It can also be oversimplified and boring and taught very poorly.
A boy rides his bicycle for 30 minutes and he travels 7.5 kilometers. How far can he travel in 3 hours?
If you do the (simple) math, with the three basic pieces of information given- an oversimplified strategy many math books employ, you’ll see that the boy traveled 45 kilometers in 3 hours.
Really? A kid rides a bike for 3 straight hours… at the same speed? Where there no hills? Didn’t he obey street signs? Did he stop for a Slurpee?
More importantly, if I don’t factor these anomalies in, is the math interesting? Is it teaching anything? Or rather, is it teaching anything meaningful?
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“Except in mathematics, the shortest distance between point A and point B is seldom a straight line.” -Anonymous
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I remember giving a bonus question in my Grade 8 class once where I told students the speed of a train that went from Vancouver, BC, Canada to Los Angeles, California, USA. I told them the distance and the time it took. Then I gave them the distance from Vancouver to Honolulu, Hawaii and asked, “If the same train traveled from Vancouver to Honolulu at the same speed as the trip to Los Angeles, how long did it take to get there?”
I have always been known for my ‘Killer’ bonus questions. This was not one of them. I had a general rule for bonus questions that if students worked to a final solution… even if it was wrong, but they showed their work, then they would get some credit for trying. Well, did I ever upset some of my students when I refused to give them any credit for the (completely irrelevant) Math that they did on the question above! The most humourous argument to gain credit was, “There could be a tunnel!” The best argument was, “I got the answer right and had time so I just did the Math anyway, do I get extra bonus marks?”
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Watch Dan Meyer‘s TEDxNYED talk Math Curriculum Makeover. Check out one of his lessons. Dan teaches Math, but more importantly, Dan teaches students to think, and to see the beauty of Math.
Math doesn’t always have an easy answer, and it shouldn’t always be about the answer. We should relish in the mysteries of Math’s beauty.
Here is a wonderful video “Nature by Numbers” on Phi, the golden ratio. A pattern found in nature and admired and appreciated for its’ beauty for centuries now!
Here is the math behind the video. Math is beautiful if we let it be so.
Hi Dave
Okay, as a former Math teacher I admit that I have struggled with ways to make Math beautiful. Interesting. Relevant, even. With some concepts it is easy to do, and with some concepts, it isn’t. Needing to “cover the curriculum” often made me feel as if I didn’t have time to explore the beauty of a concept, but truth be told… I don’t know HOW to make all these concepts beautiful. I think if there was 1 beautiful/interesting/creative lesson for each unit that I had to cover, it would be a huge success (or at least a starting point). The trick is, to find or make those lessons, and then find TIME for those lessons, because I agree, they are valuable too. Perhaps even the most valuable lessons of all. 🙂
Thank you for this post.
Cheers
E!
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Elaan,
I remember the first year you taught Math because I shared with you all of my Math Model Book resources. Also in discussions with you, I know you did some interesting projects to get your kids engaged with the curriculum. Looking back, I’d have to admit that most of what I shared, though useful to make the concepts easier to understand, where actually more about ‘covering the curriculum’ than they were about ‘making Math beautiful’. We all have our learning journeys, don’t we!
Now, more than ever, I think that rather than ‘finding time’ for these kinds of lessons we need to initiate the learning with these kinds of lessons, because as you say, these might even be ‘the most valuable lessons of all’. To teach the pattern of Phi and not show it’s beauty would be no less a disservice than reading a play and never watching it, or studying geology and never leaving the textbook. I’m really starting to see that we need to MAKE the time, rather than ‘find’ it!
I have warm ups at the beginning of class everyday that are bonus questions of the very type that you mentioned. My students call them “trick” questions but I tell them they are actually “thinking” questions. If the they get the answer right they get a bonus point, otherwise they get a 1 out of 1 just for trying the question. They actually love these no pressure questions and by the end of the year have learned to really think about the question first before thinking about the math from the textbook that they should apply.
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