Induction Squared

I came across an interesting problem recently that I gave to my students in need of enrichment.  Given a square and the ability to divide that square into smaller squares, can you divide a square into ‘n’ smaller squares. The squares do not have to be the same size. For which values of ‘n’ is this possible? For which values of ‘n’ is this impossible? Students initial reaction was to draw a square and experiment. I cannot think of a better way to begin this problem. It is organic, and contains some very speedy deductions. We begin with suspicion of …

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Maths’ True Form

I teach mathematics at the high school level, and know all about the various theories surrounding school mathematics. I can still remember the intrigue when the term “Math Wars” was introduced to me through some undergraduate reading. I immediately took to the history of my art, and found a very convoluted and bloody past. The constant pendulum between retention math, new math, back to basics, and now the new-new math is dizzying. Whenever I converse with a colleague about a new way of thinking in math education, I am sure to remind them that we are in a war. It …

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Life Without Euclid

This post has nothing to do with geometry. I guess I can’t say that exactly (because of the possible geometric representations), but I am not dealing directly with these. I am always intrigued when I think like I want my students to think. It is these moments that keep me going into the classroom hoping for new understandings. There have been times this year where students have made connections that I never have. These innocent realizations are mathematics manifested in its purest form. A similar experience happened to me this morning. I had been mulling over a problem posed by …

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Review: The Art of Problem Posing

I consider reading an essential part of my professional development. I enjoy a morning glance through a chapter or two, and like to wind down a winter’s day with a book and a cup of coffee. Sometimes reading is the only way to relax my mind at the end of a day. (Naturally, some professional literature is better at putting me to sleep than others). To this point in my young career, no book has changed my perspective on the teaching and doing of mathematics more than The Art of Problem Posing: Third Edition by Stephen I. Brown and Marion …

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Balls and Bins

One of my pervious posts mentioned the problem of the balls and the bins. I got this problem from a source on twitter that I have since forgotten. Regardless of its origin, the question has been a fun one to pose to students and colleagues alike (I even asked my in-laws with some very interesting results). For those of you who haven’t read “Practice What You Preach“, The problem is as follows: You have 8 balls, and 2 bins. 4 of the balls are Red, and 4 of the balls are White. Your job is to arrange the balls in …

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Practice What You Preach

I have already expressed my views on the value of probability within the school curriculum. When posed in a creative context, the nature of the subject leads to excellent exploration. I tell this to every class that I teach probability to, and this year my explanation caught up with me. I gave my first day lesson on a counter-intuitive problem, and then began using experimental probability to verify our results. I reserved the end of class to remind students what we were doing. I got onto my soap box and began preaching the importance of reality; although we were calculating …

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When School Math Falls Short

Warning: the following post contains algebra; I just thought I should be transparent. If three-space, divisibility, or inequalities make you queasy, please escape while you can. This afternoon, I was re-united with an old problem that I had managed to shunt into the back of my memory. Maybe because I remember it being incredibly frustrating, but (most likely) because it doesn’t fit nicely into a niche of school mathematics.  The problem is summarized as follows: You need to buy exactly 100 pets. You have exactly $100 to do so. Dogs cost $15, Cats cost $1, and Mice cost $.25. How …

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Do Teachers Play with Mathematics?

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 …

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Shouldn’t Probability be Vague?

I have always been drawn to probability because of its mysterious qualities. Maybe it is the result of the online poker fad that swept through my high school during the NHL lockout, but the calculation of odds still grasps my attention to this day. What fascinates me the most is how simple rules such as “AND” and “OR” can quickly create a mess of a situation. What begins in high school (or earlier) as a simple fraction that predicts the toss of a coin, soon balloons into factorials, combinations, Pascal’s Triangle, and Probability Density Functions. Despite the complexity of such …

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Must it Always be True?

This morning on twitter, there was a problem that I just had to solve before going out the door. It is safe to say that these types of problems are my vice. Number Theory has always held a special interest to me despite, according to G.H, Hardy, having “absolutely no practical use.” (A Mathematician’s Apology, 2001). This has all changed with the inception of encryption. I wish just to present the problem and then muse on its educational significance both for my personal learning of mathematics, and for that of my students. N is the 4-digit integer 6_9_. If these …

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