1. Language aspects of algebra Language aspects of algebra – in classroom practice and in theories of learning

2. Overview u Consider learning of first language (with which all children have gained success); u Relate this to ways in which the language aspects of algebra might be learned within a school setting.

3. Learning of first language u Young children construct their own meaning for the word “chair”, for example; u Adults do not say “This is called a chair because…”; u The meaning a child has for “chair” adapts and changes over time to fit in with the way in which the word is perceived to be used within the social community.

4. Learning of algebraic language u Students construct their own meaning for any algebraic notation, for example, 2x; u A teacher might not say “This is written 2x because...”; u The meaning a child has for 2x adapts and changes over time to fit in with the way in which the notation is perceived to be used within the social community of the mathematics classroom (e.g. when x=4, 2x might be thought of as 24 initially).

5. Why is a chair called “chair”? u A child can invent names for things but cannot stare at a chair and come up with the knowledge that this chair must be called “chair” (as indeed it is not in other languages throught the world!); u The role of adults is crucial.

6. Why is “two times a number” written as 2x? u A child can invent ways of expressing “two times a number” but cannot be expected to come up with the knowledge that it ‘must’ be written as 2x; u The role of a teacher is crucial.

7. Adult’s role u An adult (or someone who knows more of the language) needs to introduce the word “chair” within their own sentences (when and how is of significance); u Once a child begins to say the word, an adult’s response is significant.

8. A teacher’s role u A teacher (or fellow student who knows more of the algebraic language) needs to introduce the form 2x within a suitable context (when and how is of significance); u Once a child begins to say write an expression for “two times a number”, a teacher’s response is significant.

9. Adult’s use of a word u It is significant when and how a word is said. For example: –A word needs to be linked to an appropriate object/example, so it is helpful to say the word “chair” when pointing to a chair or when the child is pointing to a chair, or when some action is taking place in relation to a chair (“I am going to sit on the chair”)

10. A teacher’s use of an expression u It is significant when and how an expression is written. For example: –An expression needs to be linked to an appropriate process/object, so it is helpful to write the expression at the same time as referring to the process/object concerned (e.g. “two” is said as 2 is written... “times” is said and nothing is written… “a number” is said as x is written).

11. Adult’s feedback u Responding to a child’s attempt to say or use a word when: –the pronounciation or usage is incorrect; –the pronounciation or usage is correct.

12. A teacher’s feedback u Responding to a student’s attempt to write an expression or use/interpret an expression when: –the form, usage or interpretation is incorrect; –the form, usage or interpretation is correct.

13. Nature of feedback when a child is incorrect u Initially a child will make a sound which may only approach a ‘correct’ pronounciation, and this attempt is often accepted with enthusiasm and support (similarly in relation to usage of the word); u At a later time, an adult might offer a ‘correct’ pronounciation (or usage).

14. Nature of feedback when a student is incorrect u Initially a student may write an expression which is not conventional, and this attempt might be accepted with enthusiasm and support (similarly in relation to usage or interpretation of an expression); u At a later time, a teacher might offer the conventional notation.

15. Nature of feedback if a child is correct 1 u Initially some form of confirmation might be given; u At a later time, some form of confirmation along with a question or task to challenge and extend the child’s developing meaning (e.g. “What colour is the chair?”, “Please put this on the small chair”);

16. Nature of feedback if a student is correct 1 u Initially some form of confirmation might be given; u At a later time, some form of confirmation along with a question or task to challenge and extend the student’s developing meaning (e.g. “Someone has passed me a piece of paper with this written on: 4(2x-3) +5. What do you think they meant?;

17. Nature of feedback if a child is correct 2 u At an even later time, there may be no response - there is now an expectation that the child knows and can use this word.

18. Nature of feedback if a student is correct 2 u At an even later time, there may be no response - there is now an expectation that the student knows how to writre, use and interpret 2x.

19. Some significant aspects 1 u There needs to be a chair if the child is to start using the word “chair” appropriately; u A child needs help to link the word “chair” with chairs; u A child needs to have an interest/curiousity with a chair or have something to say about a chair;

20. Some significant aspects 1 u There needs to algebraic related situations if the student is to start expressing themselves algebraically; u A student needs help with linking written expressions with the process/objects/situations to which they relate; u A student needs to have an interest/curiousity with an algebraic situation or have something to say about it;

21. Some significant aspects 2 u A child needs to have opportunities and encouragement to practise saying and appropriately using the word “chair”;

22. Some significant aspects 2 u A student needs to have opportunities and encouragement to practise writing, or appropriately using and interpreting algebraic expressions;

23. Some significant aspects 3 u Initially a child’s and adult’s focus may be on a chair; u Later “chair” is subordinated by both child and adult to make something happen (e.g. child may want an adult to come and give them attention, and so the child may say “Daddy’s chair” to get Daddy to come and sit in a chair next to them);

24. Some significant aspects 3 u Initially a teacher’s and student’s focus may be on the appropriate writing of an expression; u Later expressions are subordinated by both student and teacher to make carry out some algebraic task (e.g. creating a procedure to draw similar shapes within logo or some other software which accepts formal notation);

25. Some significant aspects 4 u Immediate family/friends may understand a child’s mis-pronounciations and unusual usages of a word, but people outside of this immediate circle may require more sophisticated usage or clearer pronounciation of the word.

26. Some significant aspects 4 u Fellow students and the teacher may understand a student’s idiosyncratic representations and informal usage of expressions, but people outside of this particular mathematics classroom may require more precise/formal expressions and usage.