REPORT
Mathematical Communication through Proof and Representation
APEC - Tsukuba
International Conference on Innovative Teaching Mathematics
8 December 2007 - 15 December 2007
CHENG Chun Chor Litwin
Hong Kong Institute of Education
Introduction
Mathematical thinking is always an important element in mathematics curriculum.
Mathematical proof is a kind of mathematical thinking, representation, andcommunication. In this paper, we will discuss the component of mathematics
communication in the Hong Kong mathematics curriculum, how mathematics
communication can be developed in the classroom through the mathematics tasks infinding the greatest product of multiplication. And include the discussion of teachingapproaches is preferred to develop mathematics communication. The mathematicaltasks will be used in three levels of classes, primary 3, primary 4 and primary 5.
The paper tried to address the following questions on mathematical communication.

Q1

How does the Hong Kong curriculum document enhance communication ormathematical communication for students?

Q2

What are the components of mathematical communication to develop?

Q3

What kinds of approach will you prefer to develop the communication inclassroom?

Communication within the Hong Kong mathematics curriculum document
In the 1983 version of the Hong Kong Primary Mathematics curriculum, there is nospecific mention of the term “communication”. However, in the later developed TOCcurriculum (Targeted Oriented Curriculum) based on 1983 curriculum,
communication is one of the five components (investigation, concepts, problem
solving, reasoning and communication).
The term communication in the document usually refers to the generic nature, such as
discussion among students, discussion between students etc. It s not refereeing to the- CHENG Chun Chor Litwin 1 -1

mathematical nature such as proof or representation.

Components of mathematical communication
The main components of mathematics communication are representation and proof.
There are three kinds of communication of mathematical knowledge.
The first kind of communication is the internal understanding of mathematics of theindividual. It is a thinking process so that the mathematics problem and also the pathof solving the problem become a kind of coherence thinking process. With therepresentation of the solution in diagram or in words or in symbol, students obtain theunderstanding of the process.
The second kind of communication is the explanation of the solution to others. This
involves the understanding of other thoughts in understanding the questions and thesolutions. We may not understand why other people do not understand the meaning of
the problem or the solution path of the problem. One possibility is the representationof the problem and the solution. With a good representation or an acceptable
representation, students understand what other means and the content of mathematics.
The third kind of communication is internal extension of the mathematical content of
the individual, trying to extend that mathematics content that is convinced by otherand extend the difficulty of the problem or posing new problem.

Individual understandingIndividual extension

Proof and representationas communication

The mathematics tasks and the process
The aim of the paper is to discuss how primary school students communicate theirthinking through the following problems.
- CHENG Chun Chor Litwin 2 -2

Task A
Using the numbers to fill the boxes and make the greatest product
Using the set of numbers

Using the numbers

Using the numbers(1, 2, 3), (1,2,5), (2,3, 6)(1, 2, 3, 4), (1, 2, 4, 5).

(1, 2, 3, 4, 5),(1, 2, 3, 5, 6).

×

×

×

Task B
Using the numbers to fill the boxes and make the greatest product
Using numbers

Using numbers

Using numbers(1, 2, 3, 4)(1, 2, 3, 4, 5),(1, 2, 3, 4, 5, 6),(1, 3, 5, 7)(1, 2, 4, 5, 7, ).

(4, 5, 6, 7, 8, 9)

×××

Teaching approach for communication
The most important part of teaching is the selection of appropriate questions. Suchtasks allow students to express their solution through different representation. The
questions are set in the way that the answer can be observed by the pattern. Whenstudents predict their answer, they can confirm the answer through theircommunication.
Mathematics questions selected are within the mathematics curriculum, so thatstudents can perform mathematical investigations based on their knowledge onarithmetic operations. The beginning questions use some simple number so thatstudents can obtain answers easily.

- CHENG Chun Chor Litwin 3 -3

Encourage students to use generic skills such as listing and observing pattern.
Encourage students to explain their answer based on pattern and observation andexplain it through different representation.

Task A, finding the greatest product of multiple digits with one digit
Question A1：

Question A2：

Using the numbers 1, 2, 3 to fill in

Using the numbers 1, 2, 3, 4 to fill inthe boxes to make the greatest

the boxes to make the greatest

product

product××\
Listing and using pattern observation
Students are asked to list all the possible multiplications and they obtain the following
results:
1×32
31×62322×661332×23133×32131262×2

Next they are asked to find the greatest product by arranging a set of three numbers,for example, using the numbers 2, 3, 5. After some examination, many students easilyfound that the following arrangements could not produce greatest product. They are
5×3210 635270×

And they concluded that large numbers should be placed at the position P and Q.
- CHENG Chun Chor Litwin 4 -4

P×Q

When they come to question A2, using 1, 2, 3, 4 to form a greatest product in the formof× , many students can predict the greatest product through patternobservation and give the following as answer.
321412 84×Students’ logical thought on the results 321×4 = 1284
Some students found that there is a dominant effect of the larger number in certain“critical” place value. They try to compare the following multiplication with the large
number at the critical place. Hence the number in the second line could not be 1.
At one instance, when he found out that putting number 1 could not give the greatest
product, a student reckon that it will not be a possible to get greatest product if thenumber 2 is put in, and also not the number 3. Finally, they conclude that the largest
number 4 should be put at position Q. And they convinced that the following give thelargest product.
321
412 8

4×Explanation of 321×4 = 1284 is the greatest productHowever, this is not “accepted” as explanation and they need to convince others. And
they try using comparison of different product for explanation. Though many students
agree that the following arrnagment could not give largest product.
434232223344××××××- CHENG Chun Chor Litwin 5 -5The following is the representation of multiplication used in many mathematics
textbooks and difficult to see the comparison.
32×12 814421312 63×4

The teacher tries to introduce multiplication in the following format so that students
can compare different layers of number. Once it is written, many students start tofigure why the arrangement give the greatest product.
32×144213×12 0012 0080460312 8412 63321×4 = 1200 + 80 + 4421×3 = 1200 + 60 + 3

By this partition of multiplication, they can compare the sum of product layer by layer
and obtain the conclusion that 321×4 is in fact the largest.
Second explanation
For primary 5 students, some use the concepts of areas to explain why thearrangement gives greatest product. This is a pictorial representation of the above
calculation. In fact they can compare the sum of the three numbers 1200 + 60 + 3 withthe sum of the three numbers 1200 + 80 + 4.

4300

20 1

1200

80 4

3- CHENG Chun Chor Litwin 6 -400

20 1

1200

60 36Explanation 3
However, students also able to use their own representation to show that the productthey find is greatest by some logical argument. When students continue to work on
this type of questions, some have a new idea on comparison.
Question A3：
Using the numbers 1, 2, 3, 5, 6 to fill in the boxes to make the greatest
product×Many students already noticed that there are two places that the greatest number
should be put in. They are the position P and Q.
PQ×They then compare the following two products 5321×6 and 6321×5.
Many students notice that there is a common part (321) in the two products, so they
can just compare the product of multiplying the number 321×6 and 321×5.
In fact they are using distributive law to do that without noticing it, comparing (5000+ 321) × 6 and (6000 + 321) × 5.
As “5000× 6 = 6000× 5”, students know that they only need to compare the term
321× 6 and 321× 5, and obviously 321× 6 is larger than 321× 5. Hence they come tothe conclusion that “5321×6” is the greatest product.
Product 1：

5×

Product 2：

32166×- CHENG Chun Chor Litwin 7 -32157

Tasks B Finding the greatest product of two digits times two digits
Question B1：

Question B2：Using the numbers 1, 2, 3, 4 to fill in

Using the numbers 1, 3, 5, 7 to fill inthe boxes to make the greatest

the boxes to make the greatest

product

product
××

Students have tried out nearly all the possible arrangement of the numbers.

×4 32 1×4 2
3 1×2 43 1×

2 3
4 1

Students observe a pattern of arrangement of the numbers when it gives thegreatest product. They observed they need to maximize the “tenth” digits togive a greatest product. After working with the tasks A, students already have theknowledge that large number should be placed at position P and Q.
P× Q

This comes to the comparison of the values of the two products

×4 23 11302

×4 13 21312

Obviously the second arrangement 41×32 = 1312 gives a larger product.

- CHENG Chun Chor Litwin 8 -8Pattern observation
If there is 4 numbers, we need to “equally” fill in the numbers. Equally means a small
number should go with a large number, so the largest number with pair with thesmallest.
Students are then asked to notice and predict their result or explain their results. They
use the following logic
Divide the numbers into two groups. So the two large numbers 3 and 4 will go into
different group. And to compensate the value, a smaller number “1” will go with 4
and the larger number will go with 3 so as to offset the difference. Hence they try theproduct of the grouping 41 and 32.
For the next question, they list two to three possible arrangement, however, they knowthat 52×43 will be one of it as all include this product for their later comparison.
Discussion of why such arrangement gives a larger value
Students also try to explain why the difference of the two product 41×32 = 1312 and42×31 = 1302 is 10.
One of the explanations is to obtain the product line by line. This is a kindof represenation which the students need to learn and use in the discussion.
4 24 1× 3 1× 3 212004060212008030213021312(equal)( + 40)(–3 0)(equal)Students see that two of the products are the equal (40×30 = 1200 and 1×2 =2) and theother two product are 40×1 = 40 and 40×2 = 80, differ by 40 and 30×1 = 30 and 30×2
= 60. The above result in a difference of sum in 10.
Usually the multiplication in Hong Kong primary school is in the following format.
This kind of represenation unable to discuss why the first product is smaller than thesecond product.
4 24 1× 3 1× 3 21260

1230- CHENG Chun Chor Litwin 9 -9428213021312

However, other students give an explanation on the following product...
Question B2：Using the numbers 1, 3, 5, 7 to fill in the boxes to make the greatest product×Representation 2
7 3× 5 1

×7 1
5 3

He use the first arrangement 73×51 and subtract 71×53, finding that it is not easy to
do so, he compare 73×51 and 71×51, knowing that he need to add in 71×2 later on.
As 73×51 - 71×51 = 2×51, he knows the answer is 102. However, he need to subtract
the sum 71×2 = 142, so that we know that the difference of the two numbers is 142 –102 = 40. That is 71×53 is larger than 73×51 by 40.

703503500

1701503701503500

5032103This is equivalent to the format of multiplication. Some students compare the twoformats and discuss the values of the two products.

- CHENG Chun Chor Litwin 10 -10

Representation 3
The third representation is similar to representation 2. They know that the comparison
737

1× 5

1× 5

3

So they compare 7×1 + 5×3 and 7×3 + 5×1.
The teacher also uses area approach to calculate the two products and invite
students to compare the answers.
Some argument and counter examples
One student thinks that the answer should be putting the four numbers into two groups
such that the sum of each group is the same. In the problem of 1, 3, 5, 7, he think thatthe answer is 71×53, as 7+1 = 5+3 = 8.
(i)
×7
51
37+1=85+3=83763

Many students try to figure out whether the argument is valid. Some students bringout the following counter examples
The first example is using the 4 numbers 4, 5, 7, 9. They know that the greatest
product is 94×75 = 7050. However, 9 + 4 = 13, and 7 + 5 = 12. The two sums are notequal and the second sum 7 + 5 = 12 is less than the first sum 9 + 4 = 13.
The second counter example is using the set of numbers 1, 7, 8, 9, which tries to use
more extreme values. The greatest product is 91×87 = 7917. However the two sums
are 9+1= 10, and the second sum is 8+7 = 15. Contrary to the first counter example,
the second sum 8+7=15 is larger than the first sum 9 + 1 = 10.
Hence many students, after hearing the argument of the counters examples, agree thatthe first explanation is not a valid one.
(4, 5, 7, 9)(1, 7, 8, 9)
949 + 4 = 13

9- CHENG Chun Chor Litwin 11 -1

9 + 1 = 10
11

×

7

5

7 + 5 = 12×7050

878 + 7 = 157917

Using analogy for new question
Question B5：Using the numbers 1, 2, 3, 4, 5 to fill in the boxes to make the greatest
product×

Using analogy and assumption
One student uses the thinking that supposes that the number is 1, 1, 3, 4, 5.
Since the answer to 3, 4, 5 is known. They use the following result
Question ：Answer ：Using 3, 4, 5to form the greatest
product.
4×

New Question ：×

35Anaology：- CHENG Chun Chor Litwin 12 -12

Using 1, 2, 3, 4, 5 to form theUsing the previous result and fill ingreatest product.

1 and 2.
4×

35×

For the new problem, using the number 1,2 3, 4, 5 to form a greatest product in theform of×, students tries to apply the results of the above example. Assume
that the two numbers added are the same 1 and 1, then the greatest product will be4 3 15 1×

If the two new numbers are 1 and 2, then there are two possibilities for greatest
product.:(i)
4 3 1(ii)
4 3 2
5 2
5 1××

Product (i) will give a larger value than (ii) as 431×2 is larger than 432×1.
After that, they tries to find out all potential large products and confirm their results.
4×351222 412

4×352
122 032

5×

342
121 8125×341222 302

In another class of students, they have the following discussion.
New Question ：Anaology：- CHENG Chun Chor Litwin 13 -13

The greatest product for using theUsing the numbers 4, 5, 6, 7, 8, 9 tonumbers 6, 7, 8, 9 isfill in the boxes to make the greatest

×

9867product×

Hence the greatest product of 6 numbers 9, 8, 7, 6, 5, 4 is adding the new number 4,and 5.
Many students predict that the greatest product is
9 6 4
× 8 7 5843500

As they only have to consider the two cases.

×9867(i)×986754843410(ii)×986745843500

The results confirm with their prediction.
Conclusion
1
Communication and understanding depends on different representations.
2
Listing is a powerful starting point for communication.
3
Encourage students to use analogy and counter example in their mathematicalthinking.

- CHENG Chun Chor Litwin 14 -14

4
In the process of finding the greatest product, students show the following different
path of communication.

1

List all products and Pattern observation

2

Compare products of all different multiplication
Compare products of selected multiplication3

Compare products using area concepts

4

Using previous results and analogy, logical argument

5
The teaching process can add in simpler question. The following are some examples.
Question 1：

Question 2：

Fill in number 7, and 8, so that the
product is greatest.
3× 6

Fill in number 7, and 8, so that the
product is greatest.
3 4× 6 5

6
The discussion of mathematics communication is summarized in the following
framework.
Framework of three levels of mathematical communication
1

Individual understanding of the problem and solution (observe pattern andpredict answer)

2Explanation and proof through different representations such as area concepts
and layers of products

3

Individual understanding, extend further understanding through changing thesolution path or proposing new problem and their solution. (for example,changing the consecutive numbers to non-consecutive, or repeated number)- CHENG Chun Chor Litwin 15 -15