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How Craig Barton wishes he’d taught maths

98 pointsby auferstehungover 6 years ago

8 comments

joe_the_userover 6 years ago
<i>More precisely, in order to decide whether it is a good idea, one should assess (i) how difficult it is to give an explanation of why some procedure works and (ii) how difficult it is to learn how to apply the procedure without understanding why it works.</i><p>Well, teaching basic math at a commuter college years ago, it felt like the issue of &quot;teaching procedure&quot; to &quot;teaching understanding&quot; was complex. The course I was teaching was close to the end of the math requirements for a significant percentage of the students. I was very attracted to teaching ideas but this group of students essentially had the attitude that they wanted a procedure to memorize rather than an explanation, <i>not matter how complex the procedure</i>. It had a certain logic - mathematical explanation would have touched a world they were happy to and committed to leaving forever soon after this. They&#x27;d suffered through this world up this point and <i>thinking</i> about it was more painful than simply acting.<p>Which is to say, I don&#x27;t think there any easy answer for how to teach math. The failure of American &quot;new math&quot; years ago is something of a lesson in the push-pull of concepts versus concreteness as they can become ideologies in society at large.
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throwawaymathover 6 years ago
Wow, that vector space question is a great example. It’s the kind of thing that <i>should</i> be straightforward for anyone who has taken a linear algebra course, but I can also totally see students getting it wrong. This is especially the case because it’s actually very easy fundamentally (the set of all integers does not comprise a field, and so a vector space cannot be defined over it).<p>But to my recollection, most of the popular linear algebra textbooks[1] don’t spend time showing why the integers cannot form a vector space because it’s “easy.” Instead they spend time tediously walking through examples of bizarre sets defined over R and C to show which axioms are fulfilled and which are not.<p>In a similar vein to the way students might overthink the elementary probability question, I could see university students trained to disprove each of A), B), C) and D) - perhaps making a mistake along the way - instead of quickly scanning the options and picking out the one which simply isn’t defined over a field.<p>__________________________<p>1. I’m thinking of Friedberg et al, Hoffman &amp; Kunze, Axler, Strang, etc.
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keithpeterover 6 years ago
Quotes from OA that struck me as on the button...<p><i>&quot;A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later.&quot;</i><p>Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions.<p><i>&quot;I would go for something roughly equivalent [in the solving of equations such as 4x - 8 = 2x + 2], but not quite the same, which is to stress the rule you can do the same thing to both sides of an equation (worrying about things like squaring both sides or multiplying by zero later). Then the problem of solving linear equations would be reduced to a kind of puzzle: what can we do to both sides of this equation to make the whole thing look simpler?&quot;</i><p>The idea of just playing with the notation is one I fully intend to try but getting people to think in that abstract way is hard work.
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agumonkeyover 6 years ago
Math is done very very wrong, I don&#x27;t think most teachers know enough math (sorry for that dubious and bold claim).<p>As a computer guy who hates state machines and was always obsessed with math, I feel that just about everything about maths is taught wrong from the get go.<p>Just the other day I learned about something inductive function got me curious about: linear ordering of structures as proof of termination. Turns out it&#x27;s been studied in math for long: it&#x27;s called a well-order. Fine.. thing is we&#x27;re taught about linear recursion in HS .. but we have no pragmatic notion of induction except ~~ P n-1 =&gt; P n ~~ It&#x27;s so cryptically compressed that I suspect no student beside aspies and other prodigies can have the slightest clue about that. Yet it&#x27;s so important (and so obvious when shown).
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zwayhowderover 6 years ago
I skipped two years of school which also happened to be when a lot of the basics of Algebra were introduced. When I returned to school my scores were good enough to not repeat but the gaps didn&#x27;t become obvious until it was too late to fix.<p>Reading that article I now realise it was that I lacked fluency. I didn&#x27;t instinctively &quot;know&quot; how to do simultaneous equations because unlike my peers I hadn&#x27;t spent two years doing them, so I had to remember how to solve them every single time.<p>All I can say now is thankfully there is the Khan Academy which rapidly improved my mathematical understanding when I needed it.
theontheoneover 6 years ago
I consider myself pretty strong at math (in university right now) and I was stumped by the vector space question. I never considered, actually, what domain scalars should be drawn from.<p>Wikipedia says &quot;the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields.&quot;
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mncharityover 6 years ago
&gt; One question I had in the back of my mind when reading the book was whether any of it applied to teaching at university level. I’m still not sure what I think about that. There is a reason to think not, because the focus of the book is very much on school-level teaching, and many of the challenges that arise do not have obvious analogues at university level. [...] I think at Cambridge almost everyone would get this question right (though I’d love to do the experiment). But Cambridge mathematics undergraduates have been selected specifically to study mathematics. Perhaps at a US university, before people have chosen their majors, [...] More generally, I feel that there are certain kinds of mistakes that are commonly made at school level that are much less common at university level simply because those who survive long enough to reach that stage have been trained not to make them.<p>Note the &quot;I think [...] almost everyone would get this question right (though I’d love to do the experiment)&quot;. This is a familiar state. Widespread. Call it, teachers who have not yet had their &quot;oh shit!&quot; moment.<p>One of the blog comments points at Eric Mazur&#x27;s (Harvard, physics) oft-repeated talk &quot;Confessions of a Converted Lecturer&quot;. Who describes the first time he gave students a Force Concept Inventory. Worried about wasting their time with such easy questions. :) Unaware physics education research was about to become a focus of his career.<p>Many have been surprised by &quot;Minds of Our Own&quot; (1997) <a href="https:&#x2F;&#x2F;www.learner.org&#x2F;resources&#x2F;series26.html" rel="nofollow">https:&#x2F;&#x2F;www.learner.org&#x2F;resources&#x2F;series26.html</a> The short (3 min) introductory video shows MIT and Harvard students struggling to light a bulb with a battery and a wire. Full episodes are below (by clicking on &quot;VoD&quot; buttons).<p>Harvard Center for Astrophysics has both first-tier astronomy and astronomy education programs. When meeting a new CfA graduate student, I&#x27;ve a little drill, prompting for the color of the Sun, and then of sunlight. They almost always get the first wrong, and then get a conflict, often with a nice &quot;oh, wait, that doesn&#x27;t make sense does it&quot; moment. The collision of two bits of non-integrated and flawed understanding. Of the few who get it right, halfish (but small N) learned it from CfA instruction on common misconceptions in astronomy education, rather than from their own astronomy education.<p>But perhaps mathematics is doing better at robust integrated understanding than are astronomy, physics, chemistry, biology and medical school. It seems possible at least.<p>It&#x27;s not just people who have had, or not had, their &quot;oh shit!&quot; moment. Professions too. Medicine realizing that medical errors were a major cause of mortality. Realizing even cheap easy universally-approved interventions (aspirin for ER chest pain) weren&#x27;t consistently being executed. Realizing other industries had decades of experience on how to pursue quality, to which medicine had been oblivious. When the New York Times babbles about &quot;Truth&quot; and &quot;The Journalism You Deserve&quot;, I shake my head and think, there&#x27;s a field that has no clue how badly it&#x27;s doing, how much work on process quality it&#x27;s unaware of; a field that has not yet had its &quot;oh shit!&quot; moment.
jp57over 6 years ago
Soon someone from HN is going to come and remove the word ‘how’ from the headline. Why? Because clickbait!<p>Except, of course, the book really is about <i>how</i> he feels he should’ve taught math.