
How Craig Barton wishes he’d taught maths - auferstehung
https://gowers.wordpress.com/2018/12/22/how-craig-barton-wishes-hed-taught-maths/#more-6435
======
joe_the_user
_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._

Well, teaching basic math at a commuter college years ago, it felt like the
issue of "teaching procedure" to "teaching understanding" 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, _not matter how complex the
procedure_. 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'd suffered through this world up this point and _thinking_ about it
was more painful than simply acting.

Which is to say, I don't think there any easy answer for how to teach math.
The failure of American "new math" years ago is something of a lesson in the
push-pull of concepts versus concreteness as they can become ideologies in
society at large.

~~~
mncharity
This is a widespread tradeoff. I'd a conversation with a first-tier college
biology professor, about a way to give a more integrated, transferable
understanding of a topic. He liked it, but observed, my students will shortly
be taking the MCAT (high-stakes medical school entrance exam), and our time
together is limited, and the MCAT doesn't test for understanding of the topic,
only for something superficial and memorizable, so, I would be doing my
students a disservice if I reallocated time to understanding.

Perhaps early primary school is an opportunity to escape this tension. With
weaker test constraints, and more years of payoffs over which to amortize the
costs of better understanding.

~~~
kiba
My understanding of learning is that it both takes a longer time to actually
learn something useful and that we're inefficient in learning or retaining
anything because we rushed from one topic to another in our courses.

------
throwawaymath
Wow, that vector space question is a great example. It’s the kind of thing
that _should_ 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).

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.

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.

__________________________

1\. I’m thinking of Friedberg et al, Hoffman & Kunze, Axler, Strang, etc.

~~~
burlesona
As a bumbling idiot who never did university level math, I don’t understand
the question or the answer. (I did understand the probability one and knew the
right answer to that at least.) I’m trying to catch up. Would you be kind
enough to explain it?

~~~
throwawaymath
Sure.

Operations in applied linear algebra - such as matrix multiplication and
solving systems of linear equations - are formalized by the theory of vector
spaces, much like calculus is formalized through the theory of analysis.
Vector spaces are algebraic structures which axiomatize the linearity you need
to carry out these operations. If you can establish your equations exist in a
vector space, you can prove that they admit linear relations and are thus
solvable as linear systems.

More precisely, a vector space V is any set S defined over a field F which is
_closed_ under both vector addition and scalar multiplication, where the
elements of S are called vectors and the elements of the _underlying_ field F
are called scalars. The term "closed" means that for every pair of vectors _x_
, _y_ in V there exists a vector _x_ \+ _y_ in V, and for every scalar _c_ in
F and vector x in V there exists a vector _cx_ in V. There are eight axioms in
total, for things like associativity and commutativity, but those aren't
germane to this particular example. What's important is that vector spaces are
what allow you to form linear combinations of things, which is the scaffolding
you need to prove things like linear dependence and independence; whether or
not a system of linear equations has no solutions, one solution or infinitely
many solutions, etc.

Fields are the algebraic structures which formalize the elementary arithmetic
you're already familiar with over sets like the the complex numbers, the real
numbers, the rationals, etc. They are sets which are closed under "regular"
addition and multiplication. Notably, integers do _not_ comprise a field
because integers do not have multiplicative inverses. Multiplicative inverses
are the axiomatic way of establishing that in any field, division must be
possible. So concretely, there is no multiplicative inverse _1_ / _n_ for any
integer _n_. There is in the set of rationals, but not the set of integers.
Therefore integers are not _closed_ under multiplication, and they cannot
comprise a field.

Since the integers do not comprise a field, you cannot define a vector space
over the integers, because the scalars used to define scalar multiplication in
vector spaces are just elements of the underlying field. If you try to define
a vector space over a set without multiplicative closure, the vector space
cannot be closed under scalar multiplication. Among other things, linear
combinations stop being invertible (or even possible in general), and linear
relations don't exist.

So circling back to the specific question: it's asking which of the given sets
comprises a vector space. You can make all kinds of abstract vector spaces
(e.g. the set of all polynomials over a field, the set of all polynomials with
degree at most _n_ over a field, the set of all continuous functions, etc).
But if you stick with the definition of a vector space, you don't need to
tediously test each of the given sets for the eight axioms. You just have to
remember the integers don't comprise a field, so the set of all triples of
integers can't be a vector space either.

Hopefully that's clear, let me know if you'd like me to clarify anything.

~~~
MarcusBrutus
The integers are totally closed under multiplication. They lack an inverse
under multiplication that's why they aren't a field.

------
keithpeter
Quotes from OA that struck me as on the button...

 _" 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."_

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.

 _" 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?"_

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.

~~~
zazen
> 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.

I also agree fully. A little while back I did some support tutoring for
A-level maths students. The number of students who turned up who mysteriously
"had problems with longer questions"... I wish I'd known the example of
calculating the perimeter of the rectangle with fractions. That would have
really helped explain why the problem wasn't really the length of the
question, it was the fact that the student had never properly learned the
component skills separately.

Unfortunately, the problem of building impressive-looking edifices on shaky
foundations is absolutely endemic in British high-school maths teaching.
Thousands of students who never quite understood fractions are "learning"
calculus through being taught recipes, and the easier exam questions are
formulaic enough that they get through with Cs at least, without any
mathematical understanding.

The A-level statistics modules, in particular, have very impressive _sounding_
syllabuses. Students learn T-tests, Chi-squared tests, all this sophisticated
statistical machinery. If all these students really understood this stuff,
Britain would have a vast army of highly trained statisticians. But nothing of
the sort is true, of course: students are just learning a recipe for
processing numbers. I can't imagine the carnage if a statistics exam asked the
students to write an essay explaining the principle by which a T-test works.

Pardon my rant, this has been on my mind for a while.

~~~
keithpeter
Going back around the millennium or before when I last taught A level maths at
college, we had them in over the summer before term started for a two week
intensive algebra and basics course.

Seemed to help.

The original author (Tim Gowers, a Fields medallist and professor of
mathematics at Cambridge) has a totally hilarious blog post about being asked
to coach a teenager doing A level maths...

[https://gowers.wordpress.com/2012/11/20/what-maths-a-
level-d...](https://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-
necessarily-give-you/)

~~~
zazen
Thanks for linking that, it's a great read. I really should read more of
Gowers' posts.

The phrase "memory works far better when you learn networks of facts" was a
happy find - I've never been able to express that idea so concisely.

I remember discovering they'd moved "differentiation from first principles"
away to a further-maths module, as if it's a peripheral, difficult little
oddity for the keen kids to hear about. It was the surest, saddest sign that
the powers that be had given up on genuinely educating the average A-level
maths student.

~~~
mncharity
> memory works far better when you learn networks of facts

One challenge with teaching a more rough-quantitative Fermi-question-ish
introduction to sciences, is it's more sensitive to integration and
correctness of understanding. With a Trivial-Pursuit memorize and regurgitate
style of "understanding", damage from misconceptions and fragmentation of
knowledge is local. Whereas rough-quantitative reasoning benefits from being
able to... slide around the knowledge space. Jagged misconceptions and
fragmented knowledge seriously impedes the sliding. I imagine memory is
similar. Nice phrase.

------
agumonkey
Math is done very very wrong, I don't think most teachers know enough math
(sorry for that dubious and bold claim).

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.

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's
been studied in math for long: it's called a well-order. Fine.. thing is we're
taught about linear recursion in HS .. but we have no pragmatic notion of
induction except ~~ P n-1 => P n ~~ It's so cryptically compressed that I
suspect no student beside aspies and other prodigies can have the slightest
clue about that. Yet it's so important (and so obvious when shown).

~~~
RBerenguel
Induction is baked in the most “common” way of defining the naturals (Peano
axioms). IIRC, it’s the definition I got for the “proper naturals” when I was
in HS (but, my Maths teacher was a mathematician, and was who got me
interested in them).

------
zwayhowder
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't become obvious until it was too
late to fix.

Reading that article I now realise it was that I lacked fluency. I didn't
instinctively "know" how to do simultaneous equations because unlike my peers
I hadn't spent two years doing them, so I had to remember how to solve them
every single time.

All I can say now is thankfully there is the Khan Academy which rapidly
improved my mathematical understanding when I needed it.

------
theontheone
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.

Wikipedia says "the scalars can be taken from any field, including the
rational, algebraic, real, and complex numbers, as well as finite fields."

~~~
throwawaymath
Does Wikipedia actually say that? That's pretty misleading. For any vector
space V defined over a field F, V is only closed under scalar multiplication
using the scalars of F. You can't choose scalars from arbitrary fields for any
given vector space. The scalars have to be chosen from the underlying field of
the particular vector space.

------
mncharity
> 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.

Note the "I think [...] almost everyone would get this question right (though
I’d love to do the experiment)". This is a familiar state. Widespread. Call
it, teachers who have not yet had their "oh shit!" moment.

One of the blog comments points at Eric Mazur's (Harvard, physics) oft-
repeated talk "Confessions of a Converted Lecturer". 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.

Many have been surprised by "Minds of Our Own" (1997)
[https://www.learner.org/resources/series26.html](https://www.learner.org/resources/series26.html)
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 "VoD" buttons).

Harvard Center for Astrophysics has both first-tier astronomy and astronomy
education programs. When meeting a new CfA graduate student, I'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 "oh,
wait, that doesn't make sense does it" 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.

But perhaps mathematics is doing better at robust integrated understanding
than are astronomy, physics, chemistry, biology and medical school. It seems
possible at least.

It's not just people who have had, or not had, their "oh shit!" 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'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 "Truth" and
"The Journalism You Deserve", I shake my head and think, there's a field that
has no clue how badly it's doing, how much work on process quality it's
unaware of; a field that has not yet had its "oh shit!" moment.

------
jp57
Soon someone from HN is going to come and remove the word ‘how’ from the
headline. Why? Because clickbait!

Except, of course, the book really is about _how_ he feels he should’ve taught
math.

