The n-Category Café

Tim Porter wrote:

How to fight this is very difficult to know.

I think to understand these complex issues we’d need to compare what’s going on between countries, and also within different classes and social groups within countries, and also over long periods of time.

I’ve never read a serious study of math education that compared different practices worldwide over the whole world ever since the Industrial Revolution, and the effects of these different practices. Maybe such studies exist — but what I mainly see is a lot of people putting forth strong opinions and arguments based on their parochial experience. So, I’m not surprised that the math wars are dominated by fads and heated controversy.

Let’s see what’s easy to find in a couple of minutes on the world-wide web.

Okay, there’s Trends in International Mathematics and Science Study (TIMSS), which assesses the math and science skills of 9-year-olds and 15-year-olds in 60 countries. A sample graph:

Unfortunately this exercise has only taken place three times so far: in 1999, 2003, and 2007 — and the 2007 results aren’t out yet. It would be nice to collect data over a much longer time span. But anyway, here’s an easy-to-read analysis of the results, focused on ‘what’s wrong with the United States’:

• Alan S. Brown and Linda LaVine, What are science and math test scores really telling U.S.?

A few quotes:

Maybe you’ve read the headlines: “Math and science tests find fourth and eighth graders in U.S. still lag many peers,” proclaimed the New York Times. “No gain by U.S. students on international exam,” reported the San Francisco Chronicle. “Economic time bomb: U.S. teens are among worst at math,” warned the Wall Street Journal. And just in case you didn’t sense the alarm, “World crushes U.S. kids in math, science,” summarized the Boston Herald.

Of course, every commentator has a seemingly obvious solution. Spend less time motivating students and more time enforcing standards, proclaimed the Cleveland Plain Dealer. Give parents school vouchers, and open charter schools, demanded a Newark Star-Ledger columnist. Throw away the calculators, and get back to basics, insisted a coalition of conservative Californians.

There you go: instead of carefully analyzing the situation, everyone spouts off their own opinions!

But here are some interesting things you see when you look at the data a bit harder:

[Patrick Gonzales, who heads the TIMSS analysis effort at the U.S. Department of Education, says:] “When you break students into standard sociological groups — parents with college education, minorities — the gap between the top and bottom is greater within the United States than between U.S. and top-performing Dutch students. “There are significant differences between boys and girls in math and science in fourth grade,” Gonzales acknowledges. “But they pale in comparison with the differences between white and black or poor and wealthy.”

The difference between fourth-grade boys’ and girls’ TIMSS math scores is only one-tenth the difference between the U.S. and Singapore. [Singapore does best of all on this test.] The difference between eighth graders in poor and wealthy U.S. school districts, however, is 1.5 times greater than the difference between Singapore and the U.S. at both grade levels. When the same children are compared in science, the differences within the United States are four to five times greater than the differences between the U.S. and Singapore.

The gaps are even noticeable in the number of eighth-grade students who scored at the TIMSS intermediate level in math. The gross numbers show 93 percent of Singapore students and 64 percent of U.S. students reached these levels; U.S. males scored 65 percent, and females 64 percent. While 75 percent of white students rated intermediate status, only 35 percent of blacks and 45 percent of Hispanics did so. Wealthier school districts scored 86 percent, while poor districts rated 32 percent.

“Shave off the middle-class suburbs and we’re in the range of competitiveness,” says Roger Bybee, a well- known educator who heads BSCS, a nonprofit science-curriculum development organization in Colorado Springs, CO. But he also points to an issue as serious as the gap between rich and poor. “TIMSS is a curriculum-based test, and our better schools are competitive at demonstrating knowledge of the curriculum. But if we go back to innovation — critical thinking, reasoning, invention, and discovery — it’s a little less clear that we are competitive.

“The scariest part of it,” Bybee continues, “is that PISA also asked students about their educational aspirations, and U.S. students ranked second after those in South Korea. We have the highest aspirations, but we are near the lowest in terms of problem-solving skills. Our skills are not commensurate with aspirations.”

I also find this interesting:

Bybee points to the work of William Schmidt, the noted researcher whose Michigan State University center coordinates U.S. participation in TIMSS. Schmidt was the first to apply the expression “a mile wide and an inch deep” to U.S. math and science curricula that jump from topic to topic without providing a coherent picture of how the topics fit together.

Schmidt often publishes graphs that show the sequence of mathematics topics studied in first through eighth grades. The vertical axis lists math topics from simplest (whole number meaning) on top to hardest (slope and trigonometry) on the bottom. Grade levels one through eight appear from left to right along the bottom.

For the top-achieving countries, the graph shows a diagonal progression from the top left (easiest topics, first grade) to bottom right (hardest topics, eighth grade). According to Schmidt, these nations teach first and second graders whole number meaning, whole number operations, and measurement units—and nothing else.

Schmidt’s chart of three sample U.S. states shows no slope whatsoever. These states cover nearly every topic in nearly every grade. In addition to whole number meaning, operations, and measurements, first- and second-graders in all three states learn data representation and analysis, polygons and circles, estimating computations, measurement estimation and errors, 3D geometry, and patterns, relations, and functions. All told, the three states introduce 13, 20, and 30 different topics, respectively, in grades one and two.

“Compared with higher-performing countries, our curriculum is incoherent,” adds Bybee. “It lacks focus and rigor. We tend towards an emphasis on terms and facts—and I’m not opposed to facts—but we know from contemporary models of learning that students also need conceptual ideas to hang those facts on.” Francis (Skip) Fennell, president of the National Council of Teachers of mathematics and professor of education at McDaniel College in Maryland, agrees. He also underscores how hard it is to teach such a fractured curriculum.

“In my job, I get to hear the frustration of elementary classroom teachers around the country,” he says. “In 49 of 50 states, there are state curriculum frameworks, and their requirements are all over the place. They have 20 to 30 — and sometime hundreds — of objectives. How is a teacher going to achieve 100 objectives in 181 days of school?

“This sends a signal to a fourth-grade teacher who may not have a degree in math that all these 100 objectives are equally important. But they’ve never been equally important. Experienced teachers know this, but many teachers worry about covering all the topics because their students are going to be tested on them.

I would agree with some of Thomas’s comments, but the problem is not as he thinks, I believe. The details of what is or is not in a syllabus is quite often not that important as long as something of the essence of the processes remains. If students learn about logical processes of thought by using discrete math rather than Euclidean geometry they still will have some idea and intuition of how reasoning works. When, however, the syllabus does not encourage thought of any type, and just presents hoops to jump through that is far more serious. The essence of many jobs that employ mathematicians is the reasoning, exactitude, etc. that we try to build in University but if those aspects are being removed from the school syllabus then that makes the university lecturer’s task so much harder.

I have organised and given Masterclasses in the UK and there are very many enthusiastic young people who love doing maths (and at Bangor some of them would be coming 100 km to a Saturday morning session.) Those kids could THINK, and thank goodness there were some excellent teachers in that area who were encouraging them to do so. I have done similar sessions in Canada and Ireland, and in each case my impressions are the same (and are very heartening) BUT the examination system is often selling those kids `down the river’. The result of dumbing down is often to turn off the potential mathematically inclined young person and to discourage them. Their talent is seen as being not that important in society since society does not encourage thought even in elementary examinations.

The problem IS very complex. Public perceptions of mathematics are often not helped by us mathematicians ourselves, but returning to the question of mathematical knowledge and skills, as examined in public examinations, the dumbing down is at the meta-conceptual level, and not really that some concepts have been withdrawn because they are past their sell-by date!

1. Name and define the Fundamental Rules of Arithmetic.

Wow, they taught the Peano Axioms in eighth-grade Kansas classrooms?

Could any 8-grader from 1895 pass eight grade today?

That test does not cover most of the modern syllabus. Some of the more glaring omission are the entire fields of physics, chemistry and biology, which at least in Sweden is taught in grade 7-9. Also, there is nothing about foreign languages, not even Latin. And only very boring math seems to be covered, changes of units, percentages and that kind of stuff. If I remember right, at that age we mainly worked with sines and cosines, quadratic equations and simple systems of linear equations.

What surprises me is that there is no Christianity neither. I would have thought that that would be a sizeable fraction of the syllabus.

1. Give the epochs into which U.S. History is divided.

Sheesh kebabs! U.S. history is divided into epochs?!

http://www.snopes.com/language/document/1895exam.asp

Maybe they’re sneakily luring you to think that those cartons look like they hold 2 pints/1 litre, so cost around 60p. Then you’ll divide by the 7 and multiply by the 3.

Foolish you for not realising that you’re on the ‘calculator’ paper so that they wouldn’t give you something you could do in your head. £4.20 divided by 3 multiplied by 7 is much more likely.

In many subjects, mathematics included, there is much to be gained from project or course work, BUT …

I was recently in Galway and there was a visit from someone working in Mathematical Education in Dublin. She had been investigating the perceptions of mathematics of students training to be teachers in secondary schools in Ireland. What she reported was moderately frightening! The interpretation of the terms in the syllabus (and I mean here the terms relating to the purposes behind mathematics, the conceptual aims and objectives of the syllabus at quite a high level) was far from what most practitioners of mathematics at more than a very elementary level would sign up to. For instance, `applications of mathematics’ was interpreted by some of the students as `being able to do calculations’, whilst theoretical stuff was not liked because it did not involve calculations. The worry is that some of those students may end up directing course work in schools.

We have to ask some awkward questions. Ronnie Brown and I over a period of some twenty years tried to stimulate debate on some of the issues. If you have not looked through his web page material on this, it raises some good points, so look at for instance

http://www.bangor.ac.uk/~mas010/icmi944.html

or the article

What should be the context of an adequate specialist undergraduate education in mathematics?,

available at

http://www.bangor.ac.uk/~mas010/publar.html.

The first of these gives an instance of the reactions of a teacher to some mathematically able pupils in coursework.

In the last analysis, I would suggest that the proof of the pudding is in the eating…. We have to ask ourselves if the inclusion of course work as it currently is, has resulted in pupils/students gaining an increased perception of what mathematics is really about. Does it lead to better more mathematically able students? I do not prejudge the answer, but we must neither assume that the so-called advantages of course work are self evident (because we are told so by the `experts’ in education) nor that there are no advantages in coursework.

Two final points: mathematics is both a science and a creative human activity. Playing the piano or playing a game such as soccer, requires training, so does the creative activity of mathematics!

There is an old saying : those who can’t do teach, those who can’t teach teach others to teach!
(or the variant due to Mark Twain, I think, To do good is noble, to teach others to do good is nobler, … and less hassle!!)

PS. I think we did a good job at Bangor in getting some of these issues across to our students, (and there are many others in other places who have tried similar things), but we were closed down due to low recruitment (amongst a whole lot of other things)! There is no intention here of suggesting a correlation between the two facts, but just to say that as a community the issues being raised in this discussion will influence our own personal future.

Good points! When I used to teach GSCE maths as an impoverished PhD student there was no coursework. My only experience is watching my son do his this year, and to be fair there was much of value.

His project was to take see what would happen to the sum of the 5 numbers appearing in a ‘T’ shape as you transform it around a grid. The natural numbers are ordered in the grid and the width of the grid is a variable. He had to investigate the transformation to the sum on translating and rotating the ‘T’. There were some small glimpses of group theory possible if you knew where to look.

But the coursework component looks set to disappear. They seem to want to replace it by supervised versions, but I don’t see how these can run for any reasonable duration without risking the same problems.

“What hooked y’all on maths?” In my case it was an accidential look into a booklet on number theory in a bookstore causing the feeling that that’s exactly what I like. After leaving the shop, my brother burst into laughter because I was so excited in telling him what’s so interesting about it, that I forgot the book after payment in the store. School never got me interested in math (admittedly I didn’t give schools much time for that). That initial interest was stabilized by the nice content/weight ratio of math texts and the fun when texts looking enigmatic at first become understandable. The drawback of such a start of motivation is a strongly selective interest in math subjects.

So, an interesting question is whether Brazilian engineers are in some ways better than British ones because they’ve been forced to do — and perhaps memorize? — lots of math.

It’s worth noting that they only have evidence of recent skimming activity via electronic access, and it’s quite possible that they’re forgetting the skimming they used to do. I remember just over 10 years ago when the university used to still get primarily printed journals I’d walk over and skim through the journals looking for those articles worth reading in more depth. (As you get older you just naturally get the feeling “the world (and I) are going to hell in a handbasket” so you’ve got to work to avoid that bias.)

The two big effects the internet has had on me are

(1) Partly to increase worry that I’m repeating work: you can’t really keyword/concept search swathes of printed journals so I’d have taken the view that if I happen to be partly independently redoing something published in an obscure venue/conference then that’s just what happens, and that reviewers would be as unlikely to know of anything out of the way as I would. Nowadays I tend to spend much more internet time trying to find any related work (partly because reviewers may well find it).

(2) I tend to retain a vague memory of page layout/position in a manuscript/position of diagram in a papers. Because an awful lot of software is focussed on presenting stuff in a window and only viewing part of it, it’s like wearing horse blinders (I’d assume :-) ) and reduces the effectiveness of this spatial sense/memory.

Is this what is known as or has replaced the
dread Elevenses? Ah, for the good old days when there was included the question:
what is the next term in the sequence

1/4 1/2 1 3 6

scroll down if you need a few more terms

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