Learning and Teaching of Algebra Through Tasks That Encourage Algebraic Thinking

Most of us who have worked with middle-school children would agree that algebra is one subject which children find difficult to understand. What makes matters worse is that it is also a subject which is used extensively in higher classes across subjects. Several domains like computer science and engineering use algebra for their development. Algebra is very important while working with natural sciences like physics, chemistry and biology. It is also an essential tool to understand the world we live in better. Many questions about the environment and economics can be understood better with the knowledge of algebra.

For many people, the word algebra would bring back memories of alphabets like x, y, and z. But a lot of algebra can be done in primary school and middle school. Can you believe it if I say that in lower classes one can engage with algebraic problems which do not include the dreaded words, “Solve for x?”

Let us begin the discussion with the question, “What is Algebra?”

For a lot of mathematics educators, algebra is not just a subject but a way of thinking. They say that the purpose of algebra in elementary school is to develop ‘algebraic thinking’.

What is algebraic thinking? And what is not algebraic thinking?

Let us look at a simple example. The following question is asked in a classroom. 193 + 288 = ___+ 287

One student’s answer is as follows: “193 + 288 = 481 and 481 – 287 = 194. So the answer is 194.”

Another child’s answer is this: “On the left side I have 288, and on the right I have 287. 287 is one less than 288. So for the two sides to be equal, the number in the box has to be 194, which is one more than 193.”

According to you, what is the difference between these two answers?

In the case of the first student, he has looked at the numbers individually and solved both the sides separately and then equated them. But the second student has not only looked at the numbers but also the relationship between them.

What is the advantage of one over the other? Let us see what happens when one used larger number like, 2793 + 3478 = ____+ 3477

In this case, the first student would have to first add 2793 + 3478 to get 6271 and then subtract 3477 from 6271 to get 2794, which is a lot more tedious process than the first problem.

But for the second student, the solution does not change much because she has looked at the structure of the problem instead of just the numbers. If we review the two solutions we would find out that the first student is only working with numbers while the second student is also looking at the structure. This process of looking at the structure instead of just the numbers can be called ‘algebraic thinking’.

If one looks at the second student’s answer, one can see that she has found what we can call as a generalised solution to the problem. In the language of algebra, the number sentence looks something like this: a + b =[ _____ ] + (b – x) And she solves it like this, a + b = (a + x) + (b – x).

So, we can see that without actually using letters like a, b or x, this student has already started doing algebra.

Let me share my experience of doing number sentences.

We started doing these exercises with students in a vacation camp. The objective of the vacation camp was to explore students’ algebraic reasoning when exposed to algebraic ideas through tasks like numbersentences and pattern generalisation.

Some of the initial number-sentences we gave them were something like this –

3 + 8 = ___+9
15 + 29 = 20 + ___
17 + 6 = ___ + 4
52 + 13 = 12 + ___

All the students chose to solve these problems like the way the first student did, i.e., ‘3 + 8 = 11 and 11 – 9 = 2 so the answer is 2.’

All the teachers, including me, tried in various ways to encourage the students to see the structure of the number-sentences. But the students continued to solve these number-sentences in the method shown above. As one of our attempts to encourage them to look at the structure of the numbersentences, we gave them similar problems but with large numbers like four-digit numbers. Suddenly most of them started solving it like the way the second student did, i.e. algebraically.

Some students may naturally think algebraically. But being able to think algebraically is not just a ‘talent’ that a child has but a ‘skill’ a child acquires when she needs it. We as teachers can create situations or develop problems such that she gets encouraged to develop this type of thinking.

Once the students started looking at structure, they went on to generalising these number-sentences in various ways. One of the ways was the following.

Student: “All these statements will be always true.” a + b = (a + ___ ) + (b – ____ )

Teacher: “What are a and b and what is the box?”

Student: “a and b hold the same value on either side of equal to and the box refers to the same number in a number sentence.”

Number-sentences can be very important tools to work with students even in lower classes, even before ‘formal’ algebra comes in.

Like number-sentences, another tool that is very useful in encouraging algebraic thinking or reasoning in students is that of patterns.

There are different types of patterns we can use in our classrooms. Let us look at some of them.

1) Repeating Patterns: In these patterns there is an identifiable unit that repeats. Some standard examples of repeating patterns are 123123123….where 123 is the repeating unit. Another example of repeating patterns can be a combination of shapes.

2) Growing Patterns: These patterns are characterized by how one element relates to the next one. These patterns increase or decrease by a constant difference. 1,3,7,… or 1, -4, -10, …. or

3) Recursive Patterns: In these patterns the relationship between the terms defines the pattern. The famous Fibonacci numbers 1, 1, 2, 3, …. or the minimum steps required in the Tower of Hanoi problems 1, 3, 7, … are examples of recursive patterns.

The position paper on ‘Teaching of Mathematics’ in NCF 2005, emphasizes the role of patterns in elementary school algebra and the role they play in higher classes – “The identification of patterns is central to mathematics. Starting with simple patterns of repeating shapes, the child can move on to more complex patterns involving shapes as well as numbers. This lays the base for a mode of thinking that can be called algebraic. A primary curriculum that is rich in such activities can arguably make the transition to algebra easier in the middle grades.”

These patterns can offer opportunities to students for generalizations and for developing algebraic reasoning. Let us look at some questions which can be asked in the context of patterns to encourage algebraic reasoning. Let us also look at two sets of questions and study their role in encouraging algebraic thinking.

Let us look at an example of repeating patterns.

Please do answer the above questions before proceeding.

In Set 1, one can always find the answers by just writing the patterns till the 12th or the 15th position.

So, without having to actual look at the pattern and understand it one can answer the questions in Set 1. In contrast to that, for Set 2, writing the pattern till 112 can be tedious. And that might encourage the students to look at the pattern and generalize it.

If we look at the pattern carefully, we find that the repeating block is of 5 shapes. Hence, the shapes repeat after every 5 shapes. That is, the shape at the 1st position will be the same as the shape at the 6th position, which will be same as the shape at the 11th position or 106th position.

Similarly, while answering the second question in Set 2, one finds out that the 4th and the 5th positions will have the star. This means that at every 5th shape after the 4th and 5th positions there will be a star. What do you think is a better way of describing this?

While doing this pattern with some students, one student gave me a very interesting answer. I am quoting his answer, “Every position which ends with 1 or 6 will have a red rectangle, every position ending with 2 or 7 will have a blue circle, every position ending with 3 or 8 will have a brown pentagon and every position ending with 4, 5, 9 or 0 will have a star.”

Do check out what whether his answer is correct and also try to find out how he came to his answer?

After repeating patterns let us look at some examples of growing patterns.

Study the pattern given below and complete the following table:

Design	Number of sticks used
1	5
2	8
3	11
4
5

Design	Number of sticks used
15
20
25
75
n

When this question was asked to Class 5 students, these were some of the answers they gave for the last column.

• 3 + 2 + 3(n – 1)
• 2 + 3n
• 3(n + 1) – 1

One aspect of working with patterns is generalising and the other aspect is justifying your answer.

All the three answers give a generalized form for the given pattern. Try to find the justification for the first two answers given by the students.

Let us look at the third one and try to understand the student’s justification.

Student: “To solve this problem, I added a matchstick to each of the designs. So, the new pattern looked like this. Then I counted the matchsticks for every design. And I found that at every stage the number of matchsticks was 3 x (design number + 1) i.e. 3(n+1). But then I had added one matchstick for every design so I need to remove that. So, the number of matchsticks in every design is 3(n + 1) – 1.”

This strategy of manipulating patterns was not discussed in the class but the students discovered this technique herself.

Another interesting discussion that followed this pattern was discussing why the three forms given below are the same.

• 3 + 2 + 3(n – 1)
• 2 + 3n
• 3(n + 1) – 1

This discussion gave the teacher an opportunity to talk about rules of opening brackets and algebraic manipulation.

In this article we saw some situations which can give teachers opportunities to encourage students to think algebraically and help create situations in the classroom which can develop students’ algebraic thinking skills.

There are a lot of more tasks which will help in developing algebraic reasoning among students. Please do look at the resources mentioned at the end of this article for more such tasks.

Some sample tasks: https://drive.google.com/file/d/16Bw jWtozvyesLaVyVT20Qj5Bm5YUCu49/view?usp=sharing

References and Useful Readings

National Council for Education, Research and Training, 2006. Position Paper on Teaching of Mathematics by the National Focus Group, National Curriculum Framework – 2005. (Retrieved from http://www.ncert.nic.in/ html/focus group.htm)

Takker, S., Kanhere, A., Naik, S., & Subramaniam, K. (2012, October). From Relational Reasoning to Generalisation Through Number Sentence Tasks. In epiSTEME 5.

http://episteme.hbcse.tifr.res.in/index.php/episteme5/5/paper/view/162/36

Three Components of Algebraic Thinking: Generalization, Equality, Unknown Quantities http://courses.edtechleaders.org/documents/elemalgebra/rubin_diffalgebra.pdf

Usiskin, Z. (1988). Conceptions of School Algebra and Uses of Variables. In A. F. Coxford (Ed.), The Ideas of Algebra, K-12 (pp. 8–19). Reston, VA: NCTM.
https://people.math.wisc.edu/~kwon/135Spring2014/alg.pdf