Week 9

Pattern, Algebra, Pattern Pattern, Algebra

This week’s workshop brought back my secondary school memories of algebra, functions and pattern. I have learnt algebra and functions in my secondary school through the use of formula in finding the ‘unknown unit’. Frequently, what I have learnt was memorizing the formula from the textbooks and used it to solve the mathematical problems! Therefore, this ‘mindset’ has governed my initial understanding of how young kids learn pattern, functions and algebra. One big question in my mind was; do we expect young kids to solve algebraic equations and function problems?

Algebra

The journey towards my confusion as well as curiosity began when we are needed to share some thoughts and feelings that pops into our heads when we heard the word, “algebra”. The list of related words was a relief as some of them are very familiar to me. Taylor-Cox’s (2003) article also gave me clearer idea on what algebra in Early years looks like. I noticed the word ‘pattern’ was on the top on the list. But, how are patterns connected to algebra?? How can repeating, growing and relationship patterns help young kids learn algebra?

Pattern

However, one more thing that I have learnt from this workshop was patterns are not just merely the repetition of ‘red bear, blue bear, red bear, blue bear’, but it more than that. Patterns are a way for young children to recognize order and to organize their world and are important in all aspects of math at this level (Clements, 2004). As claimed by Economopoulus (1998); Heirdsfield (2007); Smith (2001) and Taylor-Cox (2003), pattern involves recognizing, describing, extending and translating patterns. The video shown in this workshop also explained the real-situation where young kids create patterns by sorting objects, putting number to the pattern, experimenting using the number and saying the number.

From the video, I can relate to ‘number sense’ that have been learnt in Week two; where through patterns, students can do split counting of multiplication. Indeed, through patterns, we can predict pattern, what comes next? What comes after AAAABBB?? Here, then through finding ‘the missing pattern’ I was be able to relate to algebraic thinking! Indeed, through repeating and growing pattern, I can relate to find the missing pattern and therefore find the solution!

Functions

Instead of pattern, functions also play an important role in algebra in early years (Taylor-Cox, 2003) and Willoughby (1997) defined functions as one of the most important and pervasive concept in Mathematics. Functions that recognize and identify how things ‘change’ in relationship to each other involve quantitative change and qualitative change (Heirdsfield, 2007 & Taylor-Cox, 2003). I have experienced learning functions during my upper secondary schooling. It involves numbers and it does relate to algebra. However, during this workshop, a conflict arouse when we were needed to give an example of functions that was not numerical! Some examples given were:

Purple cat -----à black cat

Pink dog ------à green dog

. Therefore function= Purple ---à black

Pink ----à green

Indeed, the situation became harder when one of my classmates suggested ‘red bird’. It made me wonder what colour of ‘bird’ it will become. Personally, I found this activity can result in confusion, but indeed it was a good example; as we can’t predict pattern unless we used some rules? How can it serve as a function (qualitative change)?

In addition I found the function machine was an interesting way of introducing children to generalization of patterns and functions. The activity of using a function machine in ‘connecting’ the relationship between two objects was really gave me an idea how it is ‘functioning’. However, Willoughby (1997) purposed that children need to be introduced with more concrete objects as shown in his idea of a ‘magic number machine’.

Generalization

Algebraic thinking is also related to generalization. A ‘swimming pool’ activity in this workshop has constructed my understanding of ‘looking for generalization’. Indeed, I needed to use visual thinking as it used white tiles and blue tiles. After I built the pools using the guideline provided, I needed to identify the patterns from the blue tiles and the white tiles. I documented my ‘thinking’ as I needed to predict; if there are 49 blue tiles, how many white tiles are there?? Is there are 44 white tiles, how many blue tiles are there? Here, I realized that I need to construct a formula so that I would not count them on to solve the problems. I needed to look for generalization!

Equality and inequality

Another important feature of algebra in early years is equality and inequality (Taylor-Cox, 2003). Equality or sameness is an important concept as it offers children with recognizing, defining, creating and maintaining equality. Through this workshop, I realized that ‘the concept of balance’ is important in solving algebra equations! Using a pan balance scale, the activity has lead me to understand equality as balance. As claimed by Charlesworth and Kind (2007), by exploring with a pan balance, children find that if they put certain objects on each side, the pans will balance. Not only that, to trigger my thinking, I was required to think of the way mathematical problems were solved using arithmetic and algebraic. Personally, I found that using arithmetic thinking was easier as I used to do it in number facts. I tended to rely on my ‘counting ability’ when I needed to use algebraic solutions. Even though I tried not to count it, it seemed automatic. However, on the positive side, I found that this strategy of thinking is effective for teaching young kids at the beginning, before teaching them how to count big numbers and memorise the number facts!

Final thought

Therefore, through the activities in this workshop, I agreed with NCTM (2000) that algebraic concepts can evolve and continue to develop; although the concepts discussed are algebraic. Furthermore this does not mean that young children are going to deal with the symbolism often taught in a traditional schooling, because before formal schooling they have already developed beginning concepts relating to patterns, functions and algebra in their everyday lives.

My personal philosophy

Exploring this topic has brought me some insights. From this workshop, my personal philosophy is constructed around, ‘it is never too young for kids to learn about pattern and algebra. Indeed, it is not as I thought earlier that; the young kids need to solve algebraic equations and function problems while learning algebra and pattern, but children can learn algebra through patterns. I agree with Charlesworth and Kind (2007) that algebra begins with a search for patterns. By identifying patterns, it helps children be able to make generalizations beyond the information available and therefore serves as an important component of the young child’s intellectual development in algebraic thinking. They need to be invited to look for the patterns that are all around them in their everyday world because pattern spotting and looking for rules is fundamental and at the heart of mathematics. Therefore, interesting learning experiences will help young children learn something more than traditional people did that is; learning algebra in year Six!

References

Suggested websites

• Introduction to Algebra

>> This website gives a clear view about introduction to algebra that teacher might need. It also integrated with some strategies and resources that teacher can have to teach algebra.

• Teacher's actions: Patterns supporting the development of early algebraic thinking

>> This pdf file provides very insighful teaching experiences from teachers in two classrooms 'experimenting' the relationship between patterns and algebraic thinking.