Figurate numbers

A Bridge between History and Learning of Mathematics

Tünde Kántor

Institute of Mathematics, University of Debrecen, Hungary

E-mail:tkantor@science.unideb.hu

Abstract: It is necessary to rethink the main principles of the Hungarian

mathematics teaching, to apply new methods and new contents, to renew the

training of the teachers in the spirit of Tamás Varga. Nowadays the Hungarian

mathematics teachers are uncertain in consequence of the bad PISA results. They

want to teach better, but they need some help.

In this presentation I want to share my teaching experiences and give a new

approach to the practice of mathematics instruction, connecting a problem of the

history of mathematics with the modern learning of mathematics. We present

some problems of Mayer’s Mathematischer Atlas (1745), and analyze his method

in discussing plane and space figurate numbers. We deal with some other

mathematical problems which were posed for secondary school students (Problem

AMC 10, Mason’s problem, KöMal problem, Viviani’s theorem).

Classification: A30, B50, C70

Keywords: Bruner’s representation theory, Tobias Mayer’s Mathematischer

Atlas, figurate numbers, Mason’s problem, mathematical problems for secondary

school students.

Theoretical bases

In the 21st century the teaching methods have changed. It is necessary to return

to the visual communication or to the historical roots. J. Bruner worked out new

ways in the theory of instruction. In his research on cognitive development of

children he proposed three modes of representation:

1. Enactive representation (action-based)

2 .Iconic representation (image-based)

3. Symbolic representation (language-based).

Bruner believed that the most effective way to develop a coding system is to

discover it rather than accepting it passively from the teacher. The pupils have to

construct their own knowledge for themselves. Therefore it is helpful to have

experience or illustrations to accompany the verbal information. We can find real

life examples which help in understanding the examined mathematical problem

(enactive representation). The iconic representation helps us in the right

perception of the problems, in the solving of problems, or in the stabilization of

knowledge.

The individual development of the pupils’ knowledge has common features with

the history of the mathematics. Bruner’s idea suggests that our learning process

is more effective when we are facing with a new material, if we follow a

progression from enactive to iconic and then to symbolic representation. Often in

the individual learning or in the history of mathematics a problem was first solved

in enactive and iconic representations, which later turned into the symbolic

representation. This procedure increases the efficiency of problem solving. I want

to present this through the admirable works of Tobias Mayer (Marbach, 1723 -

Göttingen, 1762). He was a self-taught mathematician, cartographer, astronomer,

Professor of Mathematics and Economics at the University of Göttingen (1751-

1762). His aim was in his work Mathematischer Atlas (Pfeffel, Augsburg, 1745)

to offer material for self-education, to construct a summary for people who want

to learn mathematics on advanced level by themselves. He provided a very

excellent example of learning mathematics by applications, based on his

experience, drawings and knowledge. He applied Bruner’s representations

hundreds of years earlier. He had a special method of the presentation: he gave

graphic descriptions of mathematical knowledge, definitions, their properties,

applications in every day and technical life. We want to present the Table XLV.

in details, the figurate and space numbers, and its return in the 21st century in the

learning and teaching of mathematics.

Plane and space figurate numbers nowadays as mathematical problems at

secondary schools

The Hungarian curriculum does not contain this theme. We met it in competitions,

in the KöMal (Mathematical and Physical Journal for Secondary Schools), on

advanced level of learning mathematics, mathematical circles. On enactive level

we meet space numbers every day at the market as egg and orange pyramids.

In 2019 at the Nagy Károly Mathematical Student-meetings (Komarno) I worked

with high school students on the topic of figurate and space number in a

mathematical circle. They were students of 9-12 classes (aged 15-18). The

students received the problems on a worksheet, except Mason’s problem, which I

gave them in the right place and they solved it by themselves.

The group of the 15 students was mixed in many ways. The participants were

primarily 10th grade students, but there were 9th, 11th and 12th grade students. Their

native language was Hungarian, but they were not only from Hungary, but also

from Slovakia and Serbia. They were all interested in mathematics, they wanted

to expand their knowledge of mathematics. However, this meant that their

mathematical background was also different. Thus, I was able to test my

hypothesis that solving problems related to figurate numbers does not depend on

age, but on mathematical background. In consequence of the heterogeneous

group, we started from the bases.

Our worksheet: Plane and space figurate numbers

Mason’s Problem (Problem 3)

Figurate numbers

We briefly discussed plane figurate numbers: triangular-, square-, rectangular,

pentagonal, hexagonal numbers (Problem 1).

We followed the historical way and Bruner's theory. The geometrical

representations of figurate numbers affords a quick insight into the structure of

these integers. So we began by drawing the dots forming the equilateral triangles,

squares, rectangles, regular pentagons and hexagons. We represented the figurate

numbers, by drawings and after it we counted the dots in specific cases, tabulated

our results, and then tried to give them a general formula.

We mentioned some theorems (Problem 2) which we proved by working together

in algebraic or geometric ways for example the following theorem: Twice of a

triangular number is a rectangular number (Problem 2a).

Mason’s Problem (Problem 3)

The students solved Mason’s Problem. We shall analyze and evaluate their works.

15 students worked on the worksheet (9th grade 3 students, 10th grade 6 students,

11th grade 3 students, 12th grade 3 students). The structure of the exercise followed

Bruner's theory.

Mason’s Problem 1 started with concrete drawing and calculation, followed by

the recognition of the rule and its application backwards. Problem 1 with drawing

and concrete calculation was solved by everyone. They noticed: 243 is divisible

by three. Regularity and the position of the 25th circle were calculated by 10

students. An error or omission occurred in the 10th grade.

Mason’s Problem 2 The calculations of specific values and right guess was made

by everyone. 10 students gave the proof. In each grade there was one student who

do not gave the proof. There were students who knew the sum- formula of

arithmetic series, or calculated it. Eight of them gave algebraic proof and two

students gave geometric proof.

Mason’s Problem 3 was related to Mason’s Problem 2. A student noticed this.

The specific values were correct for all students, the conjecture was correct for 11

students, the proof was missing, incorrect or incomplete in 5 cases.

The results confirmed our previous observations and our hypothesis:

• Correctness and quality of the solutions do not depend on the grade , it

depends on the students’ mathematical knowledge

• The work of the students was more effectively on the inactive and iconic

plane than on the symbolic plane.

Students’ solutions

Bendegúz’s (grade 9th) solution

István’s (grade 9th) solution

Bendegúz’s (grade 9th) solution

Zalán’s (grade 11th) worksheet

Space numbers

In the second part of the session, we turned to the polyhedral numbers. I changed

tactics, we started by posing specific problems from the everyday life, and from

the history of mathematics, from Tobias Mayer’s Mathematical Atlas. In

particular cases, we calculated tetrahedral numbers, cube numbers, pyramidal

numbers. We wrote some formulas without completeness, and tabulated the

results of the first ten cases. The main reason for this was that from the curriculum

of the 10th grade is missing the formula for the sum of the first n square numbers.

The first step was to solve a competition task (AMC 10, 2004. Problem 7). This

did not require any theoretical knowledge and could easily be solved by drawing

and counting.

Problem 4

A grocer stacks oranges in a pyramid–like stack whose regular base is 5 oranges

by 8 ranges. Each orange above the first level rests in a pocket formed by 4

oranges. How many oranges are there in the stack?

(A) 96 (B) 98 (C) 100 (D) 101 (E) 134

The students easily solved this contest problem. Their strategy was the following:

they made drawings of the different levels of the oranges and then counted them,

and finally added the numbers.

The solution:

S= 5 · 8 + 4 ·7 + 3 · 6 + 2 · 5 + 1· 4 = 100, so the correct answer is C.

Then we followed the historical way, namely from Tobias Mayer’s

Mathematischer Atlas (Table XLV. Fig. 5.A, Fig. 6.A, Fig. 7.A). We discussed

Figures 5.A, 6.A, 7.A. These Figures are iconic representations. In these specific

cases, the task was to calculate tetrahedral numbers, pyramidal numbers based on

figures.

Problem 5

Calculate the total number of the canon-balls on Figures 5.A, 6.A, 7.A.

Figure 5.A Figure 6.A Figure 7.A

On Fig. 5.A the canon- balls form a regular tetrahedron with regular triangle base.

On the sides of the triangle formed base there are 4 canon- balls, on the top 1

canon ball. So the total number of the sum of balls is:

S = 10+ 6 + 3 + 1 = 20.

On Fig. 6.A the canon-balls form a pyramid with square base. On the sides of the

square formed base there are 5 canon- balls, on the top 1 canon ball. Their sum is

the sum of square numbers: S = 52 + 42 + 32 + 22 + 12 = 55.

On Fig. 7.A the canon-balls form a pyramid ending in a row, and on different

levels there are rectangular numbers. On the sides of rectangle formed base there

are 4 and 6 canon-balls. They end in the top in a row with 3 canon-balls.

So their sum is: S = 4 · 6 + 3 · 5 + 2 ·4 + 1 · 3 = 50.

We discussed two more tasks. Problem 6 was a problem of Nagy Károly

Mathematical Student-meetings (Problem poser: Szabolcs Fejér, Miskolc).

Problem 6

In large stores, we often see buildings built from goods of the same size. In many

cases this can be cans. We imagine a square based pyramid built from such cans.

Numbers of the tins are squares in any row. How many tins we have if there are

25 rows? One tin is in the last row on the top.

What can we say if the base of the pyramid is triangle formed?

Problem 7 was taken from the KöMal (No. 2, 2017). It was posed for school

leaving students. This problem was about mandarins pyramids with square and

triangle base (enactive and iconic representations of space numbers). Here

students had to count space numbers and deal with some other mathematical

problems connected with them. The problem poser gave help, he gave the

formulas of the sum of the first n- square and triangle numbers.

Problem 7

a) A grocer made square and triangular pyramids from mandarins. We take a few

layers of mandarin from the top of the square pyramid. We would like to buy a

weekly portion with the same amount of mandarin for every day. Give the

numbers of pieces of the mandarins you can buy.

b) If we quadruple the number of mandarins we have bought, we can build a

pyramid twice as high as the square pyramid was originally. Prove that this

statement does not depend on how many layers of mandarin we bought at the

grocer.r

In another teaching experiment we were solving an old historical problem, the

Viviani’s theorem, with ICT tools (GeoGebra).

Viviani’s theorem: The sum of the distances from any interior point to the sides

of an equilateral triangle equals the length of the triangle’s altitude.

The students were accustomed to interact in peers. We carried out our experiments

using historical problems focusing on new technology. We dealt with an

important property of the equilateral triangle and with variations of the conditions.

We found there is a parallel between historical way and students individual

learning process. The development of concepts and proofs followed the historical

sequence. We pointed out that the teacher still plays an important role not only in

providing students with good problems, but on focusing on new technology, on

investigating how to apply dynamic software in teaching of mathematics in a

class, how the new technology influences the students’ work and their motivation.

Summary

Bruner’s idea suggests that our learning process is more effective if we follow a

progression from enactive to iconic and then to symbolic representation when we

are faced with a new material. In the individual learning and in the history of

mathematics, problems are often solved first in enactive and iconic

representations, which later turn into symbolic representation. This procedure

increases the efficiency of the problem solving using historical roots. I presented

this way at first on the admirable work of Tobias Mayer (1745), and then went on

to our everyday instruction. In another teaching experiment we were solving an

old historical problem, Viviani’s theorem and its problem-field with ICT tools

(GeoGebra).

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