The Big Bad Math

Math HomeworkWhat do you like about working here?” I asked during my interview to work at KCP Tech. I was rather struck by Vishakha’s response that she liked being able to help people learn math. She thought that it was completely ridiculous that it was acceptable for a grown adult to say “I’m bad at math” in public when nobody would ever say “I’m bad at reading.

It had not really occurred to me just how absurd that sounded. For me math was simply a tool that everyone had to learn. While in school I’d be given a formula, told to memorize and practice it repeatedly for a test, with barely a mention why it worked, or what I was learning it for. For me it wasn’t hard or scary, just boring and something I had to do.

My classmates would tell me that math would be clear later. “When you take physics and calculus together in your senior year, then all the math you have been learning will start to make sense.” But I never did take physics or calculus.

So to me math continued to look like a high-level science. A formula written on a whiteboard surrounded by steaming beakers and Tesla coils. But it shouldn’t be. Being taught math and never knowing why it works can lead to lifelong math anxiety, according to new research.

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Parents, Children, and Functions in Sketchpad

Functions are hard for students.

Students seem to master various families of functions – linear, polynomial, exponential, trigonometric, and so forth. They can graph them, evaluate them, transform them, and answer a variety of questions about them. But ask even our better students a question that’s out of the ordinary and we’re likely to be taken aback by the result.

Does there exist a function all of whose values are equal to each other? (If so, give an example; if not, explain why.)

When Marilyn Carlson asked this question of college students, only 7% of a group that completed college algebra with grades of high A (95 and above) were able to give correct examples, and 25% of a group that received A’s in second-semester college calculus gave the incorrect example of y = x. Other questions in this study gave similarly disturbing results. Shockingly, many of our best math students lack fundamental understandings of functions. [CBMS Issues in Mathematics Education, Volume 7, 1998, published by the American Mathematical Society]

There are undoubtedly many causes for students’ inability to reason about and make sense of functions. Students may lose sight of the big picture (of how one quantity depends on another) as they concentrate on mechanical exercises that emphasize fine details. They may see functions only as mathematical equations, without also recognizing functional relationships that take other forms.

Though there are no simple answers, the more students notice dependencies in the world around them and think of these dependencies as functions, the more inclined they’ll be to understand a mathematical function as a description of how one mathematical object depends on another. A bulletin board slogan would be appropriate:

Which brings me to Sketchpad. Though I’ve worked on quite a few Sketchpad activities related to algebraic functions, there’s a deeper and largely neglected connection between Sketchpad and function concepts: The essence of a construction lies in functions that connect objects to each other.

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Educational Technology: What is the Cost of Free?

I am so tired of the question “why should we buy your product when we can get the same thing for free?”

No, you can’t.

Image Courtesy of images.google.com

I get it, I really do. I was a teacher for a long time, wanting to improve my classroom with limited funds, and I was constantly on the hunt for “free”— free lessons, free stuff at conferences (books, rulers, pencils, etc.). I was also an administrator, having to make decisions about what to purchase with an ever-decreasing budget, so free options were always a consideration, although not often the choice. Because sometimes you have to weigh choices against more than just a price tag.

I could quote research here why one product is a better choice over another, or respond to other people’s thoughts on free technology (Dan Moren), but I am going to take a much more personal approach. Why would I choose a product that costs money over a similar product that is free (or just significantly cheaper)? I write as an educator—my reason for purchasing or obtaining a new ed tech resource is to improve my instructional strategies or to help my students and teachers understand and learn better. Sometimes free is the right choice, sometimes paid is the right choice. But your choice should be based on more important factors than money—which product/resource will truly help you reach your end goal of improved learning and understanding?

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Music and Math in the News

I have a memory from elementary school of a lesson on music and math that was taught by guest instructors. The instructors had us clap out poly-rhythms, with half the class clapping in two’s while the other half clapped in three’s. We progressed into more challenging sequences, coming to the realization that although the two beats were offset most of the time, periodically we all clapped together. A great lesson on common multiples—and common denominators—and the fact that I still remember that lesson oh-so-many years later is telling.

So I had mixed emotions when I read that a yearlong effort to integrate music and math into some elementary classrooms in San Francisco was successful. Happy, because it sounds like the students loved the lessons and got a richer understanding of fractions through the curriculum. But also exasperated, because in today’s education landscape, “successful” is used synonymously with  “test scores went up.” And that’s all that matters in the current climate of declining budgets and “data-based” decision making. As the article notes,

The results offered insight into how to teach math more effectively, but more than that, gave principals, parents and teachers an academic selling point for keeping music in schools despite budget cuts.

So music isn’t valuable for its own sake, but only as a vehicle for improving math test scores? Really?

Because it’s Friday, and I don’t want to start the weekend with a bummer blog post, I’ll take the positive view. More music in schools? Great. Kids having fun with fractions? Fantastic! And in case anyone needs more ammunition for keeping music in schools, here’s Arjan Khalsa making the case that “math and music are one.” Amazingly, in his five-minute Ignite presentation, Arjan shows fractions in music, contrasts Eastern and Western music, touches briefly on Sikh wedding ceremonies, and illustrates the anatomy behind singing.

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Lifelong Learning

A number of things I’ve come across lately have made me think that there may be a revolution afoot in adult pursuit of learning in this country—adults truly acting as lifelong learners. And the internet makes this possible in ways that have never been available before.

I come from a family that values education, and never stops learning. My grandma was the earliest adopter of computers I know, using a word processing program on her Commodore 64 when I was in elementary school, creating the next generation of the San Diego Floral Association’s newsletters. Even as she became quite elderly, she never stopped learning, teaching, growing, creating, and contributing. In her last years, she wrote a book, which my mom and I published after her passing. You’d be surprised how interesting the history of gardening is in San Diego! It’s a tour through events like the growth of a city around the Panama-California Exposition of 1915–1917, the creation of Balboa Park, and the increase in women becoming respected as professionals. But as I refresh my memory about those events as I link to them on wikipedia, I recall that she never connected to the internet herself; she researched the old-fashioned way.

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When Factors Put on Their Dancing Shoes

What’s the narrative? That question, so fundamental to any novel, may not sound as relevant when applied to mathematics. Take, for example, the topic of factors: 1, 2, 4, and 8 are factors of eight; 3, 5, and 7 are not. Where is the inherent drama in these relationships? In most elementary mathematics curricula, there is nothing remotely exciting or theatrical about factoring.

Not long ago, Keith Devlin, the “math guy” for NPR, proclaimed that You Can Count on Monsters by Richard Schwartz to be “one of the most amazing math books for kids” he had ever seen. At its core, Schwartz’s book is about factors and primes. Oh, and monsters, too.

Schwartz draws charming geometrical monsters for each number between 1 and 100. Each prime monster relates in some way to its numerical value. Three, for example, is represented by a triangle monster whose face, eyes, and mouth are all triangles. Five is a smiling five-pointed star. The composite monsters are more complex. They incorporate elements of those prime monsters that are their factors. So 15, whose prime factors are 3 and 5, is composed of both the triangle and star monster. The book’s progression of 100 monsters offers an engaging narrative arc by embedding the concepts of factoring, primes, and composites into the challenge of analyzing each illustration of a monster to see how it was assembled.

A "dancing" model of factors

The factors of 12 dance around it in a circle.

When I saw Schwartz’s book, I began to wonder whether there was a different narrative about factors to be told, one that would involve animation and Sketchpad. I soon settled on a plotline focused on dance. What if a number’s dance partners were its factors? A number like 24 would be incredibly popular, with dancing mates aplenty. But pity those poor primes! Aside from themselves, only 1 would consent to be their partner.

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What do you wonder? Real-world math problems are everywhere

We hear this everywhere – students should be doing “real-world” math and they should be applying what they learn in math to “real-world situations.” Textbooks and math resources advertise their “real-world problem solving” experiences and “real-life math applications.” But, as those of us who are math teachers can attest, often times the real-world examples are contrived or forced and—let’s face it—not very interesting to students. I mean, what student really cares what time two trains leaving from New York and Detroit are going to cross paths?

The Best and Cool Bridges

What do you wonder? Image courtesy of images.google.com

The idea of real-world problem solving and applications in mathematics is important. As the Common Core specifically states in the Standards of Mathematical Practice, “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” But I wonder if we often try too hard to create real-world problems when, if all we did were look around and ask “what do you wonder?” and “what do you notice?”, we would find that math problems are everywhere.

I admit—I stole the “what do you wonder?” and “what do you notice?” from Annie Fetter, an inspiring math educator at The Math Forum. Annie (@MFannie) did this amazing Ignite! presentation at NCTM 2011 on just asking students those simple questions. Her point is that students will come up with creative questions and applications about mathematics if we just let them.

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“Data Games”—Getting the Math Right in Math Games

As a parent and math educator, I’m always on the lookout for high-quality, fun apps that help my sons and other young people develop their understanding of math. There are a large number of math games for mobile devices and PCs being churned out, and many articles being written in popular ed tech blogs and parenting magazines that pick the “best” of these. My boys and I try many of these suggestions, but most of these recommended games leave me very disappointed.

My youngest son on iPad

I’m not going to call out any by name, but I estimate that 90% of them are little more than flash cards with glitzy, animated bells and whistles. This genre of games have kids just practicing their math facts in a rather dull set of drills that the child must get through to get to the “fun stuff.” Kids are rewarded with activities they might enjoy like blasting off rockets and slinging creatures around, but these activities require little math. These highly-rated games offer little more than new contexts for flash cards. As my colleague Karen pointed out in a recent post, “With computers we should be doing things DIFFERENTLY, not trying to do the same old thing with a different tool.”

Other games do better and aim to require that players use math in solving the puzzle or accomplishing the task. Some of these have seemed promising to me at first, but usually as my sons and I play them, I realize that players can win the game without actually using math, relying instead on other clues or strategies they pick up from the game environment instead.

We’ve worked hard to avoid these pitfalls of game design in Data Games, a set of free online games we’re developing at Key. To get a sense of a Data Game, you might try playing Markov, one my boys have liked.

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March Mathiness

Check out the graph below, which shows the heights of 29 people. What do you notice about the graph? What do you wonder? (Thanks, Annie Fetter!)Heights

Now, what if I told you that this group of 29 people were all women—would you be surprised? Perhaps not, if you understood the title of this blog post. I know that most of the country focuses on the men’s NCAA basketball tournament, but I want to shine a little light on the women, who play with grit and grace and heart.
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Victory Through Data Analysis!

A Data Games graph:

Go on, touch it.

This is part of our new project, Data Games. If you’ve used Fathom or TinkerPlots, that graph probably looks familiar. Well, except for one big thing: it’s sitting there live and interactive in your web browser.

But first, let’s play a game. Continue reading

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