Month: March 2017

Total 4 Posts

Teach the Controversy

Here is how your unit on linear equations might look:

  1. Writing linear equations.
  2. Solving linear equations.
  3. Applying linear equations.
  4. Graphing linear equations.
  5. Special linear equations.
  6. Systems of linear equations.
  7. Etc.

On the one hand, this looks totally normal. The study of the linear functions unit should be all about linear functions.

But a few recent posts have reminded me that the linear functions unit needs also to teach not linear functions, that good instruction in [x] means helping students differentiate [x] from not [x].

Ben Orlin offers a useful analogy here:

If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Michael Pershan then offers some fantastic prompts for helping students disentangle rules, machines, formulas, and functions, all of which seem totally interchangeable if you blur your eyes even a little.

Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines. Pick two: not all of one is like the other. A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

And then I was grateful to Suzanne von Oy for tweeting the question, “Is this a line?” a question that is both rare to see in a linear functions unit (where everything is a line!) and important. Looking at not lines helps students understand lines.

So I took von Oy’s question and made this Desmos activity where students see three graphs that look linear-ish. The point here is that not everything that glitters is gold and not everything that looks straight is linear. Students first make their predictions.

Then they see the graphs again with two points that display their coordinates. Now we have a reason to check slopes to see if they’re the same on different intervals.

Finally, we zoom out to check a larger interval on the graph.

I’m sure I will need this reminder tomorrow and the next day and the next: teach the controversy.

BTW. In addition to being good for learning, controversy is also good for curiosity.

Bonus. Last week’s conversation about calculators eventually cumulated in the question:

“Calculators can perform rote calculations therefore rote calculations have no place on tests.” Yay or nay?

I’ve summarized some of the best responses — both yay and nay — at this page. (I’m a strong “nay,” FWIW.)

1,000 Math Teachers Tell Me What They Think About Calculators in the Classroom

Yesterday, I asked teachers on Twitter about their classroom calculator policy and 978 people responded.

I wanted to know if they allow calculators a) during classwork, b) during tests, and also which kinds of calculators:

  • Hardware calculators (like those sold by Texas Instruments, Casio, HP, etc.).
  • Mobile phone calculators (like those you can download on your Android or iOS phone).

(Full disclosure: I work for a company that distributes a free four function, scientific, and graphing calculator for mobile phones and other devices.)

I asked the question because hardware calculators don’t make a lot of financial sense to me.

Here are some statistics for high-end HP and Texas Instruments graphing calculators along with a low-end Android mobile phone. (Email readers may need to click through to see the statistics.)

[table id=6 /]

You pay less than 2x more for the mobile phone and you get hardware that is between 30x and 300x more powerful than the hardware calculators. And the mobile phone sends text messages, takes photos, and accesses webpages. In many cases, the student already has a mobile phone. So why spend the money on a second device that is much less powerful?

1,000 teachers gave me their answer.

The vast majority of respondents allow hardware calculator use in their classes. I suspect I’m oversampling for calculator-friendly teachers here, by virtue of drawing that sample from a digital medium like Twitter.

734 of those teachers allow a hardware graphing calculator but not a mobile phone on tests. 366 of those teachers offered reasons for that decision. They had my attention.

Here are their reasons, along with representative quotes, ranked from most common to least.

Test security. (173 votes.)

It’s too easy for students to share answers via text or picture.

Internet access capabilities and cellular capabilities that make it way too easy for the device to turn from an analysis/insight tool to the CheatEnable 3000 model.

School policy. (68 votes.)

School policy is that phones are in lockers.

It’s against school policy. They can use them at home and I don’t have a problem with it, but I’m not allowed to let them use mobile devices in class.

Distraction. (67 votes.)

Students waste time changing music while working problems, causing both mistakes due to lack of attention and inefficiency due to electronic distractions.

We believe the distraction factor is a negative impact on learning. (See Simon Sinek’s view of cell phones as an “addiction to distraction.”)

Test preparation. (54 votes.)

I am also preparing my students for an IB exam at the end of their senior year and there is a specific list of approved calculators. (Phones and computers are banned.)

Basically I am trying to get students comfortable with assessments using the hardware so they won’t freak out on our state test.

Access. (27 votes.)

Our bandwidth is sometimes not enough for my entire class (and others’ classes) to be online all at once.

I haven’t determined a good way so that all students have equal access.


These reasons all seem very rational to me. Still, it’s striking to me that “test security” dwarfs all others.

That’s where it becomes clear to me that the killer feature of hardware calculators is their lack of features. I wrote above that your mobile device “sends text messages, takes photos, and accesses webpages.” At home, those are features. At school, or at least on tests, they are liabilities. That’s a fact I need to think more about.

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Jennifer Potier:

I work in a BYOD school. What I have learned is that the best way to disengage students from electronic devices is to promote learning that involves student sharing of discussion, planning, thinking, and solving problems. When the students are put “centre stage,” the devices start becoming less interesting.

Chris Heddles:

The restriction on calculation aids and internet connections still stems from a serious cultural issue we have in mathematics teaching — the type of questions that we ask. While we continue to emphasise the importance of numerical calculations and algebraic manipulation in assessment, electronic aids to these skills will continue to be an issue.

Instead, we should shift the focus to understanding the situation presented, setting up the equations and then making sense of the calculation results. With this shift, the calculations themselves are relatively unimportant so it doesn’t really matter how the student process them. Digital aids can be freely used because they are off little use when addressing the key aspects of the assessment tasks.

In many ways our current mathematics assessment approach is equivalent to a senior secondary English essay that gave 80% of the grade for neat handwriting and correct spelling. If this were the case then they too, would have to ban all electronic aids to minimise the risk of “cheating” by typing and using spell checking software.

If we change what we value in assessment then we can open up better/cheaper electronic aids for students.

2017 Mar 24. Related to Chris’s comment above, I recently took some sample SAT math tests and was struck by how infrequently I needed a calculator. Not because I’m any kind of mental math genius. Simply because the questions largely concerned analysis and formulation over calculation and solution.

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Problems with Personalized Learning

A reader pointed me to this interesting article in the current Educational Leadership on “personalized learning.” She said it raised an alarm for her that she couldn’t quite put into words and she asked if I heard that same alarm and, if so, what words I’d use to describe it.

I hear a few alarms, some louder and faster than others. Let me point them out in the piece.

Here we describe a student’s typical day at a personalized learning school. The setting is the Waukesha STEM Academy-Saratoga Campus in Waukesha, Wisconsin.

You could be forgiven for not knowing, based on that selection, that one of the authors is the principal of the Waukesha STEM Academy and that his two co-authors have financial ties to the personalized learning industry the Waukesha STEM Academy is a client of the other two authors [see this exchange. -dm]. What should be disclosed in the article’s first paragraph can only be inferred from the authors’ biographies in its footer. This minimal disclosure is consistent with what I perceive to be irresponsible self-promotion on the part of the personalized learning industry. (See also: “… this robot tutor can essentially read your mind.”)

(Full disclosure: I work in education technology.)

Then, in describing a student’s school experience before personalized learning, the authors write:

… [Cal’s] planner, which looked similar to those of the other 27 students in his class, told him everything he needed to know: Math: Page 122; solve problems 2—18 (evens). [..] Each week looked about the same.

If this is truly the case, if students didn’t interact with each other or their teacher at all, if they simply opened their books and completed a textbook assignment every day, every week, we really can’t do much worse. Most alternatives will look great. This isn’t a sober analysis of available alternatives. Again, this is marketing.

[Cal] began to understand why he sometimes misses some of the things that he hears in class and ands more comfort in module-based courses, where he can fast forward and rewind videos and read instructions at his own pace.

Fast-forwarding, rewinding, and pausing instructional videos are often cited as advantages of personalized learning, not because this is necessarily good instruction, but because it’s what the technology permits.

And this isn’t good instruction. It isn’t even good direct instruction. When someone is explaining something to you and you don’t understand them, you don’t ask that person to “repeat exactly what you just said only slower.” You might tell them what you understand of what they were saying. Then they might back up and take a different approach, using different examples, metaphors, or illustrations, ideally responding using your partial understanding as a resource.

I’m describing a very low bar for effective instruction. I’m describing techniques you likely employ in day-to-day conversation with friends and family without even thinking about them. I’m also describing a bar that 2017 personalized learning technology cannot clear.

His students don’t report to class to be presented with information. Instead, they’re empowered to use a variety of learning tools. Some students, like Cal, prefer step-by-step videos; others prefer songs and catchy rhymes to help them learn concepts. [..] He opens a series of videos and online tutorials, as well as tutorials prepared by his teacher.

In the first sentence, we’re told that students like Cal aren’t presented with information. Then, in the following sentences, we’re told all the different ways that those students are presented with information.

Whether you learn concepts from a step-by-step video, a rap, or a written tutorial, you are being presented with information. And a student’s first experience with new information shouldn’t be someone on a screen presenting it, no matter the style of presentation.

Because there is work students can do before that presentation to prepare themselves to learn and enjoy learning from it.

Because the video presenter treats students as though they have the same existing knowledge and prior conceptions about that information, even though those conceptions vary widely, even though some of them are surprisingly durable and require direct confrontation.

Because these video presentations communicate to students the message that math is something you can’t make sense of unless some adult explains it to you, that learning is something you do by yourself, and that your peers have nothing to offer your understanding of that new information.

I like a lot of the ethos around personalized learning — increasing student agency and metacognition, for example — but the loudest, fastest alarm in the article is this:

The medium is the message. Personalized learning is only as good as its technology, and in 2017 that technology isn’t good enough. Its gravity pulls towards videos of adults talking about math, followed by multiple choice exercises for practice, all of which is leavened by occasional projects. It doesn’t matter that students can choose the pace or presentation of that learning. Taking your pick of impoverished options still leaves you with an impoverished option.

2017 Mar 22. There are too many interesting comments to feature them individually. I’ll single out two of them directly, however:

  • Todd Gray, the Superintendent of the School District of Waukesha.
  • Anthony Rebora, the Editor-in-Chief of Educational Leadership.

The Difference Between Math and Modeling with Math in Five Seconds

Jim Pardun sent me a video of a dog named Twinkie popping balloons in the pursuit of a world record. How you train a dog to do this, I don’t know. How there is a world record for this, I don’t know either.

What I know is that this video clearly illustrates the difference between math and modeling with math.

You can’t break math. Some people think they broke math but all they did was break ground on new disciplines in math where, for example, triangles can have more than 180° and parallel lines can meet.

Our mathematical models, by contrast, arrive broken. “All models are wrong,” said George Box, “but some are useful.” And we see that in this video.

Twinkie pops 25 balloons in 5 seconds. How long will it take her to pop all 100 balloons? A purely mathematical answer is 20 seconds. That’s straightforward proportional reasoning.

But mathematical modeling is less than straightforward. It requires the re-interpretation of that answer through the world’s imperfections. The student who can quickly and confidently calculate 20 seconds may even be worse off here than the student who patiently thinks about how the supply of balloons is dwindling, adds time, and arrives at the actual answer of 37 seconds.

Feel free to show your classes that question video, discuss, and then show them the answer video. Or if your class has access to devices, you can assign this Desmos activity, where we’ll invite them to sketch what they think happens over time as well.

The difference between the students who answer “20 seconds” and “37 seconds” is the same difference between the students who draw Sketch 1 and Sketch 2.

You might think you know how your students will sort into those two groups, but I hope you’ll be surprised.

That difference is the patience that modeling with math requires.

BTW. I’m very interested in situations like these where the world subverts what seems like a straightforward application of a mathematical model.

One more example is the story of St. Matthew Island, which dumps the expectations of pure mathematics on its head at least twice.

Do you have any to trade?

2017 May 19. Steve Rein asked for the data set. Right here.