How I can tell the younger son is my child… January 28, 2012
Posted by mareserinitatis in math, younger son.Tags: math, minus, negative numbers, younger son
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The younger son is learning how to manipulate negative numbers in math. However, he was getting very irritated when listening to the ‘lectures’ yesterday. The lecture would use the term ‘minus’, as in -6 is pronounced ‘minus six’. Every time it did that, the younger boy would make some exasperated grunt and say, loudly, “Negative!”
I can only think this may be because I always call them ‘negative’. The term minus, to me, implies an operation. If so, he obviously picks up on subtleties a lot better than I thought.
Outnumbered January 5, 2012
Posted by mareserinitatis in gifted, math, teaching, younger son.Tags: math, teaching, younger son
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Today I’m going to be working with the elementary students again. This will be interesting as I completely switched gears from what we were doing before. The stuff we were doing before was fun, but as we move through the book, it looks like they need a lot of multiplication and division…which most second graders don’t have.
Today, we’re going to learn about other number systems. In particular, I’m going to have them pick a number using Indo-Arabic numerals and ‘translate’ into other numbering systems – Egyptian, Roman, Babylonian, Mayan, and Chinese. This will give us an opportunity to talk about different bases, positional numbers (i.e. the concept of place value), and how many systems don’t have a zero. (Although, there’s debate in some cases.)
After doing the prep, I’m SO glad that we don’t use the Babylonian system. Base 60?! No wonder my math professor got annoyed when we used degrees.
Thanks to the MacTutor History of Mathematics Archive for the picture!
Take that, Larry Summers! December 15, 2011
Posted by mareserinitatis in education, feminism, math, papers, science.Tags: intelligence, male variability, math
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I came across an article on the new research by Kane and Mertz which supposedly disproves the “greater male variability” hypothesis. That is, while averages for both genders are approximately the same, males have more variance in their intelligence. Thus, when intelligence tested, you’ll see more males at both the upper and lower tails of the distribution.
When Larry Summers was talking about the greater male variability hypothesis (GMVH) in his really awful speech, he was talking about those who are at least two standard deviations away from the mean. If you look at the distribution of IQ for each sex, which is what he was referring to, you can see that the ratio at the 98th percentile is approximately 2:1 male to female.
If IQ were an accurate predictor of success in academia and academics came primarily from that top 2% (neither of which are universally true), you would then expect to see approximately 2 men for every woman in those professions. Unfortunately, the ratio is much worse than that (from the perspective of women and feminist men, anyway). This very compelling evidence of social, cultural, and/or economic factors, potentially up to outright sexism coming into play when women are being considered for academic positions. The fact that it is still so far from this ratio makes me have a lot of issues with Larry Summer’s argument. Aside from all that, there is the issue that IQ isn’t the best predictor of success.
However, let’s pretend it is…or that it at least that it may be reflected in math achievement for the tests used in the study. In the study, they took variances from scores on tests like TIMSS and PISA, both of which are given internationally and used to compare various countries’ standing. Specifically, the paper examined the variance question.
To do this, we can begin by looking at the data from IQ Comparison site, which says that the standard deviation in the WISC IV IQ test was about 14.54 for men and 13.55 for women. The variance is the square of the standard deviation, giving the variance for men as 211.4116 and women as 183.6025. If you want to do a comparison, just take the ratio of men’s variance to women’s and you’ll get a variance ratio (VR) of 1.15. Keep in mind that the data this is taken from the US standardization which was used to norm the test, and it was done in the early 80s. If you want to compare that to the data presented in the paper, the US VR in 2003 was 1.11 on the TIMSS and 1.19 on the PISA. In 2007, it had dropped to 1.08 on the TIMSS (no PISA data is given). Therefore, the VR has changed.
The authors use the math testing data to do this for many countries, not just the US. You would expect that if the GMVH is true, then you would see VRs of about 1.15 from most countries and that it is constant in time. What Kane and Mertz find is that the number seem to vary a lot, but many of them have changed. That by itself gives an indication that a VR of 1.15 is not fixed and that the VR may be somewhat cultural. Further, they changed through time. Some of the VRs increased, like in Australia, and some decreased, like Japan’s.
This is the table presented in the paper:
They then attempt to find a correlation between male variance and the VR ratio. If GMVH is true, you would also expect that a higher VR ratio would be highly correlated with males having a larger variance. That’s not what they find, however. The correlation value is fairly low, and the authors state that sometimes a higher VR is actually due to poorer performance on the test by boys.
There is significantly more analysis than I’ve communicated in this post, but the gist is that they found that gender equity in economic and educational arenas were the best predictor of test performance. This gives a good indication that the GMVH is bunk – performance in math is not biologically destined.
Jonathan M. Kane and Janet E. Mertz (2011). Debunking Myths about Gender and Mathematics Performance Notices of the American Mathematical Society
The best students December 7, 2011
Posted by mareserinitatis in education, geology, math, teaching, younger son.Tags: animals, cub scouts, geology, math, younger son
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At teacher conferences a few weeks ago, my son’s teacher mentioned that she was going to be taking a short period during the day to break kids into two groups. One group needed some help with some of the more basic concepts in math, while the others seemed fairly advanced.
I got very excited, and I asked if I could come in and do some fun math stuff with the advanced group. She said she’d appreciate it because then she could focus on the other kids who needed more help with things.
Yesterday was my first shot at this. It’s only about 20 minutes of seat time once a week (along with about an hour of prep, considering I have to bring in materials). I worked with a group of six, and it was fun.
That’s the one thing about teaching college versus elementary school kids: college kids never get excited the same way little kids do. Of course, maybe it’s because you have to use a fundamentally different approach – more hands on – with little kids. On the other hand, I think you lose something with maturity. I have worked with a couple different cub scout groups, and they often have requirements to learn some geology for various badges. There is something amazing that happens when you put a group of 6-10 year old boys in front of rocks and other things they can touch. They’re fascinated with everything and seem to hang on your every word (when they’re paying attention). When you do the same to college kids, they just kind of shrug and proceed forth, maybe discussing the rocks with neighbors.
For these kids, I’m using a Mathworks book on how to be a zoo vet, and I decided to let each kid have their own animal as we work through the problems. Yesterday, we talked about building crates because we’re shipping our animals from one zoo to another. The kids were SO excited that they got their own animal. I tried to bring a variety: there were poison arrow frogs, king cobras, and piranhas for the boys and pandas and koalas and dolphins for the girls. I was pretty close: the two girls chose dolphins and koalas, and the boys mostly went for the dangerous animals. (One chose a polar bear, which is on the fringe between dangerous and cute and cuddly, IMO.)
Either way, they were really getting into building their crates. They were talking about the differences in sizes between all the animals, and it’s amazing all the movement and excitement and gestures that go into discussions among 7-8 year olds.
After the twenty minutes was up, I was exhausted. My comment about how college students never seem to get excited is exactly why I prefer to teach them: I can’t handle the energy level of really young kids all day long. I have to admit that I admire elementary school teachers for doing this. However, despite being exhausted, I was really tickled with their excitement and the fun we had. I’m looking forward to next week.
An appropriate challenge November 19, 2011
Posted by mareserinitatis in education, homeschooling, math, younger son.Tags: EPGY, math, perfectionism, younger son
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I’ve mentioned before that the younger son is doing math through Stanford’s EPGY program. In order to get into the program, he had to take an exam online to see if he qualified. Now that he’s enrolled, he gets weekly emails from his teacher talking about his progress. Most of the time, they say something like, “Keep up the good work.” I just tell the younger son that his math teacher is happy with his progress since he’s not real familiar with the concept of email. (Occasionally we’ll talk about some of the concepts she thinks need a bit more explanation.) I learned there’s a lot of other things he doesn’t quite understand…but I’ll get back to that in a moment.
Last month, he took a final exam to finish the grade he was working on. He didn’t pass, but he didn’t bomb it, either. He was a few points lower than the cutoff to go onto the next grade. It was a good thing, in retrospect.
The younger boy is a Perfectionist (with a capital P!), and it kills him to not do something perfectly. In fact, he refused to read until very recently because he couldn’t figure out all the words immediately. He was very disappointed when he didn’t pass the math exam and had to go back and redo some of the material. He retook the test a few days ago and got a very high grade. The lesson learned is that ‘failure’ isn’t death and doom…just means you need a bit more practice before you can go on. I think the practice did him some good as it seemed like he really got a better handle on things the second time around. I also think it helped him to see he isn’t expected to understand everything the first time he sees it. In other words, this is a good learning experience for the young perfectionist…one he would likely have not gotten in school given his grades are much higher there.
After finishing the test the second time, I showed him the email his teacher sent. I said that it was from his teacher at Stanford. I guess I’d never mentioned that bit before.
“My teacher is at Stanford?”
“Yeah, do you know what that is?”
“No.”
“It’s a big college that made the math program you’re using.”
“I didn’t pass the test the first time.”
“No, but that’s okay because it’s a hard math program. You just needed more practice. You wouldn’t be able to figure some of this stuff out unless you were pretty good at math.”
“Did you tell my teacher at Stanford that I’m good at math?”
“I’m pretty sure she knows.”
What’s kind of funny is that I don’t think he knows. That’s good, though, because it means he’s being challenged and not repeating work he already understands.
Another approach to multiplication November 14, 2011
Posted by mareserinitatis in math, teaching, Uncategorized, younger son.Tags: arithmetic, math, multiplication, teaching
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My younger boy has been working through multiplication, and the problems he’s doing are getting more difficult, so I decided it was time to start working with the dreaded memorization.
I’ve talked before about simply giving kids multiplication tables to work from. My younger son, however, seems like he’s a little better with memorization, so we took the following approach.
Most kids have a fairly easy time with learning to count by twos, threes, and fives. So that’s where you start. The other thing the child needs to know is how to add with carrying. If the child can do that, the rest of the tables are easy. Since counting by 2s and 3s is known, we’ll start with fours.
If you have a problem involving a 4, say 8 x 4, then you have them compute 8 x 2. Once they have the answer to that, have them double it. So basically, once they know all their 2s, they can easily obtain their fours. The same principle goes for 6s and 8s. For a multiplication problem involving a 6, they can either add the corresponding problem with twos three times or take the threes problem twice. Finally, for 8s, they can work from twos to fours, and then from fours to eights.
With the younger boy, this means that if he has a problem like 8×7, he first figures out 2×7. He doubles that answer to get 4×7, and then doubles it again to get 8×7. For him, adding things up goes a bit faster.
For nines, he uses the finger method: he holds down the finger that corresponds to the multiplier, moving from left to right. That is, if he has 3×9, he holds down the third finger from the left. To the left of that finger, he has the number of tens (in this case, 2), and to the right he has the number of ones (7). So the answer is 27.
So what do you do about 7s? Actually, given you have methods for everything else, the only one to memorize is 7×7. On the other hand, if you have a kid that sort of stuck when it comes to commutivity of multiplication, then another way to deal with it is that it’s the sum of the threes problem and the fours problem. (7×7 = 3×7 + 4×7 = 21 + 28 = 49)
Tens are usually pretty easy, so I’ll skip that one.
Eleven and twelve were learned by breaking them into two parts. First, take the number times ten and then take it times one (for eleven) or two (for twelve) and add the results. So 12×9 would be 10×9 plus 2×9.
I’m fairly certain this method would have never worked with my older boy. He has very poor working memory and ADHD, so I don’t think he was able to do a lot of this in his head (and was always resistant to writing it down). For him, I think using a multiplication table was a better approach. For the younger boy, though, who seems to enjoy working through problems and has a very good working memory, this has been a far more, and I might even say quicker, method.
You were paying attention! September 12, 2011
Posted by mareserinitatis in math, older son.Tags: math, older son
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My older boy came up to me last night. He was working on an algebra problem, and he said he thought the book had an error.
He was right. When I looked at the solution, it turned out they had only used some of the information given, while the other part was ignored. Turned out that if you used the other part, you got contradictory information. He could have chosen to use part of it, but looked at all of it and found an error.
“Good job!” I said. ”Obviously you were paying attention.”
He thought I was joking. I wasn’t. I’m proud of him for looking at all of the information and not making any assumptions…especially when it’s a subject he really doesn’t like and could have just written down and answer and been done with it.
Completely stunned August 24, 2011
Posted by mareserinitatis in education, gifted, math, younger son.Tags: gifted, math, school, younger son
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We were anticipating some issues with the younger boy starting school this year. Primarily, we have a problem: he’s already 2 years ahead in math. He’s been working through Stanford’s EPGY math program, somewhat irregularly over the summer, and he’s managed to move that far ahead. This has been kind of a surprise because he initially didn’t seem to be that gifted in math.
We decided to sit down with the principal and his new teacher and talk to them about alternatives. At first, it was fairly obvious they wanted him to be doing math with the other kids but then to add enrichment or even to go to another class. The problem is that he’s doing well with the EPGY program, so we’re very reluctant to end that. He also seems to be going at a much faster pace than we expected. Even now that he’s nearly two years ahead, he’s still only spent about 3 months to do about a year’s worth of math. Putting him in an advanced classroom that still moves at a slower pace is probably not going to be good for him.
We went in, hoping that they’d be okay with us giving him other things to do during math time. They didn’t seem real keen on the idea, and we were really reluctant to try to have him do two sets of math each day…one at a lower level or slower pace and then an additional one that’s right for him.
When it was obvious they didn’t like the ideas we suggested, I just sat there and waited for them to come up with something. Finally, the principal said he’d be willing to help supervise him in doing some sort of independent study project of his choosing during math time.
I just about keeled over.
We went from them not wanting to pull him of math to do something else to them being willing to let him do his own independent study project?!
The principal apparently used to supervise kids where they did this type of project-based learning, and I get the feeling he misses it. And I think this would be something the younger boy would love to do.
So, despite the fact that I was feeling very uneasy about what was going to happen, I think we hit the jackpot.
Linkety Link July 31, 2011
Posted by mareserinitatis in feminism, links, math, science.Tags: feminism, links, math, science, women in science
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I came across this fascinating article on the history of research in global climate change. I am greatly amused by the fact that methods used for oil and gas exploration were later utilized to validate theories on climate change. Irony.
Anyway, it’s a fascinating read: The Discovery of Global Warming
I’ve also been remiss in not posting a link to this sooner. (As you can tell, blogging hasn’t been at the forefront of my brain.) GEARS wrote two great posts on diversification in STEM fields: Diversification In Stem Fields and On Diversification: with Dr. Anna Garry and Professor Ursula Keller.
Of course, there’s a lot going on at EngineerBlogs. I wrote a post recently titled Died-in-the-wool Engineer.
For fun, you should think about whether math should be taught in schools. (And yes, the video is a spoof.)
Touch math June 10, 2011
Posted by mareserinitatis in education, math, teaching, Uncategorized.Tags: finger counting, math, memory, touch math
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I have another rant about elementary math education, but this one is slightly different.
I am completely appalled that schools still do not teach touch math.
You probably have no idea what I’m talking about because you’ve probably never heard of it.
When I was in fourth grade, we moved to a town in rural North Dakota called New Salem. At the time, the population was 2000. Now it’s half that. If you’ve ever driven through North Dakota and seen the world’s largest Holstein cow on the side of a hill, you’ve seen New Salem. (The cow, by the way, is named Salem Sue.)
Shortly after we moved there, I was sent to get some extra help in math. In fourth grade, I was still counting on my fingers. Some people term that a math disability now. However, the way this rural school saw it, there was an easy fix. I was sent to the special ed teacher for two one-hour sessions. I was, of course, feeling very ashamed about my finger counting. She said that it’s not unusual and that she had a way to fix it. She pulled out a piece of paper with an image that looked something like this (only, back then, it was black and white because I grew up in the days before they invented color):
So what the heck is this?
It’s a way of computing addition and subtraction facts.
The dots are places where you touch the number, and the dots with rings around them are places where you touch the number twice. You’ll notice that seven has a single touch and three double touches. Basically, when a child is sitting there with a math problem in front of them, they can touch the number in the designated places, and count those touches rather than their fingers.
As I got the hang of this (which was very quickly), I started being able to see the points and make computations without touching the numbers. I started developing some visual strategies for calculations. Eventually, with practice, I ended up memorizing my math facts. I’ve written before about how math facts are better memorized through practice than rote. I think this would be another great method for teaching facts, as it obviously worked for me. In fact, you can look at the Touch Math website and see that there’s a decent amount of research showing that the strategy works well both for average children as well as those with learning disabilities.
So why isn’t it being used? I’m still surprised that teachers in a small rural school district were thinking so far ahead. I’m not sure why larger schools have not followed suit in the past 30 years, and it’s really unfortunate that so many people have not heard of it.
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