3, 2, 1, and 1 circles to equal 22 inches, they figured out the area of several circles and set up proportions between the area and the number of pennies, etc. It was wonderful for them to discuss the merits and drawbacks of the different methods.

Adding the 1-inch circle 22 times! I never saw that coming. Our system closed off that path before students had the chance even to express their preference for it.

So everyone has a different, difficult job to do here, with different criteria for success. The measure of the machine-graded system is whether it makes those student ideas invisible or visible. The measure of the teacher is whether she knows what to do with them or not. Only the teacher’s job is possible now.

Featured Comments

Sue Hellman:

This doesn’t even touch the students who get questions RIGHT for the wrong reasons.

Dave Major:

Dashboards of the traditional ‘spawn of Satan & Clippy the Excel assistant’ sort throw way too much extremely specific information straight to the surface for my liking (and brain). That information is almost always things that are easy for machines (read. programmers) to work out, and likely hard or time consuming yet dubiously useful for humans to do. I wonder how many teachers, when frozen in time mid-lesson and placed in the brain deli slicer would be thinking “Jimmy has 89% of this task correct and Sally has only highlighted four sentences on this page.”