How can we pass the time?

For the past three year, we have spent New Year's Eve at the Wealthy Theater in Grand Rapids listening to Michigan supergroup, Starlight Six. They usually play three sets of music, with short intermissions between sets. During one of the breaks last year, I was looking for something to do (trying to find a problem to play with) when I noticed the light string at the back of the stage.

The string of 25 lights were hung in a way that I could see two groups of 13. 

This seemed quite appropriate given that it was 2013. And it got me wondering about what other groupings I might make with this string of lights.

I imagined using two interior anchor points (adding two more lights) in order to create three groups with nine in each group.

Making four groups meant adding three more lights. With 28 lights, each of these groups would have seven lights.

Five groups created a problem. When the four anchor lights, which were being double counted, were added to the original 25, I had a number that was not divisible by five. But six groups, with 25 (original) + 5 (anchor) lights, resulted in five lights per group. It had me wondering if other strings would be as "friendly" to various groupings or if there was something special about 25.

So I thought about a string of 26 lights. Two groups added one anchor resulting in 27 total, which is not divisible by two. Three groups added two anchors resulting in 28 total, which is not divisible by three. Four groups also didn't work. But five groups added four anchors for 30 total, and 30 is divisible by five - resulting in 6 bulbs per group.

This still left a lot of questions to explore. But the band was back on stage, so I filed this found problem away for another time.

Feel free to use it as a way to pass the time this coming year. Or better yet, find your own problem.

Why does this bother me?

First, because some of my students check in on this blog, I want to be clear that the thoughts presented here represent my own issues and are not meant to shame my students or dismiss their perspective. Second, for my 200th post, I thought I would share one of my issues with you. So, consider yourself warned.

Near the end of the semester, I often ask students to reflect on the course and write a letter advising future students on what to expect. (I wrote about it here.) This time, I asked them to narrow it down to three important points. One student wrote:

"This is an education class not a math class."

For some reason, this point really bothered me. I do not know how my student intended it (and it doesn't matter - this is about me), but I took it negatively. One of the ways I can tell I have had a button pushed is when I start getting defensive. So when I began to line up all my arguments, like soldiers on a battlefield, in order to attack this point, I knew that it was time to take a step back and consider why I was uncomfortable with this point.

It did not come to me immediately, but then yesterday I read these Tweets, 

and the linked articles (here and here, respectively). Both use reports from the National Council on Teacher Quality (NCTQ) to reinforce the narrative that U.S. efforts to train teachers are "abysmal." Yes, I know that the NCTQ teacher prep rankings are non-sense (none other than Linda Darling-Hammond makes the case) but I seem to have bought into the non-sense. Therefore, when the student wrote that my course was an "education class," I began to worry that I was part of the problem.

Most teachers can probably understand this line of thinking. Regardless of what we know about the realities of education, the message seems to be that teachers in the U.S. are to blame for all of our educational ills, whether real or imagined. Therefore, I want to do all I can to "make things better" - even if it is unwarranted.

A friend of mine once told me that if my buttons were being pushed, then I needed to move my buttons. In this case, I needed to shift my attention to aspects of the course that were preparing these future teachers for the realities of the profession; this was, after all the overarching goal of the course.

Please pardon me if this next piece seems self-serving, but I need to acknowledge some of the points other students shared:

• Keep up with the readings! They are very important towards your teaching future.
• As future teachers you understand that the main goal is helping the students learn. That is your professor's goal as well!
• You will work extensively in collaboration with your peers, as you will as future classroom teachers.
• Do not procrastinate! Otherwise you will have an overwhelming amount of work to do.
• Put a ton of effort into this class even though a lot of the things that you do won't be graded because you will get out what you put into the class.
• Take time to reflect on different aspects of class because there is a lot of useful information, and you want to be sure you have learned most, if not all, of it.

Reviewing these, there does seem to be a lack of attention to the mathematical content; this is something to consider for next time. However, as education classes go, I am satisfied that this course provided students with some of the tools necessary for success in teaching: reading research, focusing on learning, being prepared, giving less attention to grades and more on learning, and recognizing the importance of reflection. 

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.

How does it fit?

At the beginning of the semester, I wrote a post about using the history of the LEGO company as a cautionary tale for innovation in education reform. The idea came from a Diane Rhem interview with the author of Brick by Brick. But there was another idea presented in book that also caught my attention - clutch power.

When a child snaps two bricks together, they stick with a satisfying click. And they stay stuck until the child uncouples them with a gratifying tug. And therein lies the LEGO brick's magic. Because bricks resists coming apart, kids could build from the bottom up, making their creations as simple or complicated as they wanted. … it is clutch power that makes LEGO such an endlessly expandable toy, one that lets kids build whatever they imagine. (page 20)

It was the last sentence that got me thinking. How might I design learning experiences that use clutch power - allowing learners to "build whatever they imagine?" In particular, I had one of our math courses for preservice teachers in mind. It explores a variety of mathematical topics that students often see as disconnected: algebra, geometry, measurement, and data.  Was there a way to treat concepts in that course as building blocks that learners could use across the domains to build new understandings instead of simply following the instructions I provide?

I am still working on that question, but I did start experimenting with simple problems that allow learners to add their own ideas while demonstrating their content competency. The following is one problem (modified) from the final I just gave. I would be interested in your feedback.

Plot the points (2, 2) and (2, 4) on the coordinate plane. Pick two other points that could combine with these first two to be used as the vertices of a trapezoid. Use this context to develop items that would align with the following targets (be sure to justify your alignment using the indicators):

• Graph points on the coordinate plane to solve real-world and mathematical problems (5GA1, 5GA2)
• Classify two-dimensional figures into categories based on their properties (5GB3, 5GB4)
• Solve real-world and mathematical problems involving area, surface area, and volume (6GA1-A4)
• Draw, construct, and describe geometrical figures and describe the relationships between them (7GA1, 7GA2, 7GA3)
• Solve real-life and mathematical problems involving angle measure, area, surface area, & volume (7GB4, 7GB5, 7GB6)

Because this was a new assessment for most of my students, this time around I offered some example items to pick from:

1. Define the term "trapezoid" and then use that definition to identify what other names apply to your shape.
2. Find the area and perimeter of your shape.
3. Extend each side (creating 16 angles) and find all the angle measures.
4. Write your own question for this context.

I still expected them to align the item with the targets, justify their alignment, and come up with a correct response.