Now there are friends of mine who are Curry School students, and I've seen one tutor kids on Reformation history in Clemons library -- but they're going into secondary education. This elementary classroom situation is a little different. The entire thing is a simulation of a 3rd grade (math?!) classroom, with the instructor simulating the teacher and the ed students simulating...3rd grade students. The teacher to his credit, is engaging, and his rhetorical strategies seem useful.... just the activities they're doing don't seem all that intensive.
I really can't see how participating in a national healthy foods recipe competition is all that useful. (And I love food science.) I suppose it must be about the ingredients-measuring. And calendar-time planning.
Now there's another instructor (less charismatic) coming in to discuss how to teach young kids fractions; a new piece of software -- "designed for preschoolers but has been used in elementary schools sometimes up to 4th grade". I must say I like some aspects of what I see -- they give me a, "oh hey! that might be useful!" reaction. I'd keep the sound effects and the trusting kid narrative voice, but use really powerful, engaging diagrams.
I just see many ways in which their approach could be scaled so much up and made so much more intense. One thing I do see is an underestimation and a consistent underexpectation of the imaginations of children. A reminder of the kind of elementary school classroom I grew up in. (And my first grade teacher was a graduate of Harvard.) Certainly if they do mean to inspire children to discover concepts for themselves there isn't a lot of inspiration going on.
I currently do not see a lot of inspiration in the ed school students going on either, in a "imagine the possibilities!" sort of way.
Now the lecturer is showing how the software can be used, and how it can be used in Spanish -- but not showing them for example, how it might be customised or programmed to suit a lesson -- even when the lecturer was the leader of the team responsible for its development! (Though apparently other students did write the program.)
I don't know. I imagine that an elementary school teacher could be taught how to use Mathematica and write simple scripts (fun slider bars!) to create kid-friendly yet very engaging, powerful plots.
"Why do we invert and multiply? [when dividing a fraction]" goes the instructor. "Can anybody tell me?"
ReplyDeleteno hands shoot up.
is the question too simple, that it's deemed insulting?
--
One person later called it a shortcut. But I remember Ms Tai's sweet voice -- "remember dividing is the opposite of multiplying" (more or less at this stage)
and that's all I can remember. the concept however -- and how I got it -- has been in my head ever since. Every time I have to divide a fraction (lovely stuff, harmonic variables in physics), or use Kirchoff's rules, I have a flashback to that moment.
I think she proved why dividing by some fraction b/x is the same as multiplying by x/b, starting by multiplying by x/x ("clearly 1, clearly the same", she said).
a/(x/b) = ax/b = a*(x/b)
but done in such a beautiful way that made the whole class go, "oh!!"
but I don't think elementary school teachers like to use proofs.
What?
ReplyDeleteOkay, your proof has a typo. Beyond that, you can't assume the rule to prove the rule. Be very careful with what you are saying.
To your question, the teachers in the class were not insulted by too simple a question.
THEY HAD NO IDEA WHAT THE ANSWER WAS. The shortcut answer may have been right or wrong, depending on the reasoning.
Let's ask it more: why do you invert the second term, but not the first? What is it that division of a fraction means? Where have they been given a formal definition of division from which they could come up with such an answer? Heck, where have they been given an explanation for WHY multiplication of fractions works? How would you explain it? Remember not to invoke any rules you've not already justified. Since they don't have definitions for fractions, they are pretty much sunk at providing why the rules are what they are.
she multiplied by x/x (axiom of identity), and then cancelled, and then use the axiom of associativity ax/b = a*(x/b)....
ReplyDeleteso it's not using the rule to prove the rule?
I was typing in haste -- this is what happens when you backspace and rephrase in a hurry.
a/(b/x) = ax/(bx/x) = ax/b = a*(x/b)
I don't think I've used inversion to prove the rule of inversion.
On top of that, the associativity behaviour of fractions was rather eureka too -- for a P5 student. Before that it had been just a bunch of rules, that you "kinda" knew what they meant.
of course division of fractions made most sense in situations in physics, or the situation
ReplyDeletexy =k.
Actually many Cartesian situations should be taught in elementary school.
You wouldn't have to teach quadratic equations yet (the hardest problems would be ay = bx^2, where and b are rational number), or what a focal point was, etc. definitely not.
But a lot of interesting numerical relationships -- that build "intuition" come out of plotting functions.
more followup: the instructor encouraged his students to think of situations where division of fractions would be natural and necessary, but this too, resulted in too long a silence.
PV = nT(R) might be asking too much.
and it occurs to me that many of the rates problems -- "divide distance by speed to yield time" often don't seem intuitive, without dimensional analysis, where you divide by fractions.
"If I increase or decrease my speed should I complete it in more time or less?"
but the above suggestion usually doesn't pop up till middle school.
My experience is that if a person doesn't know why invert-and-multiply works, that they won't really understand it when they see it "proven" with associativity and identity.
ReplyDeleteSymbolic manipulation only has meaning after you've already done it, and done it with things where you did know the meaning.
When you present it this way, you are suggesting that division of fractions is derived from something. But what definition is it derived from?
a/(b/x) = ax/(bx/x)
requires knowing why multiplication works the way it does: why
does a/b * c/d mean (a*b) / (c*d)?
Most elementary teachers can't explain why. Because they have not worked with any definitions--no definition for a fraction, none for multiplication, none for division.
Symbolic manipulation seems like magic to most people for many reasons, but one big one is because they don't know the difference between derived results and definitions, so they don't know when they are doing things that are meaningful.
If you start by explaining whole number division so it is abstract, that can help a great deal.
For whole numbers a, b, and q, the quotient q = b/a is just another way of writing b = qa. This is the definition of division. Note, we're not deriving this using associativity and identity, we're defining division this way.
Then, given this as a definition, division for fractions should, and does, mean the same thing. Again, the definition of A/B is the fraction C so that A = CB. From that as a starting point, you need more. You need to prove that (for whole numbers k,l,m,n) for a nonzero k/l, then for every fraction m/n, there exists a unique fraction A such that
A * k/l = m/n.
It's not obvious such exists and is unique, by the way. Now, kids will have to gloss over that or be introduced to it in a very careful form, but adults who are teaching it shouldn't.
And we can prove that A, then, is m/n * k/l.
From there, you can reason to invert-and-multiply: (for whole numbers m,n,k,l)
m/n / k/l means we want a fraction C such that m/n = C * k/l, and then you can show, by careful use of the defn of multiplication and equivalent fractions, how to show what C is.
Hmm--I like to use partitive division to explain the division of fractions algorithm. Do some examples. For instance: (2/3)/(5/7). So, 2/3 of a unit fills 5/7 of a set. How much is in a set?. First you find out how much is in 1/7 of a set. You do this by dividing by 5 (because 2/3 fills 5 pieces). Then you need to multiply by 7, because you need 7 sevenths to get a whole set, so (2/3)/(5/7) = (2/3)/5*7. Now we either do some associative law stuff: (2/3)/5*7=(2/3)*(1/5)*(7)=(2/3)*(7/5) or we just do a whole bunch of problems like that and look for patterns. Bar diagrams are great for this. Not only can I teach it to (some) 7th graders this way, I am under the illusion that my el-ed majors get it too.;)
ReplyDeleteLiping Ma has an excellent discussion on division of fractions in “Knowing and Teaching Elementary Mathematics.”
ReplyDeleteMy own preferred approach is to think in terms of speeds. If you are going, say, 30 mph, obviously (!) in five hours you go 150 miles, and you can get your speed back by taking 150/5. Similarly, in half an hour, you go 15 mph, so, somehow, 15 divided by 1/2 should again give 30 mph. Work this out systematically, and you get the standard rule.
A similar approach is to insist that the units used must not matter when they obviously shouldn’t. E.g., 36 inches divided by 9 inches is obviously 4. But that is the same as 3 ft. divided by 3/4 ft. Again, work through this ideal systematically, and you get the usual rule.
I like these two approaches because they illustrate why some obvious ways in which we would like to be able to apply division to the real world will only work if we use the standard rule.
There is nothing wrong with the “inverse of multiplication” argument, but I think one could reasonably ask “*Why* does division always have to be the inverse of multiplication?” The two examples I have given help the student see that if you want speed to always be distance over time or if you want to get the same answer for division of lengths regardless of your units, then you will need the standard definition for division of fractions, whether or not you care about division being the inverse of multiplication.
Incidentally, in abstract terms, both of these approaches are based on the truth of the “cancellation rule” (see Ma for details).
These considerations are not, in fact, solely of interest in elementary-school pedagogy. Look at when the “cancellation law” works or fails in clock arithmetic (it depends on whether the divisor has a common factor with the number of hours on the clock, the so-called “modulus”), and you start to get into some interesting issues in number theory.
And, if you really want to shoot for the moon, learn about how the failure of the cancellation law for addition of vector bundles can be "fixed" by moving to K-theory, and then try to see how this connects to the Atiyah-Singer Index Theorem, and... (I hereby reach the limit of my current knowledge of math).
My point is you could write a huge monograph on the implications of all this! But I’d be satisfied if schoolteachers had actually read and understood what Ma has to say on the subject, which is very clear and very insightful.
Dave
P.S. This was one of the things that really bugged me in grade-school math, since neither the text nor the teacher clearly explained it. It took me years to figure it out for myself along the lines I briefly sketched above.
I think you all overestimate the knowledge elementary teachers have of the mathematics they teach.
ReplyDelete" If you are going, say, 30 mph, obviously (!) in five hours you go 150 miles, and you can get your speed back by taking 150/5. Similarly, in half an hour, you go 15 mph, so, somehow, 15 divided by 1/2 should again give 30 mph. Work this out systematically, and you get the standard rule."
This was not something that most of my MSMI teachers, grades 3-6, would be comfortable doing off the top of their head. The majority would be tentative at doing it, and not trust their solutions until they'd conferred with each other. They would still have trouble explaining how this was related to the rule if they were in front of a class of students.
Allison said...
ReplyDelete"I think you all overestimate the knowledge elementary teachers have of the mathematics they teach."
No, not really I'm afraid. I think my pre-el-ed students briefly, mostly, understand why you invert and multiply while I'm teaching it, but I'm sure it's lost within a week after the end of the class (it takes a lot of time for sense making with fractions to sink in). I suspect that less than 1% of the teachers who teach division of fractions really understand it. It's most discouraging. I was greatly influenced by Li Ping Ma in my attempts to understand and teach division of fractions in a way that doesn't rely heavily on algebra (I think most high school teachers understand how to get the division of fractions rule algebraically).
Also discouraging (though you all won't find it surprising) when I went looking in Connected Mathematics to see if they had any clever ways of teaching and visualizing, I found...guess what? They don't do division of fractions! (caveat: that was the first edition--I haven't checked out the second edition). What a cop out! The whole "teach things through problem solving and conceptual understanding" sounds great until you find evidence that indicates that they do this by dropping things from the curriculum that are hard to motivate. Teaching strategies should be measured by how well they teach hard things, not how well they teach easy things...
Sorry. Off topic. I need to get a blog of my own to rant on...maybe after the end of the semester.
Allison wrote:
ReplyDelete>I think you all overestimate the knowledge elementary teachers have of the mathematics they teach.
Not if you’re referring to me!
I gave up on our elementary schools and elementary-school teachers almost precisely fifty years ago when I was in first grade: the teacher was obviously a moron, and the textbooks were obviously written for morons (and this was in the “good” post-Sputnik era!). I don’t even want to reform them – I figure reforms will just end up with changes such as longer school years that lock the poor inmates up for longer than they are already locked up.
You’re no doubt correct (with reference to my examples for explaining division of fractions) in saying:
>This was not something that most of my MSMI teachers, grades 3-6, would be comfortable doing off the top of their head. The majority would be tentative at doing it, and not trust their solutions until they'd conferred with each other. They would still have trouble explaining how this was related to the rule if they were in front of a class of students.
Still, this sort of approach, and similar approaches suggested by Liping Ma, are the right way to teach fraction division, so I throw them out here for everyone here who is engaged in homeschooling or afterschooling.
And, after all, there are a tiny fraction of non-morons who are teachers, some of whom who are here at ktm. Here and there they can be tiny, little rays of light (candles in the darkness?) who actually do teach something of substance to the kids.
Dave
Incidentally, in dealing with division of fractions, we should give the constructivists their due and recognize that we *are* free to define “division” as we wish for fractions.
ReplyDeleteWhen I was in grade school, I actually wanted to know the standard algorithms before we covered them, and I carefully tried to figure out how to divide fractions. I thought about how we refer to, say, “dividing” a bag of jellybeans in half, and I realized that is the same as multiplying by 1/2.
So, I concluded that multiplying and dividing is actually the same operation for fractions!
Before anyone laughs too hard, let me point out that in all “fields of characteristic 2,” addition and subtraction are indeed the same operation. And, fields of characteristic 2 are very useful in modern computer technology: I myself hold several patents on the applications of such fields to computers and to satellite-communication systems.
So… the idea of two mathematical operations “collapsing” into one is not as crazy as it sounds.
The problem with my proposed definition for division of fractions is that it would not be useful for solving problems such as the ones I gave above. And, of course, dividing by 4/2 would not give the same answer as dividing by 2, and the graph of y = 1/x would be badly discontinuous, and so on.
So, my definition was “wrong” because it turns out not to be useful.
However, more serious issues like this crop up all over mathematics: the “natural” way to multiply matrices might be to multiply corresponding entries, just as we do for matrix addition. But this is “wrong” because we want to use matrix multiplication to serve another purpose (to evaluate serial linear transformations).
I’m currently studying a book on algebraic geometry (Kendig’s text), and the author periodically explains why some definition is the “right” definition: it generally hinges on what the uses and consequences of that definition are. I could go through numerous cases in the history of math (fibre bundles, differential manifolds, etc.) where it took a while to figure out the “right” definition.
In real math, the rules and definitions are not really “given”; rather, they are chosen on the basis of what ends up giving the most interesting and useful results.
A “teach-the-standard-algorithms” approach or even an “accept-the-standard-definitions and-prove-theorems” approach is not really teaching math, but just teaching students to be human calculators (or uncomprehending theorem provers).
All this is no excuse for failing to teach the standard algorithms (or the standard axioms and definitions in higher math), but it does mean that Liping Ma’s slogan (“a profound understanding of fundamental mathematics”) deserves to be taken seriously, and, indeed, extended to all teaching of mathematics.
Dave