kitchen table math, the sequel: fractions, again

## Thursday, January 17, 2013

### fractions, again

Wonderful post on Diane Ravitch's blog, written by a high school math teacher. The whole thing is great, but I especially love this passage:
I teach high school math. I took a break to work in the private sector from 2002 to 2009. Since my return, I have been stunned by my students’ lack of basic skills. How can I teach algebra 2 students about rational expressions when they can’t even deal with fractions with numbers?

Please don’t tell me this is a result of the rote learning that goes on in grade- and middle-school math classes, because I’m pretty sure that’s not what is happening at all. If that were true, I would have a room full of students who could divide fractions.

SteveH said...

It's interesting how often we still hear about the "rote" complaint when that method hasn't been taught in K-6 for 20 years or more. When my son was in first grade ten years ago, they were in the process of replacing (the horrible) MathLand with Everyday Math. MathLand had been around for many, many years. It only works if you can claim some sort of mysterious understanding that makes if OK to do really poorly on state tests that supposedly check for this sort of thinking.

And also, how can they blame teacher prep when curricula like Everyday Math tell teachers to keep moving and "trust the spiral". It supposedly works by definition and if kids don't do well, then it must be their fault (what more do they need than "Math Boxes"?), or the fault of their parents or society or whatever. What Works Clearinghouse says that Everyday Math shows some sign of working (relatively) or whatever. In fifth grade, my son's Everyday Math teacher had to not trust the spiral because a number bright kids did not know the times table and many other basics. She covered only 65% of the material but sent home a letter claiming victory over critical thinking and problem solving. Wow. skills and understanding covering less material! Blah, blah, woof, woof.

allison said...

Funny how it's Common Core's fault, when CCSS hadn't been implemented in 200-2009. The author has no idea what happens in a class room, but still found a way to complain about the wrong thing.

it's too bad that people who don't know what CCSS say blame it. CCSS are the best chance of good fractions standards being taught in the 50 states, and represent a significant leap ahead in minimal taught content in at least 48 of them, and still more coherent than the other 2.

but i'm here to argue with SteveH.

I see plenty of rote learning in fractionland, if by rote you mean "teach some tricks and procedures without the faintest appeal to reason or mathematics."

please find me a teacher using TERC or EM in 4th or 5th grade who can explain why multiplication of fractions means we multiply the numerators and multiply the denominators, but adding fractiona doesn't mean adding numerators and adding denominators.

They can't, so they don't. they default to the procedure.

you think they can explain division of fractions when they can't explain the above? they can't. again, they resort to procedure without reason.

the reason why students can't recall the procedures is because they checked out years before. they aren't drilled on any numeric computation at all to mastery, and none is explained either, so it's impossible for them to manage the multistep operation of fraction mult or div--e.g. they haven't multiplied numbers enough to see common factors.

i know kids using TERC who say " i know 3 x 7 is 21, and 5 x 7 is 35, but i can never remember 8 x 7."

6th graders, the above kids are. and scoring a "marginal proficient" on the state exam, too.

There's an example where rote learning, defined again as a procedure with not a faintest hint of reasoning, got them some of the table, but neither it nor TERC taught them that these three facts were interrelated, and a third derivable from the other two.

practicing the facts and procedures to mastery is a necessary but not sufficient condition for success in algebra. understanding why the procedures work is necessary condition for success, too. some kids can get far without knowing why, but most break down in hs or earlier.

Cassandra Turner said...

Allison:i know kids using TERC who say " i know 3 x 7 is 21, and 5 x 7 is 35, but i can never remember 8 x 7."

I meet teachers that don't see the common factors all the time.

Cassandra Turner said...

BTW: Here is a fear expressed by an Admin re:CCSS

"All teacher-prep programs will be issuing teacher credentials based upon the programs’ alignment to the Common Core, and each teacher’s proficiency in delivering the Common Core. In ten years, virtually all job-seeking teachers will be imbued with the new standards. They’ll all be equally afflicted with Common-Core poisoning. It will soon be nearly impossible to find a recently published textbook in any subject that isn’t aligned to the Common Core."

lgm said...

56=7x8 is the memory device taught in my district. There is more effort going into memory tricks than the real arithmetic.

Let me not forget my favorite..the 9s. Hold your fingers up, palms facing you. Need 2x9? Put your 2nd finger from the left down. Now look and count: 1 finger is up on the left side of the 2nd finger from the right. This is the number of tens in the answer. Now count the digits up on the right side of the 2nd finger from the right. This is the number of ones in the answer:8. So, your answer is 18. I suspect if a kid needs help on a state test, he can just look around at other students' finger math.

SteveH said...

"CCSS are the best chance of good fractions standards being taught in the 50 states, and represent a significant leap ahead in minimal taught content in at least 48 of them, and still more coherent than the other 2."

"Significant leap"? I don't see it. There is evidence that in K-6 it will be business as usual. Of course, it remains to be seen what, exactly, will be on the PARCC test our state will use, and it remains to be seen what the low cut-off point will be, but schools will continue to claim that increasing the percent of students who get over the low cut-off will define quality education. The world of math education will still be relative, not absolute. It didn't bother anyone at our high school open house last fall when they said that our proficiency level was 48% in math. They quickly added that this was one of the top scores in the state. Bad raw scores on a simple state test get converted into a percent who get over a low cut-off, and that gets translated into a state rank. They should give a course on how to make bad statistics look good. In any case, by high school, many parents don't look to state standards to determine if their kids are on a proper STEM track. Many of these parents know that this is also true for K-8.

About the only hope I have for CCSS is the hope that states will calibrate test results with ACT/SAT scores and AP tests. David Coleman of the College Board says he wants to do that, but he or the SAT may end up the loser. They will have to clearly quantify the nonlinear jump in results that move students from proficiency on the CCSS to STEM readiness in high school. They might even ask parents how that happened for their kids.

State test proficiencies define a different slope than the slope required to get good SAT/ACT scores and to be ready for a STEM career. The slope might be slightly higher for CCSS, but it still suffers from the same problem; it only defines a low end.

With one or two calibrated tests for most all states, we might get some data that better shows the nonlinearity or the need to define a higher STEM proficiency level in the lower grades.

SteveH said...

"I see plenty of rote learning in fractionland,..."

I guess I have to spell it out. There is still a common complaint that schools are teaching rote steps as defined by "traditional" methods even though those traditional methods have not been around in many schools for decades. Of course, the irony is that most reform math curricula end up teaching lots of things by rote. Their understanding is only a pie-chart type of understanding that cannot possibly lead to a flexible understanding of how to manipulate rational expressions.