What's the problem we're trying to solving in math education?

The way we ask a question tends to frame our solution. This was one of the most long-lasting lessons that I learned at the d.school's week-long Teaching and Learning Studio. The activity we did early in the week to introduce this idea is one that I still do with teachers. I'll do my best to recreate that experience here.

First, I need you to break into one of two groups. Each group will do a different task. Let's assign your task based on your birthdate. If the day of your birth is odd, then do the first task. If it is even, then do the second one. (e.g. I was born on the 29th, so I'd be in group one.)

To begin, click on the appropriate link provided below. It will open up a copy of a Jamboard with the task's question. You can do your work on the Jamboard (it is yours to keep) or on a separate piece of paper.

Task One (odd birthdates)

Task Two (even birthdates)

Don't read any farther until you've completed your task or you'll ruin the surprise.

Once you've completed your task, click on the other task question and consider how that question might result in different solutions and why.

The first group often comes up with a variety of creative vases. Some have multiple openings at the top. Some hold the flowers sideways. However, because of the way the question is framed, they are all obviously vases.

Very few vases show up in the second group. Instead, flowers grow on the wall or hang from the ceiling. One participant had the flowers floating in a clear container with a fan at the bottom. Sure, it might not actually be feasible, but it is a much more creative way to start seeking solutions.

IBM does a version of this activity in their

Enterprise Design Thinking Course

I ask educators to do these tasks to highlight that most of our efforts to innovate in math classrooms amount to the first task. No matter what we do, we just end up with another vase. This is one reason why Kathy and I wrote Designing Math Adventures - because we want learners to experience something more than another math lesson.

So, what's the question you'll ask to frame your next school year? I'd like to collect them in the comments. Thanks for contributing to our creative problem-solving.

What does the research really say? (Math Edition)

The American Educational Research Association (AERA) recently published a study, Which Instructional Practices Most Help First-Grade Students With and Without Mathematics Difficulties?, that caught the attention of several media sources. Many of these sources misrepresented the research with headlines like:

• Here's How Not to Teach First-Grade Math
• Drill, Baby, Drill? Study Finds Worksheets, Practice Improve Math Skills More than Fun Stuff

Other outlets were truer to the research:

• Explicit Instructions Works Best for Struggling Math Students
• Music, Manipulative Are Fun, But Basics Best for Struggling Math Students

However, these pieces still ignore some of the nuances of the research. To his credit, one of the author adds some perspective in this short video.

Still, achievement in mathematics needs to be defined. From the paper:

We used the ECLS- K’s Mathematics Test to estimate children’s mathematics achievement. ... The test was based on the National Assessment of Educational Progress (NAEP)’s specifications. (p. 6)

If your idea of mathematical achievement is simply scoring well on a standardized-test, then direct instruction probably is the most efficient way to prepare students to achieve. 

For example, when putting together a piece of furniture from IKEA, it makes sense to follow the directions provided explicitly. Why would I explore assembling our coffee table when the directions provide step-by-step instructions on how to achieve my desired outcome? How else can I be sure that my table will look exactly like the one I ordered?

Item Number: 801.762.84

And, therefore, like every other IKEA coffee table, 801.762.84.

But what if I want something unique? In this case, the directions might not apply (or even exist). My stepson made this coffee table for us out of a part of a tree that was from my childhood home. It is one of my most prized possessions.

Now, he is a talented woodworker who can (and has) put together a lot of furniture using directions. When he wants or needs to be creative, however, he can be. For me, that is true achievement.

We need to apply the same approach to teaching mathematics. Focusing solely on achievement based on content assessed through a standardized-test misses an important element of mathematical ability - thinking beyond what can be explicitly taught. Dr. Yong Zhao does a good job of addressing this disconnect in this post. So I leave you with this:

Source

What does the research say? It depends on what one is looking to achieve.

Update (7/4/14): And now an example related to teaching language arts has surfaced.

Do you ever make mistakes?

The above was on my Facebook news feed this past week. As a young adult, I remember watching Bob Ross on our local PBS station, WNMU. If you are not familiar with his work, watch this short sample.

Anyone considering making online teaching videos could learn a lot from Bob Ross. He treats the viewer as an apprentice. We get to watch him engage in an authentic act. And as he engages in this act, he describes what he is doing and why he is doing it. Let me amend what I wrote at the beginning of this paragraph: Any teacher could learn a lot from Bob Ross.

PS: When John Golden (@mathhombre) found out I was writing this post, he asked me to include the following. I do it gladly. The idea of "happy accidents" fits with my efforts at graceful imperfection.

What did you think of Sir Ken?

A former student asked me this earlier today. You see, Sir Ken Robinson visited GVSU this week (news story) and shared his message about the need for creativity to be infused in education - not just something added on. I told the student something like, "He was certainly inspiring. I left feeling passionate about being an educator and affirmed in my efforts to improve teaching."

(Watch this and see if you don't agree)

"But," I added, "I'm still thinking about how to implement the ideas he shared. What does it look like?"

The student responded, "You always ask that."

My colleague, John Golden, was also at the presentation and wrote this recap. He had the same question (no surprise for anyone who knows us). John focused on education in general, however, and I want to focus on a specific example Sir Ken gave about mathematics. There's currently no transcript or video of the complete talk so some of this is based on memory and notes which admittedly are a product of my own filter. So be forewarned.

Sir Ken talked about asking a math professor in London how the math department evaluated doctoral theses. The math professor explained that the math, obviously needed to be correct, but that this was not usually a problem. So it boiled down to two things. First, the work was something original that contributed to the current knowledge-base. And second, it was aesthetically pleasing. The math professor explained that math was a way of representing the beauty and truth of the world and that a dissertation needed to be able to demonstrate that connection. (If anyone remembers this differently, I'd be happy to make the necessary adjustments. Please leave your memories in the comments.)

As I thought about these requirements, it became clear that K-12 schools often focus on the initial point, correct mathematics, while ignoring originality and aesthetics. So what would this look like if we also incorporate Sir Ken's admonition that the later elements, associated with creativity, are not added on after the fact? I think standards-based assessments and problem finding activities are two sources to consider as ways to address these expectations, but I am open to other. What do you think?

What's your story?

This past Friday we attended the release party of the Harvest Queen cd at Kauffman Auditorium in Marquette, Michigan. (The story of how this project came to be is told here [Kickstarter Video] and here [WNMU interview].) The concert, which included traditional and new waltzes, was amazing, but it was only a portion of the overall event. Crossing Paths was intended to be an opportunity to explore various connections between local food and music.

In the afternoon, Kailin Yong conducted a violin master class for players of all ages and abilities. While my wife attended this hour-long class, I went for some local coffee. When I returned, Kailin was answering questions from those assembled on the stage. Someone asked him to explain when he started improvising in music.

Kailin responded by playing a beautiful, classical melody and then said (to the best of my memory), "I was very good at telling this story - an old Italian's story. But I wanted to tell my own story. I had been playing for years before I felt free to improvise. I do not want my students to have to wait as long as I did, which is why I had you working on improvisation today."

This reminded me of what Jonah Lehrer shared in Imagine about Yo-Yo Ma

"I was nineteen and I had worked my butt off," Ma told David Blume of The New Yorker in 1989. " I knew the music inside and out. While sitting there at the concert, playing all the notes correctly, I started to wonder, 'Why am I here? What's at stake? Nothing. Not only is the audience bored but I myself am bored' Perfection is not very communicative." (p. 86)

In both cases, expert musicians found being perfect was not enough. They longed to express themselves. They wanted to tell their own story or, at the very least, find a way to add a part of themselves into the retelling of the story of some other songwriter.

It seems to me we spend a lot of time in math class trying to get our students to retell someone else's story perfectly. The standardization of the curriculum leaves little room for learners to improvise and add their uniqueness to the mathematics. It is no wonder that most students see mathematics as a static set of rules. Whether we "tell" them the rules or have them "discover" the rules it amounts to the same thing - they are playing someone else's tune.

I am glad that I returned before violin master class was over (It turns out my compulsive timeliness can come in handy.) Thanks to Kailin's answer, I want to plan more opportunities that allow learners to improvise. I, too, do not want them to have to wait as long as I did to pick up this skill.

How do you foster creativity?

The following was on a bulletin board in the art class at the middle school where I taught.

Because I wanted my math class to be more like that art class - learners exploring the boundaries of their knowledge and skills - I posted a copy in my classroom.

Twenty years later, a copy is now thumbtacked to my office door. It reminds me that fostering creative problem solvers is one of my primary goals as a teacher of teachers. Some of the ideas may not be reasonable, but creativity sometimes requires a certain suspension of rationality.