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Here is the talk I gave at CMC-North last weekend: Video Games & Making Math More Like Things Students Like.

Students generally prefer video games to our math classes and I wanted to know why. So I played a lot of video games and read a bit about video games and drew some conclusions. I also asked my in-laws to play two video games in front of a camera so we could watch their learning process and draw comparisons to our students.

These are the six lessons I learned:

  1. Video games get to the point.
  2. The real world is overrated.
  3. Video games have an open middle.
  4. The middle grows more challenging and more interesting at the same time.
  5. Instruction is visual, embedded in practice, and only as needed.
  6. Video games lower the cost of failure.

My analysis of Khan Academy’s eighth-grade curriculum was viewed ~20,000 times over the last ten days. Several math practice web sites have asked me to perform a similar analysis on their own products. All of this gives me hope that my doctoral work may be interesting to people outside my small crowd at Stanford.

Two follow-up notes, including the simplest way Khan Academy can improve itself:

One. Several Khan Academy employees have commented on the analysis, both here and at Hacker News.

Justin Helps, a content specialist, confirmed one of my hypotheses about Khan Academy:

One contributor to the prevalence of numerical and multiple choice responses on KA is that those were the tools readily available to us when we began writing content. Our set of tools continues to grow, but it takes time for our relatively small content team to rewrite item sets to utilize those new tools.

But as another commenter pointed out, if the Smarter Balanced Assessment Consortium can make interesting computerized items, what’s stopping Khan Academy? Which team is the bottleneck: the software developers or the content specialists? (They’re hiring!)

Two. In my mind, Khan Academy could do one simple thing to improve itself several times over:

Ask questions that computers don’t grade.

A computer graded my responses to every single question in eighth grade.

That means I was never asked, “Why?” or “How do you know?” Those are seriously important questions but computers can’t grade them and Khan Academy didn’t ask them.

At one point, I was even asked how m and b (of y = mx + b fame) affected the slope and y-intercept of a graph. It’s a fine question, but there was no place for an answer because how would the computer know if I was right?

So if a Khan Academy student is linked to a coach, make a space for an answer. Send the student’s answer to the coach. Let the coach grade or ignore it. Don’t try to do any fancy natural language processing. Just send the response along. Let the human offer feedback where computers can’t. In fact, allow all the proficiency ratings to be overridden by human coaches.

Khan Academy does loads of A/B testing right? So A/B test this. See if teachers appreciate the clearer picture of what their students know or if they prefer the easier computerized assessment. I can see it going either way, though my own preference is clear.

A+ Job, CMCN14

This was supposed to be a brief preamble to a post about what I learned at a recent conference, but it ballooned into this long, glowing mash note to the conference itself. You should find some way to attend next year.

California Math Council’s conference in Monterey, CA, last weekend was the best conference PD I’ve ever experienced. Your mileage may have varied depending on your session choices (or whether you were even there) but every. single. element. fell into line for me.

  • Great evening keynote with Tony DeRose of Pixar. (Shorter version here.) I love keynotes that are just outside, but not too far outside our discipline.
  • An excellent pick of four sessions on Saturday. There were at least three great picks in every block. Painful choices. I went out for a few names I knew would be fun (Lasek, Fenton, Stadel). But I also ventured out for a name I didn’t recognize (Barlow) and learned an enormous amount about math teaching as well as about how to talk with math teachers about math teaching. I’ll share some details in a later post, which was supposed to be this post until I got all breathless about the conference itself.
  • The Ignite sessions on Saturday evening were best-in-class. They were all entertaining and interesting, which is unusual enough, but three of them drew standing ovations. Five minute talks. Standing ovations. A standing ovation off of five minutes. Don’t worry. I’ll make sure you see them later.
  • The community. I get such a charge off the crowd that assembles on the Monterey Coast annually. I walked around Point Lobos with mentors, broke bread with peers, and met lots of new teachers from local programs. One of the keynote presenters and I both gave talks we had already given elsewhere and we both noted how charged up the crowds were, how great the vibe was, relative to those other venues. No idea why, but I’ll take it.
  • The venue. Unbeatable.

So great job, California Math Council. Everybody else: be sure to sign up to present and attend next year.

A reader asked me what classroom technology she should purchase with $1,000. My response:

I’d install whiteboards on every vertical surface in the room. I’d make sure I had a good document camera. And I’d probably purchase video capture equipment, a hard drive, and a microphone so I could record my lessons. That’ll probably get you close to $1,000.

I felt clever recommending old-school whiteboards with a new-school technology grant. But then I put the question out on Twitter and everybody suggested the same purchase:

Crazy, right? What would you buy?

$1,000 isn’t nothing, but there are lots of organizations giving away that sum and more to teachers. I have it on some authority that The Mathematics Education Trust has trouble some years giving away their (fairly substantial) grants. “Not enough qualified applicants,” I was told. So get out there. Get some cash. Get those high-tech whiteboards.

BTW. I think we can trace some of this recent popularity of whiteboarding to Peter Liljedahl, an associate professor at Simon Fraser University. Liljedahl gave a presentation at the Canadian Mathematics Education Forum on whiteboards, which he called “Vertical Non-Permanent Surfaces,” which is why I’m looking forward to finishing graduate school.

tl;dr — Khan Academy claims alignment with the Common Core State Standards (CCSS) but an analysis of their eighth-grade year indicates that alignment is loose. 40% of Khan Academy exercises assessed the acts of calculating and solving whereas the Smarter Balanced Assessment Consortium’s assessment of the CCSS emphasized those acts in only 25% of their released items. 74% of Khan Academy’s exercises resulted in the production of either a number or a multiple-choice response, whereas those outputs accounted for only 25% of the SBAC assessment.

Introduction

My dissertation will examine the opportunities students have to learn math online. In order to say something about the current state of the art, I decided to complete Khan Academy’s eighth grade year and ask myself two specific questions about every exercise:

  • What am I asked to do? What are my verbs? Am I asked to solve, evaluate, calculate, analyze, or something else?
  • What do I produce? What is the end result of my work? Is my work summarized by a number, a multiple-choice response, a graph that I create, or something else?

I examined Khan Academy for several reasons. First, because they’re well-capitalized and they employ some of the best computer engineers in the world. They have the human resources to create some novel opportunities for students to learn math online. If they struggle, it is likely that other companies with equal or lesser human resources struggle also. I also examined Khan Academy because their exercise sets are publicly available online, without a login. This will energize our discussion here and make it easier for you to spotcheck my analysis.

My data collection took me three days and spanned 88 practice sets. You’re welcome to examine my data and critique my coding. In general, Khan Academy practice sets ask that you complete a certain number of exercises in a row before you’re allowed to move on. (Five, in most cases.) These exercises are randomly selected from a pool of item types. Different item types ask for different student work. Some item types ask for multiple kinds of student work. All of this is to say, you might conduct this exact same analysis and walk away with slightly different findings. I’ll present only the findings that I suspect will generalize.

After completing my analysis of Khan Academy’s exercises, I performed the same analysis on a set of 24 released questions from the Smarter Balanced Assessment Consortium’s test that will be administered this school year in 17 states.

Findings & Discussion

Khan Academy’s Verbs

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The largest casualty is argumentation. Out of the 402 exercises I completed, I could code only three of their prompts as “argue.” (You can find all them in “Pythagorean Theorem Proofs.”) This is far out of alignment with the Common Core State Standards, which has prioritized constructing and critiquing arguments as one of its eight practice standards that cross all of K-12 mathematics.

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Notably, 40% of Khan Academy’s eighth-grade exercises ask students to “calculate” or “solve.” These are important mathematical actions, certainly. But as with “argumentation,” I’ll demonstrate later that this emphasis is out of alignment with current national expectations for student math learning.

The most technologically advanced items were the 20% of Khan Academy’s exercises that asked students to “construct” an object. In these items, students were asked to create lines, tables, scatterplots, polygons, angles, and other mathematical structures using novel digital tools. Subjectively, these items were a welcome reprieve from the frequent calculating and solving, nearly all of which I performed with either my computer’s calculator or with Wolfram Alpha. (Also subjective: my favorite exercise asked me to construct a line.) These items also appeared frequently in the Geometry strand where students were asked to transform polygons.

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I was interested to find that the most common student action in Khan Academy’s eighth-grade year is “analyze.” Several examples follow.

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Khan Academy’s Productions

These questions of analysis are welcome but the end result of analysis can take many forms. If you think about instances in your life when you were asked to analyze, you might recall reports you’ve written or verbal summaries you’ve delivered. In Khan Academy, 92% of the analysis questions ended in a multiple-choice response. These multiple-choice items took different forms. In some cases, you could make only one choice. In others, you could make multiple choices. Regardless, we should ask ourselves if such structured responses are the most appropriate assessment of a student’s power of analysis.

Broadening our focus from the “analysis” items to the entire set of exercises reveals that 74% of the work students do in the eighth grade of Khan Academy results in either a number or a multiple-choice response. No other pair of outcomes comes close.

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Perhaps the biggest loss here is the fact that I constructed an equation exactly three times throughout my eighth grade year in Khan Academy. Here is one:

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This is troubling. In the sixth grade, students studying the Common Core State Standards make the transition from “Number and Operations” to “Expressions and Equations.” By ninth grade, the CCSS will ask those students to use equations in earnest, particularly in the Algebra, Functions, and Modeling domains. Students need preparation solving equations, of course, but if they haven’t spent ample time constructing equations also, those advanced domains will be inaccessible.

Smarter Balanced Verbs

The Smarter Balanced released items ask comparatively fewer “calculate” and “solve” items (they’re the least common verbs, in fact) and comparatively more “construct,” “analyze,” and “argue.”

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This lack of alignment is troubling. If one of Khan Academy’s goals is to prepare students for success in Common Core mathematics, they’re emphasizing the wrong set of skills.

Smarter Balanced Productions

Multiple-choice responses are also common in the Smarter Balanced assessment but the distribution of item types is broader. Students are asked to produce lots of different mathematical outputs including number lines, non-linear function graphs, probability spinners, corrections of student work, and other productions students won’t have seen in their work in Khan Academy.

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SBAC also allows for the production of free-response text while Khan Academy doesn’t. When SBAC asks students to “argue,” in a majority of cases, students express their answer by just writing an argument.

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This is quite unlike Khan Academy’s three “argue” prompts which produced either a) a multiple-choice response or b) the re-arrangement of the statements and reasons in a pre-filled two-column proof.

Limitations & Future Directions & Conclusion

This brief analysis has revealed that Khan Academy students are doing two primary kinds of work (analysis and calculating) and they’re expressing that work in two primary ways (as multiple-choice responses and as numbers). Meanwhile, the SBAC assessment of the CCSS emphasizes a different set of work and asks for more diverse expression of that work.

This is an important finding, if somewhat blunt. A much more comprehensive item analysis would be necessary to determine the nuanced and important differences between two problems that this analysis codes identically. Two separate “solving” problems that result in “a number,” for example, might be of very different value to a student depending on the equations being solved and whether or not a context was involved. This analysis is blind to those differences.

We should wonder why Khan Academy emphasizes this particular work. I have no inside knowledge of Khan Academy’s operations or vision. It’s possible this kind of work is a perfect realization of their vision for math education. Perhaps they are doing exactly what they set out to do.

I find it more likely that Khan Academy’s exercise set draws an accurate map of the strengths and weaknesses of education technology in 2014. Khan Academy asks students to solve and calculate so frequently, not because those are the mathematical actions mathematicians and math teachers value most, but because those problems are easy to assign with a computer in 2014. Khan Academy asks students to submit their work as a number or a multiple-choice response, not because those are the mathematical outputs mathematicians and math teachers value most, but because numbers and multiple-choice responses are easy for computers to grade in 2014.

This makes the limitations of Khan Academy’s exercises understandable but not excusable. Khan Academy is falling short of the goal of preparing students for success on assessments of the CCSS, but that’s setting the bar low. There are arguably other, more important goals than success on a standardized test. We’d like students to enjoy math class, to become flexible thinkers and capable future workers, to develop healthy conceptions of themselves as learners, and to look ahead to their next year of math class with something other than dread. Will instruction composed principally of selecting from multiple-choice responses and filling numbers into blanks achieve that goal? If your answer is no, as is mine, if that narrative sounds exceedingly grim to you also, it is up to you and me to pose a compelling counter-narrative for online math education, and then re-pose it over and over again.

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