kitchen table math, the sequel: Math appreciation

Monday, August 24, 2009

Math appreciation

One of the junior high schools in our district is an "international baccalaureate" school.

From the school's website:
International Baccalaureate Middle Years Program (IB-MYP)

The IB program has been in existence for several decades and has now reached over 2,000 schools and 124 countries.It is an internationally recognized and highly reputable program designed to fully engage students with an aim of creating a better and more peaceful world. It is catered specifically to the early puberty and mid-adolescent student. The MYP helps students develop the attitudes, life skills and knowledge necessary to participate fully in a growing and changing world.

It is vested in the ethics and values of young people, and its unique characteristics allow students to make connections between subjects, link what they learn to the real world, and reflect on their learning.


An explanation of the MYP (Middle Years Progam) Mathematics program follows. Emphasis mine.

MYP mathematics expects all students to appreciate the beauty and usefulness of mathematics as a remarkable cultural and intellectual legacy of humankind, and as a valuable instrument for social and economic change in society.

MYP Aims

The aims of any MYP subject and of the personal project state in a general way what the teacher may expect to teach or do and what the student may expect to experience or learn. In addition they suggest ways in which the teacher and the student may be changed by the learning experience.

The aims of teaching and learning mathematics are to encourage and enable students to:

recognize that mathematics permeates the world around us
appreciate the usefulness, power and beauty of mathematics
• enjoy mathematics and develop patience and persistence when solving problems
• understand and be able to use the language, symbols and notation of mathematics
• develop mathematical curiosity and use inductive and deductive reasoning when solving problems
• become confident in using mathematics to analyze and solve problems both in school and in real-life situations
• develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics
• develop abstract, logical and critical thinking and the ability to reflect critically upon their work and the work of others
develop a critical appreciation of the use of information and communication technology in mathematics
appreciate the international dimension of mathematics and its multicultural and historical perspectives.

23 comments:

Crimson Wife said...

Why is recognizing & appreciating the beauty of mathematics and instructing children how to quickly & accurately solve mathematical problems assumed to be mutually exclusive goals?

I personally hope to do both in my family's homeschooling.

And the history of math is IMHO an interesting topic, something that I make sure to include in our study of civilizations.

Tracy W said...

I don't understand the point being made here.

If you've developed the knowledge, skill and attiutdes necessary to pursue further studies in mathematics, well that strikes me as the most important things schools can teach in mathematics. Knowing that maths isn't merely a tool of Western cultural capitalist patriachical imperalists, but is part of the heritage of all humanity also strikes me as a good thing.

I'm not sure about the focus on appreciating the beauty of mathematics, or enjoying it, but I guess the goals are to enable students to do this, not to require them to do so.

MagisterGreen said...

"...expects all students to appreciate the beauty and usefulness of mathematics..."

That's right, kids, you'd better appreciate it! We don't say anything about expecting you to know it, but you'd better appreciate it, by Jove.

"...as a remarkable cultural and intellectual legacy of humankind,..."

No, no, a thousand times no. Math is not a cultural artifact. Long division in sub-Saharan cultures is no different from long division in Asian cultures or Western cultures. Nor is math any sort of "legacy" of mankind. Math is universal and is really the one point of convergence we would have with other intelligent life forms(as numerous science fiction writers have imagined).

"...and as a valuable instrument for social and economic change in society."

The amount of sophomoric, and despotic, thinking encapsulated in that brief fragment boggles the mind.

Allison said...

Tracy,

I'll have a much longer answer for you tonight, but for now, I will say that the goal of my child's middle school math program should be to prepare my child for high school math. not to give them an appreciation of math.

We can flesh that out more--the goal of middle school math is teach the students so that all students are prepared for success in high school math: that they can at least pass Algebra 2, and as many students as possible can be on a college prep calculus track.

Compare that to what they said the goal was.

Allison said...

Imagine if they had said that the chemistry program expected all students to appreciate the beauty and usefulness of chemistry.

Does that equate to students DOING chemistry? of KNOWING about covalent or ionic bonds? of Le Chatelier's principle?

Imagine the music program's goal was to expect all students to appreciate the beauty of music.

Does that equate in anyway to the mastery of reading music, composing, or performing?

Assuredly, someone who has mastery of something should be able to appreciate it. But the converse does not follow.

Anonymous said...

I would like to see music appreciation and art appreciation/history courses in middle and/or high school. Art and music are part of our cultural heritage, which I feel all should know, but not everyone has the interest or talent to create them. For the academic disciplines, appreciation is nowhere near enough.

Crimson Wife said...

I agree with Anonymous above. Not everyone has to be able to play an instrument or make a work of art himself/herself. But everyone should be able to appreciate the masterpieces of music and art. It's enough for me to be able to discuss a musical or artistic composition intelligently even if I cannot create a decent one of my own.

With math & science, however, students need both their own technical competence and an appreciation of their inherent beauty.

Math is a reflection of God's order in the universe He created. Obviously, a government-run school cannot teach that explicitly, but they can teach about "nature's order" without specifying who or what created it.

Anonymous said...

"Not everyone has to be able to play an instrument or make a work of art himself/herself. But everyone should be able to appreciate the masterpieces of music and art."

How is this different from saying "Not everyone can do math, but everyone can learn to appreciate the beauty and usefulness of mathematics"?

Children can be taught how to observe and draw realistic representational figures (e.g. "Drawing for Children" by Mona Brookes or "Drawing on the Right Side of the Brain").

Children can also be trained to recognize and produce relative pitch and to sing and play musical instruments (e.g. singing with Solfege and various instrument instructional methods).

With practice (spaced repetition!) and training one can become a reasonably good artist or musician, just as with adequate practice and instruction one can develop a good facility for mathematics.

The difference is that society places a higher value on mathematics and other academic subjects, so we expect and provide more practice and more instruction in those areas.

The fixed mind set (à la Carol Dweck, "I can't draw." "I'm tone deaf." Barbie's, "Math class is tough!"), lack of adequate instruction and failure to practice is what holds people back in many endeavors. With explicit and direct instruction and with dedicated practice one can draw well, sing or play well and do calculus.

Tracy W said...

MagisterGreen, Allison, are you reading the same set of quotes that I am?

The aims include explicit listings of being able to actually do stuff with mathematics. For example:
"develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics." And "use inductive and deductive reasoning when solving problems."

While I would change the emphasis of the list a bit, I think it's a reasonably good list of aims for mathematical education.

And MagisterGreen:
Math is not a cultural artifact. Long division in sub-Saharan cultures is no different from long division in Asian cultures or Western cultures. Nor is math any sort of "legacy" of mankind. Math is universal and is really the one point of convergence we would have with other intelligent life forms(as numerous science fiction writers have imagined).

MagisterGreen, you're the one who has got this wrong. Maths is a cultural artifact. The meaning for example of the mathematical symbol = is entirely cultural. Furthermore, you claim that long division is no different varying in culture? Ever tried doing long division in Roman numerals? Different cultures have different numbering systems that do drastically vary in effectiveness. (Luckily we can copy better ideas and are not restricted to one culture).

And as for not a legacy, precisely how much maths do you think you'd've known if you had to invent it all from scratch rather than being taught it? Archimedes knew nothing of calculus, I do. Is that because I'm smarter than Archimedes, nope, it's because I've inherited a cultural legacy that included not merely Archimedes' mathematical knowledge but Newton's too.

Mathematics is an incredible cultural and intellectual legacy of humanity.

Original: "...and as a valuable instrument for social and economic change in society."
MagisterGreen: The amount of sophomoric, and despotic, thinking encapsulated in that brief fragment boggles the mind.


Oh yes, of course, mathematics has contributed exactly zero to social and economic change. Apart presumably from all its trivial uses in engineering (which, with the invention of cheap land transport in the forms of railways and the internal combustion engine has revolutionised economies, warfare, and living patterns, to give merely one example), its use in statistics, where evidence of increased living standards and life expectancies has been used to refute Marxist economics, where statistics has been used to test the efficiency of new medications, drastically increasing life expectancies and in the case of contraceptives, causing social change. And I don't believe I've exhausted the list.

If it's sophoric and despostic to think that mathematics is a valuable instrument for social and economic change, then I am proud to be sophoric and despostic. And personally if I had kids I would prefer their teachers to teach them sophoric and despostic matters as long as they're true, rather than lying and saying that mathematics has no value in causing social and economic change.

Allison said...

Tracy, here's the whole sentence:

The aims of teaching and learning mathematics are to encourage and enable students to develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics.

That's the 7th of 10 goals, and it's the closest they've got to something concrete.

But it doesn't say the aim to teach the students to do stuff in math. It says the aim is to encourage and enable students to develop knowledge.

That's two levels of indirection away from "we aim to teach your kids math."

You have to work hard to say something that indirect. A lot of effort was put into that phrasing.

At best, their 7th goal of 10 is that they be a guide on the side to your child doing math. It's up to your child to get themselves ready for high school math. They just enable.

MagisterGreen said...

Tracy,

I think you are confusing the symbology of math with the reality of math. The fact that we use certain symbols to represent certain ideas, while another culture would use others, does not alter the underlying reality of what we are doing. Whether you use your fingers, an abacus, a calculator, or a supercomputer, 2+2 will always =4. The reality of math is universal and true, however the symbology of it may change.

To say that mathematics is a legacy of humanity implies that, absent humans, math would not exist. In that I say that you are mistaken. The reality of math would always exist but, without humans, the ability to recognize and render it into intelligible terms would, perhaps, not exist.

The awareness of math and the ability we have to use it to understand the universe is a cultural and intellectual legacy of mankind. But the fact of math is not.

As for the social aspect, I think we have a different view of cause vs effect. The advances that have been made possible, using math and other aspects of human knowledge, have come about in cultures which have allowed free, or at least less restricted, inquiry into nature, science, and such. So I would say that the culture comes first, which allows the inquiry and thus the discovery of things like advanced mathematics, which then brings about tangible results which thus further affect the society. But the society, and its fundamental underpinnings which allowed the discovery of the knowledge, has to come first. Consider: the Chinese made numerous discoveries in math, astronomy, science, and so on, and yet their culture remained essentially static for thousands of years. Likewise with Pharaonic Egypt, which accomplished amazing feats of engineering but evolved very little over the millennia. To pick up your mention of Archimedes, his discoveries were made possible by the fact that he lived in a culture, Hellenistic Sicily, that allowed him to sit around and doodle in the sand, which valued his ideas and which allowed him to test his theories and to try until he succeeded. I would propose that, should you take Archimedes and place him in a different cultural setting (say Pharaonic Egypt), he would not have been as prolific or even able to do his work.

Furthermore, you say that the ancients knew nothing of, say, calculus. Says who? Prove it. They may not have had the knowledge or ability to express it as we now do, but to say they were essentially ignorant of it is to assume facts not in evidence. Their understanding may have been simplistic, and they may have lacked a method or symbology to represent it, but we don't know that they were ignorant of it. The Romans had no number zero (0). Yet the concept of 0 was something they were aware of and worked around, even though they had no way to represent it. In other words (to come back to my first point) the reality of math was there despite the fact that the ability to represent it was not.

Allison said...

To MagisterGreen's point:

n^(p-1) is congruent to 1 mod p

is true.

(You can read this as:
if p is a prime number, and for any integer n, n to the (p-1) is congruent to 1 modulo p.)

It would still be true if Fermat hadn't proved it. It would still be true if someone else proved it using different notation. It would be true if no one on Earth has yet proved it, or even no one in the universe.

If you believe otherwise, then you've been reading post modernist philosophy too long.

Allison said...

To the bigger point:

Remember, we're talking about middle school. There is a limited amount of time to teach the kids what they need to know before high school. We're not talking about the goal of teaching something or not; we're talking about the goal of teaching THIS and therefore NOT TEACHING THAT. There's an opportunity cost to deciding to spend time teaching the beauty and cultural relevance of mathematics. Fundamentally, it leaves you less time to teach the mechanics of math.

Teaching about math as an instrument social and economic change is really about teaching about social and economic change, and teaching that is really about a political worldview espoused by Bill Ayers. It's nothing to do with teaching factoring.

Crimson Wife said...

I can be a perfectly productive member of society without being able to sing on key or draw a decent picture. Yes, it's nice to be technically competent in art or music, but not really necessary. Numeracy, however, *IS* a necessary skill for being a productive member of society. That's why math is a core academic subject while art & music are electives.

VickyS said...

I love math and appreciate its beauty and elegance. But I'm pretty sure my appreciation developed as a result of my understanding, not the other way around.

The big pedagogical mistake being made in math (and science too) is to assume that kids can't, or won't, find the subjects inherently interesting and enjoyable. To the contrary, I think the more one learns, the more fun it is. A lot of kids are checking out academically in the face of mindless "appreciation" activities.

Anonymous said...

Is anyone here familiar with David C Geary's work in "An Evolutionarily Informed Education Science"?

His thesis is that evolution has not provided the necessary scaffolding to help students with biologically secondary knowledge such as algebra and reading. These skills have a cultural origin, relatively recent emergence, and provide useful skills.

His paper explains why most students need explicit instruction, effort, and practice to acquire biologically secondary knowledge (eg- a right angle is 90 degrees) and abilities (writing, reading, algebra). Speech and an ability to read faces affected survival and simply are differently acquired.

So much of our discussions on KTM are explained by his recognition that most students need explicit instruction in math, reading, and writing to hook into the working memory and attentional control parts of the human brain that evolved to acquire such secondary competencies.

John Sweller sees Geary's work as having critical implications for instructional design. It explains why discovery learning makes sense as a hypothesis (no one has to teach most humans how to speak) but lacks any credible body of evidence supporting its efficacy.

Both these men in the last few years have articulated the "whys" behind what works in instruction and what learning is acquired easily and unconsciously and what needs worked examples.

Their work should be better known by all of us who care so deeply about childrens' learning, what works, and why.

RMD said...

anonymous said:

"So much of our discussions on KTM are explained by his recognition that most students need explicit instruction in math, reading, and writing to hook into the working memory and attentional control parts of the human brain that evolved to acquire such secondary competencies."

I strongly agree.

In addition to evolutionary explanations, we can look simply at children's reactions to instruction and determine, through observation, what works and what doesn't. The educational establishment, for all their talk in science about the scientific method, fail to use this approach to developing instruction.

The Follow Through study gave us tremendous experimental data on what works and what doesn't, and it was thrown away by the educational establishment because they "can't handle the truth".

Allison said...

I guess I'm a bit perplexed why anyone would think that "evolution has not provided" is a way to think about learning or evolution.

Evolution isn't FOR us. It doesn't care about humans. Evolution doesn't care about anything--that's an anthropomorphism. It is a process that favors diversity of species.

Of course we need explicit instruction! That's the norm! The abnormal thing is that humans evolved language acquisition that doesn't rely on instruction. It's an astonishing set of genetic mutations.

Here's what I really don't get. This is OBVIOUS to anyone who has watched children for an hour. AN HOUR! Only academics researchers find this somehow provacative. It's a lot like how researchers somehow feel the need to spend more than 10 seconds on the issue of whether humans were born tabula rosa or not. How ANYONE could think that if they watched children for an hour boggles the mind.

So I'm sure researchers see this as having critical implications for instructional design but only because they went so far in the wrong direction in the first place that they needed to spend decades figuring out instructional design after they broke it. Parents couldn't possibly have believed that math, science, writing, etc. didn't need explicit instruction.

le radical galoisien said...

I say definitely became more inspired to study mathematics (at the age of 14) after reading Simon Singh's book about Fermat's Last Theorem (that brought exposed me to people like Evariste Galois, Taniyama, Shimura, and Sophie Germain)...

At that age mind you, I had just failed math miserably in Singapore, with a final grade of 47. Now I will avoid the correlation-causation fallacy, but suddenly I found math a lot more interesting to study after that book, cuz suddenly after that my mathematic imagination took flight.

le radical galoisien said...

"Of course we need explicit instruction! That's the norm! The abnormal thing is that humans evolved language acquisition that doesn't rely on instruction. "

Not really abnormal I think ... Pinker tries to discuss this in his books... no more abnormal than elephants having a superflexible trunk that sometimes has the dexterity of human hands (and way more strength), or ants being able to carry things 50 times their own weight.

There are several possible pathways. One hypothesis I believe posits a social origin ... a socially competitive origin, that is. Politics is pervasive among all primate species, with many alliances in the ruthless ladder to prestige. That is, language did not evolve to facilitate cooperation among humans, but competition *between* humans. Alliances formed during childhood would have the most effect, partially explaining language's critical window.

The remnant of such an evolutionary pressure felt then can then be witnessed today when children spontaneously create or creolise languages; as you can imagine, being a group of children able to spontaneously generate a language (a sort of secret code) that some other group in power couldn't understand might be of considerable political advantage.

It's a fascinating hypothesis, and I think there's some value in it though there are also many other interesting hypotheses to consider.

le radical galoisien said...

And the only reason why language evolved in humans is because only humans got to the point where the natural group size was quite naturally large -- around say, 150 people. That's a large group. (You don't see many primate groups that big -- often they're restricted to 20-30 individuals.)


If you try to construct all the possible networks and relationships among 150 people, you can see how it explodes quickly.

(And it was brain power that probably imposed the upper limit on the size of primate groups. And the evolution of brain power was limited by nutrient intake. Yayy for our ancestors eating fish and bone marrow)

Tracy W said...

Alison: you have missed the earlier goals of "understand and be able to use the language, symbols and notation of mathematics" and "become confident in using mathematics to analyze and solve problems both in school and in real life".


Your initial criticism was not about the word "enable" rather than teach, though I can see the problem with that now.

MagisterGreen, you appear to be assuming that we can separate the reality of mathematics from its symbology. Yes, 2+2 will always equal 4. But there is no way to state that 2+2 = 4 without using some sort of symbology and that symbology is culturally determined.

I don't know how you can teach mathematics without using some symbols, even if it's "if I have one apple and then take another apple, what do I have?".

I agree with you that a culture needs to have other things than mathematics to change and develop. I cannot see anything in my original comment that indicates that I did not agree with that. Nor can I see anything in the original document that disagreed with it. The statement was maths is "a valuable instrument for social and economic change in society" not "maths is the sole cause of social and economic change and will always induce social and economic change regardless of what else is happening in society." Can you please point out anything in my comment or in the original post that you think that implied that either I or the other author believed that maths could cause social and economic change by itself and independently of society.

Sorry, I can't prove that Archimedes knew nothing of calculus, I can't prove a negative (outside mathematics and only then sometimes). My mistake in saying that. I will now correct it to "I know of no evidence that Archimedes knew of calculus". I don't quite understand your Roman point, you state that the Romans had no number zero, presumably you agree that we now have the number zero, how do you think this change came about?

Similarly for Allison's point about that theorem of Fermat's. Yes, it would still be true even if no one had proved it yet. But *we* wouldn't know this truth. The mathematics we as humans know would not include that one of Fermats' theorem in this hypothetical world. And, postmodernist or not, I don't know how you could expect any teacher to teach that theorem of Fermats if they didn't know of it, nor had discovered it themselves.

Tracy W said...

On thinking about it, I think there's a difference in terminology here. MagisterGreen is reading the word "mathematics" as referring to the whole world of the measurement, properties and relationships of quantities and sets (to use the answers.com definition). I, and possibly the writers of the MYP Mathematics programme, am reading the word "mathematics" as referring to human knowledge about the measurement, properties and relationship of quantities and sets, including our knowledge of better and worse symbol systems for acquiring and using this knowledge.