How does a mathematician see the world?

And how does a math teacher help learners to see the world through a mathematician's eyes?

This was a major challenge during the past semester in my Intermediate Algebra sections. Many of the students came to class expecting me to show them a procedure that they would practice until test time - when they would reproduce the procedure and promptly forget it. Nearly all of the students had seen the Intermediate Algebra content in high school (linear, quadratic, and exponential functions) but it hadn't stuck.

This was the problem. They had seen the content. Now it was time for them to use it to see the world. So I shared pictures and videos and asked them to look at them through a mathematical lens.

At one point, a student said, "Just once, I wish I could see the world through your eyes." Exactly! Unfortunately, they were often so afraid of making a mistake, of breaking the mathematical glasses, that they were not willing to even put them on. They did not know how to be playful with the math we were exploring.

So I introduced them to Yes, And ...; this is a problem finding activity that I developed using a well-know improv game where participants accept and build on the ideas of their partners. Here are the instructions for my version:

• Provide a mathematical context (often pictures or videos) but without any identified problem;
• Pair up the students and find a fun way to identify Student A;
• Student A picks one of the contexts and finds a problem to solve;
• For one minute (this is usually enough time to get started without completely solving the problem), Student A talks through his or her thinking while Student B writes as much as possible down on a piece of paper;
• After one minute, the students switch roles. I say, "Yes, and ...," to signal the switch and to emphasize that Student B ought to build on the work that was already done.
• Student B thinks out loud for one minute while Student A records the thinking on the same paper.
• After one minute, I say, "Yes, and ...," and the roles reverse again.

Yes, And ... can go on as long as the teacher wants. I found six minutes (three rounds) to be about right the first time we played the game. One student submitted this to demonstrate her engagement with the task.

I cannot claim for certain that Yes, And ... changed my students' view of mathematics or helped them to see the world through a mathematical lens. What I know is that for six minutes they played with math. It's a start.

Why do you want to learn to play the mandolin?

It was time to learn something new. That was my mindset when I signed up for mandolin lessons at Earful of Fiddle Camp. Even though I have had some musical training (piano and flute), I knew learning to play a stringed instrument was going to be a challenge. So I spent last week swatting mosquitoes and strumming strings, all so I might someday call myself a mandolin player.

Laurel and Michael
of Red Tail Ring

Oh, and as the sign says, we also got to listen to some old time music. On Tuesday, Red Tail Ring taught a workshop on harmony in the afternoon and put on a concert that night. Afterward, I asked Michael (he played the mandolin on a few songs) if he ever gave lessons. He said, "Yes." And then, perhaps noticing my age, he asked, "Why do you want to learn to play the mandolin? What are your goals?"


I responded, "Don't worry, I'm not looking to replace you." He laughed, but I could see that he was serious about knowing my musical aspirations. Perhaps his instructional approach would depend upon what I wanted to accomplish.

"I'm struggling with my fingering."

He asked, "You mean your fingers are tender?"

"They are," I said looking at the blisters on my fingertips, "but I mean creating a clean sound when I play."

He explained that the two were related. Once calluses formed, I'd be able to press on the strings with more commitment. However, we still hadn't addressed the original questions. He tried again, "Who do you want to play with or for?"

Now I got it. "Mostly with my family. My wife and son-in-law are here, too. You know, like around the campfire. Oh, and I'd like to be able to play for my grandson; songs like Puff the Magic Dragon and Itsy-Bitsy Spider." This made sense to him and we made arrangements to connect later in the summer to arrange some lessons.

At the end of the week, campers perform for the rest of the camp, friends, and family in an event called, Earful of Idol. When the group I played with finished our song, I was reminded of my conversation with Michael. I had been miserable most of the day trying to memorize the song. During the performance, I only played about half of the notes, and only half of those at the right time. My "bandmates" picked up the slack and the audience was generous with their applause, but I was ready to quit mandolin. There was no point continuing if I wasn't enjoying it.

That's when Michael's question hit me, "Why do you want to learn to play mandolin?" My purpose for picking up the mandolin was to have fun with my family. It wasn't to perform in front of strangers. Even the best audience couldn't hold a candle to this guy.

Vance wants more cowbell!

Maybe I'll be grateful I played at Earful of Idol someday - maybe not. (I do not subscribe to the "someday, you'll thank me" school of teaching.) At the very least, I gained empathy for students who "quit" math because the effort does not seem worth the experience. Whether it's music or math, learners need to find their own purpose in order to stay engaged in learning.

Finals! Huh! Yeah! What are they good for?

We just finished finals at Grand Valley. Coincidentally, this week of exams overlapped with the release of the second Hobbit movie; this might explain why these lines (modified from this poem) were going through my head Thursday as I walked around campus:

I am not trying to suggest that like the One Ring, final exams are evil and must be destroyed. However, I do wonder about the wisdom and unintended consequences from placing so much emphasis on a single assessment event.  Rarely do final exam scores make that much of a difference on final grades. Yet, there is so much stress associated with final exams, for both the students and the teachers, that it got me to wondering if these tests are worth it.

During the MTH 323 final event (we did not have a written exam), I asked the teachers to consider whether or not our school ought to have final exams and why. This is the course for preservice teachers that I designed to reflect what it looks like to be a math teacher in a K-8 school. So we created a "school" and went about developing a curriculum. Therefore, it seemed reasonable to discuss the final exam policy we would have at our school.

Grade-level groups talked about the question and then shared their thoughts and rationale. I added my two cents, but mostly these points represent the teachers' perspective.

• We need to have some sort of culminating experience as a way to identify what a student has mastered and possible weaknesses in the curriculum.
• The experience needs to be different than the traditional exams that represent a cram and purge approach that does not result in lasting learning.
• Like it or not, exams are a part of our culture and we would be doing our students a disservice if we fail to provide some experience with these types of assessments.
• Perhaps we could give the exams a couple of weeks before the end of the semester so those who need to can make revisions and the rest can extend their learning.
• Whatever we have them do needs to be meaningful.

It was the last point that resonated with me. How do we ensure that the culminating experience is meaningful - that it fosters the upper levels of the Engagement Continuum and not just compliance?

Developed by MTH 323 Teachers to monitor their engagement
Based on Engagement Taxonomy of Morgan and Saxton

At breakfast this morning, I asked a colleague what he thought final exams were good for. He said, "To see if students have synthesized the information addressed in the course." After a pause, he continued, "Of course, those are the questions they often struggle on most, which is discouraging."

"Why do we give all the other questions, then? The ones that focus on recall. Why not focus on the synthesis questions?" I asked.

"Probably because it's what we've always done," he responded. Neither of us were satisfied with this answer, but finals week was over (it was, after all, final) so there was nothing we could do about it now except think about what we might do different the next time around.

What if it's boring?

Engagement Rubric

It is the first week of the semester at GVSU and we have been focusing on engagement in my math courses for preservice elementary teachers. There are two big take-aways I hope my learners get from this week: (1) An awareness of what engagement really means (this is an old version of the activity that needs updating); and (2) An acceptance that they are ultimately responsible for their own engagement in the learning process. Given the conventional wisdom that engagement looks like compliance and that it is the teacher's responsibility to engage students, addressing these points takes effort - hence a week's worth of lessons. (Stephen Carroll at Santa Clara University calls this Flight School in his classes, and his lasts for two weeks.)

So we spend time identifying what engagement looks like for us - because it is personal. We also discuss how outside influences (Conditions of Learning) can support engagement, but that true engagement resides within the individual. And then I give them time to practice engaging with information that might not be all that interesting to them. (Please do not misunderstand. I am not suggesting Daniel Pink's TED Talk is boring. It's just disconnected from what many of my students expect in a math class for elementary teachers.)

Here's the home workshop:

Teaching Workshop (The Puzzle of Motivation)

How does this apply to education

TLW apply the engagement strategy of asking questions while watching a lecture.

Needs:            40 minutes; access to Internet; & journal

Schema Activation:  Journal Jot       [no more than five minutes]

How are engagement and motivation related?
What role does motivation play in education?

Focus: Asking Questions       [no more than five minutes]

The first day of class, we found out how engaging it can be to ask the questions that drive the delivery of information. However, we do not always have that option. For example, when we are watching a video of a lecture there is no way to interact directly with the speaker.

Yet, questions can still be a powerful tool. Perhaps it will help if we try to engage in the lecture by considering what questions the speaker is answering. Even better, we can watch the lecture with a question in mind and try to find answers in what the speaker shares.

So, as you watch the following TED Talk:

• Make a list of the questions Daniel Pink might be answering; and
• Keep this question in mind: How does this apply to education?

Activity: Engaging with a Lecture            [no more than twenty-five minutes]

Reflection:  Monitoring Our Engagement   [Remaining part of the 40 minutes]

Using the Extended Engagement Continuum (modified from Morgan and Saxton, 2006) shown below, evaluate where you were as you watched the video, why you were at this level, and what you can do next time to increase or maintain (if you are at #8) that level of engagement.

1. Disengaged: distracted by other things while doing the task;
2. Compliant: completed the task but didn’t get much else out of it;
3. Interest: being curious about what is presented;
4. Engaging: wanting to be, and being involved in the task;
5. Committing: developing a sense of responsibility towards the task;
6. Internalizing: merging objective concepts (the task or what is to be learned) with subjective experience (what is already owned) resulting in understanding and therefore ownership, of new ideas;
7. Interpreting: wanting and needing to communicate that understanding to others;
8. Evaluating: wanting and being willing to put that understanding to the test.

How did focusing on questions help (or not help) you engage with the lecture?

+++++

A final thought for the teachers reading this post: Your students may not always have teachers that are as worried as you are about student-engagement. How are you preparing them to be successful in those other classes - to recognize they have the power to be engaged regardless of the situation. I offer one approach. I would be interested in your ideas.