But forget "higher-order thinking." Let's turn to basic mathematical knowledge that every sixth-grader should know, but many of my students (more and more each semester) do not. And I know they don't know these things because I have to explain them in class. Students do not know
- what a rate is: I have more than a few students who do not understand why they cannot just add the tax rate to the item price to get the total sale price.
- basic addition and subtraction: I have more than a few students who do not understand that you subtract the cost from the revenue to get the gross profit margin, or do not know that to get the total costs, you add the fixed and variable costs.
- basic multiplication and division: I have more than a few students who do not know that they must mutiply the number of units by the unit cost to get the total cost. I have more than a few students who do not know that because the interest rate is annual, they must divide it by 12 to calculate the monthly amoritization table.
- the relationship between multiplication and division: When we start doing optimization problems in Excel Solver, I have to tell students that because Solver does not like division, they must construct their problem with multiplication instead, and I have many students who do not know or understand how to do this (I also have more than a few students who do not know that you cannot divide by zero.)
- what an arithmetic mean is: I have more than a few students who not only do not understand what a mean is, but seem unable to grasp the concept. It goes without saying that they also do not grasp any statistical concept beyond the arithmetic mean.
Ed discovered yesterday that C. doesn't have a clue what a 10% reduction in price means.
He couldn't think how to figure it, and, when Ed reminded him how to figure it (he does know the procedures), he didn't know what the answer meant.
Ed reminded him about moving the decimal point. C. moved it the wrong way and came up with the possibility that the new cost would be $350. (How many times have I told him - and had him tell me - that when you "move the decimal point" by one digit you are either multiplying or dividing by ten, depending upon which way you moved it? Many.)
When he eventually figured out that 10% was $3.50, he got confused because he thought $3.50 must be the reduced price and he knew that couldn't be right. (Thank God for small favors.)
He had no idea he needed to subtract the 10% from the original price, though he did eventually realize there was a second step. (Next question: how many times have I told him - and had him tell me - that to find out what the price will be after a 10% deduction you can either multiply the original price by 0.1 and subtract the product from the original price, OR you can multiply the original price by 0.9 and be done with it? Many.)
This reminds me of my friend's son who, in 8th grade this year, could not figure a 10% tip for a pizza delivery - not even when his mom gave him pencil and paper and told him to do it that way.
He's in the accelerated math class, too.
These kids have learned nothing.
It's a nightmare.
Speaking of which, Susan J asked the other day whether it might make more sense to start with fractions this summer, instead of percent.
The answer is yes.
I've put away Algebra 1, and I've fished out my copy of Saxon 7/6 (7th grade), which it turns out I do own, after all.
We're going to be doing Saxon bar models for the rest of the summer and then on into the school year and possibly beyond......right up to the point at which C. has fractions, decimals, and percents imprinted on his tough, leathery, little pre-teen brain.
Maybe a branding iron would do the trick.
.........................
That wasn't a very nice thing to say.
hyperspecificity in autism
hyperspecificity in autism and animals
hyperspecificty in the rest of my life
hyperspecificity redux: Robert Slavin on transfer of knowledge
Inflexible Knowledge: The First Step to Expertise
Devlin on Lave
rightwingprof on what college students don't know
percent troubles
what is 10 percent?
birthday and a vacation
Take them shopping.
ReplyDeleteWhen I was a kid, every few Saturdays, a bunch of us kids in the neighborhood would get on our bikes and ride 1.5 miles on the back streets to the drugstore, where we would ponder over whether to purchase Zot!s or Jolly Ranchers or plain old Snickers with our allowance money. No adults ever went on these trips; it was just a pack of about 10 kids aged about 9 through about 14. On our own, we had to add up our purchases and figure out the tax (it was 4% back then, so really easy) to know whether we would have enough cash. (No kids had credit cards back then.) I can remember the day I realized that if my candy purchase was 12 cents or less (which was only one or two Zot!s, I wouldn't have to pay any tax.
I think I've said this before, but my mother would take me grocery shopping, and I would amuse myself in line by adding up the purchase price and calculating the tax and telling my mom the total to the penny, before we reached the register.
I probably first learned about sales tax the summer when I was five and the ice cream truck stopped at the end of our driveway every day.
I keep reading articles that say that teens, tweens, and pre-tweens are the biggest spenders (or influences of purchasers) in the family, so why can't they calculate sales tax and percent off?
Take him with you when you shop for groceries. Ask him to figure out which is a better deal, $2.39 for 8 oz. or $3.49 for 12 oz. Ask him how much the sales tax would be on an item. Find an item that's on a percent-off sale and ask him to figure out the sale price given the full price (or vice versa). Ask him to check whether the "cents per ounce" printed on the shelf tag is correct -- I have found that frequently it is not.
Ooh, ooh, an example. I was buying watch batteries at CVS the other day for about ten watches whose batteries have run down (don't even ask why so many!). The batteries came in packs of 2 for $6.99, and I almost bought those, until I saw that they also came in packs of 1 for $3.29. I hung the 2-packs back on the rack and bought out the 1-packs. :-)
This is good advice - and I was thinking about this myself.
ReplyDeleteUnfortunately, the revelation that he doesn't know what a 10% deduction is happened after a trip to Gamestop where he was, I thought, engaging in a lengthy discussion with the clerk about the various options involved in the transaction (how much he was getting back on his card for the trade-in, how much he'd lose if we subscibed to Gamer Magazine, etc.)
Come to find out, the only person having an extended conversation about the financial options was the clerk.
That's the good news, actually.
The clerk was young - he definitely knew percents!
Unless.. he was just reading everything off the LED screen.
ReplyDeleteI need to start paying attention.
ReplyDeleteSports statistics always worked for me when I was young - and this was before sabrmetrics went mainstream.
ReplyDeleteActually, one of the things for which I'm most grateful to my math education is that it lets me appreciate the work of Bill James.
I found that the Singapore sequence of fractions, then ratio, then rate worked well for my daughter. The three are so interconnected and build upon each other that she was able to take each step without much trouble.
ReplyDeleteSingapore 6A covers percents extensively - parts of a whole as a percentage, one quantity as a percentage of another, solving percentage problems by unitary method.
ReplyDeleteAs my daughter (now 10) moved through division, fractions, decimals and percents and ratios, she wanted to treat them all as isolated procedures. In trying to help her remember and understand that a fraction is a division problem is a decimal is a percentage and that she should use whatever form was appropriate to the problem, I told her that they (fractions, decimals, division, etc.) where so intimately related that they all shared the same pair of underwear. (Believe me, no 10 year old can forget that image!) When she had trouble figuring out how percents related to fractions and decimals, I explained that the underwear they were all sharing was size 100. Percents were simply equivalent fractions, expressed as 100ths. She no longer has trouble dealing with problems that require moving between the various components - she just strips those underpants off the fraction and hands them over to the percentage.
wow - great help from everyone - though now I'm stumped....
ReplyDeleteI think I'll pull out my Singapore Math books and see whether we just need to do the whole fraction/ratio/decimal/percent sequence.
also...how should I get C. started on baseball statistics - actually, he **is** started on them; reads them every day....
but I can't tell whether he's learning from them???
I'll get Ed to think about this. They talk about the stats every morning.
This is synchronicity, because I have an article on Bill James sitting on my desk.
"Percents were simply equivalent fractions, expressed as 100ths."
ReplyDeleteThat's why understanding equivalent fractions is so crucial. An equivalent fraction is also a proportion.
Once kids realize that, say, 3/4 and 6/8 are equivalent one can play the game of finding any missing number in the equation, and voila, proportions and a bit of algebra. Ain't math easy!
That's why understanding equivalent fractions is so crucial. An equivalent fraction is also a proportion.
ReplyDeleteWell....I think he's "got" this, though nobody should bet money on my perceptions....
It's not that he can't be more fluent in these things; he can.
The "real" problem seems to be "transfer of knowledge" or generalization.
rightwingprof talks about this from time to time, I think.
I've mentioned, too, that he's had virtually no word problems - certainly no systematic work on them.
He doesn't seem to be "recognizing" that a decimal/percent problem in real life is the same thing as a decimal percent problem at Gamestop.
I told her that they (fractions, decimals, division, etc.) where so intimately related that they all shared the same pair of underwear
ReplyDeleteHe'll like that.
update: Ed liked the underpants metaphor.
ReplyDeleteHaven't had a chance to spring it on C. yet.