elementary school math-fact challenges

[chhotii]

Here's what I don't get: I've been told that the MCAS is un-timed. (Is this true?)

If this is so, why is there so much emphasis on doing the math-fact sheets as fast as possible? Timing them and so forth? For some kids, rising to the challenge will work. But there are some kids (such as Sophia, and at least one other kid you know...) for whom the timer causes so much activation of the sympathetic nervous system that it impairs both performance and learning. In a big way.

Mathematicians are not judged on speed. So why is math treated like a track event in 2nd grade?

Comments

[spinrabbit.livejournal.com]

It is true that the MCAS is un-timed. And I agree that the "mad minute" style of math-facts sheets can be counter-productive for some kids.

What they're aiming for with the race-style sheets is to create a situation that requires the kids to recall rather than re-calculate the basic facts. Because if you have to figure out, rather than remember, what 7x6 is later on when you're doing long division or simplifying equations in algebra or some such, it's going to make extra work and introduce extra sources of error. There are certainly other ways to encourage memorization though.

(I say this as someone who didn't have 7x6 absolutely down pat until sometime in high school, and more than once had to rederive the quadratic formula during a test rather than just using it like she was supposed to)

[achinhibitor.livejournal.com]

There are certainly other ways to encourage memorization though.

That's likely true, but is there a way to test for memorization?

I don't know if it has changed again, but for a while the popular method of teaching math was oriented toward rediscovering what was already known rather than toward memorizing it. That environment wouldn't encourage memorization.

[chhotii.livejournal.com]

but is there a way to test for memorization?

Probably not. Where do you draw the line between memorization and a really fast algorithm? For example, my boss, who is no spring chicken and a math wiz and the P.I. of his own lab, claims that (based on self-introspection) he doesn't actually have 6+8 memorized. When he sees 6+8, his brain does:
6+8
=4+2+8
=4+10
=14

I do this too. So I've been trying to teach Sophia to manipulate addition problems using "ten buddies" when she's stuck. But I don't really believe that my boss really needs to do this. I think that some of us just have a knee-jerk need to manipulate numbers. But how could you ever tell for sure?

[gosling.livejournal.com]

I do this too. Also, I was so terrible at the "speed and accuracy" (which was really speed) tests in the early elementary years that it was basically assumed I had no math ability at all, an assumption I did not actually think to question until many, many years later.

(I will note that my father, who was a quite competent physicist, was considerably worse at that sort of speed calculation/memorization than I was. He also essentially saw the world in mathematical terms much of the time as far as I could tell.)

[sweetmmeblue.livejournal.com]

I was terrible at it because it made my disgraphia worse. I knew the answer but when I went to write it it came out wrong.

[gosling.livejournal.com]

I had that problem pretty much entirely throughout my education. In my first year chemistry class in college I was often essentially teaching the (perfectly competent) tutor things she knew less well than I; nonetheless I was unable to pass the class because of exactly that problem. (If the tests had been untimed I could have gone through and checked my work several times until I caught all the transposed and otherwise misplaced numbers. I did fine in the untimed class assignments; they just took me a while longer than some people.)

I have wondered for close to thirty years if I have undiagnosed disgraphia.

[skreeky]

I sort of do addition that way myself, only it's visual. Base 10 numbers are like containers with 10 spaces, and the container has 6 spaces already full, leaving 4 spaces. You want to add 8 blocks (the items are always perfectly cubical blocks fitting into a rectangular tube), and 4 of them fit into the existing container, leaving 4 to go in the next container. Hence 14. Since it's visual, though, it's essentially instantaneous. I just *see* the way the numbers fit together and that's the answer.

I figure I'm just weird. I certainly never tried to explain this to an adult.

[skreeky]

I'll follow up to say that I have no such visual built into my brain for times tables, and thus I HAAAAAAAAATED them. I still don't know what 7x8 is without thinking really really hard, because I don't have a mnemonic trick for 7s or 8s. I actually have to mentally make a factor tree and multiply the 7 by the 2s until I get there. Sometimes I remember that I can do 7x9 and subtract 7. (9s are great. You subtract 1 from the multiplier for the first digit, and then subtract that digit from 9 for the second: 7-1=6, 9-6=3, so it's 63, and I didn't have to do a whit of memorizing numbers. Incidentally, 63-7 is 56, which I just did with visual tubes.)

I pretty much flunked timed worksheets.

My dad is a mathematician, and I felt much better when he told me that not only does he stink at arithmetic but that most of the "real" mathematicians he knows stink at arithmetic, and that I was going to be really good at the conceptual stuff later. And lo, I was.

[chhotii.livejournal.com]

Yeah, I used to re-derive all of physics from Newton's 3 laws on every high school physics test.

I don't think this is a bad thing. It meant that (at the time) I actually understood the material. It worked, though, because I had the confidence to know that I could do it. Without that confidence? Disaster.

[gosling.livejournal.com]

Yeah. This. Comprehension is far more important at any level of learning.

[achinhibitor.livejournal.com]

Here's what I don't get: I've been told that the MCAS is un-timed. (Is this true?)

I hadn't heard this, but I'm not surprised. The traditional system for standardized tests that allow measuring a wide range of abilities is to write tests that almost nobody is expected to finish -- speed with which people answer questions seems to be correlated with the abilities the test is intended to measure. Eventually it was discovered that one could suffer learning disabilities that put one at a disadvantage on timed tests, and so the ADA-required accommodation was that one had unlimited time on tests. This caused the growth of an industry to provide such diagnoses for the children of affluent parents, so they could take the SAT without a time limit, and so get higher scores than they would otherwise. Eventually, SAT counterattacked by devising a testing system that doesn't involve time limits.

It looks like the MCAS has some similar countermeasure and doesn't need time limits either.

[achinhibitor.livejournal.com]

Mathematicians are not judged on speed.

Well, they're judged on output of new stuff. But it's hard to be productive in one area of math without "having a good working knowledge of" the areas on which it is based. And "working knowledge" usually means "able to perform the standard activities by habit".

The example that drove this home for me was watching my brother struggle with calculus. The problem was that he hadn't learned algebra to the point where he could execute it reflexively. During any particular calculus derivation, there would be one or more places where his work was not identically the same as the book's version. If he'd really learned algebra, he would have seen at a glace that the two were equal, but he couldn't think through the algebra at the same time he was thinking through the calculus.

Unfortunately, time pressures are ubiquitous in the real world, and it seems to me that the best course is to help the kid learn how to control her emotional state when under time pressure. (How else will she be able to drive, where every task is real-time?)

[jbsegal.livejournal.com]

Different real-time tasks (and real-time expectations) generate different internal responses.

[chhotii.livejournal.com]

the best course is to help the kid learn how to control her emotional state when under time pressure...

Of course, if we knew how to do that, then that would be the best course. If you hear of any techniques for teaching control of emotional state, let me know, I'm all ears.

But I think in the case of my kid, at her age, that skill just involves some neurological maturation that she's delayed in. Meanwhile, though, while we wait for that to happen, she's learning all the wrong things such as "math is hateful" and "I suck at math".

[gosling.livejournal.com]

Schools invariably do an excellent job at teaching emotional attitudes towards various subjects and shaping a student's sense of herself as a competent (or less than competent) learner. Sadly what is learned is too often exactly what she is learning, especially for kids who have any sort of learning disability or emotional immaturity.

[achinhibitor.livejournal.com]

If you hear of any techniques for teaching control of emotional state, let me know, I'm all ears.

The only technique I know of (and there may well be others) is "systematic desensitization", exposing the subject to increasingly intense experiences of the type they have a phobia regarding, interleaved with time to emotionally recover, introspect, and learn/devise coping strategies. (ah, there is a Wikipedia page on it) I gather it helps a lot if the subject can be guided by someone who has had the same phobia but has overcome it. I've seen press reports of progressive desensitization being used for claustrophobia, and I've used it on myself to reduce my social phobias. (See Zimbardo's Shyness (http://www.amazon.com/Shyness-What-It-Is-About/dp/0201550180) for a series of exercises that are essentially that.)

Of course, no technique will work unless the subject believes it is necessary to change (for one reason or another). But to succeed in the Real World, one must adjust one's emotional processes to match the demands of the world, since the world won't do so. Ideally you can help your daughter realize this before she's 22 and hunting for her first job.

that skill just involves some neurological maturation that she's delayed in.

The question that comes to me is why don't you delay her education until she has the neurological maturity expected/demanded of children in that grade?

she's learning all the wrong things such as "math is hateful" and "I suck at math".

I guess it comes from my personal background, but I don't see these as terribly important. Certainly I was poor at arithmetic and still am (particularly multiplying 6, 7, and 8) and I still don't like it. Other parts of math I like. But I see those likes/dislikes as being intrinsic to my personality, not learned aversions. As for how one conceptualizes one's skills, presumably that is updated based on experience.

Deep under this is the idea that school is not about what one enjoys but what one must do. And an incredibly valuable skill is the ability to diligently do things that are distasteful. Certainly in my life, lack of self-discipline has been much more of a barrier to success than lack of intelligence or other inherent skills. So school is a good place to learn self-discipline. Conversely, if a child learns that raising a fuss when confronted with an unpleasant task gets one out of doing the task, the opposite lesson is learned.

[chhotii.livejournal.com]

The question that comes to me is why don't you delay her education until she has the neurological maturity expected/demanded of children in that grade?

That's a fascinating question, Dale! Clearly you do not have kids.

One reason: My understanding is that my child has to be schooled somehow, under penalty of law. Either public school, private school, or a home-school plan approved by the school system and actually followed (or enough resemblance thereto that you can file progress reports on said home-schooling plan). Parents receive a letter from the school system, to the effect that if the kid doesn't show up for school the parents will be reported to that agency of the state in charge of taking children away from bad parents and putting them into foster homes.

[achinhibitor.livejournal.com]

(Though this is retrospective by now.) The idea I had some vague inkling of was, "You could ask the school district to hold your daughter back a year on the grounds that she is not developmentally ready for this grade's work." I would expect that would not cost you extra.

This might be worth considering if you daughter is young for her grade. There seem to be significant advantages to being old for one's grade. (Consider its effect on soccer careers: http://www.nytimes.com/2006/05/07/magazine/07wwln_freak.html?pagewanted=all)

[gosling.livejournal.com]

I drive just fine, and I loathe being put under pressure more than just about anything else.

[chhotii.livejournal.com]

I didn't really learn to drive until I was almost 30.

I think it should be OK if people mature at different rates, as long as you get there eventually. Sophia wants to live in NYC or SF when she's in her twenties. Both of these cities have excellent transit systems. (Not that I won't teach her to drive if she's ready at the standard age.)

[achinhibitor.livejournal.com]

I drive just fine, and I loathe being put under pressure more than just about anything else.

I'm astonished, because driving (especially in Boston) is the most nerve-wrackingly real-time activity I've ever done.

[gosling.livejournal.com]

Being able or unable to manage emotionally is not even remotely the same as being skilled at mathematics. They appear to be being treated as virtually identical by the school, however, to the detriment of both her confidence in her mathematical ability and whatever interest in mathematics she previously had.

This problem is widespread in schools and has been for quite some time.

[achinhibitor.livejournal.com]

Being able or unable to manage emotionally is not even remotely the same as being skilled at mathematics.

When the fundamental goal is to get the child to be able to do certain things quickly, by rote, being unable to do so due to being unable to cope with the stress has the same effect as not knowing the facts. Perhaps distinguishing the two causes would lead to different teaching regimens, but I don't know of any workable regimen other than "practice it until you get it right". (More about practice later.)

As for school generally, I take the Pinkeresque attitude that schooling is for teaching people things that humans are not good at and do not enjoy learning. (Contrasted with walking and talking, which humans have as natural skills, and have natural drives to learn.) In that regard, confidence in one's learned skills naturally comes from having learned the skills.

In my case, I did not like the early years of math. I was particularly truculent over two episodes, counting between 10 and 100, and multiplying large digits (6 through 9) by large digits. I was helped over that hump by my mother sitting down with me and forcing me to learn the stuff. (When I'm at a restaurant, I try to stick someone else with figuring out the bill.) OTOH, I liked algebra and geometry, and eventually went on to get a Ph.D. in math.

But I still don't like doing arithmetic, and when I'm at a restaurant, try to stick someone else with figuring out the bill. I doubt that's because I was taught arithmetic in an incorrect manner. But that's beside the point, because school's goal is to teach me arithmetic. Enjoying it is not particularly important. Worse, learning how to learn and perform well a skill one does not enjoy is a vital skill.

One of the more revealing/depressing studies I ever read of is someone who compared the employment outcomes of people who had graduated from high school with those who got GEDs. This was in the old days, before the MCAS and other exams to ensure that one did not get a high-school diploma without, like, actually learning the stuff. Whereas the GED required you to pass a stiff examination. It turns out that despite the GED holders actually knowing more of the material, they did worse in the job market. It seems that high school trains students (or filters them) for working smoothly in regimented, bureaucratic systems ... and that is a vital job skill.

[skreeky]

See above advice from my dad: Real mathematicians stink at arithmetic. =)

[whitebird.livejournal.com]

Here's an odd way to think about it.

Being able to do something you've learned quickly and still accurately shows competence. If you try to do something you know unusually fast, you can pick up where there are hiccups in your knowledge or understanding or memorization.

This is why speed rehearsals are useful when preparing for putting on a theatre production.

Of course, this way of looking at it doesn't help dealing with the stress, but, if you concentrate on being able to look at the problem and either quickly solving it, or not, then going down the list of problems as quickly as possible in a first run and answering what you know you know first of all and then going back and working out the rest will provide you and your instructors with useful diagnostic information.

[happyfunpaul.livejournal.com]

Re "MCAS is untimed": Yes. There are times ("30 minutes left", etc.) that get read as part of the instructions, but those instructions amount to "and if you're not done, then... keep going!" At my school, we build in some extra time for everyone during MCAS tests, to make sure no one rushes, and then for the students who still aren't done, we move them to another room, and they take as long as they want, up to the end of the school day (so, basically 6 hours for a 1-hour or 2-hour test -- sadly, there is a former student of mine who often does take that long).

I don't know what "math-fact" sheets are, so I won't comment on that, and besides, other folks seem to have given interesting answers already.

I have my own opinions on how "extra time" is not the one-size-fits-all accommodation it gets treated as, and that there are interesting unintended long-term consequences of students facing fewer and fewer timed tests, but that would be too tangential, so I'll refrain. Suffice to say, for some students, extra time is quite appropriate and helpful.