Reflections on Mathematical Communicationfrom Taiwan Math Curriculum Guideline and PISA 2003
Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin
In this essay we present our thoughts on mathematical communication. We will
confine the meaning of mathematical communication in the mathematics classrooms,and argue that if it is going to be in the mathematics curriculum then it must haveclear guideline for instruction methods, teaching materials and assessments. We
discuss the role of mathematical communication in Taiwan’s government
Mathematics Curriculum documents, and seek for practical insights from PISA 2003
test problems and grading guides. Our APEC project will be designed for grade 6 or
12 year-olds, we will consult PISA 2003 (which is designed for students of age 15)
and come up with matching ideas.
Mathematical communication for the mathematics classrooms
One who lives in a society cannot avoid communication. He is either expressing
himself or tries to understand expressions of other people. Education takes theresponsibility to help children and young adults to grow into the society, so it is quite
natural that education must take some care about communications. And thecommunity of math educators has also some thoughtful discussions on mathematicalcommunication in particular, starting from 1—6 graders and extended to secondaryschools. We will learn much of the expertise most renowned mathematics educators inthis symposium. And here we report some thoughts from the local and practical pointof view.
When people talk about mathematics, when someone write about mathematics,
for instance an article of public interests on the most beautiful identity of mathematics,
that is exp(i*Pi)+1=0, or even a (Japanese) fiction on the expression doctor loved
most, and when people read the articles or fictions, they are all some manner ofcommunication and they are all about mathematics. However, this general sense ofmathematical communication is too broad to discuss in the context of mathematics
education. It might end up with pointless if we consider too much about this broad
sense of mathematical communication. On the contrary, we should consider themathematical communication as one of the competence that can be taught and learned
in a curriculum. In this sense, the mathematical communication under discussion shallbe a target or subject of mathematical teaching and learning. That means, among otherthings, it happens between teachers and pupils in a classroom, it must have instruction
methods or suggestions, it better has texts, examples, role models or teaching- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 1 -materials, and in some sense this learned ability shall be measurable.
Mathematical communication in Taiwan’s mathematics curriculum guideline
The government document for mathematics curriculum guideline (grades 1—9) has
been modified or revised quite often in the recent years. The current document wasoriginally issued in Nov, 2003, which preserved some statements about mathematicalcommunication proposed by the previous edition. The document categorized themathematics curriculum into five subjects; among them was the connection subject
which had 9 targets for mathematical communication. For instance, the first target formathematical communication was “Understand the comprehension of themathematical language, such as symbols, terms, tables, graphs and informaldeductions.” Such a statement of curriculum target was not clear enough for textbook
writers, neither for teachers and parents, to understand and to implement. Therefore in
the appendix the document composed a list of elucidation (explanations and examples)for each target.
The elucidation for each item of mathematical learning target was completedwith examples and do-s and don’t-s; we can say that they are quite clearly put andfairly straight forward to follow. Unfortunately the elucidations for the mathematicalcommunication targets were not as clearly written, and hence were not very helpful.
For instance the explanation for the first target was: “Language is necessary for thecommunication of mathematical contents. Of course one has to understand thelanguage in order to communicate.”
We think the curriculum guideline (or standard) is different from an education
policy or philosophy in the sense that the former is an operational document. It may
not be as precise as a mathematics text, but it shall be at least as precise as a legal
document. Textbooks, teaching materials, classroom activities and, most concerned,assessment problems, shall be able to refer to this document for their proper conducts.
In case of controversial situations, the document shall be the source to settle thedifferent points of view.
It is in the foregoing sense that the statements, together with their elucidations,for mathematical communication were considered not proper for an official
curriculum document. Therefore they are temporarily concealed in the most recent
revision of the document in late 2007. More knowledge, and hence more studies, arenecessary to make this subject clear enough to be considered a mathematics learning
subject or target. Our APEC project is expected to offer a little bit help.
Emphasizing the written mathematical communication
There are suggestions on the merits of oral mathematical communication that takes- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 2 -place in classrooms, either between teach and students or among students. It seems
American pupils are more verbal then ours in Taiwan (it seems Japanese pupils arequite verbal in the classrooms too). Although pupils in elementary schools are much
more active and willing to share their opinions then their elder brothers and sisters, it
is still doubtful that the oral communication could be a norm in Taiwan classrooms.
Teachers may ask students to think out loud, however, it is to certain degree against
the value of this society. We understand that teachers might welcome pupils to thinkout loud, for in that case they don’t have to make the efforts to probe the students’minds. However, pupils at young age may not be able to express their thinking
effectively, and teachers may fail to listen their minds verbally.
One of the other two concerns about oral mathematical communication is: It
takes time. Everybody knows that communication, especially the deeper ones, takestime. Public schools of California have almost twice the hours for mathematics thenthose in Taiwan. This is one of the most serious constraints for communications indepth in the classroom. According to the lesson study videos of this APEC project,Japanese teachers almost always run out of time when there are profoundcommunications in the classroom.
And finally we also concern about the teachers training. It also takes time toeducate teachers for adopting the oral communication skills, classroom management
skills and time management skills.
To conclude, we think for the recent future, it is not practical to motivate oral
mathematical communications in Taiwan classrooms. It is more practical, and wethink it is more urgent, to motivate written mathematical communications in themathematics curriculum (including assessments). It is one of the good traditions thatare vanishing. The ultimate form of mathematical writing is to write down thestatements of definitions and theorems, and to write a proof. Since the abandon ofwritten problems in national examinations, students’ abilities of writing (and alsoreading) are thought to degrade very soon.
There are not much formally to write for elementary school mathematics.
However, a good manner is of great worth for the future. One of the aspects toemphasize is the logical and meaningful recordings of arithmetic procedures. Forinstance, to write a sequence of equal signs in 17×27＝17×(30-3)＝510－51＝460－
1＝459, pupils often fail to present the equal sign (fail to write the sign or fail to keep
the quantities equal). However, in the situation of solving an equation shall be writtenwith meaning words to connect the equations. For instance, since 4x-3=5 amounts to
4x=5+3=8, by dividing 4 on both sides we have x=8/4=2. Some pupils may add
improper equal signs at wrong places, and many pupils simply write the equations
consecutively without understanding what they are deducing. Teachers’ writings on- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 3 -the blackboards or lecture notes may help a lot as role models for students to follow.
And it may not take too much effort to ask teachers adopt a better (more logical andmeaningful) way of mathematical writing.
Mathematical communication in the three phases of mathematisation
PISA 2003 uses the word mathematisation for the multi-phase process of solving
problems mathematically. The front phase involves the understanding of problem, andtransformation from reality situations to mathematical models. The inner phase ispurely mathematical, one solves a mathematically formulated problem bymathematical methods, including calculation and deduction. The final phase is tointerpret the mathematical result back into the reality.
There are ingredients of mathematical communication in the front phase. For
instance, a common problem says “What are the three consecutive even numbers withthe sum 264?” This is a mathematical problem, one needs not the transformation. Abasic level of communication ability is to link the daily language consecutive into its
mathematical meaning. And one has to understand the mathematical terms even andsum. A more interesting communication problem is when someone presents a solution
by first saying “Let n be an even number …”, and someone might ask something like
“How can you let n be an even number? What if it is odd? Why n must be even andcannot be odd?” What can a teacher do when some student asks this kind of question?
The communication involved in the inner phase is to write arithmetic and logic
expressions properly, for instance the writings demonstrated in the previous section.
Although ultimately one is expected to write a mathematical proof for thecommunication of his mathematical reasoning, it is neither necessary nor practical todevelop this level of abilities under grade 9.
Mathematical communication in PISA 2003
PISA talks about mathematical communication in some core principles of their test
design. For instance, they define mathematical literacy as the capacity of students toanalyze, reason and communicate effectively as they pose, solve and interpretmathematical problems in a variety of situations involving quantitative, spatial,probabilistic or other mathematical concepts. They try to design a test that measuresthe mathematical literacy of 15 years-olds. PISA’s idea of communication focused on
the final phase of mathematisation which, they think, involves some form oftranslation of the mathematical result into a solution that works for the originalproblem context, a reality check of the completeness and applicability of the solution,a reflection on the outcomes and communication of the results, which may involve
explanation and justification or proof.
- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 4 -In my opinion, it is not surprise to notice the economists, enterprisers andbankers based OECD takes a more practical approach for the PISA test. They caremuch about the outcome, the result and the final solution (even with compromise). I
am also glad to learn that PISA provided a practical definition of their idea ofmathematical communication by describing their problems and grading standards.
PISA divides students into six levels of proficiency. As communication belongs
to the final phase, it is supposed to be developed later then the routine operations.
There is no assumption of communication abilities for levels one and two. The
communication abilities start at level 3: develop short reports on their interpretations,results and reasoning, and gradually upgrade to level 6: precisely communicate theiractions and reflections regarding their findings, interpretations, arguments, and theappropriateness of these to the original situations. The descriptions will be more
concrete when we inspect samples of problems and grading standards.
Sample 1. A problem in the subject of quantity. Mei-Ling from Singapore waspreparing to go to South Africa for 3 months as an exchange student.She needed tochange some Singapore dollars (SGD) into South African rand (ZAR). There are threeproblems in this set.
1. Mei-Ling found out that the exchange rate between Singapore dollars and South African rand was:1 SGD= 4.2 ZAR. Mei-Ling changed 3000 Singapore dollars into South African rand at this exchange rate.How
much money in South African rand did Mei-Ling get? This is a level 1 problem.

2. On returning to Singapore after 3 months, Mei-Ling had 3 900 ZAR left. She changed this back toSingapore dollars, noting that the exchange rate had changed to: 1 SGD = 4.0 ZAR. How much money in
Singapore dollars did Mei-Ling get? This similar

question becomes a level 2 problem.
3. During these 3 months the exchange rate hadchanged from 4.2 to 4.0 ZAR per SGD. Was it in
Mei-Ling＇s favour that the exchange rate now was
4.0 ZAR instead of 4.2 ZAR, when she changed herSouth African rand back to Singapore dollars? Givean explanation to support your answer. This is

obviously a question that requires
communication. It is considered level 4.
Sample 2. A problem in the subjectof uncertainty. A TV reporter showed this graphand said: “The graph shows that there is a hugeincrease in the number of robberies from 1998 to1999.＂ Do you consider the reporter＇s statement tobe a reasonable interpretation of the graph? Give an- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 5 -explanation to support your answer.

Students are expected to say no to the reporter. If one can deny the reporter byconverting the absolute counts to relative rates, or if one points out that we need moredata of previous years to actually judge about the trend, then he is considered to havelevel 6 of proficiency. If one can say something like an increment of 8 out of 500 is
not considered huge, he is considered level 4.
Sample 3. Another problem of uncertainty which shows PISA’s idea of level 5. The
diagram shows the results on a science test for two groups, labelled as Group A and Group B. The meanscore for Group A is 62.0 and the mean for Group B is 64.5. Students pass this test when their score is 50 orabove. Looking at the diagram, the teacher claims that Group B did better than Group A in this test. Thestudents in Group A don＇t agree with their teacher. They try to convince the teacher that Group B may notnecessarily have done better. Give one mathematical argument, using the graph that the students in Group A
could use.

If one can point out the distorting effect of the outlier in the results of Group A,
or he can argue that more students pass the test in Group A, then he is considered atlevel 5.
PISA provides a set of good examples for the assessment of mathematicalcommunications. However, it aims at 15 years-olds. We think there are feasible
materials and test problems in this manner that are suitable for students of grade 6. In
our APEC project, we would like to design a lesson for teachers of 6 graders that will
effectively teach mathematical communication with test problems to measure theprogresses.

- Chang-Shou Lin, Wei-Chang Shann and Su-Chun Lin 6 -