Table of Contents
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- Array Chart
- Oldest Array Chart in Japan
- Array Charts and Multiplicative Sentences
- Properties of Multiplication
- Counting Number Using Array Charts
- Limitations of Array
- Related Instructional equipment
- Why Did I Write This Article?
1. Array Chart
This article is the sequel to "Towards Japanese Multiplication Instruction". In this article, I would like to show you array charts, the notion of array and its application to the understanding of multiplicative structures.
An array chart is a rectangular arrangement of identical objects.
We can define an array chart without using "rectangular"; it is a set of identical objects that are arranged in horizontal and vertical directions in a plane.
The symbol of a circle or a square is typically put to use as an object. If the square 1 cm on a side is adopted, then we are able to calculate the areas of squares and rectangles easily. Basically I have use of a filled circle (●) for this purpose.
The following figure is an example of an array chart.
The following figure forms a rectangular but we would not regard it as an array chart.
The following figure indicates the arrangement composed of 3 vertical objects and 4 horizontal ones, but obviously this does not belong to array charts.
An array is an important data structure frequently used in the field of computer programming. However the array in the context of programming is generally associated with a 1-dimensional arrangement, and we have to write a 2-dimensional array or a multidimensional array to express an array of arrays clearly. If the term array is based on a programming sense, then the array chart in the math education is derived absolutely from 2-dimensional arrays.
The composing objects seen in an array chart are often called dots instead of objects. Using stickers, beads, or marbles, children can make up array charts to take in their hands. The tangible aggregation is easier to compare with the ones of different sizes than the paper-and-pencil drawings of array charts.
2. Oldest Array Chart in Japan
It is interesting to find the oldest array chart in Japan or in the world.
I would like to report two charts used or introduced in Japan. One can be seen in the math textbook (so-called "green cover") published in 1935 for Japanese elementary schools. There are two array charts consisting of the buttons of clothes, in a textbook for the 2nd grade students. Both charts have exactly 5 rows, and the left part has only 1 column while the right has 6 columns. By contrasting these two arrangements of buttons, the textbook asks how many times as many buttons there are in the right as in the left, before letting the learners count the total number of the buttons. For details, visit http://homepage3.nifty.com/ooiooi/rekisikakezan.htm, written in Japanese by a professor of math education.
Another noteworthy array chart was presented by Kenzo Nakajima in 1968*1. He introduced the notion of array together with the array chart that consists of dots of 6 rows and 9 columns, for defining the multiplicative sentence 6 x 9.
The principle of the multiplication in this sense is also shown in his article. After the definition of the direct product (a.k.a. Cartesian product) is given, A x B corresponds to the lattice points, taking A={a,b,c} and B={x,y} as an example.
The explanation like this hints that array charts were not familiar with the Japanese math education in those days, namely until 1960s.
Although the children need not learn Cartesian product, the usage of sets is closely related to the modernization of math education. The modernization brought just a decade enthusiasm and confusion, I suppose. However the arrays survive now, as a tool for understanding the multiplicative structure. I have seen the examples of Cartesian products together with the illustrations of array charts in Japan and overseas.
3. Array Charts and Multiplicative Sentences
(a) One Sentence
We attempt to explore the variety of the multiplication, by finding several multiplicative sentences in the following array chart of 3 rows and 4 columns.
The most popular math sentence for this chart is "3 x 4". Interestingly enough, the formulation applies to Japanese math education where the multiplicand is on the left of the operator as well as to the classes mainly outside Japan which set out the multiplier on the left.
In Japan, the original chart is divided into four equal groups:
Since the numbers of objects in each group is 3 and the number of groups is 4, we have 3 x 4 = 3 + 3 + 3 + 3 = 12.
According to the interpretation in Mathematics for Elementary Teachers: A Conceptual Approach*2, the original arrangement is carved by means of the following disconnection.
Since the chart means 3 groups of 4 objects each, we have 3 x 4 = 4 + 4 + 4 = 12.
The convention of the notation presenting the number of rows in advance of that of columns would be useful when we show the size of matrices in high-school or collegiate mathematics.
(b) Two Sentences
The size of an array chart or a rectangular arrangement are not always described as "row x column". What is the size of the screen you are looking at? My display has the resolution of 1680x1050, where 1680 and 1050 means the width and the height, respectively. Obviously, the width is associated with the column and the height is close to the row.
Thinking of the area calculation of rectangles, parallelograms and triangles, it is more convenient to put into place "column x row" for a sentence of the area of the rectangular as well as "row x column". This idea is applicable to the formulation of array charts. Therefore, "3 x 4" and "4 x 3" are both the math sentences of the original chart.
Still, I would like to emphasize that the reason why "4 x 3" should be a correct math sentence is due to the following way of partitioning it.
At that time, the number of objects in each group is 4 and the number of groups is 3, and 4 x 3 = 4 + 4 + 4 = 12 is derived.
We can also find the two way of segmenting an array chart into equal groups through the following illustration*3:
(c) Many Sentences
Under the assumption that the multiplication is reflected in "the number of objects in each group" x "the number of groups" or is a short expression of the repeated additions, we can get further multiplicative sentences for the array chart.
Apart from "3 x 4" and "4 x 3" described above, we can gain the following approaches.
- When the number of objects in each group is 2, we have 6 equal groups, then 2 x 6 = 2 + 2 + 2 + 2 + 2 + 2 = 12.
- When the number of objects in each group is 6, we have 2 equal groups, then 6 x 2 = 6 + 6 = 12.
- When the number of objects in each group is 1, we have 12 equal groups, then 1 x 12 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 12.
- When the number of objects in each group is 12, we have only one group, then 12 x 1 = 12.
That's all. It is of course because all the divisors of 12 are 1, 2, 3, 4, 6, and 12. We have to shut off the multiplicative sentences including fractions or negatives since the array charts with such numbers cannot not be drawn.
Among them, "2 x 6" means 6 pairs shown as follows.
AABB CCDD EEFF
We can illustrate "the number of objects in each group" and "the number of groups" separately, for the common array chart. An introductory class of the multiplication*4 shows that there is more than one way of grouping which leads to "3 x 4" or "2 x 6".
(d) 1-2-M
Fascination MAXX is a song available in the video game Dance Dance Revolution and its composer is 100-200-400 after the variation of beats per minute.
Until now, we found only 1 sentence, exactly 2 sentences, and more sentences from the common array chart, and I can say "1-2-M", where M stands for "many", with the numbers of sentences. However we have to brush up these sentences as well. In other words, we would like to make sure when each sort of multiplicative sentence is effective, taking the meaning and the application into consideration.
Although "1 x 12" and "12 x 1" can be associated with the chart shown above, honestly, those math sentences looks artificial and unnatural. And then, are "2 x 6" and "6 x 2" suitable for that chart?
Indeed it will be difficult to fix the boundary which everyone agree with. I think that this involves an issue of communication; in an extreme case, we would not like to get the world where a chill can (or must) say "a common animal with four legs, fur, and a tail" or "Canis familiaris, a highly variable domestic mammal closely related to the gray wolf" in response to dog's painting or a real dog, after they learn a dog in a dictionary such as Longman English Dictionary Online or Merriam-Webster Dictionary.
We have another case using array charts by opening the commentaries of the Education Ministry guidelines*5 to page 81.
The example is intended so that the children can arrange 12 marbles ingeniously and understand a multiplicative construction of numbers. Marbles are a key instructional equipment here, since they are movable. In contrast, each array chart is fixed and unchangeable, no matter if it was drawn in print or appears on the screen.
I have to acknowledge that it is difficult to settle the correspondence of an array chart to one or more multiplicative sentences. As far as I know, there is no instructional examples found outside Japan where two formulae or more are assigned to an array display. "Row x column" is a bit more popular than "column x row".
Hereafter we use "row x column" as the multiplicative sentence for a given array, and refer to "the n-by-m array chart" as the chart of n rows and m columns.
4. Properties of Multiplication
In this section we will make sure the properties of multiplication, namely commutative, associative, and distributive laws through array charts, instead of algebraic, rigorous proofs for any multiplicative sentences among positive integers.
(a) Commutative Law
The following figure is associated with "3 x 2 = 6".
This figure means "2 x 3 = 6".
There are several ways of getting the 2-by-3 array chart from the 3-by-2 array chart:
- To move 2 objects. In general, we can transform the a-by-b array chart where a > b into the b-by-a array chart by revolving the (a-b)-by-b subarray within the original array.
- To have the whole chart turn 90 degrees to the left or right.
- To transpose the chart, just as getting the transposed matrix.
- To changing the observer's eye instead of the chart.
In either case, the number of objects stays constant. Therefore we obtain 3 x 2 = 2 x 3.
(b) Associative Law
How many object are there in the following array chart?
We can find four 3-by-2 array charts. Regarding the dots of an array, 3 x 2 = 6, as the multiplicand and the number of arrays, 4, as the multiplier, we have (3 x 2) x 4 = 6 x 4 = 24.
When closing up all the arrays, the following 3-by-8 array chart is derived. The total number is equal to 3 x 8 = 24.
In addition, we can discover the equation 8 = 2 + 2 + 2 + 2 = 2 x 4. Therefore we have 3 x 8 = 3 x (2 x 4) by substitution. Since the equal sign is the equivalence relation, we finally have (3 x 2) x 4 = 3 x (2 x 4).
This is an example illustrating that two ways of the "order-of-multiplication" yield the same answer.
(c) Simple Case of Distribution Law
Before the distribution law, I would like to show that, when the multiplier increases by just 1, the product increases by the multiplicand. In Japan, students learn this property in the second grade since it can be demonstrated through the fact (or a viewpoint) that the multiplication is the repeated addition.
The following is the 3-by-5 array chart and the total number of the objects is equal to 3 x 5 = 15.
When portioning the chart in the following manner, we have the 3-by-4 array and the string of 3 objects.
The total number is computed by 3 x 4 + 3 = 12 + 3 = 15. We therefore obtain 3 x 4 + 3 = 3 x 5.
This property can be developed inversely. That is to say, when the multiplier decreases by just 1, the product decreases by the multiplicand. We can see the feature with the array chart; when removing the string of 3 objects from the 3-by-5 array, we will see the 3-by-4 array and have the equation of 3 x 5 - 3 = 3 x 4.
By repeating the operation of subtraction, we will have 3 x 1 = 3 and 3 x 0 = 0, which are rather difficult to derive only by means of the repeated addition.
(d) Distribution Law
The following two array charts explain the distribution law:
And:
The former figure is the 3-by-5 array which is associated with 3 x 5 = 15. The latter figure consists of the 3-by-3 array and the 3-by-2 array, and then we have 3 x 3 + 3 x 2 = 9 + 6 = 15. Therefore 3 x 3 + 3 x 2 = 3 x 5.
We can do likewise to confirm the distribution law of splitting the multiplicand and that of sectioning both the multiplicand and the multiplier.
5. Counting Number Using Array Charts
This is a rectangular arrangement but not an array, and already shown.
How many objects are there? We can make several sentences.
- 3 x 4 - 2 = 12 - 2 = 10
- 3 x 4 - 1 x 2 = 12 - 2 = 10
- 3 x 3 + 1 = 9 + 1 = 10
- 3 x 2 + 2 x 2 = 6 + 4 = 10
- 4 x 2 + 1 x 2 = 8 + 2 = 10
Among these sentences, some multiplications are derived from arrays and some are based on "the number of objects in each group" x "the number of groups".
I have another applied question; what we are going to know is the total amount when the identical coins form an array. How much is the amount when every filled circle is a 10-cent coin?
If we wrote "30 cents x 40 cents = 1200 cents", then we would feel like making a lot of money. But in reality, "10 x 3 x 4 = 120" is a valid sentence and the total amount is just 120 cents.
6. Limitations of Array
Array charts are a tool for understanding multiplicative structures, but I consider that we should not heavily depend on them.
When using arrays as a visual instrument, the numbers of row and columns must be positive integers. For example, it would be difficult to make the array charts of "0 x 3" and "4 x 0" so that these two sentences can be distinguished.
The extension of numbers, furthermore, has to be taken into account. When adopting unit squares as equal objects, we can deal with continuous quantity and scale with the arrays where the row and/or the column is a fraction. However, is such an extension possible for negative numbers, complex numbers, quaternions, and so on?
Another sort of problem is found from the charts we have seen:
And:
The equation is 3 x 5 = 3 x 4 + 3. The left hand side stands for a rectangular while the right hand side consists of a rectangular and a string. And then I have to point out that the r-by-1 array chart is indistinguishable from the string of r objects. This might follow that children cannot understand the distinction between 1-dimensional and 2-dimensional quantities. But understanding the difference between areas and lengths is so important that choosing the statements of the girth of a given rectangular was on the test in an achievement test of Japan.
And, I say, let us have another look at the equality for the associative law: (3 x 2) x 4 = 3 x (2 x 4). The right multiplicative operator in the left side hand and the left one in the right are based not on the array but on "'the number of objects in each group' x 'the number of groups'", or the multiple-oriented solution. If we explained the relation only with arrays, some objects would need moving. That means a transformation of 2-dimensional arrangement into 1-dimensional one. Otherwise, we might make use of 3-dimensional arrangement together with a length, a surface and the volume.
Considering the development of classes, array charts are not always a suitable tool for showing and sharing the properties of multiplication.
7. Related Instructional equipment
We would like to reconfirm the significance of the usage of array charts by comparing with relevant instructional equipment.
(a) Multiplication Table
First of all, we are able to come across the multiplication table. It may be a good assignment to write a program that outputs the multiplication table with the numbers in an orderly manner.
Using the multiplication table, we can find the product on the cross over point of the specific row and the column, which correspond to the multiplicand and the multiplier, respectively, in Japan. Consequently, the quick reference matrix is expected to be used apart from the array charts.
(b) Kake-Wari Chart
The "kake-wari chart" is a schema for visualizing multiplicative and divisional structures, devised and put to practical use by The Association of Mathematical Instruction (Sugaku Kyoiku Kyogikai), an influential organization about the math education in Japan. This chart furnishes the row and the column with the different roles; the number of row means the quantity per unit while the columns intends the number of unit amounts.
The kake-wari chart looks like an extension of the array chart in which the object is typically a square. Plus the quantity per unit is placed to the left of the array while the number of unit amounts is indicated below.
Thanks to these accompanying items, they can express the math sentences "0 x 3" and "4 x 0" separately. In addition to drawing the charts by hand, they make movable paper-crafting kake-wari charts to apply to various multiplicative situations.
(c) Decanomial
Montessori education has developed the "decanomials" which look like array charts. A typical decanomial consists of the 1-by-1 piece to the 10-by-10 piece (the numbers before and after "-by-" vary apart). When fitting together all the pieces properly, we can form the 55-by-55 square. We can also pile the pieces together. Decanomials are designed so that the children can handle and arrange them in order to recognize the numbers and the products. Note that each piece is one-colored and no numbers are set down on any pieces.
The pieces are tiles in the most popular decanomials although some pieces are made with beads. The tile is colored according to the length; for example, the 5-by-3 tile and the 3-by-5 tile are the same in color (and in size) but different from the 5-by-6 tile. From the compound 55-by-55 square of the tiles, we can see the dogleg sequence of the color.
By using the beads, since the color depends on the quantity per unit, the 5-by-3 piece and the 3-by-5 piece are distinguished in color.
(d) Paper and Sticker
The decanomial of beads is associated with an attempt in Japanese math education. They make the array by putting round seals on a paper. Since there are different colors of the seals by columns, the two factors of the product are easy to identify. Moreover the sheets can be made use of for understanding the distribution law, not only by sticking them together but by cutting off the paper.
8. Why Did I Write This Article?
I would like to say again that this article is the sequel to "Towards Japanese Multiplication Instruction". In the previous article, I have introduced two ways of relating a word problem with an "array diagram". At some point in time, I would strongly like to focus on the array and explain its characteristics, applications, and limitations.
Most of this article is a translation of the entry dated January 25th, 2012, whereas the explanation of decanomials was drawn from the entry dated May 15th, 2012.
By the way, this article uses the array "chart" throughout the document since I would prefer it to the array "diagram".
*1:中島健三: 乗法の意味についての論争と問題点についての考察, 日本数学教育会誌, Vol.50, No.6, pp.74-77 (1968). http://ci.nii.ac.jp/naid/110003849391
*2:http://highered.mcgraw-hill.com/sites/dl/free/0072532947/78545/bensec3_3.pdf
*4:新版 小学校算数 板書で見る全単元・全時間の授業のすべて 2年下, p.21.
*5:http://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2009/06/16/1234931_004_2.pdf
