Why take the road less traveled?

I love being in the woods. It is nearly always an adventure. Even if it's a path I have been on before, the possibility of a surprise around the next bend is exciting to me. I never know for sure what I will see or hear. That's not to say that I am always comfortable with this state of uncertainty. A few weeks ago, I heard two coyotes howling up ahead as I was walking on a logging road in the Upper Peninsula of Michigan. Or there was the time I encountered an overwhelming smell that I can only describe as wet dog as I walked alone along the nature trail at Seney National Wildlife Refuge. In cases like this, I try to find a big stick to "support" me as I walk.

What keeps me coming back to these roads less travel are the sights and sounds that come from exploring something new. For example, while walking down one of the access roads at Seney, I came across this beautiful Trumpeter Swan that, true to its name, announced my presence to others in the area.

I have found that this metaphor of taking the roads less traveled also applies to my teaching practice. When I first began asking my middle school students how they would solve a problem before showing them the right way, I was quite uncomfortable with the idea that they might share something new - something for which I was unprepared. Fortunately, I had mentors that assured me it would be alright and encouraged me to explore.

The first time I recall letting my eighth-graders lead the way was when we began a section on dividing fractions. I put three-fourths divided by one-half on the board and asked for a volunteer to share how they thought we ought to calculate the result. Here's essentially what happened (requires ShowMe log in).

Because the student had not used "invert-and-multiply" to calculate the quotient, I considered the effort incorrect and tried to determine what went wrong. First, the student was finding common denominators, so there was a chance the student was confusing this procedure with the one for adding and subtracting fractions. Then the student divided across, which resembles the procedure for multiplying fractions. Clearly, the student was mixing up the various rules for fraction computation and I would need to be explicit about the differences. But something was bothering me. The answer the student came up with using this mish-mash of methods was correct. I attributed it to the numbers I had selected and decided to try the method with another pair. It worked again. I was at a loss.

I don't remember what I did with the students, but I do remember exploring this approach more and finding out that it always works and why. This experience hooked me. Now I am not suggesting that the student was doing anything other than trying to apply various rules and happened upon a solution.  But I might not have ever heard or seen this approach if I had not been open to following this unfamiliar path.

If you are still uncomfortable with exploring the "wilderness" with your students, then take solace in knowing that you do not need to go too far off the regular roads to experience something wild. Simply being alert to students' thinking while covering familiar ground can allow for new ideas to be uncovered. Just like when I was able to film this Peregrine Falcon on the bike path near our house, it only takes being aware of something new and prepared to see where it goes.

When will we ever use this?

A couple of weeks ago, this comic showed up on xkcd.

Forgot Algebra

As a math teacher, I can relate to the sentiment of this comic. People regularly boast to me about how bad they are at math (especially algebra), and still, they are successful in life. Much like the young woman in the comic, these people seem to believe that they were lied to about the utility of math, and they are resentful about it. Maybe lied is too strong a word, but perhaps we do misrepresent the purposes behind the math taught in schools.

Because I was teaching some algebra to middle school students twenty years ago, I remember how I responded to the question, "When will I ever use this?" Therefore, I have a theory about what might have been going on in Miss Lenhart's Algebra classes in the late 80s/early 90s. I imagine it might have looked like one of these four scenarios (depending on what she tried before).

Has much changed? I mean, besides the fact that we now add cell phone contract problems into the mix. For the most part, students remain unimpressed as we tie ourselves in knots to demonstrate how they will need the math we teach in everyday life. That is why I suggest that when asked by students, "When will we ever use this?" that teachers respond truthfully.

I don't know when or if you will ever need this particular concept. It depends on what you do with your life and what technological advances are made in the future. But you know what you will need to be able to do, regardless? You will need to problem solve. You will need to think critically (reason and prove). You will need to be able to communicate quantitative thinking to others. You will need to use representations to support your thinking and share your thinking. And you will need to make connections in order to consolidate your understanding. Mathematics is a discipline that provides opportunities to practice and strengthen all of these skills. So, as we solve for x, I want you to monitor your thinking because that's what's really important.

That is how I learned to respond to my middle school math students. The NCTM Process Standards gave purpose to my math lessons, and the students bought into it. It worked for me because I finally believed in what I was teaching.

How will you organize your classroom of the future?

This past Wednesday was the last day for my Teaching and Learning Middle Grade Mathematics course. It was also the end of our unit on teaching mathematics, which meant developing an anchor chart to represent our current understanding (much like we did for doing math and learning math). Instead of having them summarize the content, I asked them to consider what their future classroom might look like given what we knew about doing, learning, and teaching mathematics.

The following video sums up the activity. This group of preservice teachers considered the role communication (learners placed in groups) and technology would play in their classroom but then they stopped and thought about their future learners. The result was surprising to me (the reason I like open-ended activities like this).

Cynically, I might think that they were trying to avoid the work, but they had already done the work. The final vision of their future classroom represented an authentic concern about meeting learners where they are at. The other groups agreed and I felt good about what they had learned this semester.

We spent the rest of the class watching a TED Talk and discussing the future of education. It wasn't the end-of-the-year party that a lot of classes were having, but it was a celebration. I love my job!

Where do you stand on teaching mathematics?

Today my Teaching and Learning Middle Grades Mathematics course begins their unit on teaching mathematics. This follows units on doing mathematics and learning mathematics. We begin this unit by taking a simile survey that asks respondents to complete the following:

Ideally, a mathematics teacher is like a(n):

a. Coach

b. Doctor

c. Entertainer

d. Gardener

e. News Broadcaster

f. Orchestra Conductor

• Choose the simile that you believe best describes a mathematics teacher and explain your choice.
• Choose the simile that does the worst job of describing a mathematics teacher and explain your choice.
• Is there another simile that does a better job than these of describing a mathematics teacher? If there is, then what is it and what makes it better than these?

This survey follows similar simile surveys done for doing and learning mathematics. The doing mathematics unit had integers as one of the topics, so I asked my learners to rate the "doing" similes from -5 (the worst) to 5 (the best). We created a life-sized number line as a way to share and compare our choices. 

Rational number was the mathematical focus in the learning unit. For this survey, learners assigned their worst "learning" simile a 1 and then decided how many times better each other simile was than the worst one. The life-sized number line went from 0 to 1 this time. Learners computed the following ratio for each simile: simile rating to best simile rating. (i.e. The best simile was a one.) Again, we shared and compared our choices by standing on the number line.

The teaching mathematics unit concentrates on geometry. Today, I will ask them to stand in one of four corners representing strongly agree, agree, disagree, or strongly disagree for each simile. It is certainly a simpler process than the others but it often results in some interesting discussions.

Ideally, a mathematics teacher is like an entertainer. Where do you stand?