Drill-Based Math Instruction Diminishes the Math Teacher as Well

Emma Gargroetzi posts an astounding rebuttal to Barbara Oakley’s New York Times op-ed encouraging drill-based math instruction. Gargroetzi highlights two valid points from Oakley and then takes a blowtorch to the rest of them.

I haven’t been able to stop thinking about her last sentence since I read it yesterday.

Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.

I’m perhaps more hesitant than Gagroetzi to judge intent. Lots of teachers were, themselves, victimized by drill-based instruction as students and may lack an imagination for anything different. But I’m absolutely convinced that a) we act ourselves into belief rather than believing our way into acting, and b) actions and beliefs will accumulate over a career like rust and either inhibit or enhance our potential as teachers.

A math program that endorses drills and pain as the foundational element of math instruction (rather than a supporting element) and as a prerequisite for creative mathematical thought (rather than a co-requisite) inhibits the student and the teacher both, diminishing the student’s interest in producing that creativity and the teacher’s ability to notice it.

Teachers need to disrupt the harmful messages their students have internalized about mathematics. But we also need to disrupt the harmful messages that teachers have internalized as well.

What experiences can disrupt the harmful messages teachers have internalized about math instruction? Name some in the comments. I’ll add my own suggestions later tomorrow.

2018 Aug 25. I added my own suggestion here.

Featured Comments

Faye calls out the process of learning content and pedagogy simultaneously:

Many mathematics teachers do not have the mathematics content knowledge that they need themselves. The Greater Birmingham Mathematics Partnership has found that teaching teachers mathematics using inquiry based instruction results in increased content knowledge for the teachers and a change in their beliefs about how and what all children can learn, i.e., acting themselves into changed beliefs.

Chris:

Math teachers circles (www.mathteacherscircle.org/). They provide the space for math teachers to be mathematicians (in the same way a lot of the arts teachers I know are still practicing artists).

Another Chris echoes:

It wasn’t until I was asked to think about mathematical tasks and ideas for my own understanding that I could ask the same of my students. And then, it was unavoidable…there was no going back.

William Thill elaborates:

But when I can tap into the emotional and intellectual highs that emerge from playing with cherished colleagues, I am more likely to “set the buffet” for my students with more open-ended exploration times.

Martha Mulligan:

… watching yourself teach on video is a great experience to disrupt harmful messages about math instruction, like talking too much as the teacher. I know that many math teachers feel the need to provide the most perfect, refined, rehearsed explanation so that students can see what they are supposed to see in the way they are supposed to see it. I certainly felt (at time still feel?) that way. That practice diminishes the students’ roles of sense-making on their own. But watching a video of myself teaching was one of the most humbling things I’ve done and it changed my practice so much. I also watched them among other trusted teachers from whom I learned so much. Having time to stop a video, talk about, reflect on it, etc is very powerful. Even seemingly simple things like wait time and teacher movement/positioning can look very different than what we imagine we look like.

Alexandra Martinez calls out the limitation of reading narratives and watching videos of innovative teaching:

I think the most powerful way to disrupt teacher’s own experiences and expectations is new creative experiences with their own students. The evidence and reflection can support teachers in seeing what is possible. If we ask teachers to imagine what is possible through narrative, they won’t always believe it. But when they see their own students speaking and thinking as mathematicians, that evidence disrupts their established belief systems. So I’d say observations, modeling, Coteaching, pushing in, PLC planning with lesson study can all potentially do this.

Be sure, also, to check into Chris Heddles’ a/k/a Third Chris’s dissent:

I’m going to go against the grain and admit that I use drill as a prerequisite (or at least an opening activity) with many of my students.

Learning the Wrong Lessons from Video Games

[This is my contribution to The Virtual Conference on Mathematical Flavors, hosted by Sam Shah.]

In the early 20th century, Karl Groos claimed in The Play of Man that “the joy in being a cause” is fundamental to all forms of play. One hundred years later, Phil Daro would connect Groos’s theory of play to video gaming:

Every time the player acts, the game responds [and] tells the player your action causes the game action: you are the cause.

Most attempts to “gamify” math class learn the wrong lessons from video games. They import leaderboards, badges, customized avatars, timed competitions, points, and many other stylistic elements from video games. But gamified math software has struggled to import this substantial element:

Every time the player acts, the game responds.

When the math student acts, how does math class respond? And how is that response different in video games?

Watch how a video game responds to your decision to jump off a ledge.

Now watch math practice software responds to your misinterpretation of “the quotient of 9 and c.”

The video game interprets your action in the world of the game. The math software evaluates your action for correctness. One results in the joy in being the cause, a fundamental feature of play according to Groos. The other results in something much less joyful.

To see the difference, imagine if the game evaluated your decision instead of interpreting it.

I doubt anyone would argue with the goals of making math class more joyful and playful, but those goals are more easily adapted to a poster or conference slidedeck than to the actual experience of math students and teachers.

So what does a math class look like that responds whenever a student acts mathematically, that interprets rather than evaluates mathematical thought, that offers students joy in being the cause of something more than just evaluative feedback.

“Have students play mathematical or reasoning games,” is certainly a fair response, but bonus points if you have recommendations that apply to core academic content. I will offer a few examples and guidelines of my own in the comments later tomorrow.

Featured Comments

James Cleveland:

I feel like a lot of the best Desmos activities do that, because they can interpret (some of) what the learner inputs. When you do the pool border problem, it doesn’t tell you that your number of bricks is wrong – it just makes the bricks, and you can see if that is too many, too few, or just right.

In general, a reaction like “Well, let’s see what happens if that were true” seems like a good place to start.

Kevin Hall:

My favorite example of this is when Cannon Man’s body suddenly multiplies into two or three bodies if a student draws a graph that fails the vertical line test.

Sarah Caban:

I am so intrigued by the word interpret. “Interpret” is about translating, right? Sometimes when we try to interpret, we (unintentionally) make assumptions based on our own experiences. Recently, I have been pushing myself to linger in observing students as they work, postponing interpretations. I have even picked up a pencil and “tried on” their strategies, particularly ones that are seemingly not getting to a correct solution. I have consistently been joyfully surprised by the math my students were playing with. I’m wondering how this idea of “trying on” student thinking fits with technology. When/how does technology help us try on more student thinking?

Dan Finkel:

I think that many physical games give clear [evaluative] feedback as well, insofar as you test out a strategy, and see if you win or not. Adults can ruin these for children by saying, “are you sure that’s the right move?” rather than simply beating them so they can see what happens when they make that move. The trick there is that some games you improve at simply by losing (I’d put chess in this column, even though more focused study is essential to get really good), where others require more insight to see what you actually need to change.

Orchestrate More Productive Mathematics Discussions with Desmos Snapshots

Let me describe a powerful teaching tool we just released and the company values that compelled us to build it.

First, let’s acknowledge that statements of values are often useless. Values are only useful if they help people make hard decisions. Our company values should (a) help educators decide how we’re different from other math edtech companies, (b) help us decide how to spend our limited time in the world. So here is one of our values:

We believe that math class should be social and creative – that students should create mathematics in every form and then share those creations with each other and their teachers.

Many other companies disagree with those values, or at least they spend their limited time in the world acting on different ones. For example, many other companies think it’s sufficient for students to create multiple choice and numerical responses to express their mathematical thinking and to share those responses with a grading algorithm alone.

Our values conflict, and the result is that other companies spend their time optimizing adaptive grading algorithms while we spend our time thinking about ways to provoke mathematical creativity that algorithms can’t grade at all. We may both work in “math edtech” but we are on very different paths, and our path recently led us to a very thorny question:

What should teachers do with all these expressions of mathematical creativity that algorithms can’t grade?

Let’s say we ask students an interesting question about mathematics or we ask them to define a relationship and sketch its graph. That’s good math, but the teacher now has dozens of written answers and sketches that their computers can’t grade.

Other math edtech software offers teachers scarce insight into the ways students think mathematically. We offer teachers abundant insight which is a different kind of problem, and just as serious. We’ve spent months building a solution to this problem of abundance and we likely would have spent years if not for one book:

Mary Kay Stein and Margaret Smith’s Five Practices for Orchestrating Productive Mathematical Discussions.

Smith and Stein describe five teaching practices that promote student learning through summary discussions. Teachers should (1) anticipate ideas students will produce during a task or activity and then (2) monitor student work during class for those ideas and others that weren’t anticipated. Then the teacher should (3) select a subset of those interesting student ideas, (4) sequence the order of their presentation, and then help students (5) connect them.

In our classroom observations of our activities, we noticed teachers struggling to select student ideas because there were so many of them streaming from the students’ heads into the teacher’s dashboard. Sometimes teachers would make a note about an idea they wanted to select later, but when “later” came around, the student had already developed the idea further. So then we saw teachers take screenshots of that idea and paste them into slide software for sequencing. Smith and Stein’s recommendations are already ambitious and our software was not making it easier for teachers to enact them.

So we built “Snapshots.”

If you see interesting ideas at any time during an activity, press the camera icon next to it.

Then go to the “Snapshots” tab.

Sequence the ideas by dragging them into a collection.

Add a comment or a question to help students connect their classmates’ ideas to the main ideas of the lesson.

Then press “Present.”

We tested the tool ourselves during a summer school session in Berkeley, CA, and also with teachers around the country. What we’ve noticed is that students pay much more attention to discussions when the discussion isn’t about a page from the textbook or a worked example from the teacher but about ideas from the students themselves.

It’s the difference between “Let me tell you about a really useful strategy for multiplying two-digit number” and “Let me show you some useful strategies from around the class for multiplying two-digit numbers. They’re all correct. Decide which seems like less work to you.”

Here are some of our other favorite uses from the last month of testing.

Match the diagram to the expression.

Which of these answers are equivalent? How do you know?

Values help us all decide how to spend our limited time in the world, and nobody feels those limits quite like classroom teachers. Teachers frequently, and with good cause, evaluate new ideas and innovations by asking, “Does my class have time for this? What will we have to skip if we do this?”

Your decision to spend your limited class time talking about your ideas, your textbook’s ideas, or your students’ ideas is a loud expression of your values. Students hear it. We hope your students hear how much you value their mathematical creativity, explicitly in your words and implicitly in how you spend your time. You bring those values. We’ll keep working on tools to help you live them out in your classroom every day.

My Month Teaching Summer School & The Curse of Content Knowledge

I begged some middle and high school teachers in Berkeley, CA, to let me teach summer school with them this month.

Three reasons why:

  • I knew some of my professional muscles were atrophying, and I can only strengthen them in the classroom.
  • I knew our ideas at Desmos benefit enormously when we test them regularly in classrooms.
  • I knew that for me (and for everyone on my team at Desmos, FWIW) classroom teaching is psychologically satisfying in ways that are impossible to reproduce anywhere except the classroom.

So I rotated between four classes, helping high school students with mathematics that was at their grade level and below, for the most part using Desmos activities.

This was my longest continuous stretch of classroom teaching since I left classroom teaching nearly ten years ago and I learned a lot.

Two truths in particular would have been very hard for me to understand ten years ago.

One: content knowledge is such a curse.

The more math I understand and the better I understand it, the more likely I am to evaluate student ideas for how well they align with mine.

“Which one doesn’t belong?” we asked the class on an opener.

  1. 5x – 5 = 20
  2. 5x = 25
  3. 5x – 15 = 10
  4. -5x + 10 = -5

One student said that B didn’t belong because “it’s the only one with two variables.”

I knew this was formally and factually incorrect. 25 isn’t a variable. It became very tempting in that moment to say, “Oh nice – but 25 isn’t a variable. Does anybody have any other reasons why B doesn’t belong?”

Instead, the teacher and I called a time out and talked in front of the class about the sense the student had made, rather than the sense she hadn’t yet made.

“There are two of something in B. Does anybody know a name for it?”

My content knowledge encourages me to evaluate student ideas for their alignment to my level of understanding rather than appreciating the student’s level of understanding and building from there.

You can see that tendency in some of the responses to this tweet:

Those students understood the absolute difference between the denominator and the numerator (two shots missed) but not the relative difference (two shots missed when you took 38 is better than when you took 20). They needed more experience at a particular level of mathematics.

Perhaps you and I both know a formal algorithm that would help us get an answer to this question (eg. calculating common denominators; calculating a percentage) but simply explaining that algorithm would conceal some very necessary mathematical work under the attractive sheen of correctness. Explaining that formal algorithm would also tell students that “The informal sense you have made of mathematics so far isn’t even worth talking about. We need to raze it entirely and rebuild a different kind of sense from the foundation up.”

I blundered into those moments periodically in my month teaching summer school, most often when I understood my own ideas better than I understood the ideas a student was offering me and time was running short. In each instance, I could tell I was contributing to a student’s sense that her ideas weren’t worth all that much and that math can’t be figured out without the help of a grownup, if even then.

Two: content knowledge is such a blessing.

I was able to convert my mathematical content knowledge from a curse to a blessing every time I convinced myself that a student’s ideas were more interesting to me than my own and I used my content knowledge to help me understand her ideas.

(Shout out to grad school right there. If nothing else, those five years cultivated my curiosity about student ideas.)

Here is a truth about my best teaching I learned last month in summer school:

Make yourself more interested in the sense that your students are making rather than the sense they aren’t making. Celebrate and build on that sense.

Celebrate it because too many students feel stupid and small in math class (especially in summer school) and they shouldn’t. The teacher time out helped us understand the student’s thinking, but try to understand what it’s like for a student to hear the big people in the room take her ideas so seriously that they’d bring the class to a stop to discuss them.

Build on that sense because it’s more effective for learning than starting from scratch. This is why analogies are so useful in conversation. Analogies start from what someone already knows and build from there.

I don’t think I understood that truth when I left the classroom a decade ago. My content knowledge was high (though in many ways not as high as I thought) and I was less curious in understanding my students’ ideas than I was in the attractive sheen of correctness.

All of which makes the real tragedy of my month teaching summer school the fact that I’ll likely have to wait until next summer to put this experience to work again.

BTW. Max Ray-Riek’s talk 2 > 4 is a beautiful and practical encapsulation of these ideas. Watch it ASAP.

Featured Comments

Ann Arden:

After eighteen years, it’s becoming very apparent that I’m not very helpful as a teacher if I can’t/don’t understand the way a student is making sense of something.

Pam Rawson:

Whenever I find myself going down the road of trying to “fix” a student’s thinking, I pause and then ask a question like, “What do you mean by …” or “Can you say more about …”

Annie Adams:

This past year, I was only teaching our refugee population in middle school. Since moving away from California, I hadn’t really encountered the needs of language learners, and people with interruptions in their education in a while. My muscles, too, have atrophyed. Luckily, I have spent many hours learning from the #iteachmath community, using visuals to illicit information, and subtracting the clutter in problems to open up scenarios for discussion. I thought this would be great, because I just wanted then to know they can solve problems. I learned so much about what I did not know about student needs, and about how students approach problems that are unfamiliar to them, when they can’t express themselves fully, and when they are trying to build on the few things that are familiar in their toolkit. This empathy with our students is something we all do daily, but naming it and focusing in it, rather than our own agenda, is the complicated and powerful design of teaching.

Marc Garneau:

Your second truth is where I applied my most energy. I put in way more time, most of my time, into figuring out what sense they were making, and helping them to realize the same for themselves. For most, it was at least half-way through, if not three-fourths, for them to begin seeing what my goal for them was. They began caring for their learning, and caring for each others’ learning!

The Four Questions I Always Ask About New Technology in Education

A tweet where someone asks my impressions about Graspable Math.

A tool called Graspable Math found an audience on Twitter late last week, and a couple of people asked me for my opinion. I’ll share what I think about Graspable Math, but I’ll find it more helpful to write down how I think about Graspable Math, the four questions I ask about all new technology in education. [Full disclosure: I work in this field.]

1. What does it do?

That question is easier for me to answer with basic calculators and graphing calculators than with Graspable Math. Basic calculators make it easy to compute the value of numerical expressions. Graphing calculators make it easy to see the graphical representation of algebraic functions.

Graspable Math’s closest cousins are probably the Dragonbox and Algebra Touch apps. All of these apps offer students a novel way of interacting with algebraic expressions.

Drag a term to the opposite side of an equality and its sign will change.

Move a term from one side of the equation to the other.

Double click an operation like addition and it will execute that operation, if it’s legal.

Click to perform an operation like addition.

Drag a coefficient beneath the equality and it will divide the entire equation by that number.

Drag to divide by a coefficient.

Change any number in that sequence of steps and it will show you how that change affects all the other steps.

Change a number in one place in the sequence of steps and it will change it everywhere else.

You can also link equations to a graph.

Connect the equation with a graph.

2. Is that a good thing to do?

No tool is good. We can only hope to figure out when a tool is good and for whom and for what set of values.

For example, if you value safety, an arc torch is a terrible tool for a toddler but an amazing tool for a welder.

I value a student’s conviction that “Mathematics makes sense” and “I am somebody who can make sense of it.”

So I think a basic calculator is a great tool for students who have a rough sense of the answer before they enter it. (ie. I know that 125 goes into 850 six-ish times. A basic calculator is perfect for me here.)

A graphing calculator is a great tool for a student who understands that a graph is a picture of all the x- and y-values that make an algebraic statement true, a student who has graphed lots of those statements by hand already.

A basic and graphing calculator can both contribute to a student’s idea that “Mathematics doesn’t make a dang bit of sense” and “I cannot make sense of it without this tool to help me” if they’re used at the wrong time in a student’s development.

The Graspable Math creators designed their tool for novice students early in their algebraic development. Is it a good tool for those students at that time? I’m skeptical for a few reasons.

First, I suspect Graspable Math is too helpful. It won’t let novice students make computational errors, for example. Every statement you see in Graspable Math is mathematically true. It performs every operation correctly. But it’s enormously helpful for teachers to see a student’s incorrect operations and mathematically false statements. Both reveal the student’s early understanding of really big ideas about equivalence.

In one of their research papers, the Graspable Math team quotes a student as saying, “[Graspable Math] does the math for you – you don’t have to think at all!” which is a red alert that the tool is too helpful, or at least helpful in the wrong way.

Second, Graspable Math’s technological metaphors may conceal important truths about mathematics. “Drag a term to the opposite side of an equality and its sign will change” isn’t a mathematical truth, for example.

Move a term from one side of the equation to the other.

It’s a technological metaphor for the mathematical truth that you can add the same number (3 in this case) to both sides of an equal sign and the new equation will have all the same solutions as the first one. That point may seem technical but it underpins all of Algebra and it isn’t clear to me how Graspable Math supports its development.

Third, Graspable Math may persuade students that Algebra as a discipline is very concerned with moving symbols around based on a set of rules, rather than with understanding the world around them, developing the capacity for conjecturing, or some other concern. I’m speaking about personal values here, but I’m much more interested in helping students turn a question into an equation and interpret the solutions of that equation than I am in helping them solve the equation, which is Graspable Math’s territory.

These are all tentative questions, skepticisms, and hypotheses. I’m not certain about any of them, and I’m glad Graspable Math recently received an IES grant to study their tool in more depth.

3. What does it cost?

While Graspable Math is free for teachers and students, money isn’t the only way to measure cost. Free tools can cost teachers and students in other ways.

For instance, Graspable Math, like all new technology, will cost teachers and students time as they try to understand how it works.

I encourage you to try to solve a basic linear equation with Graspable Math, something like 2x – 3 = 4x + 7. Your experience may be different from mine, but I felt pretty silly at several points trying to convince the interface to do for me what I knew I could do for myself on paper. (Here’s a tweet that made me feel less alone in the world.)

Graspable Math performs algebraic operations correctly and quickly but at the cost of having to learn a library of gestures first, effectively trading a set of mathematical rules for a set of technological rules. (There is a cheat sheet.) That kind of cost is at least as important as money.

2018 Aug 8. Elizabeth Hernandez writes in the comments:

One thing I might add to the section about cost is that it is so important to find out how student data is being used. Resources that are labeled as “free” often make students and teachers pay with their data. That is unethical if the vendor doesn’t provide information about what data they collect and how it is used. Graspable Math is a no-go for me because I can’t find their terms of use or privacy policy. The only information I saw about data collected was one vague sentence when I click login.

4. What do other people think about this?

I spent nearly as much time searching Twitter for mentions of Graspable Math as I did playing with the tool itself. Lots of people I know and respect are very excited about it, which gives me lots of reasons to reconsider my initial assessment.

While I find teachers on Twitter are very easily excited about new technology, I don’t know a single one who is any less than completely protective of their investments of time and energy on behalf of their students. Graspable Math may have value I’m missing and I’m looking forward to hearing about it from you folks here and on Twitter.

BTW. Come work with me at Desmos!

If you find these questions interesting and you’d like to chase down their answers with me and my amazing colleagues at Desmos, please consider applying for our teaching faculty, software engineering, or business development jobs.

2018 Jun 11. Cathy Yenca pulls out this helpful citation from Nix the Tricks (p. 54).

An image showing the page from Nix the Tricks.

2018 Jun 11. The Graspable Math co-founders have responded to some of the questions I and other educators have raised here. Useful discussion!