3. Reflective thinking to Barwell’s article

Integrating language and content: Issues from the mathematics classroom

I agree to Barwell’s opinion that language and content can be seen as reflexively related, for example, through solving word problems. The various aspects of thinking mathematics and learning language are all closely interwoven, with attention to written form, to problem genre, and to mathematical structure. Barwell uses his analysis for two ESL students’ dialogue during word problem solving to demonstrate that there is a link between attention to mathematical structure (i.e. content) and attention to written form (i.e. language), including the correct use of words, word spelling, verb tense (grammar), which comprise important parts in language learning. His article enlightens me with a way of how to embed language learning into math learning and teaching, the way which will be beneficial for my future bilingual teaching in Shanghai.

Over the past ten years, Chinese Education Ministry has been engaged in making efforts to reform Chinese English instruction due to the fact that Chinese English instruction prior to 1998 showed a big failure in educating students to be English proficient. So, after investigating and observing the bilingual education in Singapore, Hongkong, Canada, China realized that to integrate English learning into other subjects involving math, physics, chemistry, PE, art, etc in formal school education, is an essential method to improve Chinese English teaching whose main goal aims to make Chinese students more proficient at English use and communication in the world because of the globalization. Shanghai municipal government encourages all schools to carry out bilingual teaching, which has already become an urgent task in accordance with the economic rapid development since 2000. Here, we need to note that in Shanghai (China), we say ‘bilingual teaching’ instead of ‘bilingual education’. The term ‘bilingual’ in Longman Applied Linguistic Dictionary is defined as:

(A person who) knows and uses two languages.

In everyday use, the word bilingual usually means a person who speaks, reads, or understands two languages equally well (a balanced bilingual), but a bilingual person usually has a better knowledge of one language than of the other.

　　For example, he / she may:

　　a) Be able to read and write in one language but speak and listen in another language.

　　b) Use each language in different types of situation, eg. one language at home and the other at work.

c) Use each language for different communicative purposes, eg. one language for talking about school life and the other for talking about personal feeling.

‘Bilingual Education’ means:

The use of a second or foreign language in school for the teaching of content subjects.

　　Bilingual education includes three models as below:

　　a) The use of a single school language which is not the child’s home language. This is sometimes called an immersion bilingual model.

　　b) The use of the child’s home language when the child enters school but later a gradual change to the use of the school language for teaching some subjects and the home language for teaching others or alternately use both home language and a second language in a subject teaching. This is sometimes called maintenance bilingual model.

c) The partial or total use of the child’s home language when the child enters a school and a later change to the use of the school language only. This is sometimes called transitional bilingual model.

(Wang, 2004)

Several influential educations, who have been long-term engaged in Chinese bilingual teaching field, including Wang, differentiated the concept of ‘education’ and the concept of ‘teaching’. Based on their discussion, ‘Education’ involves a series of planned activities influencing learners’ moral education, intellectual education, and physical education according to certain educational expectations. However, ‘teaching’ means the instructional interaction between teachers and students. From the above definitions, we can see that the concept of ‘teaching’ is narrower than the concept of ‘education’. Meanwhile, terming ‘bilingual teaching’ is more consistent with the current situation of Chinese language education. Our current situation discovers a fact that we lack sufficient bilingual teachers with good English proficiency whilst our students are generally identified as quite limited English proficiency. In so doing, we need to ground our bilingual teaching on the maintenance bilingual model as meaning to that we still have to depend on our home language to support students’ mathematical content understanding. All Chinese students are considered as ESL learners but different from the ESL learners in English spoken countries, because Chinese students only learn English at schools. If we integrate English learning and math content learning, we must note that both the understanding of math terminologies, concepts, problem structures and meanings in English and the ability to translate from English to mathematical symbols are hurdles to Chinese students. In the meantime, as far as I’m concerned, Chinese students as ESL learners need a math learning context in which they can practice and improve reading, listening, writing, speaking skills of English simultaneously rather than merely focus on one skill.

If I go back to Shanghai in the near future, I think that I will incorporate bilingual teaching into problem-based learning context where students can start language and math learning with carefully selected real-life problems. From my perspective, these problems arising from real life involve more than word problems. Over the past 40 years, many research findings demonstrated that students in PBL can become more active learners instead of passive recipients in the traditional environment. These problems are beneficial for helping students not only improve their reading skill but also enlarge their vocabulary reservoir subsuming both general English and mathematics English words and enrich their knowledge, including history, social development, etc about western countries. . They embrace the mathematics structures featured by word problems, so students also need to analyze math structures expressed in English and learn how to translate English words and sentences to mathematical symbols and expressions. They are ill-structured, so students need to seek for more information via different methods which can help students develop various learning skills, including problem- solving skill, critical thinking skill, cognitive skill, and etc. In PBL, students are usually divided into small groups of 5-6 in each. In the small groups, students can have more opportunities to participate in both verbal and written practices of English through searching the meaning of math technique words, comparing the meanings between general English words and math words, analyzing the problem sentence structures, explaining solution processes, describing conjectures, proving conclusions and presenting arguments. Students in the small groups can have less shyness and fear to speak English and share mathematical ideas more fully than in the big-size class.. If students have questions about what they are thinking and learning, the teacher, in PBL, as a facilitator, can help them dispel the puzzles and explain ‘why’ and ‘how’ questions either in English or home language to facilitate their understanding of math content and English language knowledge. At the final step of each lesson, each student will be encouraged to present their assimilation and applications of new knowledge (i.e. solving new similar problems) both verbally and in written form which purport to help them not only consolidate their math learning but also pratice their speaking and writing skills of English. In addition, I need to point out that, from my perspective, it’s not easy to be a blingual teacher, who should not only have sufficient math knowledge background but also be very competent at English language.

The above discussed is some of my immature thoughts for bilingual teaching in Shanghai. I’m still thinking about a better approach to make my future biligual teaching more efficient in accordance with the current language abilities and situations of Shanghai students.

Wang, B. H. (2004). Bilingual Education and Teaching in China. Journal of Education, Shanghai Education Press.

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.

1. Reflective thinking to Halliday's article

Halliday’s paper: Sociolinguistic aspects of mathematical education
As a person who speaks English as a second language, I really understand the importance of a language, especially when you are far away from your home country, using a second language to study, work, communicate with local people in a foreign country. If you do not have enough proficiency in English, for example, in Canada, you will feel to be isolated by the local society. Language is not only a tool that we can draw on to express our ideas explicitly or interpret others’ opinions but also is an art, for example, how to use the exact words to convey your meaning precisely, how to use the language to show our respect, politeness without breaking local people’s customs when we talk to them. These are my personal experiences in the use of English in Canada. Although I had been learning English for more than 10 years when I was still in Shanghai, sometimes I found the English I learnt in China is different from the English I’m using now. So, in my view, to learn a second language needs a good language context, the same as the one in which we learn Chinese.

In this article, Halliday conferred the old word “register” with new meaning in a language context. He defined: “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.” To be honest, this is the first time that I know the meaning of register in a language context. I think I need more readings to help me understand the meaning of “register” further.

Halliday mentioned that the development of a new register of mathematics will involve the introduction of new “thing-names” through reinterpreting existing words, creating new words, borrowing words from other languages, calquing, etc. This let me think how we should use the language in our math teaching. In my previous teaching, I sometimes employed new or reinterpreted words which are in accordance with my students’ daily language, the prevalence of the society, etc, as my classroom verbal interaction with my students.

I’m interested in knowing that Chinese favors calquing as meaning to creating new words in imitation of another language. I suddenly realized that it’s exactly true. In Chinese, we have a lot of exotic words imported from western countries. As for these exotic words, we use Chinese words which sound like the same pronunciation of English, such as cement (shui men ting), chocolate (qiao ke li), email (yi mei er), etc.

What I’m impressed with reading is the part of talking about the uniqueness of the mother tongue. When I speak mandarin or Shanghai dialect, I can express the same thing in different ways. That’s exactly true. When my students had difficulty in understanding one way of interpreting math concepts, I can immediately use several other alternatives with the same meaning to explain them. However, when I use English to teach math, I have to admit that I really have encountered the dilemma the author described in his paper. Halliday said: “the teacher who is forced to teach in a language other than his mother tongue has at his command only one way of saying something.” As a teacher, I know that language is very vital in our teaching. If I can not speak a second language freely, it’s hard for me to develop the ability to say the same thing in different ways, to predict what the other person is going to say or add new verbal skills.

Finally, I agree to the author’s opinion: the more informal talk goes on between teacher and learner around the concept, relating to it obliquely through all the modes of learning that are available in the context, the more help the learner is getting in mastering it. Mathematics teaching definitely involves lots of language conveying the understanding of concepts and problems. That means language is a key factor leading to students’ efficient final understanding of math knowledge. Therefore, it’s necessary for us to study the relationship between language and math education. The benefit from the study is how we can use language to facilitate and enhance students’ math learning.

Self-introduction

Hi, everyone,
I'm Julia Lan Dai. I come from Shanghai, China. I had been teaching mathematics at a senior high school in Shanghai for consecutive ten years before I immigrated to Canada. I landed in Vancouver in August of 2006, I have been living here for a year and a half. I am a student from CUST department with mathematics education in major. This is the last course for my master's degree. Since English is a second language to me, this course involving studying the language between language (English) and math education will be challenging to me. Hope everybody can succeed in our pursuit.
Anyway, nice to meet you all.
Cheers,
Julia