In my last post, I mentioned my top ten list of 'problems'. I decided to call it my top ten 'issues' because I wanted to frame them in the positive - not what's wrong, but how to do it right. However, this is still a list of the top ten problems, because most of this is done wrong in most classrooms.
And here they are:
1. If you're going to teach math, you need to enjoy it.*
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they'll be willing to really sweat.
2. If you’re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I'm still learning about multiplication myself. Over at Axioms to Teach By, I said (back in September), "You don't want the product to be 'the same kind of thing'. ... 5 students per row times 8 rows is 40 students. So I have students/row * rows = students." Owen disagreed with me, and Burt's comment on my last post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to use a compass and straightedge to multiply. I'm looking forward to playing with that. I think it will give me new insight.
3. Games are to math as books are to reading. Let the kids play games (or make up their own games) instead of "doing math", and they might learn more math.
Denise's game that's worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice. Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four.
4. Students are willing to do the deep work necessary to learn math if and only if they’re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. (Maybe you'll get to read him here soon.) I'm not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons.
5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: "The secret key to mathematics is pattern."
U.S. classrooms are way too focused on procedure in math. It's hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)
See George Hart for the artform. I don't know who to recommend for the language angle. Any recommendations?
6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, ... And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)
7. Math itself is the authority - not the curriculum, not the teacher, not the standards committee.
Read Math Mojo – you can’t want kids to do it the way you do. You have to be fearless, and you need to see the connections.
8. Real mathematicians ask why.
If you’re trying to memorize it, you’re probably being pushed to learn something that hasn’t built up meaning for you. See Julie Brenna's article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, ...
Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it?
9. Earlier is not better.
The schools are pushing academics earlier and earlier. That's not a good idea. If young people learn to read when they're ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray's recent post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that's all it took.
In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA. Although some people disagree about the success of this experiment, there is nothing published which contradicts his evidence. I'd like to find more information about how this project ended.)
10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn't be last, but when I look at the list, they all seem important. I guess this isn't a well-ordered domain. ;^) Textbook Free: Kicking the Habit is an article by Chris Shore on getting away from using a textbook. (After clicking the link, click on 'Articles'.) I've been duly inspired, and will report in the fall about how it goes for me in my classes to teach without a textbook. See dy/dan on being less helpful (so the students will learn more).
31. Multiplication is not (just) repeated addition, it’s much richer than that.
Wait. I said that already... (I warned you, it's just not in my top ten.)
What do you see as the biggest issues or problems in math education?
____
*I know, top ten lists are supposed to start at number ten to keep the suspense up. But the suspense is gone - I already told you my top two in my last post. And I can't help it, I just have to start at the top.
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I am completely, 100% down with your list. I just with I knew how to convince the people who decide what we teach (and when and how.)
ReplyDeleteI seem to remember a few wild-eyed radicals talking about some revolution and changing the world. I think Colleen started it.
ReplyDelete(And Maria posted a manifesto in the comments to Colleen's post.) ;^)
ReplyDeleteSue,
ReplyDeleteI like the list... although I think if you get #1, then #2 is not an issue... and I would maybe insert 8b) A mathematician asks "What if..."
Ooh, I like that! (8b)
ReplyDeleteUnfortunately, #2 is needed. I've met people who genuinely like math now, but haven't the passion for it, or the commitment to it, to learn it more deeply. And they end up thinking a lot of things are done the way they are 'just because' - and trying to tell students that's how it is.
I think some of the folks who complain about reform curricula might have in mind teachers who are now having fun playing with manipulatives with kids, but not doing anything to move them forward.
Nice post Sue. I am sharing this with my colleagues.
ReplyDeleteAnd thank for the mention. I strongly believe that the use of grades in school is one of the number one contributors to student apathy towards learning.
I can't wait to guest blog with you.
For #5, I might suggest Ian Stewart? He plays with language (bad puns) and math, and at least some of his books show that math is a language of abstraction.
ReplyDeleteAlso, re "Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons." I think it's much like AP classes here: the grading dynamic really changes when it's teacher + students vs external exam, instead of students vs teacher-generated exam. In Japan (and China) I think a lot of the pressure is about doing well on external assessments such as college admissions tests, so teachers and students are on the same team.
This is the one thing that I think might be good about all the standards-based testing students are put through: in at least some places, those external exams can be used to get the teacher and students on the same team when before they might have been adversaries.
Well, there's a difference between doing the best we possibly can with the crappy standards we've been handed, and getting to work with quality curricula in the first place. Don't you think? There's only so much pushing back and revolting I can do when I'm held accountable to a test of an unreasonable amount of age- and math-sophistication-inappropriate material.
ReplyDeleteYes, Kate, I agree with you totally. (I was being flippant before. Sorry if it felt like I was ignoring your real concern.)
ReplyDelete>I just wish I knew how to convince the people who decide what we teach (and when and how.)
There's no easy answer to that. If it were one person, maybe you could go in and get them to see all this. But it's not; it's a huge bureaucracy. So the question is, how do we change the huge institution we know as school? And that's a hard one.
For me, it seems like a first step is getting math teachers who understand how big the change needs to be, and then all together we can organize and pressure the system. I'm not at all clear on how to do something like that successfully.
And maybe even that language (organize and pressure) is wrong. Maybe the change will be more organic.
I think our 'revolution' posts may point to ways the internet can effect huge change. It's not just about how teachers teach, it's about how students interact with the system (school).
But you are stuck in a rotten system. When they require you to teach too much, you cannot do right by the students.
At my college, I have more flexibility, but there are still very rotten aspects: Why does a math course need to have intermediate algebra (alg II) as a prerequisite to count as college level? I say it's because we're using math as a filter, in a way that's not necessary, and that's ugly.
You are a woman after my own heart and mind, Sue. I've known that from my first read of your blog. You say the things that need saying, and that I have said many times before in various settings.
ReplyDeletePreach it, Sister!
(blushing)
ReplyDelete>I think it's much like AP classes here: the grading dynamic really changes when it's teacher + students vs external exam...
ReplyDeleteGreat point! I wouldn't go so far as to call it a positive aspect of the testing, but I can see what you're saying there...
I do think there's a cultural difference, too.
I'm not sure what my top 10 would be, but let me point out some of yours that I especially like:
ReplyDelete9. Earlier is not better. I wish there were more discussion of this! It's a constant battle to slow things down, to dig deeper, to give kids time to play with the math, to gain understanding. Which leads me to
3. Games are to math as books are to reading. Playing with math really means playing with math. I would pair this with problem solving (in the sense of working on unusual, longer, problems)
2. To teach math you need to know it deeply.
And then I think we diverge, or maybe I would phrase things differently.
Arithmetic is rich. I think that doesn't match you exactly. But that might be on my list.
Jonathan
jd2718
love it. thanks for the nod. for multiplying with compass & straightedge one need only construct certain similar triangles. right triangles at that so it's even easier. bust out a couple compasses and an interlocuter; i guarantee you'll find out *something* interesting. nothing says spring like repeated addition. yours in the struggle. owen thomas (MathEdZine). http://vlorbik.wordpress.com/
ReplyDeleteon (5)... specifically on
ReplyDelete"the language of math"...
the handbook of math'l discourse
by charles wells is terrific.
(bonus point: it cites me.)
its author (charles wells)
does a blog. ot.
pat b (of this very thread)
ReplyDeletehas also worked extensively
on vocabulary (follow his link).
Owen neglected to point to his own post, written in response to this, which was too long to fit in a comment. (You'll want two windows open.)
ReplyDelete"4) is outright false or i’d've never
finished my thesis…" got me chuckling.
Thanks, Owen! :^)
My top ten: of the day, and in no particular order.
ReplyDelete1 - There isn't enough user-generated content or "making your own math artifacts." In other words, severe dis-balance between giving and taking, especially for children.
2 - The general tendency is to have an extremely impoverished list of topics for young kids. If I had a penny for every list consisting of counting and shapes entirely...
3 - Speaking of age, tying content level to kids' ages is an unbelievably silly idea. Just say "No!"
4 - Kids are barred from adult tasks, in all forms. Even seeing a professional at work, up close, is a rarity, let along long-term apprenticeships or short-term internships. The younger the kids, the harder it is for them to work. Wrong, wrong, WRONG!
5 - Impoverished example spaces are prevalent throughout. Say, many curricula are using one positional system (decimal) and only two positions in it (tens and hundreds) to teach the idea initially. As a result, each example is taking an inordinately long time to learn, with shallow understanding.
6 - Forgetting we still live in meatspace bites. Kids need to move every 10-15 minutes, and eat every half an hour or so while doing math. Also, tactile comfort objects, smells, and sounds make a difference. Oh, and while we are discussing this, learn to teach every major topic through whole-body movement.
7 - Lack of group design is widespread. If you work in a space containing more than one kid, pretty please do tasks that work better in groups vs. individually. If you can't design such tasks, don't work in such spaces. Hint: worksheets usually work better individually. And by "individually" I mean "one person in a room with one teacher."
8 - Math 2.0 where art thou?
9 - Modern literacies are often nonexistent: programming, gaming, visual, social media, computational.
10 - Stop ignoring research. Those in any way related to education, especially policymakers, who don't know first year undergraduate math methods course terminology will thereby be made into my literature review assistants for scholarly article writing. Database training included.
ten for maria.
ReplyDelete1 - There isn't enough user-generated content or "making your own math artifacts."
equations, most likely, first.
but wait. zero-th.
by-hand copies of the *symbols*
for the material at hand.
"the student learns essentially
nothing until the student's
pencil makes marks on the page"
is a pretty good first approximation
a lot of the time... or anyhow,
i'm far from the only teacher
given to *saying* stuff like this.
i've got plenty to say, too, *about*
this but i'm hoping for a list of ten
in under 2^12 characters (for a little
longer; i've begun to despair already
at least a little though if you want
to know the truth).
"unions" should look different from "u" 's
as an example more or less at random.
*our medium is handwriting.*
first-and-a-half.
out-loud discussion of and...
second.
...written sentences *about*
those equations. written
at leisure without the
instructor (or fellow student).
third.
similar or exact versions of such equations,
repeated, or, much better of course,
improvised, in a "public" setting
with small or, slightly better i
suppose, large *groups* of fellow
students. oral presentation of
the sentences themselves is not
only okay here but much to be
preferred (the board should not
be littered with sentences).
the "correctness" of the sentences
should nonetheless be at issue
throughout the presentation.
said "correctness" is to refer
explicitly to "code"...
utilizing (hey! ed jargon!)
the symbols from our step zero.
it does not escape my attention
that the "artifacts" created by
the student presentations i here
imagine are scribbles of chalk
on a board, soon erased. so be it.
...
leaving some out...
sixth
yick, computer code.
seventh
student-designed exercises,
exam templates, lesson plans...
eighth
songs and other verse, games,
comics and other graphics,
something to astonish even me.
ninth
blogs.
tenth
fanzines.
Maria, I loved your list.
ReplyDeleteOwen wrote:
>"the student learns essentially
nothing until the student's
pencil makes marks on the page"
Maybe for higher math, but not at all for young kids. The mathematical issues they're working on don't usually require pen(cil) and paper.
My son is thinking so much about what I'd call place value these days. "60 and 60 is 120, right?" "Yep." No writing - at home, anyway. Lots of mathematical thought.
I've long been puzzled by your emphasis on "the code". Maybe someday I'll get it... :^)
My son is thinking so much about what I'd call place value these days. "60 and 60 is 120, right?" "Yep." No writing - at home, anyway. Lots of mathematical thought.
ReplyDeleteI've long been puzzled by your emphasis on "the code". Maybe someday I'll get it... :^)
---sue v.
maybe today!
the "places" of "place value" are
places *in* certain symbol strings!
it sure doesn't matter that you
*speak* of such strings without
having actual *written* code
in front of your actual eyes...
that's not what i'm always
going on about at all...
60+60=120
presumably gets its interest
from 6+6=12,
together with, right,
the "place value" concept...
*as it manifests in base ten*.
now of course you and your kid
don't have to have spoken of
bases-other-than-ten for
the essential *role* of "ten"
in discussions of place value
to have become quite clear
all around.
"what's so special about ten?"
i can now imagine asking
some kid of the same age
if i were lucky enough to
know any...
and i'd sure enough expect
(maybe with a *little*
stack-the-deck prompting
from me) pretty soon to
start hearing about the
role of *zero* (in, again,
certain symbol-strings).
and when our conversations
*without* written work begin
to break down... and if we
still *care*... why then,
we'll break out some *pencils*
and take a look:
"what do you mean, *precisely*?".
we've been talking about code all along.
tangent.
calculating with numbers
is the very *model* of
one-right-answer-ism:
3*4=13
is just flat-out wrong.
and this is our greatest strength.
in principle, anything worth
talking about passionately
in a math class should have
the *same* character:
there *is* a right answer
if we could only find it.
in order to have this happen,
we have to agree on things.
we *can't* agree... and be
*sure* we agree... and be *right*...
without certain so-called "rigorous
definitions": marks on paper
(generally; otherwise
*verbatim verbal formulas*
memorized syllable-for-syllable
[mostly... i don't seek a
"rigorous" definition of "rigor"...
"one is *this* many"
and its ilk (so-called "ostensive
definitions") are all the rigor
we can *get* sometimes]).
generally the "rigor" one speaks of
is... i think... pretty *close* to the
being-able-to-calculate-it-out-like-a-computer
thing i spoke of (with reference to
elementary arithmetic) a moment ago.
and this comes from "code".
again. our power in mathematics
comes to an amazing extent from
being-able-in-principle to emulate
some doesn't-know-anything-*but*-code
*machine*.
now i'm as much of a luddite as the next
guy, if the next guy figures the wrong turn
was somewhere around "domesticated animals".
but one *glaring* benefit of computers
in math ed is that students will work
for *hours* on getting code letter-perfect
(if they know no human being can see
their failures happening), that wouldn't put
in five *minutes* of homework on paper
without getting so frantic about each
"move" that they fall apart before even
getting started. it's that "interactivity".
this used to break my heart but it's true.
if schools were for clarity,
command-line programming
would begin in about first grade.
it's much *easier* than almost
any other thing you can do
with a computer (which is why
it emerged much *earlier*
than the hugely-user-unfriendly
[from a "code" point of view]
*graphical* interfaces that
erased it from the national
consciousness in around 1984).
(somebody mention "logo".)
math *is* hard.
but it's much easier than anything else.
because we've got *all* the certainty.
(programming on this model
is of course a subset of math).
ot
it sounds like one-right-answer is part of what drew you to math in the beginning. for me, i was less one-right-answer than "the teacher can't tell me i'm wrong, and i can tell her she's wrong (if she is)." heady power for a kid.
ReplyDelete3*4=13 isn't wrong if you're in base 9. i love finding what's right in a wrong answer. it's a good use i've put my arguing skills (lawyer for a daddy) to as a teacher. i can say "it sounds like you're thinking about this other sort of problem." i can show a student the right thing they did among the wrong thing(s).
programming would be a draw for some kids. papert saw it that way, too. i don't think it's "the answer" for all kids. and can you imagine a teacher saying "your program is wrong" because it doesn't match the model 'right answer'? yikes!
it sounds like one-right-answer is part of what drew you to math in the beginning. for me, i was less one-right-answer than "the teacher can't tell me i'm wrong, and i can tell her she's wrong (if she is)." heady power for a kid.
ReplyDeletethis looks like, not only the same
ballpark, but the same foul line.
(our answers match almost exactly;
here again is "math is the authority".
it looks to me... and it may just be
another kid-with-a-hammer
obsession-of-the-day... like
to the extent that we differ at all
in discussions around "can there
really be foundations?", one big
issue is *context dependence*.
"is it *always* such-and-such"
asks the student.
and we have an opportunity to
do *philosophy* of math:
"hold on... there is no 'always'...
what type of object *is*
blahblahblah?"
[student stares dumbfaced as if
you have betrayed them: you're
"getting all mathy" on them].
now you've gotta make a decision.
go back to plain english and confusion
(usually this is the *right* option,
so don't get me wrong here)
or remind them once again...
and make it stick a little better...
that by "always" one means something
like
"every monic trinomial"
or
"every continuous function"
or
"all the equations in the exercise set"
or
some particular thing
with
dammit, a *name*.
and that we can't mean what we say
until we can say what we mean and
that this occurs much less often than
people think but it's easily remedied
if you'll just take some bloody care
once in a while like a math geek
and that one of our best tricks is
specifying the universe of discourse.
3*4=13 isn't wrong if you're in base 9. i love finding what's right in a wrong answer. it's a good use i've put my arguing skills (lawyer for a daddy) to as a teacher. i can say "it sounds like you're thinking about this other sort of problem." i can show a student the right thing they did among the wrong thing(s).
a great trick indeed.
so i'll put my mojo on it.
"your answer would be marked 'wrong'
*in the context*
of a [graded] base-ten calculation"
("... but *we* know... even if your
silly dont-blame-me-those-are-the-rules
*grader* doesn't know or care... that
we're clever enough, not only to
understand what's wrong
*in base ten context*
can be made right by
*changing* the context to base 9...")
programming would be a draw for some kids. papert saw it that way, too. i don't think it's "the answer" for all kids. and can you imagine a teacher saying "your program is wrong" because it doesn't match the model 'right answer'? yikes!
only for the ones who want to know how computers work (loops and logic); right. the "authority" here typically comes from "what happens when you run the code" (and not from the teacher's pet peeves as happens all too often in elementary math); this is a Good Thing.
"most of this is done wrong in most classrooms" Agree! If you really want to teach kids math you need to be creative enough and not just stick to standard kind of teaching. Make it more interactive and challenging through different activities and games.. Thanks for the lists.
ReplyDeleteSue, I understand why Dr. Marty directed me to your blog!! I'll need to come back to this post and leave comments... Wow! Say it Loud and Proud....
ReplyDeleteWe're from Singapore and as you know there is much talk about the country's success in Math Olympaid and the like.
ReplyDeleteHowever, I would like to share that the way math is taught here is darn boring. Typical scenario is:
Teacher teaches concept, show method to solve problem and after that it's just lots and lots of drills and practice to ensure you get your A's for exams.
I see so many interesting ways that you guys make math fun for the kids and I wish this can be done here.
The problem with the way math learning is done here is that they pile up so much in the syllabus at the primary level. Only the smart ones can cope well, the rest just work like hell. Also, there is so much emphasis on mathematical heuristics which I feel, some kids are not quite ready for it at a young age. So in the end, they just get as much exposure as they can from practicing and memorise all the diff methods. That is not learning to think, it's just learning how to be a robot and react to a problem as quickly as possible so you can finish the exam paper in time.
My daughter is 11 and a gifted child. Many times she discovers ambiguity in the questions in the teacher's worksheets. I told her this is a good learning situation and she can discuss with your teacher. I'm dismayed that the teacher just tell her to use a particular interpretation because that fits the answer. She refuse to admit that the question has a problem. That is ridiculous. I feel all learning should be interactive between kids and teachers and I"m so so disappointed the teacher just kill a kids curiousity like that.
I wish I had some advice to offer you, but I don't know anything about how life works there, and wouldn't be able to say anything useful to you.
ReplyDeleteGood luck.
Hi, Sue, as I dig deeper in your blog I keep finding worthwhile posts.
ReplyDeleteI think I've found the reference you'd asked for:
this video by James Tanton explains the U-shape of the general quadratic:
All Quadratic Graphs are U-shaped
Umm, where did I ask for a reference? I'm sure I did, but I'm not remembering...
ReplyDeleteHere you are:
ReplyDelete"Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (This was a recent blog post somewhere. I can't find it, though. Can someone point me to it?)"
Duh. Right up there. I'll edit now. Thanks.
ReplyDeleteNice article highlighting the reasons for math phobia and the solutions to end it. I totally agree with you on this. People (both children and grown-ups) need to understand that math is not boring, it is recreational. This will make learning math a happy and positive experience. (http://myblogxpedition.blogspot.com/2014/03/math-can-be-recreational.html)
ReplyDelete