Month: August 2018

Total 3 Posts

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.