Get Posts by E-mail

Archive for the 'tech contrarianism' Category

You may have heard that San Jose State University's recent partnership with Udacity ended with MOOC-enrolled students passing courses at much lower rates than their on-campus cohorts. Lots has been said about these results (Phil Hill has a good round-up of the coverage) but there's one line that deserves more coverage:

When students did get to the online programs, even navigating the computer systems could be daunting. One of the questions that tutors were frequently asked was how to do exponential notation on a computer.

Again we find computers are not a natural medium for doing mathematics. There's nothing intuitive about pressing Shift + 6 to write an exponent, no inherent connection between the idea and the action. This isn't true for computer science, where the medium is perfectly suited for the course. Or even for English composition, where typing words is only one intuitive abstraction away from writing them with a pen.

I'd wager 90% of people reading this already know how to type an exponent on a computer. They believe it's easy enough to teach and I don't think they're wrong. But this is only one instance of a problem with a lot of reach. Notation makes math difficult on a computer. But notation also makes math more powerful and interesting. That tension will be very difficult to resolve and, so far, online math providers have generally resolved it in favor of the computer at the expense of math's interest and power.

In our relentless transition from classroom-based math to computer-based math, these SJSU-Udacity results offer us a chance to pause and ask ourselves, "What's now missing?"


Computers Are Not A Natural Medium For Doing Mathematics

2013 Jul 26. Okay, taking friendly fire on Twitter, I posed this challenge:

Use a computer to compose a clear proof that a triangle's midsegments create similar triangles and send it to me for assessment.

My guess is you'll find the process a lot less annoying and a lot more clear when you pick up a pencil and some paper.

Featured Comments

David Lippman:

But on the other hand, the reality is that if our students use math any time later in their life, there’s a really good likelihood it will involve a computer, whether it’s using Mathematica to solve complex problems, doing computer programming, or just entering formulas in Excel. There is value in learning the notation for entering formulas in a computer, and it provides an valuable side benefit of reinforcing proper syntax to ensure proper order of operation.

Mr. K:

I’ve slowed down on the Euler Project problems, but last I checked, I was just short of 200. Those problems require a computer to do the math, but you don’t do the math on the computer. I have a notebook (ink on paper) dedicated to that that has a couple hundred pages filled with notes and drawings – that’s the math.

I'm used to seeing pedagogy manifest itself in lesson plans and classroom observations and curriculum and videos. It's interesting, now, to see pedagogical decisions manifest themselves in web design and code also. For example, here's some Javascript from Khan Academy's box-and-whisker plot exercises.


Head over to the exercise. Complete a couple. What pedagogical mistake has Khan Academy made in the highlighted lines? How would you fix it?

Don't get put off by the code. If you've taught box-and-whisker plots, you can sort out the issue here.

Previously: What We Can Learn About Learning From Khan Academy’s Source Code.

[via Travis Olson]

Featured Comments

Brian lands it:

This code will always generate 15 data points, and these points will not have any outliers (outside 1.5 * (Q3 – Q1)), so students can just pattern match and drag the lines to the 1st, 4th, 8th, 12th, and 15th places once they’ve sorted the data. It’s kind of fun the first time.

Dan Anderson piles on:

Agree with Brian. Always 15 data points? Never have to deal with “having two medians”? Ever? The data is between 0 and 15 (never -40 to -30, never 100 to 1000, never 0.80 to 1.15)? No outliers? Always starting with the data and making a box-and-whisker, never using the box-and-whisker to make conclusions?

Peter Franza picks on a different issue:

I think the largest error is the reliance on random numbers to provide a set of assessments that test an actual set of knowledge.

Random number generators are great for creating a large set of problems that are all basically the same, but in my experience you can provide better assessments/examples with a much smaller set of questions that are designed to illustrate the concept.


Others have danced around it, but the fundamental flaw (as in some, but not all, Khan exercises) is that you get


seven straight times, without any change in structure or difficulty, even though the underlying task has a huge variation in structure and difficulty.

Ben Alpert responds from Khan Academy:

I’ve updated the exercise so that it now includes anywhere from 8 to 15 points, so students are forced to deal with two middle numbers, both in finding the median and in finding the quartiles.

Christopher Danielson posts a video of Good Teaching:

My eyes tear up watching this sequence. I am neither kidding nor exaggerating. It gives me hope for quality classroom instruction in elementary mathematics. Be sure to notice the transition to a new task at the 4-minute mark, and how the teacher deals with the struggle that occurs at the 6-minute mark. Also please look in the kids’ eyes. Watch their body language and their waving hands. Watch them think.

You cannot flip the first instructional activity because it involves adapting instruction in response to student ideas, and it involves students justifying their thinking to the teacher and to each other.

You can’t flip that.

Leave your comments at Danielson's blog.

Tiny Math Games

Jason Dyer writes a very important post highlighting Tiny Games, a listing of games you can play quickly, almost anywhere, with only limited materials. He then pivots to ask about tiny math games.

Could one make an all-mathematics variant — mathematical scrimmages, so to speak?

His post, and Tiny Games, are important because they reject an article of faith of the blended learning and flipped classroom movements, that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.

For example, here's John Sipe, senior vice president of sales at Houghton Mifflin Harcourt, talking about Fuse, their iPad textbook:

So teachers don’t have to “waste their time” on some of these things that they’ve always had to do. They can spend much more time on individualized learning, identifying specific student needs. Let students cover the basics, if you will, on their own, and let teachers delve into enrichment and individualized learning. That’s what the good teachers are telling me.

This is a bad idea. People don't mind practicing a sport because playing the sport is fun. It's easy to tell a tennis player to practice 100 serves from the ad side of the court, for instance, because the tennis player has mentally connected the acts of practicing tennis and playing tennis. The blended learning movement, at its worst, disconnects practice and play.

Take multiplication of one- and two-digit numbers for instance.

If you need to learn multiplication facts, one option is to watch a video and then drill away. Or we can queue up all that practice in a tiny math game that'll have students playing as they practice:

Pick a number. Say 25. Now break it up into as many pieces as you want. 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would work. Now multiply all those pieces together. What's the biggest product you can make? Pick another. What's your strategy? Will it always work? [Malcolm Swan]

Easy money says the student who's practicing math while playing it will practice more multiplication and enjoy that practice more than the student who is assigned to drill practice alone.

Jason Dyer helpfully highlights two examples of tiny math games, Nim and Fizz-Buzz, but he and I are both struggling to define a "tiny math game." The success of the Tiny Game Kickstarter project indicates serious interest in these tiny games. I'd like to see a similar collection of tiny math games. Here's how you can help with that.

1. Offer Examples of Tiny Math Games

This may be tricky. We all have games we play in math class. What distinguishes those games from "tiny math games?"

2. Help Us Define "Tiny Math Games"

This may be a better starting point. I'll add your suggestions to this list. Here are some seeds:

  • The point of the game should be concise and intuitive. You can summarize the point of these games in a few seconds or a couple of sentences. It may be complicated to continue playing the game or to win it, but it isn't hard to start.
  • They require few materials. That's part and parcel of being "tiny." These games don't require a laptop or iPhone.
  • They're social, or at least they're better when people play together.
  • They offer quick, useful feedback. With the multiplication game, you know you don't have the highest product because someone else hollers out one that's higher than yours. With Fizz-Buzz, your fellow players give you feedback when you blow it.
  • They benefit from repetition. You may access some kind of mathematical insight on individual turns but you access even greater insight over the course of the game. With Fizz-Buzz, for instance, players might count five turns and then say "Buzz," but over time they may realize that you'll always say "Buzz" on numbers that end in 5 or 0. That extra understanding (what we could call the "strategy" of these tiny math games) is important.
  • The math should only be incidental to the larger, more fun purpose of the game. I think this may be setting the bar higher than we need to, but Jason Dyer points out that people play Fizz-Buzz as a drinking game. [Jason Dyer]

What can you add to our understanding of tiny math games?

2013 Apr 17. Nobody wanted to tackle the qualities of tiny math games, which is fine since you all threw down a number of interesting games. I'll be compiling those on a separate domain at some point soon.

Featured Tweets

Jason Dyer elaborates on his contribution above.

Other teachers go in on the lie that students need basic skills before they can do anything interesting in their disciplines:

2013 Apr 24. Jason Dyer elaborates in another post.

SAL KHAN: You're in tenth grade Algebra class. The teacher asks the student to do like six problems. "Oh my god." They're groaning. "This is the meanest guy on the planet." And then three hours later we're in wrestling practice and the coach says, "I want you to do fifty pushups followed by running three miles followed by another fifty pushups." And they're like, "Yes, sir. Yes, sir. Push me harder. I want to collapse." I mean, literally, sometimes people would collapse, they were willing to work so hard.

CHARLIE ROSE: So what's the difference?

It goes without saying that if you're Sal Khan or anybody else in the drill-based math proficiency software business, you should give Charlie Rose's question a lot of thought. Kids generally like sports practice a lot more than math practice so there's huge risk and huge reward here.

I don't think Khan's answer is wrong exactly but it's as though Charlie Rose asked him about the appeal of ice cream on a hot summer day and Khan enthused for a few minutes about the taste of the sugar cone. Sugar cones are tasty but there's a lot more to say there.

So let me load the question up on a tee and invite you all to swing away:

What makes sports practice satisfying and how is sports practice different from math practice?

Featured Comments


In short: Do quadratics for 3 hours a day and you are a nerd with no life. Hit batting practice for 3 hours a day and get the girl and go to the party on the weekend.

Pete Welter:

Sports practice leads to an actual game – a chance for the results of that hard work to be put to use. And that game will probably happen next week. The payoff for school math is far enough off – years – for most students to be forever. See what happens if you have a wrestler doing just pushups for 5 or 10 years without using it.


Speaking as a track coach / math teacher, I really think the biggest difference is that the kids signed up for it in sports. they knew what they were getting into, and if they want to quit, they are totally able to do so. Ask gym teachers if they have 100% participation in their classes.

Chris Robinson:

A lot of us grow up watching sports with our friends and families. Not a lot of us grow up talking and doing math. For better or worse, sports holds a significant place in our culture.

Dan Anderson:

Immediate and clear feedback. I can make that fade away basket now because I tried it 50 times from the same spot and have hit the last 10 in a row.

Mandy Jansen:

Sports — The practice gets put to use, and there is a real payoff or outcome to work toward.

Kelly O'Shea:

First, there are plenty of kids who feel the inverse (love math practice, hate sports practice). Why do we not think that’s true? Math is required, sports are not.


Recently, in a polynomial unit, my kids had to multiply a binomial by a trinomial a few times (like three times). There was also a problem of a 7-term polynomial being multiplied by a trinomial. I told my students that this problem was “challenge by choice” but that I really wanted them to try it – because it was kind of fun in an algebra geeky sort of way. It was really my subversive way to get them to understand the distributive property and that “FOIL” isn’t the best acronym. About 90% of my students tried the hard (or tedious) problem; a bunch did it correctly and most of them came really close.

Ultimately, it’s about a sense of accomplishment – whatever the practice is for.

Nathan Kraft:

We try much harder in sports practice because we’re preparing for a game. In most math classrooms, there is no “game”. There’s just practice.

Justin Lanier:

Hard-nosed and repetitive skills work in sports is essential and beneficial. In math, it’s a chore if you “get it” and unhelpful if you don’t.




Sorry. This sounds like a typical US American discussion to me.


I think another angle would be “growth mindset”. Students believe that they will get better at a sport with that kind of focused practice.

Santosh Zachariah:

Try running a season’s worth of sports practice without a “scrimmage” and see what happens.

Gary Stager:

A trumpet player, actor, cellist, dancer who is improving, does so not just through repetitive button pushing or by being yelled at by a coach/math teacher. Productive practice requires intrinsic motivation, reflective ability, attention to detail and a cognitive clarity that connects often the tiniest of personal goals to prior knowledge and past experiences. Understanding results from such clarity. Also, one cannot discount the value of relevance in this equation.

People do not get “good” at something by doing it 10,000 times without caring about it, finding personal value in the activity or by making connections to something bigger than yourself.

« Prev - Next »