Peer reviewed article
Peter Gould's research provides a discussion on the need for understanding Asian cultures and learning.
The fundamental objective of ‘Australia in the Asian Century’, the White Paper released in 2012, is for all Australians to have the opportunity to acquire the skills and education they need to participate fully in a strong economy and a fairer society. The need to strengthen education and training is stated many times in the 25 national objectives. For example, under Building capabilities we have the following goal:
- By 2025, Australia will be ranked as a top five country in the world for the performance of our students in reading, science and mathematics literacy and for providing our children with a high-quality and high-equity education system (p. 21).
The top five countries referred to are determined by rankings on the triennial Programme for International Student Assessment (PISA). PISA is an assessment of 15-year-olds in three domains: reading literacy, mathematics literacy and science literacy. As with any assessment across multiple domains, it is difficult to establish a clear rank-order. The current top five performing countries in PISA according to Australia’s Commonwealth Parliamentary Libraryare Finland, Hong Kong, Shanghai, Korea and Singapore. Of these countries, four of the five relate directly to the belief that the 21st century will be the century of Asian dominance on the world stage.
The 2009 PISA results identified Shanghai-China as first ranked in reading, mathematics and science. However, the list of the top five countries in 2009 PISA on the mathematics scale sees a change to the group mentioned above, with Chinese Taipei entering the list and Finland dropping to sixth place. This produces an all-Asian top five in mathematical literacy.
The rank-order of Australia’s performance in PISA may not be the most effective measure of our educational output. Comparing rank-order over time (as stated in the goal) is only feasible if the same countries are participating. To understand the limitations of the rank-order as a measure, it is necessary to appreciate the changing nature of the countries participating in PISA. The number of countries taking part in PISA has effectively doubled since 2000. In 2000, only 32 countries took part in the study. By the 2009 PISA study, 65 countries participated, including for the first time Shanghai and Singapore, two of the top five. In 2012, additional countries such as Vietnam have been included in the list of countries taking part in PISA.
Many different reasons have been offered for the high performance of East Asian education systems (for example, Jensen, Hunter, Sonnemann & Burns, 2012). Rather than simply attributing success to rote learning, some international research (NCES, 2003) is suggesting that classroom lessons in Hong Kong, for example, require greater deductive reasoning. If this is correct, how do students develop this capacity for reasoning and is it possible to strengthen students’ reasoning in Australia?
The role of symbols in learning to reason
Reasoning, the capacity to consciously make sense of things, is considered to be a defining characteristic of human nature. Reasoning requires more than being able to associate two ideas such as smoke and fire. It requires the ability to create and manipulate a system of symbols (Atkin, 2006). One example of such a system of symbols and signs is language.
Language is an intricate code and its role in transmitting thoughts and sustaining human culture has led to its privileged place in the school curriculum, particularly in reading and writing. Although spoken and gestural forms of language are part of the normal development of human beings, writing is not. Written language requires instruction and conscious practice to master.
Writing systems represent words. Sometimes writing uses symbols that correspond more or less to the sounds within words (as in English), sometimes the symbols correspond to syllables (as in Japanese Kana) and sometimes the symbols correspond to morphemes (as in Chinese – a morpheme is the smallest meaningful word element. Unlike English where a morpheme can have several syllables, most Chinese morphemes have a single syllable).
In seeking to understand the high-level of performance of our Asian neighbours, particularly in mathematics literacy, it is helpful to compare the role written and spoken language play in English compared with languages based on Chinese (Galligan, 1993, 2001). The recently adopted Australian Curriculum: Mathematics(ACARA, 2012) provides an opportunity to do this. For example, the expectation of the Foundation year (Kindergarten in NSW) in the Australian Curriculum in counting is:
Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point.
If we compare counting words in English to standard Chinese (Table 1), we can appreciate why English speakers struggle with the mixed heritage of our counting words. To
learn to count to twenty, English speakers must master learning in order, twenty unique words. Not so Chinese speakers. Armed with the knowledge of the number words from one (y) to ten (shí), all of which are single syllable words, a Chinese speaker can count to ninety-nine (jiŭ shí jiŭ ).
Some linguistic attributes of a language such as Chinese may allow for a lower cognitive load and more efficient processing than in English. This can be most helpful in early childhood where children are learning to read. For example, the word thirteen in Chinese is 十 三 (ten three) and the word twenty is 二 十 (two ten). This greater clarity and consistency of counting words in Chinese reduces the confusion that occurs for young learners in English between number words like thirteenand thirty. The greater transparency of meaning in Chinese is even more striking in the use of ordinals. Counting ‘one, two, three’provides little or no linguistic assistance to generating the ordinals ‘first, second, third’. However, in Chinese, the ordinal ‘third’is 第三(dì s n) where the symbol 第 translates roughly to ‘sequence’or ‘order’. So fifth and sixth would be 第五 and 第六.
Students learning in Chinese have many opportunities to become familiar with the written and spoken counting words. For example, the word for Tuesday is 星期二 (day two or starperiodtwo) and for March is 三 月 (three month). Even if you are not fluent in standard Chinese it is not difficult to determine that April is 四 月 (Kelly, Miller, Fang, & Feng, 1999). The 2nd February can also be written in Chinese as 2月2日, which translates to two month, two day. Today’s written Chinese can include Hindu-Arabic numerals.
The use of number words in Chinese is also apparent in naming geometrical shapes. The triangle is 三 角 形 (three corner shape) and the quadrilateral is 四 形 (sì bi n xíng) (four side shape). Although many people have difficulty remembering the name of a seven-sided polygon in English [heptagon], in Chinese its name is 七 形 (q bi n xíng) (seven side shape). Polygon in Chinese is 多 形 (du bi n xíng) for many side shape.
Not only is a Chinese reader supported through a transparent use of the numbering system, what constitutes a reasonable assessment question must vary between native English speakers and Chinese speakers. For example, the Australian Curriculum: Mathematics expects students to name and order months in Year 2 (ACMMG040). How much easier is it to do this in Chinese than in English?
The names of fractions are also more consistently generated and are easier to interpret in Chinese than those in English. Ordinal numbers in Chinese are not confused with fraction names the way they are in English (eg the third third). To construct a fraction in Chinese, the denominator is written first, followed by 分之 (parts of) and then the numerator. Consequently, three-eighths is written as 八分之三 and two-thirds becomes 三分之二.
English creates fraction names by using the ordinal term for the denominator and then making it plural (as in three-eighths). Unfortunately, the difficulty of dealing with the sequence of ordinal names in English is well known (Miller, Major, Shu, & Zhang, 2000). Even the Chinese equivalents of denominator 分母 (fraction mother) and numerator 分子 (fraction child) are more descriptive of the relationship between the two components of a common fraction than their English or Latin-based counterparts.
Consistent and transparent language may make some concepts more accessible than they would otherwise be, but does it really impact on learning mathematics?
Does it create a measurable difference?
Comparing learning across cultures is not easy. League-tables alone do not provide the information we seek to understand what is or what is not possible. So let us instead narrow our question to comparing learning to count in Chinese and English before the impact of schooling. In 2005, Miller, Kelly and Zhou reported the results of a longitudinal study of learning to count in Chinese and English. They found that although there was little difference in the performance of 2-year-old children learning to count in Chinese or English, between 3 and 4 years, the course of acquisition of counting began to diverge. Four-year-olds in China made very rapid progress in generalising their counting procedures once they could count to approximately 40 compared to English speakers in the United States (Figure 1).
The linguistic representation of mathematical concepts in ordinary language can affect the ease of acquisition of these concepts. Chinese speaking students start school being able to count further and faster (Chinese counting words are all single syllables) than their English speaking counterparts.
Going beyond the known
Learning to reason is integral to learning mathematics. Reasoning is also a clear focus of teaching mathematics in Japan (Sawada, 1997). The creation of the Australian Curriculum in Mathematics has also brought reasoning to the fore.
The curriculum seeks to ‘… ensure that all students benefit from access to the power of mathematical reasoning’(ACARA, 2012). Indeed, reasoning is one of the four proficiencies that operate across all of the content described in the curriculum. Yet if teachers are often able to recognise mathematical reasoning when they see it, many are less confident about knowing how to develop mathematical reasoning in their students.
One challenge of developing mathematical reasoning in students is that there is more than one kind of mathematical reasoning. Within the new Mathematics K-10 syllabuswhich incorporates the Australian Curriculum, the Stage 4 outcome related to reasoning is a ‘student recognises and explains mathematical relationships using reasoning’(MA4-3WM).
As an example of how students are expected to recognise and explain mathematical relationships, I will draw on a form of deductive reasoning described as algebraic reasoning.
Unfortunately, whenever the term algebra is used, most people think only of working with symbols like x. This omits a major component of the history of algebraic thinking, sometimes described as ‘rhetorical algebra’. From the time of the ancient Babylonians to the 16th century, algebraic problems and their solutions were frequently composed solely of words (Kaput, 2008). Rather than restricting my interpretation of algebra to manipulating symbols, I understand algebraic reasoning to be thinking logically about unknown quantities and the relationships between them.
Imagine sitting in a classroom where you can see the calendar for the month displayed but with some of the dates covered. A yellow strip has been placed vertically over three numbers and a red strip placed horizontally over it to form a cross (Figure 2).
The teacher asks the question,
Which has the larger total; the numbers covered by the yellow strip or the numbers covered by the red strip? Why?
While you ponder this question, the teacher says that first she wants you to convince yourself of the answer and then to convince a friend.
You may establish your answer by determining the values of the covered numbers and carrying out the additions.
This will certainly convince your friend. But then the teacher moves the ‘cross’created from the yellow and red strips so that it is centred at a different spot on the calendar, and asks the questions again. Do you expect the new question to have the same answer as to which group of three numbers has the larger total? As we seek to explain our answer to the general problem, we are engaged in using algebraic reasoning.
We return to the class the next day to find that instead of the calendar, the focus of the lesson is a 100-chart with a square marked on it in red (Figure 3).
The teacher asks,
What is the sum of the numbers in the square?
Your first impulse is to reach for a calculator but the teacher is not encouraging the use of calculators. These numbers are of greater magnitude than the ones on the calendar but they also suggest a number of patterns. You break up the numbers and jot down your thinking (Figure 4).
Can you convince yourself that it will always work? Can you convince a friend? Could you convince someone who did not want to believe it? This is the essence of mathematical reasoning, or more specifically, algebraic reasoning.
The new Mathematics syllabus provides many opportunities to develop students’ deductive reasoning more deeply. The challenge remains to make use of those opportunities.
Enabling all students to benefit from learning to access to the power of mathematical reasoning relies on more than teaching additional content. Developing mathematical habits of mind takes time and skilful curriculum planning. The implementation of the Australian Curriculum: Mathematicsthrough the new Mathematics K-10 syllabus provides an opportunity to focus on valuing mathematical reasoning. It will not be a simple task and, no doubt, many will proffer the belief that mathematics is only about obtaining the right answer, quickly. Those who expound this belief may sadly have been denied the opportunity to access the power of mathematical reasoning.
Will setting the goal of being a top five country in PISA help us become a ‘clever country’? As a country we should have high expectations for our future. As a sport loving country, perhaps we could learn from athletes when it comes to setting goals. Emil Zatopek, who pioneered the interval training method, was perhaps the greatest distance runner in Olympic history. When it came to setting goals, Emil Zatopek’s advice was simple.
You can’t climb up to the second floor without a ladder. When you set your aim too high and don’t fulfil it, then your enthusiasm turns to bitterness. Try for a goal that’s reasonable, and then gradually raise it.
In seeking to reach higher educational goals, investigating the role of language, culture and symbolic processes provides fertile ground to start to realise the objectives of Australia in the Asian Century.
References and further reading
Atkin, A. 2006, ‘Peirce’s theory of semiotics’, Stanford Encyclopedia of Philosophy.
Atkin, A. 2009, ‘Peirce’s theory of signs’, Stanford Encyclopedia of Philosophy, Spring 2009, accessed 20 January 2018.
Australian Curriculum, Assessment and Reporting Authority (ACARA) 2012, Australian Curriculum: Mathematics, accessed 12 April 2013.
Galligan, L. 1993, ‘An exploration of the different structure of the mathematics register in English and Asian languages: Some consequences for the teaching of ESL in preparatory programmes’, in B. Atweh, C. Kanes, M. Carss & G. Booker (eds), Contexts in mathematics education: proceedings of the 16th annual conference of the Mathematics Education Group of Australasia, MERGA, Brisbane, pp. 275-280.
Galligan, L. 2001, ‘Possible effects of English-Chinese language differences on processing of mathematical text: A review’, Mathematics Education Research Journal, vol. 13, no. 2, pp.112-132.
Jensen, B., Hunter, A., Sonnemann, J. & Burns, T. 2012, Catching up: learning from the best school systems in East Asia, Grattan Institute.
Kaput, J. 2008, ‘What Is algebra? What is algebraic reasoning?,’ in J. Kaput, D. Carraher & M. Blanton, (eds), Algebra in the early grades, Lawrence Erlbaum Associates, New York, pp.5-18.
Kelly, M. K., Miller, K. F., Fang, G. & Feng, G. 1999, ‘When days are numbered: calendar structure and the development of calendar processing in Chinese and English’, Journal of Experimental Child Psychology, vol. 73, pp.289-314.
Miller, K. F., Kelly, M. K. & Zhou, X. 2005, ‘Learning mathematics in China and the United States: cross-cultural insights into the nature and course of mathematical development’, in J.I.D. Campbell (ed.), Handbook of mathematical cognition, Psychology Press, New York, pp. 163-178.
Miller, K. F., Major, S. M., Shu, H. & Zhang, H. 2000, ‘Ordinal knowledge: number names and number concepts in Chinese and English’, Canadian Journal of Experimental Psychology, vol. 54, no. 2, pp.129-140.
Miller, K. F. & Stigler, J. W. 1987, ‘Counting in Chinese: cultural variation in a basic cognitive skill’, Cognitive Development, vol. 2, pp.279-305.
NCES — US Department of Education National Center for Education Statistics 2003, Teaching mathematics in seven countries: results from the TIMSS 1999 video study, accessed 20 January 2018.
Sawada, D. 1997, ‘Mathematics as reasoning–episodes from Japan’, Mathematics Teaching in the Middles School, vol. 2, no. 6, pp.416-421.
Keywords: mathematics literacy; teaching examples; deductive reasoning; PISA
How to cite this article: Gould, P. 2013, ‘Learning in the Asian Century’, Scan, 32(2)