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I'm really wishing for a national curriculum right now--or at least a series of textbooks that goes K-10 instead of K-6 and another for 6-8.
I'm doing math with a friend's daughter. She's homeschooling, so I get to pick the book, and we're using a Singapore text. Our most recent section is converting units of area: cm2 to m2 and that sort of thing. So, I thought to myself, this is the US, we should also do problems with in2 and ft2--I'll go look through the elementary and middle school texts in my library (I have 2-3 full series of each sitting outside my office door at work). Guess what? None of the books teach the topic at all. Aargh! Everyone is assuming someone else is teaching it, and no one is.
The first time I saw anything about area conversions was in the Saxon Math books I used to teach myself math a few years ago.
I learned inch-to-centimeter and inch-to-foot-to-yard conversions in school, but I learned nothing about area or volume conversions, and I continue to find volume conversions slightly confusing.
Speaking of Saxon Math, I've been thinking I need to get back to my books. I had almost finished the second book in the high school series when I was diverted to whatever I was diverted to. At the time, I was finding logs difficult to deal with in the "shuffled" organization of Saxon Math, especially since I wasn't studying every day.
Speaking of logs, I was emailing with Barry G. earlier today and I compiled a list of all the topics I was never, ever exposed to in high school math or college statistics.
I'll post tomorrow when I'm at my other computer.
The list is too long to remember off the top of my head.
16 comments:
The thing that usually trips people up about volume conversions is that they stop converting too soon.
Consider converting 1 cubic foot to cubic centimeters. There are 30.48 cm/ft. It is common to take 1 ft^3 x 30.48 cm/ft and stop, but you've only converted 1 of the three ft to cm (so your units would be 30.48 cm*ft*ft). You need to cube the conversion unit, so:
1 ft*ft*ft x (30.48 cm/ft) x (30.48 cm/ft) x (30.48 cm/ft) = 28316 cm^3!
There are US versions of the Singapore primary math books. They don't drop the metric stuff -- they add the customary measure stuff on *in addition* to the metric.
In my state, I think that unit conversions are in the 6th grade science curriculum to shift the burden off of an already jam-packed math curriculum.
I tried to post this already, but I think blogger ate my comments, so trying again. Apologies if it shows up twice.
One of the classes I teach every year is "Geometry for Elementary Teachers." Last year, on the final exam, I included the following multiple-choice question:
A trapezoid has bases 14 in. and 22 in., and a height of 20 in. It's area is…
a. 720 in^2
b. 30 ft^2
c. 2.5 ft^2
d. 3080 in^2
I included responses (a) and (d) as distractors, to catch people who tried to memorize a formula without understanding anything about what it means. (a) comes from using A = (b1 + b2)*h, which produces an area larger than the rectangle that encloses the trapezoid, so cannot be right. (d) comes from using A = (b1*b2)*h / 2, which ought to have units of (length)^3, so also cannot be right.
(b) is also a distractor, meant to catch people who convert from square inches to square feet by dividing by 12 (forgetting that one square foot contains 144 square inches, not 12 square inches). The correct answer is of course (c).
Would you like to know the distribution of results? I am almost afraid to post them.
Also, apologies for the superfluous apostrophe in "it's" above. I am usually much sharper than that.
I would love to see the distribution. Also like to see your syllabus, and what you think the hardest part is for the teachers.
Before you post the distribution - I'm going to guess.... I'm guessing that a & b were the most common answers (not sure which I think would be more common). I wouldn't expect very many people to make the error that leads to d. I'm going to guess <25% had the correct answer...
But what I wonder is, did the students write their work out or do the calculation in their heads? I tend to think that there might be fewer errors if they were required to present their work.
Out of 32 students, 7 chose a, 20 chose b, 3 chose c, and 2 chose d. Yes, that's right: 62.5% of the student chose b. This is after we spent a significant portion of a class period going over several homework problems including variations of "how many square inches in a square foot"?
Allison: I can't post my syllabus right now, but we use the textbook by Billstein. Like any text, it has its strengths and weaknesses.
Michael,
So what do you attribute the 62% to?
Was it that you fundamentally couldn't convey what ft^2 means is square ft, is ft*ft? Is it that in a test situation, they resorted to procedural grasping at straws? Was it the familiarity of the wrong answer compared to 2.5 sq ft, a decimal (Or the shrink in the order of magnitude? Teachers used to do trades in the summer (at least when they were men)...do they have so little actual knowledge of how many inches on a side gives you 2.5 ft?)
something else?
It is so painful to watch a teacher use a method without comprehending, clinging to it, while you're standing there telling them that you're "tricking" them, giving them a problem where that method will fail them (and therefore fail their students, too.) You hope at least the self awareness kicks in so that they can see that's what their students will do, too.
Is your course a preservice course (or inservice?)
I would love to see the distribution. Also like to see your syllabus, and what you think the hardest part is for the teachers.
I was never properly exposed to units before college. I learned some common conversions, but never a general approach. I remember having to convert inches per second into furlongs per fortnight in college. It's really very simple. Actually, I got that in engineering, not math.
I like to tell kids that you can treat units just like factors and manipulate them like numbers. If you have 5 miles per hour, then that is the same as (5*miles)/hr. If you have 12 inches = 1 foot, then (12*inches)/(1*foot) = 1 and you can multiply that times anything because you can multiply anything by one. Also, (1*foot)/(12*inches) = 1. You can do this for any conversion factor. I have a well-worn units conversion booklet within arms reach where I can find conversions between things like kilowatt-hrs and gram-calories.
A notational issue is that you might have units called ft-lbs, but it really should be thought of as ft*lbs. If I have a weight of 100 lbs at the end of a 5 ft (half length) seesaw, then the moment would be 500 ft-lbs. It's better to think of this as 500*ft*lbs.
Then if you divide this number by 2 ft, then you could see this as:
(500*ft*lbs)/(2*ft)
or, moving factors around,
500/2 * ft/ft * lbs = 250 lbs
ft/ft = 1. You can manipulate units just like factors.
If you have 360 in^2, then that is the same as
360*in*in
If (1*ft)/12*in) = 1, then I can do this:
(360*in*in)*(1*ft)/(12*in)*(1*ft)/(12*in)
moving factors around and cancelling, I get
360/(12*12)*ft*ft
= 2.5 ft^2
You need something like this when you get to complicated units like viscosity, which might be defined using:
kilogram per meter-second
which is kg/(m*sec)
You can actually find it defined as:
kg m^-1 sec^-1
or
kg * m^-1 * sec^-1
You can bring the units from the denominator to the numerator by changing the sign of the exponent, just like a factor.
I don't know why units are not discussed more in the lower grades. They can be very helpful for simple DRT problems. The units help to see if you are doing the problem correctly. If you get confused about whether you need to multiply or divide, the units can often tell you what to do.
Allison,
Preservice.
I am not sure how to account for these results. Obviously part of me worries that it is my fault: teachers always feel some responsibility for their students' successes or failures. And part of me is defensive, wanting to blame it on decades of bad intellectual habits ingrained in the students.
I am certain that most students came up with the answer 360 in^2, saw that was not listed as a choice, and thought "Oh, I have to divide by 12 to convert inches to feet", and did not think any further about it. I think if I had explicitly asked students "How many square inches in a square foot"? most would have gotten it right. I even suspect that if I had explicitly asked "Convert 360 in^2 to square feet" most would have gotten it right. But because the conversion was not the obvious focus of the problem (ostensibly this was a question about calculating the area of a trapezoid; the units conversion was just an en passant move), nothing in the problem cued them to think carefully about the units.
Which, of course, is precisely why I wrote the problem as I did. Teachers need to be thinking about details even when they are not double-underlined and highlighted in red.
I can tell you that the same question will appear on my final exam this semester, too, and I will let you know (in about a month) how the results come out.
Well, I think the "didn't think further" answer is probably right. I think it matches my experience working with teachers in inservice. The fundamental problem seems to be that teachers don't think mathematically. They don't think about what the definitions require, what is implied, what must follow from what they know. And a course or two just doesn't overcome that lack of thinking mathematically, and as teachers, we who think mathematically have not figured out how to lead them there. So part of it is our fault, part is the bad habits. The question is: how simple can we make it to think mathematically? What as teachers of teachers can we actually do, and how much of it is hard work and time on their part? I don't know.
So, if you don't think about definitions in this case, you don't think about what the unit definition means, you don't think about what units you need to convert to, don't think beyond the focus of "I'm supposed to be DOING something right now!" Their students have the same problem.
Just waking them up to see how much they do this, and how much their students do is probably the first real step in getting teachers to realize how much thinking mathematically matters. But even that is difficult work.
I recall my introduction to unit conversions when I took chemistry as a sophomore in high school. Early on in the course, our instructor helped us get on the right track by teaching us about "mole island" and giving us practice problems. The light bulb went on for me when I realized that the unit designations themselves could be manipulated algebraically, and as long as I kept track of what units I needed to eliminate and which I needed to have in my final answer, I would never again be confused as to whether multiplication or division was required to perform the unit conversion.
Haven't read the comments, but I realize my post is unclear: when I said that I learned nothing about area or volume conversions, I meant that I learned nothing in school.
Saxon Math covered them.
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