If you've read this blog at all in
the past few months you know that I'm working on a book called Singapore Math Figured Out for Parents, and I do almost all of the math tutoring
for Tutoring Colorado, my wife's tutoring business. Lately, too, I've noticed my
emotional investment in my life as a math tutor deepens with time.
So here's some more about
Singapore Math, math tutoring, and math instruction. This one noodles through
some ideas that I'm pretty sure I need to put into the book, possibly more
diplomatically (so if this angers or offends you, this would be a helpful time
to send hate mail).
And as I often do, I'm starting
off with seven little stories.
§
1
On break from an English comp
class at the College of Last Hopes, I was talking with one of my Adult Disadvantaged Learners about math in
general, because she'd been struggling in Pre-algebra. It's the class that the
College of Last Hopes offers to students whose math skills are somewhere south
of sixth grade. She told me about having
had to help her now-adult son with division when he was in fourth grade or so; he
was in one of the several curricula where they teach factoring before they
teach long division. Since she didn't understand factoring but knew the answer
could be gotten by dividing, she taught him the familiar algorithm for long
division. He then taught it to several of his classmates, somehow leading to a
general parental demand for long division right now. The principal eventually
intervened to tell the teacher to skip over that factoring stuff and teach long
division since that was what parents wanted.
When the kids hit fractions, where
factoring is often the quick and easy way, things imploded. Not only was it now
necessary to go back and learn factoring (without reference to the fundamental
relationship between division and fractions), but in their battle against
factoring earlier that year, a large number of kids (and parents, beginning
with the one who was now my student) had become convinced that factoring was
innately evil and that the cruel teacher was going to force it on them. Fractions
ended up deferred till the next year when a more old-fashioned teacher taught
it as a set of arbitrary, memorizable rules.
My student was very satisfied that
she had helped to "make sure they taught my son basic math and that's all
you need and that's all they should teach. You don't need fractions for nothing
anyway. It's just like that factoring thing, it don't make no sense and you
don't need it. I should know. My whole family's always been good at math."*
2
Next story is about a much better
student. We'll call him Adam, since he's a composite, like every case from
tutoring I talk about here; the real kid behind the composite is actually at
least three kids. (That adult student in the first story, who was "good at
math" without being able to do much of it, was quite real and individual,
however. Kids get a pass into anonymity but a grandmother gets to own her
folly. Them's, as they quaintly say, the rules).
Adam, a pleasant nine-year-old who
was recovering quickly from major conceptual math problems, grumbled that now
that he could do math, he just wished he was "good at it." This
surprised me because once we got him through the block he turned out to be
conspicuously talented, with better concentration and work ethic than most kids
his age. If I had to pick a tutee who was "good at math" he'd be at
the top of the list; nor do I stint on praise with young kids, so it wasn't
like he'd never heard that he was doing well before.
A little inquiry revealed that
"good at math" was what his friends Brian and Claudia were. Both of
them were apparently very quick at the algorithms and usually accurate.
Now, I know the school those kids attend, and the curriculum they teach
there. After months of tutoring, Adam knew a great deal more math, and
understood it better, than most of his peers.
"Well," I said,
sympathizing, and trying to understand why he thought his fellow students were
ahead of him, "some people are just lucky enough to remember patterns
really well the first time and always get them right afterwards, and they do
get right answers very quickly."
"They don't get right
answers," Adam corrected me. "I get right answers more often than
they do."
"I thought you said they were
good at math and you don't feel like you are."
This led to a more detailed
account of the peers he envied, establishing that, "They always remember
to write the little numbers to the left and above the original numbers, and
they cross them out left to right, and they get all the 'neat and complete'
points."
"But they don't get the right
answer?"
"Sometimes they forget things
or get them backwards like I used to do. I can get the right answer, but Brian
always knows how to write everything down so it looks just like in the
book. And Claudia knows a bunch of
rhymes for how to remember what order to do things in. I wish I was good at
math like that." He looked back at the page he had been working on. "I
forget. Do I add or times next?"
3
I had been struggling to steer another
tutee -- I guess we'll call this one Darcy, she lives down in Composite City near
Adam -- into a better understanding of fractions. I've been using "rectangle
models" because that's the clearest presentation of fractions I know once
a kid is old enough to get the concept. For example, here's a video of a pretty good use of a rectangle model to explain why you have to get to a common denominator to add fractions.**
Chances are you were shown the
simple versions of rectangle models while you were learning; if your teacher
was somewhere in the better half of math teachers, some time was spent
explaining what it was about and why you were being shown it. (The less-good
half tends to do it because it's in the lesson plan, but doesn't talk as much about
why it works or what we can learn from it).
It's just the familiar business of
"so if the whole box means one, we draw a line through it and each of the
smaller boxes is one half. And we write that to show we have one of the pieces
we got by cutting it into two parts ... now here's a box cut into three pieces,
and we're going to talk about two of the pieces, how do we write that? Yes, 2/3
..." and so on.
Rectangle models are very often
used to introduce the most basic ideas about fractions, and then abandoned just
as the kid gets to the hard parts. But the good Reverend Thomas Vowler Short,
almost 200 years ago, actually developed them for teaching pretty much the
whole of fractions, all the way up through fractions made up of expressions,
and fractions and ratios in elementary algebra.
They still can be a wonderfully clear view into how complicated problems
in fractions actually work. Kho Tek Hong incorporated many aspects of them into
the bar modeling methods in Singapore Math, and many other math teachers use
them too, especially for occasions when just looking at the numbers seems to be
producing nothing but confusion. (For example, here's a pretty good one that
uses rectangle diagrams to begin the explanation of dividing whole numbers by fractions, a much tougher topic for most kids.
*** )
And Darcy was pretty much in the
same situation as her predecessors going back two hundred years; rectangle
models were lifting the fog from fractions. After two sessions of rectangle
models practice, she'd reached the point of consistently being able to draw any
straightforward fraction problem as a rectangle model. She could then either
find the answer directly from the model, or see what operations she needed to
do on the fractions.
Of course, over time, that second
pathway would become the natural one. Eventually Darcy would no longer need to
draw the model to think clearly about the problem, or could draw it in her head
instantly if she ever needed it. (Rather like the way most people learned the
Alphabet Song, and some still occasionally need to sing it to themselves to
alphabetize things, but most just know alphabetical order.) In short or via
Short, Darcy eventually learned how the algorithms for fractions worked, and
thus she had a clear idea of when to apply which ones, and to recreate or
correct any of the algorithms she might temporarily forget.
Darcy knows fractions. Now it's just a matter of practicing what she
knows until it's easy and automatic for her to do fractions. But today she's very discouraged.
"I hate all this thinking.
It's a waste of time. I wish you'd just tell me what to write where," she
sighs. "Or that they would just give us directions about how to do each
problem on tests. I just want to get the answer and go on to the next
problem."
4
If you worry at all about math
teaching in the US -- and I can't imagine you've read this far if you don't --
then no doubt you've seen this bit of second grade homework, which went viral
on the Right Wing Kookoo-Bird Web, crossing over to the general web as well.
 |
| Alas, according to the Stuck Clock Principle, there are places here where the guy is absolutely right. |
As is usual with things in the Right
Wing Kookoo-Bird Web, it's misidentified, facts have been distorted to alarm
naive readers, and the actual situation is rather different from what Glenn
Beck made of it.
Nonetheless, this is not at all an
unusual parental response, or an irrational one, and the explanation offered to
the parent was not much of an explanation. Furthermore, as you'll find in Sarah Garland's actually-fair-and-balanced article,
the homework really is badly designed for its intended purpose, the intended
purpose was inappropriate, and it's hardly a surprise that the parent couldn't
discern it.
What I want to draw your attention
to, though, is that in the face of the inexplicable assignment (or, being fair,
the assignment that could have been explicated but is still pretty badly done),
the first thing a parent does is reach for the good old reliable centuries-old
algorithm. And this is a parent who is well-acquainted with and thoroughly
grasps math himself (he wouldn't last a day in his job if he didn't).
If you do go over there to look
over the full story, read the comments, as they illustrate what I'm talking
about almost as well as the story itself.
5
From the comments on a Washington Post article about math anxiety, which quickly (d)evolved into a quarrel about Common Core:
Attack: better approach to math? You mean like forcing kids to draw 18 balloons with 5 circles each, then counting them, to reach 90, rather than just letting do the much easier task of memorizing 18*5=90?And riposte: Yes, good example. That way when they reach algebra/geometry/statistics/calculus they already understand how to think conceptually about math. Rather than students who think, "I can't possible memorize this abstract stuff!" you get students who can solve complex problems using the logic they learned solving simple problems.
Both sides sort of stabbed past each other here, but I don't
think either can be blamed for the way they missed the target; it was pretty
dark in there for everyone.
To the attack, we might say: The point of having students
calculate 18*5 is not that we don't know the answer, or even that they don't. We ask them to do it so that they will learn the
math. Knowing the answers is not knowing
the math. Knowing the fast way to
the answers is knowing a little bit of math. But knowing why that's an answer
and what it means -- that's math. And a picture is what is needed for a kid who
isn't too sure about what a number is or means yet. You don't need it now, and
neither will the kid when s/he's thirty.
But conversely, neither of you was born knowing what numbers are or how
they work.
And to the riposte, we might say: Yet the response also
misses the point: in the tutoring business, I've seen only a couple of students
at most who showed any trace of trying to get through all of math by memorizing
procedures. (The more common problem by far, as with Forrest, is that a student thinks the procedures are causal, like magic spells that
make the answer true, rather than revelatory, i.e. simply revealing what was
always true.)
I have seen more of the memorize-a-long-cookbook approach among
the ADLs, who are in a sense a population selected for having difficulty with
mathematics, but even there it's scarce. If the problem were just people trying
to memorize a complicated cookbook instead of learning math, we could give them
all a good shake, tell them what real math is, and have the problem solved before
the weekend.
The problem is that for many people, brute-force meaningless
memorization is actually more attractive than understanding math. People are not
trying to get through math that way because they don't know any better. They're
trying it because they know they like it better. And that's a much harder
problem to solve.****
6
Some of the same arguments are
played out at a much higher level (for some reason most of the trolls failed to
show up, or perhaps were whacked down) is in Leah Libresco's piece in TheAmerican Conservative. Libresco is talking about it from the teacher's perspective, and she's sharp
and clear, and several of her fellow teachers, who show up in the comments, also
get it and know how it works.
The comments also feature some of
the most useful kind of commenter for a piece about a hard idea:
honest-and-not-stupid people having a hard time seeing what it's about.
There are also some trolls and
sloganeers, of course. One apparently cannot hold their numbers to zero.
But overall, in that piece, people
are talking about understanding, and it makes a much better conversation, or at
least one less irritating to read.
The reason for including
Libresco's article here, though, is a point she makes in passing a few times,
picked up by several commenters and bulldozed irritably over by others:
The best way for a kid to get to
clarity about a concept is not necessarily the way the kid will do the related
problem later as an adult. This is hardly a surprise; it's the way learning a
complex skill that you will be using for years often works. Phonics produces
more proficient readers, but proficient adult readers rarely sound words out.
Many good cooks started out with a well-edited cookbook, measuring everything
and following directions exactly, but nowadays they just grab the right
ingredients and tools and turn the stove on. A ski instructor friend tells me
that the long journey through intermediate from just-qualified-as-intermediate
to almost-advanced is mostly moving out of knowing tricks to get down the hill
and into just skiing.
But it's also quite clear that for
many people in that conversation, procedural proficiency is all there is to
math. They keep wandering back to
"all you do is just..." as a sort of touchstone or mantra, no matter
how many earnest and respectful voices tell them that that's not
"all", it's not "just" that, and that what you
"do" is often beside the point.
7
Older readers have probably seen
the "dishonest bellhop" problem, especially because Ripley's Believe
It or Not! popularized it decades ago: three men rent a room for $30, and after
they've gone up to their room, the desk clerk notices that that room was a $25
room, so he sends the bellhop upstairs with the $5 to give to the men. The
bellhop, being dishonest (that's why we named the problem after him), only
gives each man $1.00. So now each man
has paid $9 for the night, $27 in all, and the bellhop has a $2 unauthorized
tip, and that's $29. But they paid $30. Where's the extra dollar?
Newer readers may have seen this
version of the same problem: You want a shirt that costs $100. You borrow $50
from your mother and $50 from your father. When you get to the store you find
the shirt is on sale for $97. So you buy the shirt, return $1 to your father
and $1 to your mother, and perhaps because you are secretly a bellhop, pocket
that last dollar. So effectively you borrowed $49 per parent, and pocketed one,
which adds up to $99. Where's the extra dollar?
(You can tell which problem is
newer because in one of them a hotel room is $30 and in the other a shirt is
$100...)
The quick answer is that if you
draw a little table in either case and ask where the money came from and where
it went, you'll see that the money into the problem ($30 from the 3 men; or
$100 from the parents) equals the money coming out ($25 in hotel cash register,
$3 in refunds, $2 in graft; or $97 in store cash register, $2 returned, and $1
in your pocket). Those correct solutions are treating an equation as an
equation, not as a puzzle with a double line that means "write your answer
here." The reason they fool so many people is that so few actually think
in equations. (For a much better,longer, and clearer exposition of this see the Mathemagician's blog. )
And fool people they do. Presented
as a puzzle to college or high school students, I'd say maybe 1 in 50 who have
not seen the trick before will get it. Even more amusing, most students can be
shown one of the puzzles, be taken in by it, have it explained, even be able to
work the trick themselves ... and will then fall for the other version of the
same trick the week after. The trick is irresistible to many of them: there's a
procedure and an answer, so you do the procedure and the answer is right.
Right?
§
Fundamentally, all these stories
show how extraordinarily strong in everybody, but particularly in children, is
the tendency to look for a known algorithm with clearly remembered steps to
just execute without reference to meaning. Mostly, ordinary people confronted
by math want to know what to do, and then do it. Give kids a "what to do"
and, as long as they can remember it, they'll do it forever.
I strongly suspect that one reason
that people understand better if they learn why-before-how is that all they
really want to know is "how." If you show them "how" first,
they've gotten what they came for, and they'll tune the rest out, no matter how
many advantages you can explain to knowing the "why." (And of course,
explaining the value of knowing "why" to an eight-year-old isn't always
possible; it's not a very "why" age. But as with any ability or
skill, if you're ever going to be able to do it at all, you have to start as
soon as you can, and long before you're good).
This "pull of pattern"
shouldn't be a surprise. It is not uncommon in many other situations. You can see it with people who have made hundreds
of cakes from mixes but would get nervous about making one from scratch,
doodlers who draw the same drawings over and over, and readers who read only
one very restrictive genre. There are chess players who only open with the
king's pawn, guitar players who only play the Carter lick, writers who put a
topic sentence at the beginning of every paragraph, and ballroom dancers who do
the same sequence of base steps and variants over and over without really
listening to the music. Beginning realistic drawing students often have to
struggle to get over "I know how to draw eyes" (or lips, or shoes, or
hands) and learn to draw what they see rather than what they have a
prefabricated pattern for.
The pull of the pattern is so strong, almost inescapable, because so many of our basic life skills are just such patterns. We
don't necessarily want to ride with a taxi driver who tries to take every fare
from the airport to the convention center by a different route, let alone one
who is constantly experimenting with new ways to turn or brake.
Five
hundred, or a hundred, or even fifty years ago, most people who needed to do
anything with numbers only needed a few of the simple patterns (often not even
all of them), and another almost-as-simple meta-pattern to tell them which
pattern to deploy when. But the calculator and the computer have killed the
jobs that only required simple math -- along with a vast realm of jobs that
didn't require math.
The minimum math your kids will
need for a good job -- or just to understand what is going on in the world
around them -- is much more advanced than it used to be. Once, you learned long
division because it was needed by people in business to make sure they weren't
selling below cost, teachers to figure grades, and electricians to balance a
load. Nowadays spreadsheets and specialty software do all that -- but now the
kid needs to know long division because it's one of the earliest points in math
where the possibility of alternate strategies, and the need to go back, start
over, and guess ahead enters into it, and those are all meta-skills that will
be essential in learning the much higher level math they do need.
Unfortunately,
the human brain remains wired so that patterns pull just as strongly as they
did back when patterns were all you needed.
It takes effort to push people away from just learning those patterns
and stopping there.
It takes effort to push kids away
from patterns in Singapore too. The drill schools there -- after school mass
practice at arithmetic facts and simple algorithms -- are quite common, and
really popular with parents. Quite
likely, especially when they were starting out, many parents thought the only
thing going on in the drill schools was the drill, and to this day, in the not
very good drill schools, that is sometimes the only thing they actually do.
But in the better drill schools, a long generation of
emphasizing "why" in the classroom and in homework has had its
effect; the drills are not just recitations of the answers, but also are
attentive repetitions of the ideas behind things. The students don't just say
"fourteen times fifteen is two hundred ten" or work that out on a
whiteboard while mentally reciting "put them in matching columns, put down
zero, carry two ..." and so on. Rather, they say something like
"fifteen is one and a half times ten, so we can rearrange the problem into
one and a half times fourteen times ten, half of fourteen is seven, so one and
a half times fourteen is twenty-one, times ten makes two hundred ten."
They
might then be taken through the drill another way, reciting, "the factors
are five, three, seven, and two, regroup to five times two is ten, three times
seven is twenty-one, ten times twenty-one is two hundred ten." They're
practicing two slightly different algorithms that quickly yield the right
answer -- but they're also consciously reminding themselves of commutativity,
association, distribution, and partial products while they're doing it, and
they're internalizing that the right answer is always the same, but there are
many different valid ways to get there, which is the essential principle behind
including strategy in their number sense.
The kids hate it, though. So do many teachers. Tutor
manuals at drill schools have big underlined notes saying, "Do not merely
repeat right answers. Recite the whole process all the way through every
time." My guess is that, good for them or not, the students would really
rather just be told what to do, do it, and be done.
§
A hidden advantage of memorizing the
traditional algorithms, sticking to them, and avoiding all that "why"
stuff is that it's a reliable way to keep math from getting into any other part
of life. Math produces insights into why things are the way they are, suggests
which other ways are possible, dismisses some ideas as impossible, draws
attention to perceptions about the order of things, makes the sciences
accessible, and makes people smart in a way that will not work out well for
people who need a population which is gullible and compliant enough to stay
hoodwinked.
The advantages of being really
good at real math (as opposed to quick at arithmetic algorithms) is the opening
up of whole new dimensions on the world. That requires the courage to allow our
children in general, and your child specifically, to go beyond us, to have
intellectual horizons wider and more varied than ours. Not everyone wants that:
the loss of family solidarity, the collapse of the secure position of the
elders being always right, the fear of eventually being judged by adult
children who really do know and see more -- or of not being able to share much
of a world with the grandkids after they are small -- all of these are real
fears.
It's the same fear people have
about sending the first generation to college, or about learning to read
(including the fear of having to learn to read themselves, to keep up with the
kids). For that fear, all I can say is
that we all know that acting from our courage is better for the kids than acting
from our fears, and that it is the right thing to do. Furthermore, a family
that stifles its best brains, to keep them at home, is also throwing away the
possibilities you can see in what is probably the most pro-education whiskey commercial ever.
§
More than one parent who has
considered putting their kids into tutoring with me, after asking about my
approach, has nodded, and asked some version of:
Now is Singapore Math the one with the bundles of sticks, or the one with poker chips?
Is Singapore Math the one where they draw circles around things?
Is Singapore Math lattice multiplication or regular multiplication or something else?
And of course, at that point I
know I have not been communicating very well.
The fault is almost certainly mine, but I offer, as a feeble defense, the
sheer difficulty of shaking the grip of procedure on most people's idea of math
instruction.
The real answer, which I am trying
to learn to give well orally, is that Singapore Math can be used to teach any
procedure that works, and usually, somewhere in the world, it is. For many
topics the student will learn some procedures/algorithms that are slow and
cumbersome at first because it's easy to see how they work and why they always
arrive at the right answer. But the thing the student is supposed to learn from
that is not to divide by drawing circles and counting, or to multiply by
drawing diagonal lines or laying out product matrices. What the student is learning is the why behind every algorithm: that all
multiplication of numbers too large or too complicated to memorize is done by
computing partial products and adding them, which works because of the
distributive property.
The student who understands that
overall principle thoroughly will not get lost or have memory problems with
whatever algorithm he or she eventually learns. That student is likely to
immediately see why one multiplies for area or volume but adds for perimeter,
why least common denominators are needed for adding and subtracting but not for
multiplying and dividing fractions, how long division works, and eventually how
factoring a polynomial is a fast way to find its roots.
But to find the why, the student
has to look for it, which means learning to seek it. And when a trusted adult
in the student's life dismisses the why in favor of the how -- which is what
"All you do is just ..." means -- and invites the kid to leave the
difficult path of understanding the way up the mountain, in favor of a quick
tram ride to the right answer that gets the kid off the hook, very few kids
will resist that offer.
When you offer "all you do is
just..." to them -- or even push it on them, as I've seen some parents do
-- you're turning them off the path of eventual real, deep, lasting success so
that they can have the right answer on tomorrow's homework, hand it in, and forget
everything.
Do you really want to teach your
kids to give up the richly successful but difficult long term process of really
learning real math, in favor of getting done early and having more time for
video games?
§
*She wasn't exactly one of my star students in English comp, either, by
the way.
**I don't do it quite the way this guy does. Many roads lead to the
kingdom, some of which have alternate routes, shortcuts, and interesting
scenery, so the exact route tends to be highly individual, especially for
one-on-one tutoring.
*** Again, that's only a beginning.
The next step is to note that the total number of pieces will be the
denominator of the fraction times the whole number, so you multiply those two;
and then that the divisor of this number of pieces will be the numerator of the
original fraction; and thus arriving at the invert-and-multiply rule. Which,
you then demonstrate with a slightly more complicated drawing (so you wait
until they get the simple one) works for all dividends, not just whole numbers.
And just to repeat the point once again, the subject here is not "how to
do fraction problems" but "what is going on when you do fraction
problems" -- a very different subject.
**** Humble analogy (or humbling one, considering how true it is for
me): I have done research for, and written specifications for, two different
sets of dieter-assistance software over the years. I am also fat. It's not that
I don't know it would be better for me to "eat food, mostly plants, not
too much." It's that the promise of wearing a smaller shirt in a couple of
months has one hell of a time competing with the certainty of a pizza tonight.
One reason so many problems are hard to solve with education is the rarity of
problems that are solved by knowing better.
***** In fact if you teach math, you should know the Mathemagician;
lots of good things in his toybox!
