## June 4, 2008

### Dumbing Down

#### Posted by David Corfield

Mathematics exams for 16 year olds are getting easier, it is claimed. It’s fairly easy to check for yourself. Take a look at the Arithmetic, Algebra and Geometry papers from 1959 and compare with a contemporary specimen GCSE paper.

Even though the contemporary paper is one for ‘higher’ level students, this is taken by a larger proportion of 16 year olds than the old ‘O’ level. But this surely cannot be enough to justify the limited ambitions of the contemporary syllabus.

My son, who last week finished his one hour 45 minute paper with an hour to spare, was fascinated by the 1959 paper which makes you have to think. He was also surprised to find that the contemporary international version of the GCSE is more demanding than the home version.

Posted at June 4, 2008 10:51 AM UTC

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### Re: Dumbing Down

Some years ago I was a member of the Education Committee of the London Mathematical Society. We wrote to the Minister of Education (it is irrelevant which one it was) about the lowering of standards. We received a reply, if I remember correctly, from some official saying that there was nothing to worry about since Her Majesty’s Inspectors (HMIs) had just looked at all the examinations and had come to the conclusion that there was no dumbing down. The letter included a reference to a report by the HMIs which we had not heard of.

Someone obtained a copy of the report. It said that the A-level (i.e. advanced level) examinations had not decreased in standard over a 20 year period (or some time scale like that.) I do not agree with their overall results as I have a certain knowledge of the exams in other subjects, BUT they did say that there was one exception: maths. The problems they identified were twofold. One was that proof had disappeared completely and the other was that in no question was there more than one intermediate step in the reasoning. So the examinations omitted two of the essential aspects of mathematical thought!

In otherwords the examinations had become recipe based. The same thing is identifiable in other subjects (In French, pupils are encouraged to learn but the je’ form of verbs i.e. the first person singular because that is all that can realistically be examined!)

How to fight this is very difficult to know. It is not worldwide but from earlier discussions in the café aspects of it are current in the USA.

Heigh ho! Progress!!!!

Posted by: Tim Porter on June 4, 2008 2:08 PM | Permalink | Reply to this

### Re: Dumbing Down

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’:

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.

Posted by: John Baez on June 4, 2008 5:09 PM | Permalink | Reply to this

### Re: Dumbing Down

I’m less interested in how consistently a country can train its youth to perform some tricks. With Tim, I care more about whether we let at least some children glimpse something of what it is to think mathematically. To be in a position to answer the 1959 exams you would have had to have come closer to this.

Not that exams are everything, but it’s staggering how much they dictate the way students are taught.

Posted by: David Corfield on June 4, 2008 8:57 PM | Permalink | Reply to this

### Re: Dumbing Down

What needs to be induced is the faculty Polanyi describes:

The manner in which the mathematician works his way towards discovery by shifting his confidence from intuition to computation and back again from computation to intuition, while never releasing his hold on either of the two, represents in miniature the whole range of operations by which articulation disciplines and expands the reasoning powers of man. (Personal Knowledge, 131)

The worry is that we don’t seem to be aware of the need to increase children’s acceptable solution time, with which more meaningful discovery can take place.

Posted by: David Corfield on June 5, 2008 9:56 AM | Permalink | Reply to this

### Re: Dumbing Down

“the need to increase children’s acceptable solution time” - Thanks for the link! A recent wired issue about memory and learning mentioned a similar needed change in teaching: “The most motivated and innovative teachers, to the extent they take current performance as their guide, are going to do the wrong things,” Robert Bjork says. “It’s almost sinister.”

Posted by: Thomas Riepe on June 5, 2008 12:03 PM | Permalink | Reply to this

### Re: Dumbing Down

Already Socrates lamented about the lazy youth. I’m not sure it is more true today than it was then.

It is true that some skills have been lost, e.g. the ability to carry out paper-and-pencil calculations. My children once saw what kind of problems I had to solve in fourth grade, and they were quite impressed. OTOH, they are better than I was at doing calculations without paper and pencil. Moreover, they know more about computers than I did at their age (and in some respects more than I do today). And they start with English in first grade, not in fourth grade like me.

When I was a postdoc one of the old-timers, who did his PhD in the 1930s, complained that the grad students didn’t learn how to solve difficult problems in classical mechanics and electromagnetism, because they spent time on quantum theory instead. Perhaps some knowledge has to go if you want to add new stuff.

Posted by: Thomas Larsson on June 4, 2008 2:37 PM | Permalink | Reply to this

### Re: Dumbing Down

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!

Posted by: Tim Porter on June 4, 2008 3:13 PM | Permalink | Reply to this

### Re: Dumbing Down

A propos “dumbing down”, some months ago I was forwarded an email from a friend of my parents. It may be somewhat specific to place and time, but you get the idea.

• What it took to get an 8th grade education in 1895

Remember when our grandparents and great-grandparents stated that they only had an 8th grade education? Well, check this out. Could any of us have passed the 8th grade, in 1895?

This is the eighth-grade final exam from 1895 in Salina, Kansas, USA. It was taken from the original document on file at the Smokey Valley Genealogical Society and Library in Salina, KS, and reprinted by the Salina Journal.

8th Grade Final Exam: Salina, KS - 1895

Grammar (Time, one hour)

1. Give nine rules for the use of capital letters.

2. Name the parts of speech and define those that have no modifications.

3. Define verse, stanza and paragraph.

4. What are the principal parts of a verb? Give principal parts of “lie”,”play”, and “run”.

5. Define case; illustrate each case.

6. What is punctuation? Give rules for principal marks of punctuation.

7 - 10. Write a composition of about 150 words and show therein that you understand the practical use of the rules of grammar.

Arithmetic (Time, 1.25 hours)

1. Name and define the Fundamental Rules of Arithmetic.

2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold?

3. If a load of wheat weighs 3942 lbs., what is it worth at 50cts/bushel, deducting 1050 lbs. for tare?

4. District No. 33 has a valuation of $35,000. What is the necessary levy to carry on a school seven months at$50 per month, and have $104 for incidentals? 5. Find the cost of 6720 lbs. coal at$6.00 per ton.

6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.. 7. What is the cost of 40 boards 12 inches wide and 16 ft. long at$20 per meter?

8. Find bank discount on $300 for 90 days (no grace) at 10 percent. 9. What is the cost of a square farm at$15 per acre, the distance of which is 640 rods?

10. Write a Bank Check, a Promissory Note, and a Receipt.

U.S. History (Time, 45 minutes)

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

2. Give an account of the discovery of America by Columbus.

3. Relate the causes and results of the Revolutionary War.

4. Show the territorial growth of the United States.

5. Tell what you can of the history of Kansas.

6. Describe three of the most prominent battles of the Rebellion.

7. Who were the following: Morse, Whitney,Fulton, Bell, Lincoln, Penn, and Howe?

8. Name events connected with the following dates: 1607, 1620, 1800, 1849, 1865.

Orthography (Time, one hour)

1. What is meant by the following: Alphabet, phonetic, orthography, etymology, syllabication.

2. What are elementary sounds? How classified?

3. What are the following, and give examples of each: Trigraph, sub-vocals, diphthong, cognate letters, linguals.

4. Give four substitutes for caret ‘u.’

5. Give two rules for spelling words with final ‘e.’ Name two exceptions under each rule.

6. Give two uses of silent letters in spelling. Illustrate each.

7. Define the following prefixes and use in connection with a word: bi, dis, mis, pre, semi, post, non, inter, mono, sup.

8. Mark diacritically and divide into syllables the following, and name the sign that indicates the sound: card, ball, mercy, sir, odd, cell, rise, blood, fare, last.

9. Use the following correctly in sentences: cite, site, sight, fane, fain, feign, vane, vain, vein, raze, raise, rays.

10. Write 10 words frequently mispronounced and indicate pronunciation by use of diacritical marks and by syllabication.

Geography (Time, one hour)

1. What is climate? Upon what does climate depend?

2. How do you account for the extremes of climate in Kansas?

3. Of what use are rivers? Of what use is the ocean?

4. Describe the mountains of North America.

5. Name and describe the following: Monrovia, Odessa, Denver, Manitoba, Hecla, Yukon, St. Helena, Juan Fernandez, Aspinwall, and Orinoco.

6. Name and locate the principal trade centers of the U..S.

7. Name all the republics of Europe and give the capital of each.

8. Why is the Atlantic Coast colder than the Pacific in the same latitude?

9. Describe the process by which the water of the ocean returns to the sources of rivers.

10. Describe the movements of the earth. Give the inclination of the earth.

Posted by: Todd Trimble on June 4, 2008 5:25 PM | Permalink | Reply to this

### Re: Dumbing Down

1. Name and define the Fundamental Rules of Arithmetic.

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

Posted by: Blake Stacey on June 4, 2008 6:59 PM | Permalink | Reply to this

### Re: Dumbing Down

Blake wrote:

The eighth-grade final exam from 1895 in Salina, Kansas, USA wrote:

1. Name and define the Fundamental Rules of Arithmetic.

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

Peano only did his thing in 1889, so I doubt. It was probably something more like the definition of a commutative ring, or field.

Things have been going downhill ever since the beginning of time. Once even bacteria were smarter than we are; now look at them.

Posted by: John Baez on June 7, 2008 6:35 AM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: Thomas Larsson on June 4, 2008 7:00 PM | Permalink | Reply to this

### Re: Dumbing Down

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

No. As I said, this test is specific to time and place. It’s an interesting snapshot, though. [Perhaps I should say that the first two sentences of my post are mine, but the rest is not.]

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.

Keep in mind that much of America during that time was in a primitive state. Latin and fundamental science would have been for the boys back east, in Boston (not Salina, Kansas). I take it that the math section covered topics one would need to know to carry on business in that time and place.

Although the US has always been a nation of religious people, the law of the land would have forbidden actual testing of Christian precepts in American public schools, I’m pretty sure.

Posted by: Todd Trimble on June 4, 2008 7:13 PM | Permalink | Reply to this

### Re: Dumbing Down

It wasn’t until the 1963 Supreme Court decision Abington School District v. Schempp that school-imposed Bible readings were ruled unconstitutional.

Posted by: Blake Stacey on June 4, 2008 7:33 PM | Permalink | Reply to this

### Re: Dumbing Down

Okay. Maybe not the law of the land; maybe this was a matter left up to individual states or counties (although I said ‘testing’, in this case to certify passing the eighth grade, not ‘reading’ – the distinction might matter here). To be investigated further…

Posted by: Todd Trimble on June 4, 2008 11:03 PM | Permalink | Reply to this

### Re: Dumbing Down

Thomas Larrson said, “What surprises me is that there is no Christianity neither. I would have thought that that would be a sizeable fraction of the syllabus.”

My first reaction to this was to wonder what assumptions make this surprising to you, but then I realized that if someone had asked me (an American) before I read the exam whether I thought there would be any questions on religion, I would have had no idea. I think US religious history is a really complex thing, probably much more complex than the usual story of cycles of Great Awakenings.

Posted by: James on June 4, 2008 10:19 PM | Permalink | Reply to this

### Re: Dumbing Down

Christianity was certainly an important part of the curriculum in now-atheist Sweden back then. When I started school in the mid 1960s we had still morning prayer, but it might be because my teacher approached retirement. Prayer definitely disappeared from public schools with the 1968 rebellion, and prescriptive Christianity was replaced by descriptive religion as a school subject.

Posted by: Thomas Larsson on June 5, 2008 9:03 AM | Permalink | Reply to this

### Re: Dumbing Down

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

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

Posted by: Bruce Bartlett on June 4, 2008 8:05 PM | Permalink | Reply to this

### Re: Dumbing Down

Posted by: Aaron on June 5, 2008 2:16 AM | Permalink | Reply to this

### Re: Dumbing Down

6. Michael buys 3 cartons of milk.
The total cost of 3 cartons of milk is £4.20
Work out the total cost of 7 cartons of milk.

Something doesn’t seem right here…

Posted by: Bruce Bartlett on June 4, 2008 8:20 PM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: David Corfield on June 4, 2008 9:19 PM | Permalink | Reply to this

### Re: Dumbing Down

Mmm, no doubt you are operating on a level of wit here greater than my maximum.

I did notice it was the ‘calculators allowed’ paper. I am disturbed.

What on earth?!

Posted by: Bruce Bartlett on June 4, 2008 11:15 PM | Permalink | Reply to this

### Re: Dumbing Down

It said

Michael buys 3 cartons of milk.
The total cost of 3 cartons of milk is £4.20

What is the total cost of 7 cartons of milk?

It is interesting to note the complete redundancy of the first sentence and the lack of full-stop at the end of the second sentence.

Is it likely that they thought it would have been less comprehensible without the first sentence and potentially confusing to have a full-stop after a number?

Posted by: Simon Willerton on June 5, 2008 12:17 AM | Permalink | Reply to this

### Re: Dumbing Down

Bruce Bartlett wrote:

Something doesn’t seem right here…

and then:

What on earth?!

I don’t understand what you’re so perturbed about. Aren’t they just asking us to take 4.20 and multiply it by 7/3, like David said? Am I missing something?

Posted by: John Baez on June 5, 2008 5:58 AM | Permalink | Reply to this

### Re: Dumbing Down

I am with Bruce. This is a strange question. I MIGHT be able to figure out the cost, but I would not buy ANY milk from this guy.

Posted by: Alex Hoffnung on June 5, 2008 7:51 AM | Permalink | Reply to this

### Re: Dumbing Down

Now I really am puzzled as to what the difficulty might be, regardless of the political wisdom of continuing this discussion. Could I trouble either Bruce or Alex to explain precisely what it is they object to about the question? I ask because an explanation could be helpful for my teaching. Of course I fear that I’m falling victim either to a joke or a misunderstanding. Nevertheless, in the interest of clarifying the issue, I will put down my answer: 9.80.

Posted by: Minhyong Kim on June 6, 2008 2:44 PM | Permalink | Reply to this

### Re: Dumbing Down

Well, I thought the issue was one of price as I confess that I don’t know how much milk costs – I don’t buy it very often and don’t look at the price. So I decided to pick a random supermarket online (at which I do not shop) and check out the price of a four pint carton there. Using Real World Data${}^{TM}$ the question would then become the following.

Simon buys 3 cartons of milk from Sainsburys.
The cost of 3 cartons of milk is £3.44.

What is the cost of 7 cartons of milk?

You may use a calculator on this question if you wish.

Posted by: Simon Willerton on June 6, 2008 3:52 PM | Permalink | Reply to this

### Re: Dumbing Down

Simon buys 3 cartons of milk from Sainsburys.

The cost of 3 cartons of milk is £3.44.

What is the cost of 7 cartons of milk?

You may use a calculator on this question if you wish.

Easy: £7.44.

I was told by a student yesterday that in GCSE maths, if you get the right answer you get all the marks, regardless of whether or not you showed any working. Apparently only if you get the wrong answer is the working taken into consideration. This helped me to understand - perhaps - why it is so hard to persuade my undergraduate students to justify their answers in exams.

Posted by: Eugenia Cheng on June 7, 2008 2:39 PM | Permalink | Reply to this

### Re: Dumbing Down

Eugenia gets top marks for saying

Easy: £7.44.

As she got the right answer she didn’t have to show her working. However, for those of you who can’t see where the answer comes from, I should say that the price is £1.44 for one carton and £2.00 for two cartons.

This could actually be an interesting question to give to students to demonstrate hidden assumptions: here there is the assumption in the question that price is a linear function of quantity, something that is clearly not necessarily the case.

Posted by: Simon Willerton on June 7, 2008 7:02 PM | Permalink | Reply to this

### Re: Dumbing Down

Sheffield scores zero.

One entrant was too shocked at the simplicity of the question to answer. The second copied down the question incorrectly (compare this with the original). The third copied the second’s miscopy, thus rendering the question unanswerable because of nondivisibility by 3.

The second then has the temerity to question the presuppositions of the problem, encouraging a dangerous lack of confidence in mathematics, in education and indeed in the whole moral fabric of the nation, founded as it is on the certitude of gauging objectively the merits of its citizens.

Poor show!

Posted by: David Corfield on June 8, 2008 11:09 AM | Permalink | Reply to this

### Re: Dumbing Down

Well, originally I just thought that was a really low-level question for 16 year-olds. Off-hand it just seemed more like a question for 11-year olds, but maybe I’m just totally out of touch. One should bear in mind this is the “calculators” paper, and also that these 11-year olds are the same people that are already so adept at computers and the internet, etc. Surely if they can navigate the complex world of Facebook, they can type into their calculators “4.20/7*3=”. At that age I recall typing in far more complex numerical algorithms to determine who “liked” who, etc.

Posted by: Bruce Bartlett on June 7, 2008 12:01 AM | Permalink | Reply to this

### Re: Dumbing Down

Woops as you can see I would have failed this exam by falling prey to the Corfield conspiracy. I should have said “4.20/3*7=”… I think :-)

Posted by: Bruce Bartlett on June 7, 2008 12:05 AM | Permalink | Reply to this

### Re: Dumbing Down

Oh, I see. It was the nature of your objection that I completely misunderstood. I thought the question was perhaps ambiguous in some subtle way that I couldn’t see as has sadly happened in the past when I was designing exams. As far as your real point goes, I obviously sympathize and, hence, my belief in unanimity is vindicated!

Posted by: Minhyong Kim on June 7, 2008 8:17 AM | Permalink | Reply to this

### Re: Dumbing Down

Complains about confusing and flawed exam problems (despite a preceding 16-layer controll procedure by expert committees) in Germany are reported here .
Perhaps part of the problems results from a general detachement of teaching experts from school reality? E.g. last week a leading research group asked a friend for ideas because unexpected problems endanger a big study on analphabetism - somehow the analphabetic pupils refused to read and answer the questionaires. My friend first thought they might be joking, but now assures that it’s not so.

Posted by: Thomas Riepe on June 5, 2008 11:28 AM | Permalink | Reply to this

### Re: Dumbing Down

Are you just worried, Bruce, about how expensive the milk seems to be? They could be very big cartons.

Do other countries have this where you discover how ‘in touch’ politicians are by asking them the price of a pint/litre of milk? It’s amusing to see their nervous look as they expose themselves to potential ridicule.

Posted by: David Corfield on June 5, 2008 8:59 AM | Permalink | Reply to this

### Re: Dumbing Down

What do you mean? The price looks perfectly reasonable by London standards. (I must be homotopic to one of those politicians.)

Having had considerable contact with professionals in math education, my feeling is that this very discussion does not bode well for those of us who like to claim a degree of unanimity in the research community :=) …

Posted by: Minhyong Kim on June 5, 2008 1:00 PM | Permalink | Reply to this

### Re: Dumbing Down

By ‘unanimity,’ I meant ‘interpretative unanimity.’ (I hope that notion makes sense.)

Posted by: Minhyong Kim on June 5, 2008 1:10 PM | Permalink | Reply to this

### Re: Dumbing Down

David wrote:

Do other countries have this where you discover how ‘in touch’ politicians are by asking them the price of a pint/litre of milk?

Yes. And in 1992, newspapers mightily mocked George Bush the First for telling reporters that he was amazed by seeing bar code scanner in a grocery store. They’d been around for many years; clearly he was out of touch.

But, my research has just revealed that this widely reported story was much exaggerated.

Posted by: John Baez on June 7, 2008 2:06 AM | Permalink | Reply to this

### Re: Dumbing Down

What I particlarly enjoy about these questions is that they always feature a named protagonist. I miss this aspect of school exams! It added a bit of emotional impact. Longer, multi-stage questions could become quite dramatic.

Most finals questions for my undergraduate course were written in the usual dry style, except for a very recognisable few, which were all written by the same professor. One in particular sticks in my mind — an optics question — which began, “A disgruntled pedestrian gazes at a distant car headlight through the weave of his umbrella…” I think all exam questions should be written in this sort of light-hearted way, it helps to relax you.

Posted by: Jamie Vicary on June 7, 2008 3:59 PM | Permalink | Reply to this

### Re: Dumbing Down

In the UK there has been a move from exam based assessment to mainly coursework based assessment during the last twenty years. Because of the emphasis on coursework assessment, they have dumbed the exams down to compensate for the fact that they are assessing other qualities in candidates.

I.e., the exams are less demanding now than they were before, because less emphasis is placed upon them. You can’t expect students to today still do as much exam preparation as they did decades ago, because they are now being assessed mainly on time-consuming coursework projects to determine the GCSE or A-level grades. This leaves them less time to prepare for the exams, which obviously have to be dumbed down to compensate.

However, the idea is that the overall assessment should still require as much work as before, only now much of the effort is for coursework projects, not exam preparation. So comparing recent exams to those from the 1960s is missing the point about the changed assessment.

One effect is that many more students are taking these courses and getting higher grades than before, so there is a kind of ‘grade inflation’ problem, whereby universities and employers see too many applications from candidates with the same high grades to be able to distinguish the candidates easily. Hence the reason why the A* grade was introduced to supplement plain A grade at A-level.

Posted by: nige cook on June 5, 2008 4:28 PM | Permalink | Reply to this

### Re: Dumbing Down

I wrote a very witty and erudite comment earlier but sadly the gremlins ate it instead of posting it to the café. Fortunately I saved it first, but upon reflection I decided that in addition to its innate humo[u]r it suffered from a surfeit of verbosity.

It is interesting to see that this chestnut has returned. Anyone who wants to see more examples from the coffers of the BBC may be interested in a few more links.

I would like to introduce a new direction to this debate. Discussing standards can be interesting, but rarely seems to get very far and often degenerates into anecdotage.

In most countries math[s] is regarded as a core subject. Imagine that that was up for renewal this year. What case would you make for keeping maths as a core subject? If you had to trim it, what would you retain?

(If your reaction to this is “That could never happen” then be warned. It happened to French. You could be next.)

When someone (proudly) says to you, “Oh, I was never very good at maths at school.” – and after you’ve gone through the whole court process for GBH – how do you respond? This article equates being bad at maths with being bad at reading. Do you agree? I may not say it proudly, but I would readily say that I wasn’t very good at the subject English when I was at school (and as for my grades in German … well, if you want real proof of dumbing down then the fact that I got an ‘A’ in German GCSE should do it. By the way, where is that train station?). Actually, let’s take that out of parentheses. What was the point of my learning German? At the moment, the main reason for German is to confuse me completely when I try to learn Norwegian (For those who’ve been following, I got a ‘B’, which I’m very pleased with). So why the big reaction when someone says that they were never very good at maths?

And following on from that, what exactly do they mean? What weren’t they good at? If I say that I wasn’t very good at English at school then I probably don’t mean that I can’t read; in fact, I quite enjoy reading. Rather, I mean that dissecting a text is not something that I’m all that bothered about. I’d rather just read it, enjoy it, and move on to the next book. Am I a Philistine? Should I be cast out into the outer wilderness? Probably, but not for that!

What hooked y’all on maths? For me, I wasn’t all that bothered about maths until my 3rd year at University. That’s when I got hooked and that’s also when I would say that I started learning real mathematics. Admittedly, I went to a pokey little university most of you wouldn’t have heard of so we probably had a bit of a kookey syllabus. I know why I think maths is great, but I also know that about 99.99% of the world’s population would think that I was completely off my trolley for thinking so. What about the rest of you? Especially the “non professional” mathematicians that hang out here.

Why are we so keen to inflict maths on schoolkids? What have they ever done to us?

Or is it that we were all teased for being geeks and just want revenge.

Posted by: Andrew Stacey on June 5, 2008 9:06 PM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: Tim Porter on June 6, 2008 8:05 AM | Permalink | Reply to this

### Re: Dumbing Down

Interesting you mention playing the piano. A Russian mathematician recently told me that the trouble with being the doctoral supervisor of home students at a middling-good British university is that it is like being in a conservatoire having to train musicians who have practiced all through their teenage years on slightly out of tune instruments.

Posted by: David Corfield on June 6, 2008 9:07 AM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: David Corfield on June 6, 2008 9:24 AM | Permalink | Reply to this

### Re: Dumbing Down

“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.

Posted by: Thomas Riepe on June 6, 2008 9:50 AM | Permalink | Reply to this

### Re: Dumbing Down

Thomas wrote:

The drawback of such a start of motivation is a strongly selective interest in math subjects.

As long as you can survive school, it may be okay focusing on what you’re really interested in. After all, if you follow your interests where they lead you, you’ll keep getting interested in new kinds of mathematics: it’s all a connected whole.

I’m actually glad I saved some elementary topics for my old age. It’s really fun to learn simple things, and more so when you already know lots of other stuff. For example, I always avoided algebraic geometry and number theory, but now I’m studying the basics of those subjects and enjoying them immensely. When a little cohomology or topos theory enters the picture, it doesn’t faze me — but I barely understand Galois theory or how to solve a cubic equation, and it’s wonderful when something like that starts to make sense and fit into the ridiculously elaborate framework I’ve already erected in my head. If I’d been forced to learn them at a younger age, it wouldn’t have been nearly as fun.

Hmm, for some reason I’m reminded of this Woody Allen quote:

In my next life I want to live my life backwards. You start out dead and get that out of the way. Then you wake up in an old people’s home feeling better every day. You get kicked out for being too healthy, go collect your pension, and then when you start work, you get a gold watch and a party on your first day. You work for 40 years until you’re young enough to enjoy your retirement. You party, drink alcohol, and are generally promiscuous, then you are ready for high school. You then go to primary school, you become a kid, you play. You have no responsibilities, you become a baby until you are born. And then you spend your last 9 months floating in luxurious spa like conditions with central heating and room service on tap… larger quarters every day… and then Voila! You finish off as an orgasm!

Posted by: John Baez on June 7, 2008 2:39 AM | Permalink | Reply to this

### Re: Dumbing Down

All this discussion reminds me that mathematicians are very willing to find fault in the preparation of students by others but not so willing to discuss fundamental issues on the aims of the teaching of undergraduate courses. A number of issues are discussed in articles on my page which also includes a final (1987) examination paper from the HOUYHNHMS UNIVERSITY STAFF COLLEGE, MATHEMATICS DIVISION, as well as a recent article ‘Promoting mathematics’ including the radical suggestion of promoting mathematics to students, so they can act as ambassadors for the subject, equipped with the language not only to do certain things but also to discuss the value and methodology of mathematics.

By contrast, in 2006 I met a clever Bulgarian research mathematician who got a scholarship to a famous USA University. I asked how it was, and he said 3 years of hell!’ A student from Oxbridge wrote that she and her friends were scarred by the difficulty and inaccessibility of the courses - but they all did well in their careers.

A common attitude is that the design of mathematics courses is entirely about content, how much can be crammed into students. And students continually prove they are not as good at the subject as their teachers!

Academics in the social sciences are shocked to learn that a mathematics course usually contains no information on research methodology!

So please let’s not waste time knocking the others, as we can do little about what they do, but we can do something about the design of the courses we ourselves put on.

As Einstein wrote in another context:

It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little…

Let us have a real discussion of the ‘justification and usefulness’ of our courses.

I feel one aim is to encourage a love of mathematics and the communication of its contribution to culture and civilisation, and an understanding of how it advances. How should this be done, and what proportion of the course should this take? Who should teach it?

One student wrote to us to say she was shocked to hear of the closure of Bangor maths as when she was there she loved every minute! So we must have done something right (for the students)!

Ronnie Brown

Posted by: Ronnie Brown on June 13, 2008 10:37 PM | Permalink | Reply to this

### Re: Dumbing Down

Not to shock anyone, but the 1959 exam is on par with what a student aiming at top engineering universities in Brazil are supposed to master. In fact, anyone aiming for any course where there’s a lot of competition (say, Medical School, my case). My high school Mathematics curriculum alone consisted of 10 books of which 2 books were for Geometry, each over 300 pages, chock-full of theorems and demonstrations and I knew them by heart at one point in my life. It’s either that, or you didn’t make the grade for Medical School (not that Medical School uses any Geometry…)

What a shame for the U.K.. I hear they’re shutting down Physics labs all over too…This doesn’t bode well.

Posted by: gg on June 14, 2008 6:14 AM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: John Baez on June 14, 2008 7:13 PM | Permalink | Reply to this

### Re: Dumbing Down

That sounds analogous the the question as to whether Russian scientists and engineers were better than American and British because they had (in the USSR days) less access to computers, and so were forced to solve math problems analytically by pencil and paper.

Posted by: Jonathan Vos Post on June 15, 2008 2:57 AM | Permalink | Reply to this

### Re: Dumbing Down

…whether Brazilian engineers are in some ways better than British ones…

Tricky to tell we have so few of them.

Posted by: David Corfield on June 15, 2008 6:21 PM | Permalink | Reply to this

### Re: Dumbing Down

What the Internet is doing to our brains

Posted by: Thomas Riepe on June 17, 2008 11:39 AM | Permalink | Reply to this

### Re: Dumbing Down

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.

Posted by: bane on June 17, 2008 12:28 PM | Permalink | Reply to this

### Re: Dumbing Down

In spite of my general laziness in following the discussions here, I was recently reminded of this old thread because my older son is now in the fifth year. Everyone around is talking obsessively about secondary schools. So I lifted myself out of my usual torpor long enough to look into some practice exams for entry into grammar schools, intended for children 10-11 in age. I was quite favorably impressed! Perhaps my concerns about education are ill-founded. Here are two questions I just picked out at random from the maths exam.

——————————————

Q13. The ratio of men to women in the town choir is 4:6. There are 40 singers in the choir. How many of them are women?

Q11. A spinner has an equal chance of landing on any of the numbers:

40 56 84 96 64 74

What is the chance it will come to rest on a number that is a multiple of 8? Write your answer as a fraction in the lowest possible terms.

——————————————

Perhaps some people with more experience in the British system would care to comment?

Posted by: Minhyong Kim on March 20, 2009 12:13 AM | Permalink | Reply to this

### Re: Dumbing Down

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

\
\
\
\
\
\
\
\

12 24 30

Posted by: jim stasheff on March 20, 2009 1:39 PM | Permalink | Reply to this

### Re: Dumbing Down

I’m not sure where these papers I looked at stand in relation to the system as a whole, present or past. There were a number of questions of the sort you pose, at least one of which I couldn’t solve after a minute or so of thought.

Posted by: Minhyong Kim on March 20, 2009 3:54 PM | Permalink | Reply to this

### Re: Dumbing Down

why not?

Posted by: kim on March 20, 2012 2:12 AM | Permalink | Reply to this

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