Mr. McRae’s Triple Gift

This is a story from long, long ago. It is about Mr. McRae, who was our grade 4 teacher, at Macleod State School. We have written about Macleod before, and we have written, briefly, about Mr. McRae before, in regard to the moon landing:

I still have vivid-grainy memories of watching Armstrong’s first steps. A random few students from each class in Macleod State School were selected to go to the library to watch the event on the school’s one TV. I was not one of the lucky few. But Mr. Macrae, our wonderful Grade 4 teacher, just declared “Bugger it!”, determined which student in our class lived closest to the school, and sent out a posse to haul back the kid’s 2-ton TV. We then all watched the moon landing, enthralled and eternally grateful to Mr. Macrae.

He was that kind of guy. No-nonsense and intelligent and cultured.

The year he taught us, Mr. McRae was new to Macleod. He had just appeared on the playground before the first class of the year, tall and commanding. Rumour had it that he had played Under 19s for the Richmond Football Club, making Mr. McRae just shy of a Greek god. (The actual Greek god was, of course, Carl Ditterich.) He was a standard and excellent teacher. Firm, disciplined and disciplining, but kind, and with a calm and intelligent air of bemusement. He was the boss, but a thoughtful and unpredictable boss. Hence, our class getting to watch the moon landing. And, how else to explain the boxing match?

One day, Mr. McRae inadvertently started a harmless play-scuffle between two students. He then decided the dispute should be settled by a proper boxing match in front of the class. Once, of course, a kid had been sent home to fetch a couple pairs of boxing gloves. We can’t remember whether we lost, although we remember we didn’t win. In any case, neither of us had a clue how to box, and so the match was followed by Mr. McRae giving the class an impromptu lesson on technique. This was, to explain it a little, the era of Lionel Rose and Johnny Famechon and TV Ringside.

That’s all by way of background. The story we want to tell is of a mathematics lesson.

One Friday afternoon, Mr. McRae introduced his grade 4 class to Pythagoras’s theorem. Or, at least, to Pythagorean triples; we can’t specifically remember the triangles, or anything, but undoubtedly \boldsymbol{3^2 + 4^2 = 5^2} made an appearance. Why he showed us this, God only knows, but Mr. McRae ended the class with a challenge: find more triples. Our memory is that the specific challenge was to find a certain number of triples, maybe three, maybe five.

We have no idea what Mr. McRae hoped to achieve with this challenge, but we remember pondering, aimlessly, hoping to find triples. Eventually, by smart persistence and dumb luck, we stumbled upon the trick: doubling a triple gives a new triple. So, \boldsymbol{6^2 + 8^2 = 10^2}, and so on. With this kid-Eureka insight, we then happily spent the week-end doubling away.

Come Monday morning, Mr. McRae asked for the class’s triples. We proudly went to the blackboard and wrote up our largest creation. By memory, it was something in the millions. So,

    \[\boldsymbol{1572864^2 + 2096152^2 = 2621440^2}\]

or thereabouts. And then Mr. McRae uttered the fateful words:

“Let’s check it!”

There were the inevitable groans from the class, and the little Archimedes hero of the story was more popular than ever. But, Mr. McRae was the boss, and so we all set down to multiplying, including Mr. McRae himself. And, ten or so minutes later, the class collectively started to conclude … the equation was wrong. Yep, Little Archimedes had stuffed up. Which led to more fateful words:

“Let’s find the mistake!”

More groans, more multiplying, and eventually the error was found. By memory, after quite a few doubles, somewhere in the mid thousands. And, satisfied, Mr. McRae led the class on to whatever he had been planned for that day.

What is the moral? We have a reason for telling the story, beyond a simple tribute to a great, memorable teacher. We think there are morals there. We’ll leave it for the reader to ponder.

23 Replies to “Mr. McRae’s Triple Gift”

  1. My first instinct was this may have been many students first exposure to “doing Mathematics”.

    But I’m sure there is a lot more hidden in the details.

  2. You know Marty, this kind of story is dangerous. It sounds exactly like the kind of story a champion of ACARA would tell.

    I would like to invite any such champions to tell us what this story means to them.

    1. Glen, it’s obviously not ACARA’s style to engage in public debate. They are too Lord Almighty to stoop to that.

      Im not quite sure why you think ACARA types might tell this story, although it is a pretty good Thematic Apperception Test.

      1. Agreed on the TAT comment.

        Here we go: CLEARLY your beloved teacher was an early adopter of the constructivist approach to teaching mathematics, even incorporating elements of social constructivism. I see problem-based learning, facilitation over direct instruction, investigation, and inquiry. CLEARLY this demonstrates the inherent superiority of this approach to teaching mathematics: this man was a great teacher, and he used this method. DEEP LEARNING is evidenced by this sticking in your mind for so long, and the SHARING of this story with us here on the blog.

        It would be better if a true believer tried arguing this though. My heart isn’t in it.

        1. In a 21st century maths class the students wouldn’t be doing any multiplying or calculating, it’d just be pushing buttons.

            1. Continuing in my earlier mode, I’d point out that this can be solved by specifically instructing students on how to perform complex calculations through the use of technology. In fact, this story can be brought into the modern classroom through the miracle of technology.

  3. I recall that we had boxing at the school fete when I was in grade 3. I’m not sure if they have boxing at schools anymore.

    Your story made me think of my own experience that a well-chosen problem can grip a class. Not something to do in every lesson – but it is wonderful to watch. The other day I was talking to my Year 9 students about periodic functions and asked them to tell us some phenomena that are periodic.

    One student piped up that you see it in the hands of a clock. It was a good answer. I could not help but ask, “At what times in the day are the minute and hour hands at right angles?” Well, discussion took off from all quarters. It was wonderful to “see” (it was an online lesson).

    1. I’m not sure Marty’s example builds on what was already happening in the class, which (I’m guessing) is kind of the point.

      1. Thanks, RF. As I wrote, I have no idea why Mr. McRae set that challenge, or even told us about Pythagorean triples. Certainly it wasn’t with any expectation of monster triples, much less checking them. Probably Mr. McRae had no clear idea, more than “here’s an interesting thing to think about”.

        As for the point of the story, I’m honestly not sure. One point, and the reason I chose to write about it now, is SRK’s observation, that the past is a foreign country. I’ll come back to this aspect in a future post. But, beyond that, I like the story exactly because I don’t know what it means, and some obvious spooned on meanings will be dead wrong. (Which I guess is Glen’s point.)

        1. I was thinking about *that* post today when re-reading the new VCE study design at some length and thinking about just how much has changed since the 1960s.

          Some topics, oddly enough, *look* the same upon a surface reading of the study design, but the difference in how a topic such as differential calculus is assessed now compared to then is really interesting.

          There seems to be a lot more “stuff” in the curriculum at the moment and a lot more 1 and 2 mark questions on exam papers.

          I’m also not convinced calculators are entirely to blame; they certainly aren’t helping but I do wonder if there aren’t other forces at work here, too many fingers pulling on too many strings.

          1. Certainly, calculators are not the only cause, and probably not the first cause, or primary cause. But they magnify whatever awfulness is around.

  4. Mr McRae

    I’ve always enjoyed the way ‘clearly’ is used in mathematical exposition and articulation. It’s a kind of secret pleasure, a frisson almost, granting privileged access to insight and understanding. For those in the know, an entry pass with a wink and a nod.

    I miss that when I read here the contribution (with or without irony) that Mr McRae’s story clearly demonstrates the inherent superiority of problem-based learning.

    All that is clear to me is that the Pythagorean triple challenge posed by Mr McRae fully engaged and had deep impact on Marty.

    Nowhere is there anything to tell us to what extent Mr McRae used activities like this to teach Mathematics to the Grade 4 students at Macleod. All Marty has said is that Mr McRae was a standard and excellent teacher.

    This contributor would like to think that means he used and was skilful with a range of pedagogical approaches. Ranging, in the terms we see so often, from problem-solving and investigation to direct and explicit instruction.

    A moral then might follow: effective, high-quality instruction is a function of a variety of instructional methods, the weighting of which, in the hands of an accomplished practitioner, is a nuanced and complex matter.

    1. Thanks, Ian. I *think* Glen was playing Devil’s Advocate.

      Mr. McRae was a great teacher, but was this an example of great teaching? It was memorable, and the episode in its entirety had a deep impact on me, but I don’t know the challenge per se was that meaningful for me, and I think it was pretty much meaningless for all my classmates. And I was already mathsy: I didn’t need it.

      One thing I’m sure of: Mr. McRae would laugh out loud at any attempt at deep interpretation of what he was doing there, or in general. I am absolutely sure that his was instinct teaching, tempered by an intelligent and bemused empathy.

    2. Sorry Ian — didn’t mean to mislead. I was just fooling around. This kind of story does remind me of typical anecdotes used when impressing upon young students in education courses the importance of modern approaches to education, especially mathematics education. So, I really would honestly like a true proponent of that approach to education to argue for it. I’m not it.

      1. Half serious question – does there exist any empirical research on whether the vividness of a memory of how/when one was taught something is correlated with how well one has learned that something?

          1. Not sure if this is what Glen had in mind, but when education academics lean on such anecdotes as a justification for “progressive teaching”, they might think there’s a common cause: how much attention one gives to the lesson.

            1. SRK, there’s an obvious difference between remembering a lesson and remembering the content of the lesson. I don’t think the former is important, except inasmuch that it might indicate the teacher having engaged the students enough that the students paid attention and did some work. So, weak circumstantial evidence.

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