THE BIG IDEAS
Contradictory to the opinions of many, algebra is much simpler than most people perceive. As a follow on from pre-number (discussed in week 5), algebra simply describes the relationship of a pattern. As mathematics is based on a language of pattern and order, this begins at the very beginning.
Helping children learn mathematics (2012) discusses this on page 367 where it states:
"The powers necessary for algebraic thinking are being used by children as soon as they leave the womb. The babbling of babies in their cribs is not only the foundation for language but can be interpreted as mathematical (algebraic) terms such as associativity through their pauses or ordering of sounds. The contention here is that arithmetic is not the necessary precursor to algebra, but that both may be most effectively developed together. The development of cognition moving from physical experience to the remembered or imagined to the symbolised as words or arcane mathematical symbols is a very natural, intuitive progression. Thus, the power of the very young as they engage with these natural activities can be harnessed by 'encouraging students to express perceived generalities, relationships, connections, properties, and so on"
This means, as a teacher I need to provide my students with many opportunities to utilise algebraic skills, including:
1.
Recognising the pattern
2.
Describing the pattern
3.
Repeating or copying the pattern (A student
needs to see the relationships to do this)
4.
Growing, extending or continuing a pattern
5.
Replacing missing elements of the pattern
6.
Translating the pattern
THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
There are 3 concepts that fall under algebra. These concepts include:
1.
Patterns and functions
2.
Equivalence and equations
3.
Patterns, sequences and generalisations
Concept
The general concept for algebra is a statement of a relationship in a pattern. This can be introduced very simply through a game of 'Echo' (also known as Copy Cats) where a pattern is created using voice repetition - for example, the teacher will say "red, blue", student 1 will repeat "red, blue", student two will then say "red, blue" and so on. In this game, children learn patterns as well as repeated sequences.
The skill is being
able to determine what’s missing in a pattern. This skill can be practiced by growing a pattern, creating a pattern or stating the
relationship between the elements of a pattern. This can be practiced with the 'Echo' game as mentioned above or through other activities suck as using concrete objects such as blocks, draw or paint patterns on large sheets of paper or digitally using computer games or iPad apps. Another interesting method of practicing patterns is using music and rhythm based games such as the activity in the video provided.
Strategies
The strategies for algebra directly relate to the thinking strategies involved with the pre-number concepts covered in week 5 as pre-number concepts provide the prerequisite knowledge to commence a study of number – which is based on a language of pattern and order. These concepts were:
- Determining attributes
- Matching by attributes
- Sorting by attributes
- Comparing attributes
- Ordering attributes
- Patterning
THE LANGUAGE MODEL
The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning.
Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects.
Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract.
Mathematics language - Moving away from 'stories', this stage introduces mathematical terms.
Symbolic language - This stage introduces symbols.
THE LANGUAGE MODEL FOR PLACE VALUE
THE MISCONCEPTION
One misconception students may have is seeing the pattern as a block that is being repeated rather than individual elements. For example, in the pattern ABBCABBC, the student will recognise the sequence being repeated is ABBC however the student will not see each letter as individuals. This means that the student will not have the ability to replace missing elements in the pattern, for example AB_C. To remedy this, I would do an activity using a story. This story would be accompanied with concrete materials - such as blocks. An example story is:
Once upon a time there was a train with lots of carriages. The carriages had a special order, it was: red, blue, red, blue red, blue. One day a big gush of wind came along and blew off some of the carriages and now its our job to put it in the right order again.
Whilst saying the story, I would visually demonstrate it using the concrete materials, then the student can place the 'carriages' back where they go in the pattern and we can begin again with an increasingly difficult pattern.
THE ACARA LINK
Algebra is first introduced in the foundation year.
Strand: Number and Algebra
Substrand: Patterns and Algebra
Content descriptions: Sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns with objects and drawings
Elaborations:
- observing natural patterns in the world around us
- creating and describing patterns using materials, sounds, movements or drawings
Scootle resource ideas:
1. Cake time is a two minute video that can be used to introduce creating patterns as well as replacing missing elements.
2. Monster Choir: Look and listen is one of three apps in a series that focuses on patterns. This app addresses recognising, repeating, extending and translating patterns.
THE RESOURCES AND IDEAS
- Pattern worms use a pipe cleaner and coloured beads to allow young children to create their own patterns.
- The felt icecream cone board allows students to copy patterns
THE TEXTBOOK SUMMARY
- An algebraic perspective can enrich the teaching of number in the middle and later primary years, and the integration of number and algebra ... can give meaning to the study of algebra in the secondary years.
- Children can learn the language and symbols associated with algebra as they are learning about numbers.
- The australian curriculum calls for students to use mathematical models to represent and understand quantitative relationships, to represent and analyse mathematical situations and structures using algebraic symbols, and to analyse change in various contexts.
- Recording of equations during routine problems can become more algebraic as the students become more comfortable with the concepts by rather than drawing the object, drawing a symbol to represent the object. For example, a rectangle rather than a drawing of a truck (Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al., 2012).
THE REFERENCES
ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458
Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home
Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/
Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.
YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/
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