Since my introduction to the twitterverse and blogosphere, I have been on the lookout for like-minded individuals who share my passion for the teaching and learning of mathematics. I have met numerous people who document their best strategies, and have already been very helpful to me. One such community of learners is the #mathchat gang that meets once a week (and re-opens discussion at a more European friendly time later in the week) to discuss a topic or theme in math education. Although it is often tough to express pedagogical beliefs in 140 characters or less, the conversation is incredibly fruitful. It was during one of the “mathchat”s that I was struck with a particularly convicting, and ironic, realization.
The topic of the conversation was:
“How do I promote deep, productive and creative mathematical play?”
and soon comments, ideas, opinions, and examples were pouring in from all corners of the globe. After about 20 minutes, the general consensus began to shift toward the definition of mathematical play, and more practical ideas on how to initiate it. You only have to hang out with teachers for a single Professional Development day to realize that they crave practicality. If a concept laces theory together with practice, it is quickly devoured. (Such is the partial aim of this blog). I noticed the conversation getting more and more impassioned as teachers began to take ownership; then a funny thought struck me:
Teachers seem very worried about initiating mathematical play for our students, but how often do math teachers engage themselves with “deep, productive, and creative” mathematics?
I am a self-professed math nerd. I spare no chance to initiate a mathematical conversation with my colleagues–often to the point of exhaustion. Despite my love for recreational mathematics, I almost always return to the knee-jerk reaction of relating what I am doing to two things:
1) The curriculum
2) The students
I am not saying that thinking about these two things should create guilt in the reader, but teachers seem to have the the inescapable urge to apply, apply, and apply. From their admittance to a College of Education, all focus changes. We become experts of the mathematics between the covers of our textbooks, and this is the exact phenomenon that many teachers are trying to disband in students. Math teachers need to take time to sit and marvel in the intricacies of their discipline, or risk becoming proficient only at the limited scope of math in today’s textbooks.
I work with a teacher who has seemingly freed himself from the necessity to only partake in mathematical activities that can become direct lessons. (Again, not necessarily a bad thing, as teachers may discover different perspectives and outcomes from authentic, mathematical play) I admire him greatly as a teacher and learner. You see, Mr. C. got into teaching a lot later than most of us. He experienced different vocations where he honed his skills in many mathematical, and practical, ways. When he discovered his love for education, he became a teacher, but never lost that spirit of investigation for investigation’s sake. It should be noted that Mr. C.’s pathway to education also gives him perspective that I have found extremely valuable in my young career.
After the mathchat concluded, I reveled in the irony of the situation for a while. The fact that math teachers could talk about mathematical play for close to an hour but never let the topic sway onto their own exploits in mathematical play was astonishing. I then looked back on a question put out by the folks at @NCTM and set out to play.
The topics of primes has intrigued me since my introduction to number theory a few years ago. They seem to be front and center on so many problems in mathematics. I started with this problem:
“Can the sum of 2 primes be prime?”
and set off to play. When my students set off to play, I ask them to make two lists: one of observations about the problem, and one of facts that may relate to the problem. I created these two lists about this problem:
Primes are usually associated with multiplication.
Prime numbers are almost always odd.
We are going to work only with positive counting numbers.
The first couple trials work.
2 is the only even prime.
Adding 2 odds always gives an even.
Adding an odd and an even gives an odd.
From here, I set out to “play with primes”. The first step that helped me was the fact that 2 was the only even prime, so it would become valuable with creating new ones in the future. Also, the first couple of trials including 2 created primes. (2+3=5 , 2+5=7) I then narrowed my focus by determining that all odd primes when added to one another would be even, and thus not prime. Initially, I used a standard number theory proof to show this, but mathematical play pushed me in another direction–a direction that would be crucial to the expansion of my perception of the problem.
I began to think of numbers as geometric bodies. We have “perfect squares” and “triangular numbers” so why not “trapezoidal” or “rhombic” numbers? Every odd number can be seen as an even with an extra point. This is the only point that has no ‘partner’ so to speak. I represented 7 this way on my page:
7 = :::.
So every odd prime, would have one left-over “dot”. When added, these two dots would line up and create a pair. That would symbolize an even (or rectangular) number. I then tried to fit my new representation of numbers (that has somehow been lost since I began primary school mathematics) onto the question of adding primes. We know that one of the 2 primes must be 2, and now we know that if the other prime can be arranged as a rectangle of dots with only 2 missing, than their sum would become a perfect rectangle. Essentially, any prime partner for 2 that needed exactly 2 dots to “complete the rectangle” would yield a non-prime sum. I encourage you to try the same representation system. The fact from it becomes:
Any number that can be arranged as a perfect rectangle is not prime, but divisible by both of its dimensions.
That led me to my conclusion. The sum of two primes could be prime as long as:
1) One of the primes was 2
2) The other prime was NOT of the form: an+b,
‘n’ is a natural number
‘a’ is a natural number > 2
‘b’ = a-2
i.e. 7 = 3(2) +1 w/ 1=3-2; so 7+2 is not prime.
but, 11 can’t be written as “an+b”, so 2+11 is prime.
This was my happy way to put my learnings back into my niche of number theory. Test this hypothesis out; I did. I continued to play around and refine my definition. This result only raised more enticing questions to play with; each of them became further and further away from the curriculum. I had to suppress my urge to find a student connection, so I could grow deeper in the learning of mathematics. Do Trapezoidal numbers exist? What about the sum of 3 primes? How many combinations of 2 primes are there? All of these have varying degrees of difficulty. The point is not to get caught up in practicality for students. Teachers need to understand that their learning of mathematics (often through exploration and play) is directly connected to their teaching of mathematics–even though all their thoughts may not fall neatly into a lesson plan.