Versatile Mathematical Thinking in the Secondary Classroom Mike Thomas The University of Auckland

Overview A current problem Versatile thinking in mathematics Some examples from algebra and calculus Possible roles for technology

What can happen? Why do we need to think about what we are teaching? –Assessment encourages: Emphasis on procedures, algorithms, skills Creates a lack of versatility in approach

Possible problems Consider But, the LHS of the original is clearly one half!!

Concept not understood Which two are equivalent? Can you find another equivalent expression? A student wrote… …but he factorised C!

Procedural focus

Let For what values of x is f(x) increasing? Some could answer this using algebra and but… Procedure versus concept

For what values of x is this function increasing?

Versatile thinking in mathematics First… process/object versatility—the ability to switch at will in any given representational system between a perception of symbols as a process or an object

Examples of procepts

Lack of process-object versatility (Thomas, 1988; 2008)

Procept example

Effect of context on meaning for

Process/object versatility for Seeing solely as a process causes a problem interpreting and relating it to

Student: that does imply the second derivative…it is the derived function of the second derived function

Visuo/analytic versatility Visuo/analytic versatility—the ability to exploit the power of visual schemas by linking them to relevant logico/analytic schemas

A Model of Cognitive Integration conscious unconscious

Surface (iconic) v deep (symbolic) observation “                            ”         Moving from seeing a drawing (icon) to seeing a figure (symbol) requires interpretation; use of an overlay of an appropriate mathematical schema to ascertain properties

External world Internal world external sign ‘appropriate’ schema interpret Interact with/act on

Booth & Thomas, 2000 We found                e                Schema use

Example This may be an icon, a ‘hill’, say We may look ‘deeper’ and see a parabola using a quadratic function schema This schema may allow us to convert to algebra

Algebraic symbols: Equals schema Pick out those statements that are equations from the following list and write down why you think the statement is an equation: a) k = 5 b) 7w – w c) 5t – t = 4t d) 5r – 1 = –11 e) 3w = 7w – 4w

Surface: only needs an = sign All except b) are equations since:

Equation schema: only needs an operation Perform an operation and get a result:

The blocks problem

Solution

Reasoning

Solve e x =x 50

Check with two graphs, LHS and RHS

Find the intersection

How could we reason on this solution?

Antiderivative? What does the antiderivative look like?

Task: What does the graph of the derivative look like?

Method easy

But what does the antiderivative look like? How would you approach this? Versatile thinking is required.

Maybe some technology would help Geogebra

Representational Versatility Thirdly… representational versatility—the ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with representations

Representation dependant ideas... "…much of the actual work of mathematics is to determine exactly what structure is preserved in that representation.” J. Kaput Is 12 even or odd? Numbers ending in a multiple of 2 are even. True or False? 12 3 ? 12 3, 34 5, 56 9 are all odd numbers 11 3, 34 6, 53 7, 46 9 are all even numbers

Representations can lead to other conflicts… The length is 2, since we travel across 1 and up 1 What if we let the number of steps n increase? What if n tends to  ? Is the length √2 or 2?

Representational versatility Ruhama Even gives a nice example: If you substitute 1 for x in ax 2 + bx + c, where a, b, and c are real numbers, you get a positive number. Substituting 6 gives a negative number. How many real solutions does the equation ax 2 + bx + c = 0 have? Explain.

16

Treatment and conversion (Duval, 2006, p. 3)

Treatment or conversion? 25

Integration by substitution

Linking of representation systems (x, 2x), where x is a real number Ordered pairs to graph to algebra

Using gestures Iconic – “gestures in which the form of the gesture and/or its manner of execution embodies picturable aspects of semantic content” McNeill (1992, p. 39) Deictic – a pointing gesture Metaphoric – an abstract meaning is presented as form or space

The task

Thinking with gestures

Creates a virtual space

perpendicular

converging

The semiotic game “The teacher mimics one of the signs produced in that moment by the students (the basic sign) but simultaneously he uses different words: precisely, while the students use an imprecise verbal explanation of the mathematical situation, he introduces precise words to describe it or to confirm the words.”

Why use technology? It may be used to: promote visualization encourage inter-representational thinking enable dynamic representations enable new types of interactions with representations challenge understanding make conceptual investigation more amenable give access to new techniques aid generalisation stimulate enquiry assist with modelling etc

It depends on how it is used… Performing a direct, straightforward procedure Checking of (procedural) by-hand work Performing a direct procedure because it is too difficult by hand Performing a procedure within a more complex process, possibly to reduce cognitive load Investigating a conceptual idea Thomas & Hong, 2004

Task Design – A key Features of a good technology task: students write about how they interpret their work; includes multi-representational aspects (e.g. graphs and algebra); considers the role of language; includes integration of technological and by-hand techniques; aims for generalisation; gets students to think about proof; enables students to develop mathematical theory. Some based on Kieran & Drijvers (2006)

Task Can we find two quadratic functions that touch only at at the point (1, 1)? Can you find a third? How many are there?

Task–Generalising Can we find the quadratics that meet at any point (p, q), with any gradient k?

Task–Generalising Can we find two quadratics that meet at any point (p, q), with any gradient k? Extending a task by A. Harradine

One family of curves

Another task Can we find a function such that its derived (or gradient) function touches it only at one single point? For a quadratic function this means that its derived function is a tangent

How would we generalise this? Consider And its derived function

Solution So these touch at one point

For example Extension: can you find any other functions with this property?….

Newton-Raphson versatility Many students can use the formula below to calculate a better approximation of the root, but are unable to explain why it works

Newton-Raphson

Why it may fail

Newton-Raphson When is x 1 a suitable first approximation for the root a of f(x) = 0?

Symmetry of cubics

Generalising 180˚ symmetry In general (a, b) is mapped to (2p–a, 2q–b) Hence (p,q) (a,b) (x, y) = (2p-a,2q – b) e.g. q – y = b – q

Solving linear equations Many students find ax+b=cx+d equations hard to solve We may only teach productive transformations But there are, of course, many more legitimate transformations

10–3x = 4x+3 10= 7x+3 10–x= 2x+3 7=7x 8.37–x= 2x =x productivelegitimate

Legitimate: 10–3x = 4x+3

Teacher comments “I feel technology in lessons is over-rated. I don’t feel learning is significantly enhanced…I feel claims of computer benefits in education are often over-stated.” “Reliance on technology rather than understanding content.” “Sometimes some students rely too heavily on [technology] without really understanding basic concepts and unable to calculate by hand.” GC’s “encourage kids to take short cuts, especially in algebra. Real algebra skills are lacking as a result”

PTK required of teachers Pedagogical Technology Knowledge (PTK) –teacher attitudes to technology and their instrumentalisation of it –teacher instrumentation of the technology –epistemic mediation of the technology –integration of the technology in teaching –ways of employing technological tools in teaching mathematics that focus on the mathematics Combines knowledge of self, technology, teaching and mathematics (Thomas & Hong, 2005a; Hong & Thomas, 2006)

Teaching implications Avoid teaching procedures, algorithms, even with CAS—using CAS solely as a ‘calculator’ reinforces a procedural approach Give examples to build an object view of mathematical constructs Encourage and use visualisation Provide, and link, a suitable number of concurrent representations in each learning situation Encourage a variety of qualitatively different interactions with representations

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