The whale problem again:
A whale swims 40 miles in 1 1/4 hour. How far does he swim in 1 hour?
The math mom I mentioned solved this problem mentally in a couple of seconds, then explained that "1 1/4" is 5 parts.
Linda writes:
What is apparent to someone with good number sense is that 1 and 1/4 is 5/4, that is five of something called "one fourth." There's no work to show, though you could explain that if five somethings equal 50, then one something is 10 and so four somethings are 40.
Since the numbers are easy, you can do it in your head; no need for visual models or bar models or algorithms. But then, I was an algebra person, not a geometry person.
It works just as well when the whale swims 37 miles in 16/13 hours except you probably can't compute (13)(37/16) without writing it down or asking the calculator. Knowing what you need to compute, however, is exactly the same.
I have to say....it's pretty sad that I still can't put it this way (or, rather, don't think to put it this way).
Of course Linda is right; 1 1/4 is 5/4. That's what a person with a good number sense "sees" or, more accurately, grasps.
Remembering back to when I first started reteaching myself K-12 math, seeing that 1 1/4 is 5 one-fourths was an obstacle. I had good fraction knowledge for an American, I think; I could create a correct word problem for 1 3/4 ÷ 1/2, the famous challenge given to 22 math elementary school math teachers in
Liping Ma's book. I had no math phobias, SAT math scores were good, took statistics in college.... in short, I am a math literate person, certainly for the U.S., and was before I started studying K-12 math.
An aside: hmmm... It's interesting that I keep using the word "see." I'm going to assume that means I personally do need to "see" this as opposed to "grasp" it.
Interestingly, as I've moved into algebra 2, I don't feel this way about many of the topics I've encountered there (practically all of which are new to me). Will have to give this some thought. At some point, the abstractions of math came to seem natural or "real" to me. At least, I think they did.
Back on topic: In spite of the fact that my own understanding of fractions was perfectly appropriate to my needs, I had trouble with the idea that 1 1/4 is five
one-fourths. The "one-fourth" is the unit;
the 5 is multiplier and 1/4 is the multiplicand. I remember my neighbor explaining it to me one night in the context of another question. (Wish I could remember it now. I may be able to find it. She was trying to explain something about fractions, I think, by pointing out what Linda has just pointed out --- and I stumbled over the explanation.)
I think bar models are a way of teaching the kind of number sense Linda is describing. Which is why C. is going to carry on doing the bar models in 3rd grade Primary Mathematics.
Sybilla Beckmann's textbook for students in education school teachings bar models, as does
Parker and Baldridge's text. (I've found Parker and Baldridge much easier to study and understand than Beckmann's book, but that may not be a fair comparison since I began at the beginning of P&B, and only dipped into Beckmann.)
procedural "versus" conceptual knowledgeGiven my experience of (relearning) fractions, I now feel strongly that schools
must teach the addition, subtraction, multiplication, and division of fractions to mastery in all kids, bar none.
I think, too, that schools should give kids word problems to solve using all four operations -- and a good number of these word problems should be "real world" word problems, by which I mean the actual real world, as opposed to the constructivist real world, which seems to consist mostly of roller coasters and hay balers (see below).
Non-GATE kids should be given fraction word problems about things like measuring a room for tile or cutting fabric from a bolt to sew a dress (I don't care if the kid actually sews or not. Sewing is something he/she might do one day, especially if he/she has kids and sends them to a public school where the curriculum is project-based.)
This approach, which I think is what I probably had, may not give non math-brain kids a good conceptual understanding of fractions. It didn't give me a good conceptual understanding.
But it did give me
all the foundation I needed to use fractions in adult life, and to pick up the study of math later on, when I wanted to.
I now believe that non-GATE kids (not sure about the GATE kids) should be taught bar models, too. No matter what curriculum or approach a teacher/school is using, consistent teaching of bar models should be included.
Cassy T has mentioned that 3rd grade is the big year for Singapore Math; that's the year bar models are introduced. (pls correct me if I'm wrong)
I would ideally have U.S. kids work through either the
3rd grade Challenging Word Problems book, or work all the bar model problems in
Primary Mathematics 3.
the hay baler problemA post from Barry a couple of years back:
Here's a problem that appears in IMP for 9th grade It is known as the "Haybaler Problem" “You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4 and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90 and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know.”
David Klein, a mathematics professor at California State University at Northridge comments on the problem. “The process of solving this problem made me resentful of the stupidity and pointlessness of it. There is nothing ‘real world’ about it. It is completely inappropriate for kids who likely have not been taught how to solve simultaneous linear equations, or exposed at most to two equations in two unknowns. If I had been given such problems at that age, I think that I would have hated math.”
Consistent with much of the philosophy of “real life math”, the goal of the exercise is to explore strategies and to be able to write about it. This is made apparent by the “student guide” that accompanies the problem. It is essentially a scoring sheet, containing categories, with points awarded for each, such as “Restate the problem in your own words” (4 points); describe all the methods you tried before reaching your solution(s) (4 points); describe the process that lead to your solution(s) (4 points); describe all assistance provided and how it helped you (2 points); state the solution (2 points); describe why your solution(s) is correct, include all supporting data (6 points). Out of a total of 50 points, only 2 are given for the solution. In fact more points are given for describing why the solution is correct.
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