What is Algebra?

(Click on image to enlarge)

This graphic represents the concepts which, as a result of my reflection following week 12's workshop, I believe are integral to the teaching and learning of algebraic thinking in the early years. Each cog represents a key concept highlighted in QSA (2004). The choice of interconnected cogs is to demonstrate that knowledge and understanding in any one of these areas can affect the knowledge and understandings of another. It is important to note that the size of each of the cogs is not representative of the importance of these concepts. Furthermore, the boxes and arrows show the “central ideas promoted in the national algebra standard for young children” (NCTM, 2000 cited in Taylor-Cox, 2003, p. 15) and how each of these ideas relate to the concepts highlighted in QSA (2004). The rest of this reflection will concentrate on expanding on these concepts and ideas, and identifying some implications for how to they should be taught and learnt in the early years.

Patterns

According to Economopoulos (1998) patterning activities have long been accepted in early childhood educational contexts as a part of the early mathematical curriculum. However, often little consideration is given to why patterning activities are valuable and how they actually contributes to children's development of mathematical understandings. Partly as a result of this lack of consideration, many early childhood teachers seem to focus on recognising patterns, creating patterns, and "what will come next?" type activities.

As was discussed in the workshop, understanding patterns is important mathematically in that they are used to form generalisations and predictions based on mathematical reasoning. We discussed that most children (and many adults) seem to have a natural interest in forming patterns, particularly repeating patterns, but that does not mean that they understand and can make generations about the rule by which a pattern is formed, or that they are able to use the pattern to predict missing variables.

Economopoulos (1998) suggests, in order to generalise and make predictions about patterns, children must "move from looking at a [repeating] pattern as a sequence of 'what comes next' to analysing the structure of a pattern...seeing that it is made up of repeating units" (p.231). Therefore questions such as "What part repeats?" or "What comes here?" (see the blog-entry on train tracks game) and "Why do you think it's that?" can be useful to extend children's learning about patterns. Also useful are activities involving the translation of patterns, describing patterns, identifying similarities and differences between patterns (Economopoulos, 1998). These help children develop their understanding of the function of patterns as a means of making predictions, and develop mathematical reasoning abilities to justify predictions they make.

The approach of Economopoulos (1998) is also supported by Papic (2007) who highlights that although young children are capable of developing understandings of complex patterns (beyond repeating patterns), merely exposuring children to patterning experiences does not guarentee that they will come to see patterns as structures that repeat or grow predictably. This ability to analyse the structure of patterns will help children develop other algebraic concepts such as functional thinking (Papic, 2007).

Equivilance and Equations:

In the week 12 workshop, we discussed the idea of balance or equivilance being a difficult concept to explain to young children. According to Warren, Benson and Green (2007) in a study of children's understanding of the equal sign, many students mistakenly thought it meant "here comes the answer" (p.151). Using resources such as a pan balance can help children develop this understanding of balance which is a important part of algebraic thinking. For example, if I had two identical objects in either side of the pan balance, both sides are equal, or balanced. However, if I take one object from one side, the two sides are no longer equal, or are unbalanced. If I then remove the same object from the other side, the two sides should again be balanced or equal to each other again.

This idea is linked to the development of mental computation skills which we discussed earlier on in this subject (see mental computation blog-post for more details).

Functions:

According to Warren, Benson and Green (2007), a function describes the relationship between one element of a set of data with another unique element of another set of data in that the value of the first element consistently changes into the other according to a particular rule.

Functional thinking contributes to mathematics learning as it assists children to develop understandings of the inverse relationship between the operations of addition and subtraction; multiplication and division.

The sequence of learning argued for by Warren, Benson and Green (2007) is as follows.
1. Learn how to "follow a change rule";

2. Learn to "follow a backtracking";

3. Identify a change rule;

4. Identify a backtracking.

For more reflections on identifying a change rule, please read the blog-post entitled "Button Triangles"

For lesson ideas for qualitative functions please read "The train game" "Word riddles".

For lesson ideas for quantitative functions please read "function machines".

Balancing Teddies

The Problem:

(Click on the photographs to enlarge them)

If one yellow teddy balances with two green teddies...




And one green teddy and one yellow teddy together, balances with six red teddies...





Then what balances with two red teddies?

The Solution:

Well if one yellow teddy equals two green teddies; and one green teddy and one yellow teddy together equals six red teddies, then you can replace the yellow teddy with two green teddies. So when you add these two green teddies to the one green teddy that is already on the balance pan, then three green teddies must equal six red teddies.




Once the teddies are put in this arrangement, then it is easy to see that one green teddy is the same as two red teddies.



So what balances with two red teddies?


One green teddy!

What did I learn from doing this activity?

I learnt that to solve this problem, I had to apply algebraic thinking in that I had to figure our how the two facts presented in the problem related to each other - I had to identify the relationship between the two rules. Once I saw the connection, I saw that the yellow teddy on the scale could be replaced with two green teddies, the rest of it was easy to solve visually. From this activity I learnt that algebraic thinking is not limited to manipulating numbers and abstract symbols, but that in some instances visual and even concrete representations can present easier solutions. This activity taught me that using visual and concrete reprentations does not necessitate a trial and error strategy or approach. I also learnt that being able to reflect on the process of how I reached my solution, explain my thinking processes to someone else, then documenting it visually (in a way that made sense) were all difficult to do because it required a much deeper appreciation of the mathematical reasoning in order to justify my solution.

If I were to provide opportunities for children to investigate this activity, I would provide the relevant materials and set the scales up similar to the picture above. I would also allow time for the children to manipulate the material, contemplate their own reasoning before asking them to share how they got their solution and justify their processes.