Sunday, January 20, 2008

2. Reflective thinking to Pimm's article

Speaking mathematically---Communication in mathematics classrooms (David Pimm)

When I was reading Pimm’s paper, I went back to Halliday’s paper from time to time to review the meaning of a register of mathematics. I wanted to avail myself of reading these two articles to help me understand the meaning of register in a language context. In Halliday’s definition, a register is a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings. (Halliday, 1978,pp195). Pimm summarized that registers have to do with the social usage of particular words and expressions, ways of talking but also ways of meaning. (Pimm, 1987, pp108) Pimm’s summary of a register of mathematics helps me understand the term “register” further and better. Language itself is developing with new technology, ideas, products created. Mathematics itself is also a language, in which there are many words whose meanings are different from daily ordinary language. So, a register is also a place in which we can store and develop words with their meanings, expressions, structures, disregarding whether they have already been developed or are being developed or undeveloped. Halliday also illustrates some ways of developing a register of mathematics, such as reinterpreting existing words; creating new words and borrowing words from other languages, the ways that English favors; calquing words from other languages, the way that Chinese favors, etc. Although I was quite interested in knowing the various ways of how to develop a register of language, especially knowing about the way how Chinese create new words, I have to admit that I misunderstood the meaning of calquing initially. Your comment helped me realize my misunderstanding. I have never thought that “jing ji” comes from the Greek root word of economy meaning “household management”, democracy (min zhu) literally originates from the Greek root “demos” and “cracy”. I realize that I’m devoid of systematic knowledge concerning the roots of English words. I even have never thought that it is necessary for a Chinese to study how Chinese language developed, but your comment invokes my interest to study the relationship between Chinese and other languages, because I realize that the accurate and rich use of language is very important in our teaching.

In my view, English has the difference between ‘ordinary English’ and ‘mathematics English’. So does Chinese. Pimm also mentioned: there are some special terms, such as quadrilateral, parallelogram, hypotenuse, which are unlikely to encounter often in our daily language. As to those words, such as product, factor, function, reciprocal, face, degree, power, radical, legs, rational, etc, they are borrowed from everyday English but with different meanings in mathematics. Although students can refer to general meanings in ordinary English when they touch these words in mathematics classes, they still will get confused the special meanings of mathematics with the general meaning. One of my tutoring students has made several attempts to understand the meaning of a power in mathematics. She always confused the power with the exponent or was not able to recall the word power when she was asked to express the form of qk, because what she could recall naturally is the general meaning of power in everyday English. For ‘product’, what students can suddenly think of is the item either sold in the market or produced in the factory rather than the result of multiplying two or several numbers or objects. So, I totally agree to Pimm’s opinion: “the failure to distinguish between ordinary English and mathematics English can result in incongruous errors and breakdowns in communication (Pimm, 1987, pp88).”

I was so impressed with reading some examples with regards to the word meanings in mathematics that Pimm has illustrated. These examples evoke my resonance through reflecting my own teaching. The first one is the term ‘clockwise’. This is a very good example demonstrating that the everyday meaning of a word is helpful for students to understand its meaning in mathematics. When I taught trigonometric ratios in grade 10, I found that some of my students had difficulty in understanding ‘clcokwise’, ‘counter-clockwise’ or were not able to turn an angle clockwise or counter-clockwise. When I asked them to refer to their watches to see how the watch works, they told me that they couldn’t see clockwise on their watches since their watches are digital. In this sense, if a mathematics term loses its everyday referent, it will become a difficult knowledge point in students’ learning. The second one is the ‘if…then…’ format which almost all mathematics propositions fall into. I have never related it to the mathematics register with which it carries the force of logical implication. I always thought the structure ‘if…then…’ has become a rule when we compose a mathematics proposition before. Right now, I’m wondering if the structure (or these two words) can show a real logic relationship between the conditions and results of a mathematics proposition. The third example is the problem concerning diagonals in polygons. To be honest, in my previous teaching in Shanghai, I have never realized that students might have difficulty in understanding the meaning of ‘diagonal’ in mathematics. I found that the understanding of the term ‘diagonal’ is a problem to grade 7-8 students in Canada when I was tutoring some local CBC students who can not speak Chinese. When I asked them to explore the respective number of diagonals in different types of polygons, such as triangles, quadrilaterals, pentagons, hexagons, etc, they initially counted the number of diagonals by joining two vertices disregarding whether these two vertices are adjacent or not adjacent. Although they know the diagonals are the line segments connecting two vertices, their understanding is not precise. I have to admit that for the term ‘diagonal’, English is not a good language which can help students understand the exact meaning through referring to the word itself. In contrast, ‘diagonal’ in Chinese is much easier for students to understand its definition either in polygons or in polyhedrons. ‘Diagonal’ is translated into ‘对角线(dui jiao xian)’. Based on the two Chinese characters ‘对角(dui jiao)’ which means opposite angles in English, students can readily understand that this line segment needs to join two opposite angles (two opposite vertices) rather than two neighboring vertices. So, I think this is why Chinese students seldom have trouble in understanding the concept of ‘diagonal’. This is the language difference between Chinese and English. Of course, I admit that in some mathematics cases, English is much easier for students to learn math. For example, Chinese students have to memorize most mathematics notations imported from western countries through understanding English. Otherwise, they have to memorize them mechanically. Most Chinese students have such questions: why do we need to use f (the initial letter of ‘function’) to express a relation between x and y since you will never see f in Chinese words or pinyin: ‘han shu’, expressing function; why do we need to use log to express a logarithm or a logarithmic function since we can never see log in the Chinese translation ‘dui shu’, why do we need to use sin, cos, tan, cot, sec, csc to express 6 trigonometric ratios, etc.

In Pimm’s paper, he pointed out a fact that I’ve heard again and again after I came to Canada. When examples of figures are drawn to illustrate or merely invoke a concept, the orientation is seldom randomized, and many pupils seem to include the particular orientation in their concept (Pimm, 1987, pp85). Even in my tutoring job, I found my students only regarded an isosceles or equilateral triangle as a triangle. If the polygon has three sides which are not equal to each other, it’s not a triangle. They didn't realize that a scalene triangle is also a triangle. As a matter of fact, I seldom heard the same story in Shanghai. If the teacher only demonstrates some specific orientations of a polygon when they explain the concept verbally, in my view, it’s not weird for us to see that students are not able to understand it accurately. I think the problem not only involves the language interpretation of the concept of a triangle but also includes the teaching methodology. (Maybe I’m wrong.)

The last point that I want to talk about is the role of metaphor in mathematics language. Pimm mentioned that analogy and metaphor come to mind as powerful linguistic techniques for creating new meanings (Pimm, 1987, pp93). Both of them offer means by which the less familiar may be assimilated to the more familiar, by viewing the former in terms of the latter. It’s exactly true. I also often tried my best to use analogy and metaphor to explain math concepts in my teaching. Or simply saying, the means of relating something new or unfamiliar to something that students have already known is really helpful for students to learn math., especially when students learn some abstract new concepts and solve some new difficult problems. To change something unfamiliar to familiar and to transform from the abstract to the concrete need analogy or metaphor. For example, we can teach students how to solve the equation sin2x+3cosx-3=0 (cos2x-3cosx+2=0) based on the easy quadratic equation x2-3x+2=0. This is the means of analogy. We can use a room corner which connects three walls (one is the ground, the other two can be two vertical adjacent walls) who are perpendicular to each other to help students understand the 3-dimensional coordinate system, use fishing to express the process of expanding the product of two binomials, use SOH, CAH, TOA to help students memorize the trigonometric ratios, and relate ‘take away’ to subtraction, ‘have more’ to addition. I think that they are the means of metaphor, but they are the extra-mathematical metaphors as what Pimm defined. I think we may use extra-mathematical metaphors quite often in our teaching. There is not only one way of metaphor to express one certain thing. Different people may create various personal metaphors as long as they find them beneficial for them to understand mathematics. As to the structural metaphor, this is the first time I heard it, although I’m quite familiar with the example—a complex number is a vector, cited by Pimm. A complex number as a vector is a very important knowledge part when we teach complex numbers in grade 10, but I always regarded it as the geometric meaning of a complex number instead of a structural metaphor. Meanwhile, from this example, we can see the importance of using structural metaphor in math teaching, which can help students relate one knowledge fragment to another relevant fragment. In so doing, students can transfer the existing knowledge to new knowledge or contexts and also view math as a complete learning subject in which knowledge points relate to each other rather than separate from each other. However, it’s challenging for a teacher to use metaphors including extra-mathematical and structural metaphors appropriately and with high quality in his/her teaching. This ability is also the one that I need to develop for my teaching.


Susan said...

Julia -

Nice work! You have written quite a lot. (Don't feel that you always have to write long blog either -- just a few paragraphs is good too if there's lots of interesting thought and commentary in it.)

I really like the way you have made personal connections with this reading, including examples from Chinese and English and examples from your own teaching both here and in China.

I'm interested to hear that you are becoming increasingly aware of word construction and word origin in Chinese and English. I think this is a very useful kind of awareness in trying to understand culture and education -- so much of history and culture is embedded in words and language. Please keep blogging about interesting word connections you discover as the course goes on!

I realy like your example of "product" as an ambiguous word in math and your students' lack of experience with "clockwise" clocks. Very interesting too to compare the Chinese and English words for "diagonal". It's a bit like the Norwegian students who wonder why we ask (in Norwegian translation) "What do meat-eaters eat?", which in the original English was, "What do carnivores eat?" With "diagonal", the very word in Chinese gives lots of useful information and helps clarify the concept.

I agree with you that it's not only language itself but also teaching skills that make a huge difference for kids' math learning. Nonetheless, it's important to be able to recognize the effect language has on learning, and to plan lessons that acknowledge that.

Thanks -- great blog!

Charles Wells said...

Some of the things you talk about on your blog, particularly concerning language and mathematics, are covered in my book "A Handbook of Mathematical Discourse" (Charles Wells, 2003, Infinity Press). You should also read the book by Kay O'Halloran on mathematical discourse, which is based on Halliday's work.