Algebra

Hi everyone.

We meet again this time talking about new topic, Algebra. Before we go further, lets look at what is algebra and what does it look like in the early years.

First, what is algebra?

Algebra is the generalization of arithmetic ideas where “unknown values and variables can be found to solve problems” (Taylor-Cox, 2003; p.14). It is also the relationship between quantities and events is explained using rules (Origo, 2008). Two of the most important aspects of algebra are patterns and functions.

How is algebra in the early years?

Introducing algebra during the early years is important as it provides essential foundation for ongoing and future mathematics learning (Taylor-Cox, 2003). Hence, there is a needs for interconnected concepts in the early childhood mathematics program which are developmentally appropriate and very applicable to early childhood education(Taylor-Cox, 2003).

In algebra, pattern is defined as an arrangement of shapes or numbers that repeat or change in a predicatble manner (Origo, 2008, p. 75). Three types of patterns we looked at through the tutorial were: repeating patterns, growing patterns, and relationship patterns. According to Papic (2007), patterning involves the development of children's spatial awareness, sequencing, ordering, comparison and classification. Furthermore, pattern also provides a basis for algebraic thinking in term of teaching regularity and repetition in motion, colour, sound, position, and quantity, and be involved in recognising, describing, extending, transferring, translating and creating patterns. Thus, in assisting the children to learn pattern, it is broken down into seven stages:

• Look for patterns
• Participate in building a pattern
• Copy a pattern
• Create a pattern
• Extend a pattern
• Find the missing element
• Translate a pattern

Repeating pattern is a sequence or set that is repeated over and over. The core is the smallest part of the pattern that is repeated. It should be repeated three times. A term is each place or position in the sequence. 

Using the example above, teachers could integrate the seven stages in assisting the students to understand pattern. First, when looking at pattern, teachers should ask students to identify the core  to understand what is being repeated. In the example above, students will identify 1,2 as the core since it is repeated more that three times in this pattern. Automatically, from the core, students could identify repeats as the term. By recognising the core and the term, students could recognise the patterns. For example, looking at the pattern below, students will identify the pattern as two purple triangles one green circle, two purple triangles one green circle, two purple triangles one green circle, two purple triangles one green circle.

Furthermore, teacher could the example to extend students' understanding on pattern. Since students already able to identify the core and term, teachers should encourage the students to practise and continue the rule. Teachers could continue extending the pattern to 10th terms and remove the 7th term and 8th term by asking students to fill both of the terms with a correct pattern. After that, teachers might ask the students to guess the 15th or 20th term without having to extend the patterns. This important to evaluate students' understanding since a study found that, some children may be able to copy and extend patterns, but they may not necessarily identify a pattern as a unit of repeat (Papic & Mulligan, 2005). In other word, students must know how to translate the pattern which means changing the appearance of an element while maintaining the arrangement of the elements (Origo, 2008, p. 75).

Growing pattern occur when elements increase or decrease. Below are the example of object, number and shape growing pattern.

Using aspects of the 7 stages, tasks can be set for students to better understand gowing patterns.  These may include:

• "What is the core in this pattern?"
• "Describe the pattern?"
• "What would the 5th term look like?  6th term?  10th term?"
• "How would you translate this pattern?"

Relationship pattern occurs between two or more sets of number. It is inter-linked and affect each other. The example is shown below.

Students can be given the opportunity explore this pattern. Comparing the 2 columns, students should look at the following aspects of the 7 stages:

• "What is the relationship between figure and match sticks?"
• "Describe the pattern?"
• "What would the 4th term look like?  10th term?  20th term?
• "How would you translate this pattern?"

In order to understand the relationship between the two tables, students could expand the table into a numeric line pattern as below:

Functions are rules that describe the relationship between two sets of numbers (Origo, 2008, p. 43). One set of numbers is the values of the independent variable (input),where the other set is the values of the dependent variable (output). Functions recognise how "things" change in relation to each other. In the tutorial, we learnt that teachers could start exposing students to understand function by describing words or pictures. Through the "function machine", students could  clearly illustrates how we start with something, and end with something different.

The example above uses a concrete object to explain to students how does function machine works. It starts with a red circle and it turns into red triangle when it comes out from the function machine. The second example starts with dark blue circle and ends with dark blue triangle when it comes out from the function machine. From here, students could guess what will comes out (output) from the function machine if it starts (input) with green circle. Similarly, what do we start with (input) if it will comes out (output) as a light blue triangle? From this activity, students will realise that the function machine only change the shapes without changing their colours.

As the student moves through the years, functions will begin to move from the concrete to the abstract, and finally, standard prepresentation (Willoughby, 1997, p. 314). Next, teachers use numbers to illustrate a standard representation.

In the example above, it starts with number 2. It goes through the function machine and comes out as number 5. As for the number 4, when it goes to the function machine, it comes out as number 9. When it comes to number 7, it comes out as number 15 from the function machine. Here, students realise that every number that goes into the function machine will be doubled and add one. Therefore, teachers could ask students to explain the process and the output for number 8. 

Another aspect of function is equal as balance-equalities and inequalities. Equality (shown by the symbol =) occurs when two or more objects or expressions are identical in a particular attribute that can be counted or measured (Origo, 2008, p. 35). Inequality (shown by the symbol ≠) occurs when 2 or more expressions are not equal (Origo, 2008, p. 35).

In the tutorial class, we learnt to understand the topic through scale. This is because through scale, students are able to visualise and experiment the concept of equality. 

In the example above, on the left side, we have a set of 4 yellow balls and eight green balls. Otherwise, on the right side, we have three yellow balls. By looking at the equation 4+8=3+?, we know that both of the side are equal.  Therefore, with a clear understanding in equality,students will solve the problem by solving the left side first. By adding 4 and 8 students will comes out with an answer 12. As both of the sides are equal, on the left side 12-3 and the students will get 9 as an answer. Thus, students will come out with an equation (4+8=3+9).

Other examples of equation carried out in the tutorial class can be seen below.

Q: What is the equation derived from this scale?

A: (10+6=7+9)

Q: Is that equal or not and give reason to your answer.

A: It is equal because both of the sides have the same answer which is 16.

Q: What is the equation derived from this scale?

A: (7+5=6+6)

Q: Is that equal or not and give reason to your answer.

A: It is equal because both of the sides have the same answer which is 12.