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Mathematics for Teaching: some issues, some reflections PETER GALBRAITH University of Queensland Australia

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2 Learning mathematics for teaching ● There is a widely held belief that one’s perception of teaching and learning mathematics is influenced by one’s conception of mathematics…one’s conceptions of what mathematics is, affects one’s conception of how it should be presented. (Paper A for review 2005) ● The experience that participants had as learners of mathematics framed the way they perceived themselves as teachers of mathematics. (Paper B for review 2005)

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3 Some history Although these students had been ‘exposed’ to the calculus and limits for some four years…most of the students were incapable of formulating a simple argument in analysis, but more seriously possessed almost no intuition on the subject. To many of them mathematics was (and probably still is) a formalists dream come true, involving the use of routine algorithms applied to meaningless symbols to deduce routine answers to even more meaningless questions... After twelve years of schooling followed by two years of university they had all but accepted the mindless mathematics that had been thrust upon them. Few enjoyed mathematics, most simply wished to get their degree and get out – as schoolteachers… [Gray, 1975](ESM) In attending module after module, students tended to ‘memory dump’ rather than to retain and build a coherent knowledge structure... Their presumed examination strategy resulted in such a fragile understanding that reconstructing forgotten knowledge seemed alien to many taking part. [Anderson et. al., 1998](IJMES&T) Reflections on Content

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4 Mechanical Item : x 2 – ax +12=0 represents a family of equations. Four members of the family are obtained by giving ‘a’ the values 5, 6, 7 and 8. For what values of a can the equations be solved by factorising the left-hand side? A. 5 onlyB. 7 and 8C. 6 and 7D. 8 onlyE. none Interpretive Item : Which of the following could be the equation of the graph shown? A. y = (x - 2) 2 (1 - x) B. y = (2 - x) 2 (1 - x) C. y = (x - 2) 2 (x - 1) D. (x - 1) 2 (x - 2) E. none of these Constructive Item : The equations of two graphs are y = 3/x and y = x Obtain a cubic equation whose solution gives the x-coordinate of the point(s) of intersection of these two graphs. How many positive roots does this equation have? [Galbraith & Haines, 1999] (IJMES&T) Undergraduate (mis)understandings

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5 Mechanical - items Proficiency in a cued procedure. Interpretive – items Retrieve and apply conceptual knowledge Constructive – items Apply concepts and procedures introduced by the solver Conjecture: On performance M > I >C Results (N=423): Proportion correct M(0.41) > I(0.30) > C(0.19) Item types

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6 Australian Study (Queensland, Sydney, Western Australia) Focus: Existence and persistence of mathematical misconceptions (e.g) Interest is in comparative responses across contexts: (a) √4 = ? (Correct: UG = 40% ; PG=26%) (b)Noting that evaluate (Correct: UG =29%; PG = 29%) (c) Sketch the graphs of (i) y = (4 – x 2 ) and (ii) x 2 + y 2 = 4 2 for (a) -1 for (b) upper semicircle for (C) (i) School leaver and graduate comparisons

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7 ● Robustness from year to year ● conflicting answers live happily side by side without creating the least curiosity or challenge. ● Misconceptions among school leavers survive three or four years of undergraduate mathematics relatively unscathed Misconceptions, misguided and underdeveloped methods, unrefined intuition tend to remain; assignments, corrections, solutions, tutorials, lectures and examinations notwithstanding. [Gray] Some persistent themes in responses

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8 A theoretical framework: Community of Practice ● A joint enterprise defined and guarded by participants ● Contains routines, words, tools, ways of doing things etc that characterise the community ● For mathematics essential activities include: Conjecturing Defending Proving and disproving Abstracting Justifying Generalising Problem solving Communicating ● Community members engage actively in these pursuits – deep learning implies that learners need to do so as well ● There are consequences for both teaching and assessment (Many references - a philosophical bases include Social Constructivism and Deep and Surface learning)

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9 ● Mathematician sees alternatives - e.g. different but linked representation of function Graphical Geometric SymbolicNumerical Mackie, D. (2002). Using Computer Algebra to encourage a Deep Learning approach to Calculus. ICTM2, Herniossis, Crete. y x f(x) = a x or dy/dx = kx x 2 x An example - multiple representation

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10 1. VIGNETTES Find the oblique asymptote of Student A: By division As x so the asymptote is y = x - 3; Student B: Dividing N and D by x, = As x , so the asymptote is y = x – 1. Students A and B asked for resolution. Approaches: Alternative presentation formats

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11 Alternative formats (cont) ● Adaptable for purposes of teaching, classroom discussion, tutorial sheet problems, or assessment items. ● Supporting Assessment in Undergraduate Mathematics (SAUM)

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12 Alternative formats cont

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13 Alternative formats cont

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14 Alternative formats cont

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15 Alternative formats cont Khan : A and B are multiplicative inverses within the set of matrices of the form: and the usual laws of addition and multiplication apply. George: That is not correct as singular matrices do not have multiplicative inverses. ● Decide which statement you agree with and defend your choice by mathematical argument 9. DEFEND

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16 ● How to rattle the conceptual cages of intending teachers? ● What tasks will engage fundamental concepts and procedures so as to confront understanding and misconceptions? ● What is practically feasible? ● Pressures of dollarship versus scholarship in designing courses for a multidisciplinary group? ● Inclusion of rich alternative formats within the setting of problems in coursework and test situations? Implications for Teacher Education

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17 Reflections on Mathematical Modelling ● Henry Pollak (1969) - most applications of mathematics aren’t! Pretend problems Example A:The height of a species of hardwood is given by y = x 2 /20(1-x/60), where y is in metres and x is the time in years after the sowing of the seeds. What is the latest time at which such a tree should be harvested? Problems of Whimsy “The function of such problems…provide comic relief in the Shakespearean sense, and probably do a lot of good – although not as applied mathematics”. (Pollak, 1969) Two meatballs roll off of a pile of spaghetti and roll toward the edge of the table. One meatball is rolling at 1.2 m/s and the other at 0.8 m/s. They fall off the table and land on a $5000 Isfahan carpet. If the table is 1.2 m high, how far apart from each other do the meatballs land? (2004 source)

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18 More whimsy Pythagoras was sitting in calculus in the 80-degree weather of mid-April, wishing he were at the beach. While daydreaming, his terrible case of senioritis took over and his grade quickly began to plummet. When he got his final report card, he saw his grade had decreased. The amount it decreased is equal to the volume of the solid bounded in the first quadrant by y = 2 – x 3, revolved about the x-axis. If he started the 4th quarter with a 69, by how much did his grade decrease and will he pass the class with a 60 or better ? (April 2005)

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19 Models of modelling 1. Modelling as Vehicle “The curricular context of schooling in our country does not readily admit the opportunity to make mathematical modeling an explicit topic in the K- 12 mathematics curriculum. The primary goal of including mathematical modeling activities in students’ mathematics experiences typically is to provide an alternative – and supposedly engaging – setting in which students learn mathematics without the primary goal of becoming proficient modelers…Acknowledging this curricular context, we recognize that extensive student engagement in classroom modeling activities is essential in mathematics instruction only if modeling provides our students with significant opportunities to develop deeper and stronger understanding of curricular mathematics.” [Zbiek & Conner, 2006] (ESM) Modelling stays within the classroom - it plays second fiddle to other curricular purposes.

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20 Models of modelling cont 2. Modelling as Content “Starting with a certain problematic situation in the real world, simplification and structuring leads to the formulation of a problem and thence to a mathematical model of the problem…It has become common practice to use the term mathematical modelling for the entire process consisting of structuring, mathematising, working mathematically and interpreting, validating, revisiting and reporting the model.” [Blum, et.al., 2003; 2007] (ICMI STUDY 14) The real context plays an essential role in both the construction of a model - and the evaluation of its worth. The process cannot live entirely within the classroom.

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21 Modelling as curve fitting 3. Modelling as Curve Fitting Given data at 4-weekly intervals for the number of minutes of daylight (sunrise to sunset) for given locations find functions fit the data … For Melbourne (2004) we obtain the equation, based on properties of solstices, and length of the solar year. Using the periodic regression facility of a TI 83 calculator gives a technically closer fit as: !! [From a senior secondary course problem, 2005] Fundamental questions of philosophy arise.

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22 A potent example G I Taylor and the New Mexico atomic test of 1945 (Pedley 2005) ● Using only dimensional analysis and published photographs, inferred the strength of the atomic bomb tested in the New Mexico desert in ● Blast wave radius (R), could depend only on time (t), energy released instantaneously (E), and density of air into which the wave is expanding (ρ). ● Inferred (school mathematics only is needed) R = C (Et 2 /ρ) 1/5 where C is a dimensionless constant. ● Used a series of public photographs showing R against t, and a log-log linear plot to estimate C and E. ● Contacted the American authorities “ I see the bomb you tested had a power equivalent to about 17 kilotons of TNT”. (Result – apoplexy!!!)

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23 A contemporary example – Supersize me

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24 Supersize Me – Structured Questions

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25 Why modelling for teachers? ● Mathematics has a life beyond the confines of classrooms ● Structural goals – facilitating integration of content and methods of mathematical argument ● Teach how to access and use the mathematical techniques learned for addressing problems in the world. ● Beyond a simple consumer and purveyor of the knowledge of others ● Facility to engage with contemporary issues (including social) e.g. Morgan Spurlock – Supersize me. [ w n = w n-1 + (I - 24 w 0 )/7700, where ‘I’ represents the average (known) daily intake of calories. Constants (that estimate calorie ~ kg equivalences of foods, and calories consumed by various forms of exercise) are available from internet sources.]

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26 Implications for teacher education ● Mathematics Programs - Intensifying demands for service courses -- Diminishing resources- Traditional culture: - A focus on modelling (formulation, solution, reporting) looks different from traditional tertiary content (e.g. a listing of theorems or range of specific topics and techniques). (ICTMA group) ● Education Programs - Issue of Personnel - Issue of crowded pedagogy programs – what can be fitted in? ● A challenge is to create some space in both?

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27 Thank You

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