What's your vision?

From Black Elk Speaks

After the long winter of waiting, it was my first duty to go out lamenting. So after the first rain storm I began to get ready.

I recently heard Kent Dobson talk about this idea of "lamenting" - what is often thought of as a "vision quest." [I apologize in advance if I get some of the details wrong in this post. A lot of what I'm writing is a combination of my memory of and the connections made during his talk. Please let me know in the comments if anything needs correcting.] 

In the Lakota tradition, when a seeker comes of age, he or she goes off with an elder to "cry for a vision." After some preparation, the seeker is left alone to lament the current state of the community and seek answers in the form of a vision. From time to time, the elder looks in on the lamenting to advise and support the seeker. When the lamenting is complete, the seeker returns to the community and shares the vision. This is an important rite for the community because without new ideas, a community withers and dies.

As I begin a new year of supervising student teaching, I want this ritual to inform my work. I hope to play the role of the elder supporting coming-of-age teachers as they experience (and lament) the current circumstances in math education and seek new answers. I will listen to those answers and take them seriously because in many ways our profession is withering. And I will help the student teachers to share their visions with the larger math education community, so that these seekers might contribute to the development of our profession.

Thank you in advance for welcoming these seekers and helping to interpret and implement their visions.

Lessons Learned in Tanzania - Pole Pole

Start of our climb up Kilimanjaro

While teaching in Tanzania taught me a direct lesson about being resourceful, there were also some indirect lessons associated with my time in Africa. Probably the most powerful came while climbing Mount Kilimanjaro. We did not climb the entire mountain, only part of the Marangu Route, but this brief hike (ascending more than 2,500 feet over 5 miles and then back down),  made a lasting impression. In particular, the guides' exhortations to "pole pole" (Swahili for go slowly) got me thinking about our constant efforts to improve education.

The guides wanted us to go slowly for a couple of reasons. The first was to ensure that we did actually make it to our destination - the Mandara Hut. The climb is steep and plenty of people do not make it because they expend their energy early on or because of some injury incurred from inattention to the climb.

This reminds me of the point made by Stigler and Hiebert in The Teaching Gap that our efforts to improve education are often too frenetic and unfocused to be sustainable. Instead, we need to consider slow, purposeful (sometimes subtle) shifts that ensure we reach our final goal - student learning.

The other reason to take the climb slowly was so we did not miss anything along the way. Although this portion of the route is mostly forest, without any scenic views, that does not mean there is nothing to see. It is easy to miss some of the flora or fauna if one moves too quickly or without intention.

The same thing happens in education. We do not take time to notice the flowers let alone stop and smell them. I know my own teaching experience is much more positive when I focus on enjoying the journey rather than simply getting to the destination.

So I left Mount Kilimanjaro with a bracelet to commemorate the climb and the lesson.

Hopefully it will remind me to take my teaching practice slowly - to make it sustainable and enjoyable.

Do the numbers matter?

One of the issues I have with some of the Khan Academy videos is how he seems to select numbers without any planning. To be fair, he is not alone in this approach of using numbers in examples without considering the impact these numbers will have on learning. Many of the novice teachers I work with struggle with the same view that any number will do. Fortunately, reading The Teaching Gap and exploring Lesson Study introduce our teachers to the importance of being thoughtful about the numbers they use throughout a lesson.

But what happens when the numbers come from a textbook? Can these numbers be trusted? During an observation, a teacher used the following problem from Holt McDougal Mathematics Course 1.

When the teacher asked students how they tried to answer the questions, they provided three different approaches.

Many of the students originally did the same thing in both cases - divided by four. One student suggested that because the area was given for the first square that they only needed to divide 81 by two. This proved to be a turning point in the discussion as many of the students backtracked on their first instinct and said they needed to find a number that multiplied by itself resulted in 81. Perhaps they had missed "an area" the first time.

Thinking that he had addressed the issue, the teacher asked, "Which figure has the longest side?" The student he called on responded that the second square had the longer side. Then the teacher asked the student, "How much longer?" to which he got the response of 0.75 feet. Clearly, this student had missed part of the discussion, but that is not what I am concerned with here.

Why had the textbook authors used 84 feet for the perimeter of the second square? Most middle school teachers could tell you that 20.25 feet would be a common mistake among their students. So why have a number (84) that makes the first part of the answer correct even when the work is done incorrectly? I would have used a perimeter of 80 feet so that both parts would have been wrong if the anticipated error had been made. Am I wrong in this choice? This is a genuine question to which I would appreciate your feedback.