(September 14, 2017) This week we talked about the Google PageRank algorithm. I facilitated it in the same way that Dr. Emile Davie Lawrence did at the recent National Math Festival in DC. I was fortunate to be in the audience during her presentation, which was recorded and available for viewing here: http://nationalmathfestival.org/photos-videos/. In Dr. Lawrence’s session, she remarkably interacted with hundreds of people as the audience answered her questions to learn the algebra behind one aspect of the algorithm: backlinks. Because you can view the mathematics in her presentation, this report will focus on the pedagogical decisions I made with our students instead.
Dr. Lawrence used conventional algebraic notation including variables with subscripts and matrices. I wanted to know how comfortable our students would be with this while also keeping their minds wrapped around the definition of an algorithm, so I put a series of numbers on the board: 0,1,1,2,3,5,8,13,…. (This is not something she did.)
Several students quickly identified this list as the Fibonacci series. I asked
- Is this an algorithm?
- Could the number 4 be on the list?
- What is the rule?
- How can you express the rule symbolically?
We discussed, and ended up with the conventional notation for this on the board. We have such a wide range (6 years*) of age and experience in the group that it didn’t surprise me that this algebraic notion was old news to some students and totally unfamiliar to others. We did a quick calculation or two then I reminded everyone of the big picture in this course: algorithms, their applications, and their misapplications.
WHERE DOES THAT PAGE ORDER IN SEARCHES COME FROM?
The class brainstormed what they knew about how Google comes up with the list order. They didn’t know much, but the rest of the class raised their eyebrows at how much one student knew about browsing in incognito mode. Then I gave Dr. Lawrence’s example and we worked through it. We ended up with a graph theory graph on the board. This is not a traditional coordinate-plane, xy-axis type of graph. This is a graph of a network with edges and vertices.
OUR DIVERGENCE FROM THE PRESENTATION
Dr. Lawrence’s presentation used both graph theory and the notation of a system of linear equations with variables and subscripts. Our discussion was juggling the math in the exact same way as Dr. Lawrence. The graph notation was easy for everyone to follow. The equations, though, were not. Once they were on the board, the most experienced students were smiling and nodding but some of the least experienced wore deer-in-the-headlights expressions. Hmmmmm…. what to do?
Since it seemed that everyone understood what was going conceptually, the only issue was the notation. We had to (A) tell the same story without variables, (B) work through some simpler variable scenarios to aid in comprehension for some students, or (C) keep going with this notation with only some people understanding. I thought quickly about the emotional state of the students. Option (A) would work for everyone if only I knew of a way to do it. Option (B) would leave some students bored, and maybe even resentful of being in a class with people who hadn’t seen this notation before. Option (C) would dig a deeper hole of confusion and maybe even anxiety for some other students.
Fortunately, I got very lucky. First of all, M said, “I don’t understand!” relieving some of the tension in the room. Second, I somehow saw a way to do option (A). Phew! I realized that we could use numerical calculations without variables and mark those results directly on the arrows on the graph.
My 20/20 hindsight tells me I should have anticipated this problem before class and have an alternative approach to the problem in my metaphorical back pocket. But I didn’t. I’m feeling grateful that something occurred to me on the spot. I also wished I had talked to the students about the importance of acknowledging their own feelings/reactions in math. We also could have talked about the different ways people react emotionally to math problems. (One student told me later that working with symbols makes her feel good, that they make her feel smart.)
I am happy that the students got to enjoy the delight of an unexpected mathematical result (ask your children, or watch the video!). If you do watch the video, know that we didn’t get through all of it, and will pick up next time at the part where we come up with the probability distribution and test it.
Looking forward to continuing with this problem next week! I do plan to continue to present the material with the algebraic notation, since familiarity will increase comfort and usability for the younger students and will be respectful to the older students. I expect to face the above pedagogical dilemma again and again. This will be fun! (Those of you who know me will know that I am being serious, not ironic.)
*Why, you may ask, do we have such a wide range of ages in one group? The answer is that this wide range insures us enough enrollment to have a big enough group for meaningful and energetic mathematical conversation and collaboration. We have 9 students, which allows for many perspectives and insights.