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## Computer Feedback That Helps Kids Learn About Math and About Themselves

Students are receiving more feedback from computers this year than ever before. What does that feedback look like, and what does it teach students about mathematics and about themselves as mathematicians?

Here is a question we might ask math students: what is this coordinate?

Let’s say a student types in (5, 4), a very thoughtful wrong answer. (“Wrong and brilliant,” one might say.) Here are several ways a computer might react to that wrong answer.

1. “You’re wrong.”

This is the most common way computers respond to a student’s idea. But (5, 4) receives the same feedback as answers like (1000, 1000) or “idk,” even though (5, 4) arguably involves a lot more thought from the student and a lot more of their sense of themselves as a mathematician.

This feedback says all of those ideas are the same kind of wrong.

2. “You’re wrong, but it’s okay.”

The shortcoming of evaluative feedback (these binary judgments of “right” and “wrong”) isn’t just that it isn’t nice enough or that it neglects a student’s emotional state. It’s that it doesn’t attach enough meaning to the student’s thinking. The prime directive of feedback is, per Dylan Wiliam, to “cause more thinking.” Evaluative feedback fails that directive because it doesn’t attach sufficient meaning to a student’s thought to cause more thinking.

3. “You’re wrong, and here’s why.”

It’s tempting to write down a list of all possible reasons a student might have given different wrong answers, and then respond to each one conditionally. For example here, we might program the computer to say, “Did you switch your coordinates?”

Certainly, this makes an attempt at attaching meaning to a student’s thinking that the other examples so far have not. But the meaning is often an expert’s meaning and attaches only loosely to the novice’s. The student may have to work as hard to understand the feedback (the word “coordinate” may be new, for example) as to use it.

4. “Let me see if I understand you here.”

Alternately, we can ask computers to clear their throats a bit and say, “Let me see if I understand you here. Is this what you meant?”

We make no assumption that the student understands what the problem is asking, or that we understand why the student gave their answer. We just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them.

“How can I attach more meaning to a student’s thought?”

This animation, for example, attaches the fact that the relationship to the origin has horizontal and vertical components. We trust students to make sense of what they’re seeing. Then we give them an an opportunity to use that new sense to try again.

This “interpretive” feedback is the kind we use most frequently in our Desmos curriculum, and it’s often easier to build than the evaluative feedback, which requires images, conditionality, and more programming.

Honestly, “programming” isn’t even the right word to describe what we’re doing here.

We’re building worlds. I’m not overstating the matter. Educators build worlds in the same way that game developers and storytellers build worlds.

That world here is called “the coordinate plane,” a world we built in a computer. But even more often, the world we build is a physical or a video classroom, and the question, “How can I attach more meaning to a student’s thought?” is a great question in each of those worlds. Whenever you receive a student’s thought and tell them what interests you about it, or what it makes you wonder, or you ask the class if anyone has any questions about that thought, or you connect it to another student’s thought, you are attaching meaning to that student’s thinking.

Every time you work to attach meaning to student thinking, you help students learn more math and you help them learn about themselves as mathematical thinkers. You help them understand, implicitly, that their thoughts are valuable. And if students become habituated to that feeling, they might just come to understand that they are valuable themselves, as students, as thinkers, and as people.

BTW. If you’d like to learn how to make this kind of feedback, check out this segment on last week’s #DesmosLive. it took four lines of programming using Computation Layer in Desmos Activity Builder.

BTW. I posted this in the form a question on Twitter where it started a lot of discussion. Two people made very popular suggestions for different ways to attach meaning to student thought here.

## The American Time Use Survey Is “Poetry, in Data.”

The American Time Use Survey is a fantastic data set. You can find out how many more hours per day women spend on household activities than men. You can identify the time of day that the majority of Americans wake up.

You can also determine the amount of time we spend with certain groups of people in our lives from childhood to late adulthood. For example, here are graphs of the amount of time we spend with friends and with co-workers.

Fantastic graphs, right? But will students think they’re fantastic? Will they learn from the graphs? How can you effectively introduce your students to the American Time Use Survey?

I use three strategies every time. You can read about them below and experience them in this new free activity from me and my colleagues at Desmos.

First, a meta-strategy:

I don’t allow myself to rest for a second in the false comfort that this is a “real world” context, and per se, interesting to students. Contexts are never “real” or “unreal.” They don’t exist in a vacuum. Contexts become real when teachers invite their students to interact with them in concrete and personal ways.

Here are three invitations I extend to students basically any time I’d like them to experience a graph as real.

1. I invite students to contribute their own data.

The graph represents a group of people’s concrete and personal experiences: time spent with friends, co-workers, and partners. I ask students to contribute their own data so the quantities and relationships become more concrete for them as well.

2. I invite students to sketch their own graph before seeing the actual graph.

This invites students to share their own knowledge about the quantities and relationships. Students have ideas about how many hours people spend with friends throughout their lives. We should invite them to express those ideas with a graph.

I also place their own data from (1) on the graph. This extends an even more personal invitation to students and gives them an anchor for their graphing.

“That’s you on there, friend. Do you think American 15-year-olds spend more or less time with their friends than you? Okay, graph it!”

3. I invite students to reflect.

Jim Coudal called these graphs “Poetry, in data.” So I ask students to tell us which graph is most poetic and why. We’ve built up a lot of steam in the activity, and this question helps release it. It allows us to elicit from students the personal observations that haven’t yet found a home in our activity.

I posted this activity on Twitter and the majority of people said this was the most interesting graph to them.

People wrote:

It is sad to me that once we are old enough to have free time to spend with friends, we spend more time alone.

I wonder if the loneliness is by choice.

Alarming lack of social opportunities for seniors.

There is so much interesting research coming out about the impact of loneliness on people’s health.

How can we change this?

So consider the invitations you extend to students. In many curricula, those invitations are impersonal and abstract. “What is the value of the co-workers graph for a 75-year-old?” That’s a question that invites students to reflect on an adult’s knowledge of graphs and the context.

“What would your data look like? What do you think the graph looks like? Why?” These are questions that invite students to interact with the graph in personal ways, to inhabit the graph as if it were their own.

Featured Comment

As I’m feeling mighty alone personally (even though there are folks in the house) and professionally (electronically just isn’t the same) during the current “Stay at Home” situation, this data definitely evokes some poetry for me.

## Desmos Is Also a Curriculum Company Now

If you knew me as a classroom teacher, you knew I was very, very cranky about the ways many math textbooks treated students and mathematics, how they failed to celebrate and build on student intuition about mathematical ideas, how their problems were posed in ways that hid their most interesting elements, how they were way too helpful.

So it’s been a joy to get to do something more active about that problem than write cranky blog posts, to get to team up with some fantastic teachers, designers, engineers, and funders all continuously interrogating their assumptions about education, design, technology, math, and society, all to create what I think is …

This is it.

Call off the search.

You found it.

[extremely Oprah voice] You get a debt of gratitude! You get a debt of gratitude! You get a debt of gratitude!

Aside from my enormous gratitude to the fantastic team I work with daily, I’m especially grateful to two groups:

• The authoring / publishing team at Illustrative Mathematics / Open Up Resources who created and openly licensed a fantastic math curriculum, one which is the foundation of our own work. They dropped a massive gift on the math education community (or a hydrogen bomb from the perspective of the K-12 math publishing industry) and we were extremely happy to pick it up and build on it.
• You. I’m talking about the folks who have been reading this blog, commenting on my posts, critiquing my ideas from day one. Your thoughts and mine are all tied together and run all the way through this curriculum.

This blog has been quieter over the last few years for reasons that are predictable — family, Twitter, the death of blogs, etc. — but also because, for the only time in my career, I haven’t been able to write about my work.

That changes today and I’m very excited to collaborate with you folks once again on the work that matters to me most. It won’t be at its best without you.

## The 2010s of Math Edtech in Review

EdSurge invited me to review the last decade in math edtech.

Entrepreneurs had a mixed decade in K-16 math education. They accurately read the landscape in at least two ways: a) learning math is enormously challenging for most students, and b) computers are great at a lot of tasks. But they misunderstood why math is challenging to learn and put computers to work on the wrong task.

In a similar retrospective essay, Sal Khan wrote about the three assumptions he and his team got right at Khan Academy in the last decade. The first one was extremely surprising to me.

Teachers are the unwavering center of schooling and we should continue to learn from them every day.

Someone needs to hold my hand and help me understand how teachers are anywhere near the center of Khan Academy, a website that seems especially useful for people who do not have teachers.

Khan Academy tries to take from teachers the jobs of instruction (watch our videos) and assessment (complete our autograded items). It presumably leaves for teachers the job of monitoring and responding to assessment results but their dashboards run on a ten-minute delay, making that task really hard!

Teachers are very obviously peripheral, not central, to the work of Khan Academy and the same is true for much of math education technology in the 2010s. If entrepreneurs and founders are now alert to the unique value of teachers in a student’s math education, let’s hear them articulate that value and let’s see them re-design their tools to support it.

## “If something cannot go on forever, it will stop.”

Economist Herb Stein’s quote ran through my head while I read The Hustle’s excellent analysis of the graphing calculator market. This cannot go on forever.

Every new school year, Twitter lights up with caregivers who can’t believe they have to buy their students a calculator that’s wildly underpowered and wildly overpriced relative to other consumer electronics.

The Hustle describes Texas Instruments as having “a near-monopoly on graphing calculators for nearly three decades.” That means that some of the students who purchased TI calculators as college students are now purchasing calculators for their own kids that look, feel, act and (crucially) cost largely the same. Imagine they were purchasing their kid’s first car and the available cars all looked, felt, acted, and cost largely the same as their first car. This cannot go on forever.

As the chief academic officer at Desmos, a competitor of Texas Instruments calculators, I was already familiar with many of The Hustle’s findings. Even still, they illuminated two surprising elements of the Texas Instruments business model.

First, the profit margins.

One analyst placed the cost to produce a TI-84 Plus at around \$15-20, meaning TI sells it for a profit margin of nearly 50% – far above the electronics industry’s average margin of 6.7%.

Second, the lobbying.

According to Open Secrets and ProPublica data, Texas Instruments paid lobbyists to hound the Department of Education every year from 2005 to 2009 – right around the time when mobile technology and apps were becoming more of a threat.

Obviously the profits and lobbying are interdependent. Rent-seeking occurs when companies invest profits not into product development but into manipulating regulatory environments to protect market share.

I’m not mad for the sake of Desmos here. What Texas Instruments is doing isn’t sustainable. Consumer tech is getting so good and cheap and our free alternative is getting used so widely that regulations and consumer demand are changing quickly.

Another source told The Hustle that graphing calculator sales have seen a 15% YoY decline in recent years – a trend that free alternatives like Desmos may be at least partially responsible for.

You’ll find our calculators embedded in over half of state-level end-of-course exams in the United States, along with the International Baccalaureate MYP exam, the digital SAT and the digital ACT.

I am mad for the sake of kids and families like this, though.

“It basically sucks,” says Marcus Grant, an 11th grader currently taking a pre-calculus course. “It was really expensive for my family. There are cheaper alternatives available, but my teacher makes [the TI calculator] mandatory and there’s no other option.”

Teachers: it was one thing to require plastic graphing calculators calculators when better and cheaper alternatives weren’t available. But it should offend your conscience to see a private company suck 50% profit margins out of the pockets of struggling families for a product that is, by objective measurements, inferior to and more expensive than its competitors.

BTW. This is a Twitter-thread-turned-blog-post. If you want to know how teachers justified recommending plastic graphing calculators, you can read my mentions.