1. What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical Thinking

2. Euclidean geometry Conceptual Embodiment Space & Shape Geometry Platonic Mathematics The Physical World Practical Mathematics

3. Symbolic Mathematics Euclidean geometry Conceptual Embodiment Symbolic calculation & manipulation Space & Shape Counting & Number Arithmetic Algebra Calculus Trigonometry Graphs Geometry Limits... Matrix Algebra Blending Embodiment & Symbolism Platonic Mathematics The Physical World Functions Practical Mathematics

4. Symbolic Mathematics Euclidean geometry Axiomatic Formalism Axiomatic Geometry Conceptual Embodiment Symbolic calculation & manipulation Space & Shape Counting & Number Arithmetic Algebra Calculus Trigonometry Graphs Geometry Limits... Matrix Algebra Blending Embodiment & Symbolism Axiomatic Algebra, Analysis, etc Set-theoretic Definitions & Formal Proof Platonic Mathematics Formal Mathematics The Physical World Functions Practical Mathematics Set Theory & Logic

5. Symbolic Mathematics Axiomatic Formalism Symbolic calculation & manipulation Platonic Mathematics Formal Mathematics The Physical World Practical Mathematics Embodiment Embodied Meaning increasing in sophistication from physical to mental concepts Symbolic Meaning increasing power in calculation using symbols as processes to do & concepts to manipulate [procepts] Formal Meaning as set-theoretic definition and deduction Blending embodiment & symbolism to give embodied meaning to symbols & symbolic computational power to embodiment

6. Mathematicians can live in the world of formal mathematics. Children grow through the worlds of embodiment and symbolism. Mathematics Educators try to understand how this happens. Fundamentally we build on what we know based on: Our inherited brain structure (set-before our birth and maturing in early years) Knowledge structures built from experiences met- before in our lives.

7. The terms ‘set-before’ and ‘met-before’ which work better in English than in some other languages started out as a joke. The term ‘metaphor’ is often used to represent how we interpret one knowledge structure in terms of another. I wanted a simple word to use when talking to children. When they use their earlier knowledge to interpret new ideas I could ask them how their thinking related to what was met before. It was a joke: the word play metAphor, metBefore. The joke worked well with teachers and children: What have you met before that causes you to think like this?

8. Inherited brain structure (set-before our birth and maturing in early years) Examples: Recognising the same object from different angles. A Sense of Vertical and Horizontal. Classifying categories such as ‘cat’ and ‘dog’. … triangle, square, rectangle, circle. Practising a sequence of actions (see-grasp-suck). … used in counting … … column arithmetic … … adding fractions … … learning algorithms … Set-Befores May be performed automatically without meaning

9. Current ideas based on experiences met before. Examples: Two and two makes four. Addition makes bigger. Multiplication makes bigger. Take away makes smaller. Every arithmetic expression 2+2, 3x4, 27÷9 has an answer. Squares and Rectangles are different. Met-Befores Different symbols eg and represent different things.

10. Two and two makes four Addition makes bigger Multiplication makes bigger Take away makes smaller Met-Befores … works in later situations. … fails for negative numbers. … fails for fractions. … fails for negative numbers. An algebraic expression 2x+1 does not have an ‘answer’. Later, by definition, a square is a rectangle. Different symbols can represent the same thing. Some are helpful in later learning, some are not.

11. In practice, when we remember met-befores, we may blend together ideas from different contexts. For instance, numbers arise both in counting and measuring. Counting numbers 1, 2, 3, … have much in common with numbers used for measuring lengths, areas, volumes, weights, etc. But they have some aspects which are very different... Met-Befores blended together

12. Natural numbers build from counting 12345... The number track... The number line builds from measuring The number line... 12345 Discrete, each number has a next with nothing between, starts counting at 1, then 2, 3,.... Continuous, each interval can be subdivided, starts from 0 and measures unit shifts to the right. Blending different conceptions of number 0...

13. Blending counting and measuring Does the difference between number track and number line matter? The English National Curriculum starts with a number track and then uses various number-line representations to expand the number line in both directions and to mark positive and negative fractions and decimals. Doritou (2006, Warwick PhD) found many children had an overwhelming preference to label calibrated lines with whole numbers, with limited ideas that an interval could be sub-divided. Their previous experience of whole number dominated their thinking and the expansion from number track to number line was difficult for many to attain.

14. Increasing sophistication of Number Systems Language grows more sophisticated as it blends together developing knowledge structures. Blending occurs between and within different aspects of embodiment, symbolism and formalism. Mathematicians usually view the number systems as an expanding system: Cognitively the development is more usefully expressed in terms of blends. N F Z QRC

15. To show 1 + 2 + 3 +.... + 978 + 99 + 100 = 5050 Argumentation can use embodiment and symbolism

16. Different knowledge structures for numbers The properties change as the number system expands. How many numbers between 2 and 3? None Lots– a countable infinity Lots more – an uncountable infinity None (the complex numbers are not ordered) N Q R C A mathematician has all of these as met-befores A learner has a succession of conflicting met- befores

17. From Arithmetic to Algebra The transition from arithmetic to algebra is difficult for many. The conceptual blend between a linear algebra equation and a physical balance works in simple cases for many children (Vlassis, 2002, Ed. Studies). The blend breaks down with negatives and subtraction (Lima & Tall 2007, Ed. Studies). Conjecture: there is no single embodiment that matches the flexibility of algebraic notation. Students conceiving algebra as generalised arithmetic may find algebra simple. Those who remain with inappropriate blends as met- befores may find it distressing and complicated.

18. Embodied Symbolic Blending different conceptions of number Real Numbers as a multi-blend Formal R A mathematician can have a formal view A student builds on Embodiment & Symbolism

19. From Algebra to Calculus The transition from algebra to calculus is seen by mathematicians as being based on the limit concept. For mathematicians, the limit concept is a met- before. For students it is not. A student can see the changing steepness of the graph and embody it with physical action to sense the changing slope. slope + slope zero slope –

20. Calculus Local straightness is embodied: You can see why the derivative of cos is minus sine

21. Calculus Local straightness is embodied: You can see why the derivative of cos is minus sine The graph of sine upside down.... slope is – sin x

22. Calculus Local straightness is embodied: E.g. makes sense of differential equations...

23. Reflections As we learn, our interpretations are based on blending met-befores, which may cause conflict. Learners who focus on the powerful connections between blends may develop power and flexibility, those who sense unresolved conflict may develop anxiety. Mathematicians have more sophisticated met- befores and may propose curriculum design that may not be appropriate for learners. It is the job of Mathematics Educators (who could be Mathematicians) to understand what is going on and help learners make sense of more sophisticated ideas.