# 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.

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

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."

# Teacher Data Dashboards Are Hard, Pt. 2

Can you help me shuffle my thoughts on teacher data dashboards?

The Current State of Teacher Data Dashboards

Generalizing from my own experience and from my reading, teacher data dashboards seem to suffer in three ways:

• They confuse easy data with good data. It's easy to record and report the amount of time a student had a particular webpage open, for instance, but that number isn't indicative of all that much.
• They aren't pedagogically useful. They'll tell you that a student got a question wrong or that the student spent seven minutes per problem but they won't tell you why or what to do next beyond "Tell the student to rewind the lecture video and really watch it this time."
• They're overwhelming. If you've never managed a classroom with more than 30 students, if you're a newly-minted-MBA-turned-edtech-startup-CEO for instance, you might have the wrong idea about teachers and the demands on their time and attention. Teaching a classroom full of students isn't like sitting in front of a Bloomberg terminal with a latte. The same volume of statistics, histograms, and line graphs that might thrill a financial analyst with few other demands on her attention might overwhelm a teacher who's trying to ensure her students aren't setting their desks on fire.

If you have examples of dashboards that contradict me here, I'd love to see screenshots.

We Tried To Build A Better Data Dashboard

With the teacher dashboard on our pennies lesson, the Desmos team and I tried to fix those three problems.

We attempted to first do no harm.

We probably left some good data on the table, but at no point did we say, "Your student knows how to model with quadratic equations." That kind of knowledge is really difficult to autograde. We weren't going to risk assigning a false positive or a false negative to a student, so we left that assessment to the teacher.

We tailored the dashboard to the lesson.

We created filters that will be mostly useless for any other lesson we might design later.

We filtered students in ways we thought would lead to rich teacher-student interactions. For example:

• If a student changed her pennies model (say from a linear to a quadratic or vice versa) we thought that was worth mentioning to a teacher.
• We made it easy to find out which students filled up large circles with pennies and which students found some cheap and easy data by filling up a small circle.
• We made it easy to find out which students had the closest initial guesses.

These filters don't design themselves. They require an understanding of pedagogy and a willingness to commit developer-hours to material that won't scale or see significant reuse outside of one lesson. That commitment is really, really uncommon for edtech startups. It's one reason why the math edublogosphere gets so swoony about Desmos.

Contrast that with filters from Khan Academy, which read, "Struggling," "Needs Practice," "Practiced," "Level One," "Level Two," and "Mastered." Broadly applicable, but generic.

We suggested teacher action.

For each of those filters, we gave teachers a brief suggestion for action. For students who changed models, we suggested teachers ask:

For students who filled up large circles, we suggested teachers say something like:

A lot of you filled small circles with pennies but these students filled large circles with pennies. That's harder and it's super useful to have a wide range of data when we go to fit our model.

For students who filled up small circles, we suggested teachers say something like:

Big data help us come up with a model, but so do small data. A zero-inch circle is really easy to draw and fill with circles so don't forget to collect it.

Even with this kind of concise, focused development, one teacher, Mike Bosma, still found our dashboard difficult to use in class:

While the students were working, I was mostly circulating around the classroom helping with technology issues (frozen browsers) and also clarifying what need to be done (my students did not read directions very well). I was hoping to be checking the dashboard as students went so I could help those students who were struggling. The data from the dashboard were helpful more so after the period for me. As I stated above, I was very busy during the period managing the technology and keeping students on track so I was not able to engage with what they were doing most of the time.

So we'd like to hear from you. Have you used the pennies task in class? Have you used the dashboard? What works? What doesn't? What would make a dashboard useful – actually usable – for you?

Tom Woodward, arguing that these platforms are tougher to customize than the usual paper-and-pencil lesson plan:

The other piece I worry about is the relatively unattainable nature of some of the skills needed for building interesting/useful digital content for most teachers. I really want to provision content for teachers and then be able to give them access to changing/building their own content. While many are happy consuming what’s given, there are people who will want to make it their own or it will spark new ideas. I hate the idea that the next step would be out of reach of most of that subset.

And there’s Eric Scholz looking for exactly that kind of customization:

I would add a “bank” of variables at the top of the page that teachers from when building their lesson plan the night before. This would allow for a variety of objectives for the lesson.

While many adaptive systems propose to help students along the way, they are often mis-interpreted as summative assessments, through their similarities to traditional grading terms and mechanisms.

There could/should be some value to a dashboard that guides formative synchronous action but it’d have to be really low on cognitive demand.

# Teacher Data Dashboards Are Hard, Pt. 1

A study published earlier this year on teacher data dashboards, summarized by Matthew Di Carlo:

Teachers in these meetings were quite candid in expressing their opinions about and experiences with Dashboard. One factor that arose with relative frequency was an expressed concern that the Benchmark tests lacked some validity because they often tested material the teachers had yet to cover in class. A second factor that was supported across the focus group discussions was a perceived lack of instructional time to act on information a teacher might gain from Dashboard data. In particular, teachers expressed frustration with the lack of time to re-teach topics and concepts to students that had been identiﬁed on Dashboard as in need of re-teaching. A third concern was a lack of training in how to use Dashboard effectively and efﬁciently. A fourth common barrier to Dashboard use cited by teachers was a lack of time for Dashboard-related data analysis.

Khan Academy intern Josh Netterfield, in June 2013, on Khan Academy's coach reports:

Currently over 70,000 teachers actively use KA in their classrooms, but few actually use coach reports. Already we’ve seen how the right kind of insights can transform classrooms, but some of the data has historically been quite difficult to navigate.

Generally speaking, the student data available on the Khan dashboard was impressive, but it also was challenging at times for the teacher to figure out how best to synthesize and use all the data – a key future needed if teachers are to maximize the potential of blended learning

Screenshots from a video of Khan Academy's recent redesign of their coach reports:

Karen Head, on her "First-Year Composition 2.0″ MOOC:

Too often we found our pedagogical choices hindered by the course-delivery platform we were required to use, when we felt that the platform should serve the pedagogical requirements. Too many decisions about platform functionality seem to be arbitrary, or made by people who may be excellent programmers but, I suspect, have never been teachers.

[via Jonathan Rees]

Just to remind everyone of the context of my statement. We asked that certain parameters in the coding be changed (like the one governing how much we could penalize students for not doing an assignment) and were given the answer that the penalty number was “hard coded” into the program. The tech support person couldn’t understand why it was a big deal to us. To be fair, I couldn’t be made to understand why it was a big deal to change the parameter from a fixed number of 20 to a range of 0-100, but I seem to remember from my basic undergrad programming class that it isn’t a big deal to do this. Of course, in the end, I’m just an English teacher. :-)

# Computers Are Not A Natural Medium For Doing Mathematics, Ctd.

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?"

Previously

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.