professional development course 08/09 advance in curriculum studies and teaching

Title

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Professional Development Course 08/09

Advance in Curriculum Studies and Teaching Methods (Mathematics)

What do we and our students do in mathematics lesson?Mathematical Tasks for Teaching and Learning

Arthur LeeJan 10, [email protected]

http://web.hku.hk/~amslee/allggb/

http://web.hku.hk/~amslee/it07/

http://web.hku.hk/~amslee/math07/

MLT

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http://homepage.mac.com/msalee/mlt/

Exploring the Space of Teaching

topics activities

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Mason & Johnston-Wilder (2006) pp.4-5

The basic aim of a mathematics lesson is for learners to learn something about a particular topic. To do this, they engage in tasks. By 'tasks' we mean what learners are asked to do: the calculations to be performed, the mental images and diagrams to be discussed, or the symbols to be manipulated. ...

The purpose of a task is to initiate activity by learners. In such activity, learners construct and act upon objects, whether physical, mental or symbolic, that pertain to a mathematical topic. This activity is intended to draw learners' attention to important features, so that they may learn to distinguish between relevant aspects, or recognise properties, or appreciate relationships between properties. (based on Christiansen & Walther, 1986)

Mathematical Topics, Tasks and Activities

connections

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Beyond particular topics

Mathematics is traditionally divided into domains such as arithmetic, algebra, geometry, and data handling. Topics are often allocated to one of these domains: arithmetic, algebra, geometry and data handling.

There are two problems with this approach. Firstly, some topics do not fit easily into one of the categories. For example, coordinate geometry is both algebra and geometry; similarly, measures are used in both arithmetic and geometry.

The second problem is that, by separating topics in this way, mathematics can degenerate into a large collection of techniques and vocabulary, with connections between the topics being ignored.

Many textbooks accentuate this disconnection by the way they organise different topics on successive pages 'so learners won't get bored'. However, Askew et al. (1997) looked at teachers of classes which do well in standard assessment tasks. They found that most significant feature of the teachers was the richness of the teachers' awareness of how mathematical topics fit together.

Mason & Johnston-Wilder (2006) pp.20-21

themes

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Mason & Johnston-Wilder (2006) p.21

Rather than trying to fit topics into categories, it can be helpful to think in terms of some pervasive mathematical themes that serve to unify topics which might otherwise appear to be disparate.

Freedom and ConstraintMathematical Themes

Invariance in the midst of changeExtending and Restricting MeaningDoing and Undoing

1.2.3.4.

block 4

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Johnston-Wilder, S. & Mason, J. (eds.) (2005) Developing Thinking in Geometry London, Open University & Paul Chapman Publishing

Mathematical Themes

freedom & constraint

doing & undoing

invariance & change

extending & restricting

◆◇◇◇◇

Structure of a Topic

awareness & absences

harnessing emotion

training behaviour

◆◇◇◇

Learners' Powers

imaging & expressing

specialising & generalising

conjecturing & convincing

organising & classifying

◆◇◇◇◇

Pedagogic Constructs

stressing & ignoring

dimensions of possible variation

manipulating-getting-a-sense-of-articulating

do-talk-record

enactive-iconic-symbolic & different worlds

see-experience-master

structure of attention

what makes an example exemplary?

didactic transposition

◆◇◇◇◇◇◇◇◇◇

Pedagogic Strategies

scaffolding & fading

learner-constructed examples

teaching techniques

say what you see

same & different

another & another

in how many ways?

turn a doing into an undoing

diverting attention in order to automate

◆◇◇◇◇◇◇◇◇◇

topic

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Preparing to teach a topic

http://www.ncetm.org.uk/Default.aspx?page=22&module=enc&mode=100&enclbl=Structure+of+a+Topic

structure

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Preparing to Teach a Topic ...

3 strands to understanding or appreciating any mathematical topic:

cognitive, affective and enactive components,

aka awareness, emotion and behaviour.

cognitive

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Cognitive (awareness): What images and associations are part of this topic? What connections with other topics? [see also concept images] What other topics are needed in this topic? What are the obstacles that learners often encounter? What misconceptions often appear? What previous awarenesses can be made use of?

affective

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Affective (emotion): what were the root problems which turned into this topic, which this topic resolves? Historically, where did this topic come from? Why is it in the curriculum? In what other contexts might this topic be encountered or be relevant?

enactive

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Enactive (behaviour): what patterns of language are used in this topic? What technical terms are used? How are they related to everyday usage of the same or similar words and phrases? What techniques are part of this topic. What sorts of tasks are used to assess learner competence? What problems might really probe understanding?

LGE

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Learner-Constructed Examples

Learners can be given opportunities to express their creativity and to make choices for themselves by being asked to construct objects meeting certain constraints. In the process, learners display some of the dimensions of possible variation and associated ranges of permissible change of which they are aware. Asking them to construct another and another ... may prompt them to explore the boundaries of their confidence and so extend the range of examples of which they are aware and with which they are confident ...

Johnston-Wilder & Mason (2005) p.255

references

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Johnston-Wilder, S. & Mason, J. (eds.) (2005). Developing Thinking in Geometry. London, Open University & Paul Chapman Publishing.

Mason, J., Graham, A. & Johnston-Wilder, S. (2005). Developing Thinking in Algebra. London, Open University & Paul Chapman Publishing.

Graham, A. (2006). Developing Thinking in Statistics. London, Open University & Paul Chapman Publishing.

Untitled

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Watson, A. and Mason, J. (2005) Mathematics as a constructive activity: learners generating examples. Mahwah, N.J.: Lawrence Erlbaum Associates.

Mason, J. & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks. St. Albans: Tarquin.

Untitled

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Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education. New York, NY: RoutledgeFalmer.

http://library.hku.hk/record=b3465301 [netLibrary, HKUL]

changes

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In recent years pupils in the high school on the average are noticeably more immature than they were ten years ago. The subject matter also of geometry has been changed, and to a certain extent vocationalized and humanized. These changes call for modifications in class-room methods of teaching geometry, in order to obtain maximum results. It is hoped that the suggestions which follow may help to meet the new situation.

changes

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While the mind of the average high school pupil is immature and occupied with many interesting concrete things such as movie, kodak, and automobile, it is also alert, eager, and quick to grow when interested. Hence the best general course to follow in teaching such pupils is to give them at the outset, a large amount of simple and easily appreciated work. This arouses natural growth processes in their minds, so that in time, and often without serious effort, they develop the power to do more difficult work and form an active appetite for it.

Godfrey

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eprints soton ac uk 2003 Fujita-1

Fujita, T. & Jones, K.(2003) The Place of Experimental Tasks in Geometry Teaching: Learning from the Textbooks Design of the Early 20th Century. Research in Mathematics Education, 5, (1&2), 47-62.

Fujita, T. & Jones, K. (2002) The Bridge between Practical and Deductive Geometry: Developing the 'Geometrical Eye'. In Cockburn, A. D. and Nardi, E. (eds.) Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (PME26). Norwich, England, PME, 384-391.

eprints soton ac uk 2002 Fujita-1

Jones, K. and Fujita, T. (2002) The Design Of Geometry Teaching: learning from the geometry textbooks of Godfrey and Siddons, Proceedings of the British Society for Research into Learning Mathematics, 22 (2), 13-18.

eprints soton ac uk 2002 Fujita

Fujita, T. (2001) The Use of Practical and Experimental Tasks in Geometry Teaching: A Study of Textbooks by Godfrey and Siddons. In Winter, J. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 21(3), 31-36.

bsrlm org uk 2001 Fujita

Study of Early Geometry Textbooks

scrap

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main

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geometrical eye: the power of seeing geometrical properties detach themselves from a figure

Godfrey & Siddons (in Fujita & Jones)

question the strict distinction between experimental and deductive geometryplace of practical and experimental tasksdual nature of geometry: a theoretical domain vs an area of practical experience

●

●

●

●

chasm

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Fujita & Jones (2003)

transition

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In a number of countries, the early stages of geometry in schools comprise practical activities such as the drawing and measurement of geometrical figures. Later stages of schooling are then devoted to deductive geometry.

While this is somewhat in line with the van Hiele (1986) model of learning in geometry, the relationship between practical and deductive geometry remains unclear, and, in particular, the transition between them is one of the major concerns in the study of the teaching of geometry.

Fujita, T. & Jones, K. (2002) p.389

altitude

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1921 Godfrey

comment

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Fujita & Jones (2003)

chord

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1921 Godfrey

comment

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Fujita, T. & Jones, K. (2002) p.389

old books

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Geometry Books from The Internet Archive

Durell 1921

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Suggestions on the Teaching of Geometry

F. Durell (1921)http://www.archive.org/details/suggestionsontea00durerich

Willis 1922

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p.50Parts of Triangles http://www.archive.org/details/planegeometryexp00willrich

Plane Geometry; experiment, classification, discovery, application ..

Willis (1922)

triangle angles

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p.51

Properties of the angles of a triangle

triangles

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pp.63-64

Campbell 1899

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http://www.archive.org/details/observationalgeo00camprich

Campbell (1899)

prism

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triangles

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Hedrick 1916

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http://www.archive.org/details/constructivegeom00hedriala

Constructive Geometry; exercises in elementary geometric drawing

E. R. Hedrick (1916)

Untitled

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p.25

Untitled

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p.25

Untitled

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p.26

Untitled

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p.26

Untitled

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p.26

Kerr

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http://www.archive.org/details/constructivegeom00hedriala

J. G. Kerr

square it

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Square It!

http://nrich.maths.org/public/search.php?search=geoboard

Other Geoboard Activities in nrich:

http://nrich.maths.org/public/viewer.php?obj_id=2526&part=2526

Untitled

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http://www.standards.dcsf.gov.uk/secondary/framework/maths/fwsm/soefrc

© Crown copyright 200800366-2008PDF-EN-01

Pupils should learn to: As outcomes, Year 7 pupils should, for example:

218 The National Strategies | Secondary Mathematics exemplification: Y7

GEOMETRY AND MEASURES Transformations and coordinates

Use coordinates in all four quadrants Use, read and write, spelling correctly:row, column, coordinates, origin, x-axis, y-axis… position, direction… intersecting, intersection…

Read and plot points using coordinates in all four quadrants.

Given an outline shape drawn with straight lines on a coordinate grid (all four quadrants), state the points for a partner to connect in order to replicate the shape.

Plot points determined by geometric information.For example:

On this grid, players take turns to name and then mark a point in their own colour. Each point can be used only once.

Game 1 The loser is the !rst to have 3 points in their own colour in a

straight line in any direction.

Game 2 Players take turns to mark points in their own colour until the

grid is full. Each player then identi!es and records 4 points in their own colour forming the four corners of a square. The winner is the player who identi!es the greatest number of di"erent squares.

The points (–3, 1) and (2, 1) are two points of the four vertices of a rectangle.

Suggest coordinates of the other two vertices. Find the perimeter and area of the rectangle.

Plot these three points: (1, 3), (–2, 2), (–1, 4). What fourth point will make: a. a kite? b. a parallelogram? c. an arrowhead? Justify your decisions. Is it possible to make a rectangle? Explain why or why not.

Use ‘plot’ and ‘line’ on a graphical calculator to draw shapes.

Draw your initials on the screen.

Draw a shape with re#ection symmetry around the y-axis.

In geography, interpret and use grid references, drawing on knowledge of coordinates.

secondquadrant

fourthquadrant

thirdquadrant

firstquadrant

y-axis

x-axis

The origin is the point ofintersection of the x-axiswith the y-axis.

123

–3

0–3 –2

–2

–1–1

1 2 3

p.218

nrich

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Virtual Geoboard, Tilted Square, Square Coordinates



Thornton

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Thornton, S. J. New approaches to algebra: have we missed the point?. Mathematics Teaching in the Middle School v. 6 no. 7 (March 2001) p. 388-92 [http://library.hku.hk/record=b2014009~S6]

388 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

CU R R I C U L U M M O V E M E N T S I N T H E United States and Australia, characterized bysuch documents as Curriculum and EvaluationStandards for School Mathematics (NCTM

1989) and A National Statement on Mathematics forAustralian Schools (AEC 1991), have challenged theconventional view of algebra as formal structure, ar-guing that algebra is fundamentally the study of pat-terns and relationships. Increased emphasis hasbeen given to developing an understanding of vari-ables, expressions, and equations and to presentinginformal methods of solving equations. The empha-sis on symbol manipulation and on drill and prac-tice in solving equations has decreased (NCTM1989).

Has the net effect of these changes been merelyto replace one kind of procedural knowledge withanother? This article looks at three approaches toalgebra: (1) a patterns approach, in which studentsare asked to generalize a relationship; (2) a sym-bolic approach, in which students learn to manipu-late algebraic expressions; and (3) a functions ap-proach, which emphasizes generation andinterpretation of graphs. This article examines thenature of thinking inherent in each approach andasks whether any or all of these approaches are, in

themselves, sufficient to generate powerful alge-braic reasoning.

The Patterns Approach, or “Matchstick Algebra”

THE PATTERNS APPROACH TO ALGEBRA IN THEmiddle school is typified by the matchstick patternshown in figure 1. Faced with this problem, stu-dents almost invariably describe the rule as “add 3.”Most students look at the table of values horizon-tally, observing that each time a square is added, thenumber of matches needed increases by three.Well-intentioned teachers often help students find ageneral rule from this observation, saying, for exam-ple, that if one adds 3 each time, the rule is of the

STEVE THORNTON, [email protected], is direc-tor of teacher development at the Australian MathematicsTrust, University of Canberra, Australia 2601. His inter-ests include mathematical rigor and enrichment for talent-ed students.

New Approaches to Algebra: Have We Missed the Point?

S T E P H E N J. T H O R N T O N

Fig. 1 Matchstick pattern

Examine the following pattern, complete thetable, and find a rule that shows how the num-ber of matches (m) depends on the number ofsquares (s).

Rule: m = ________

s 1 2 3 4 5 100

m 4 7 10

Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

VOL. 6, NO. 7 . MARCH 2001 389

form m = 3s + k, and suggesting that students try afew numbers to determine the value of the constant.

The students regard this approach as goodteaching because it helps them obtain the correctanswer. The teacher is similarly reinforced in thebelief that he or she is acting in the students’ bestinterests, because the students are able to find therule for this pattern and, perhaps, even a generalrule for other linear cases. The ability to find theserules is, arguably, a useful skill, but do the studentsunderstand any more about the nature of algebrathan if the subject had been introduced in a formal,symbolic way? Students who use this heuristic tofind the constant and thus the general rule have, inreality, looked at the specific rather than the gen-eral. They have not necessarily acquired any well-developed notion of the general nature of the pat-tern but have merely learned a procedure todevelop a correct symbolic expression. The alge-braic essence of the problem is absent.

The Matchstick Pattern Problem is not aboutfinding a general rule. The answer to the problem,

that is, the rule itself, is unimportant. The problemis really about alternative representations. It is a vi-sualization exercise in which different ways of look-ing at the pattern produce different expressions. Vi-sualizing the pattern in different ways and writingcorresponding algebraic relationships help stu-dents understand the nature of a variable and be-come familiar with the structure of algebraic ex-pressions. This particular pattern can be visualizedin at least four different ways (see fig. 2).

Writing down the number pattern in a table, an ac-tivity commonly found in textbooks and on work-sheets, does not help students visualize the generalityinherent in the matchstick constructions. A muchmore constructive approach is to ask students to buildone element of the pattern physically and explain howit is put together, not in terms of numbers but in termsof its underlying physical structure. The different alge-braic structures then have direct physical meanings.

Numerous other visual approaches to algebra arepossible (Nelsen 1993). For example, students couldbe asked to visualize the pattern shown in figure 3in different ways so as to generate a relationship be-tween the number of shaded squares (b) and thelength of the side of the white square (n). Again, atleast four different representations are possible(see fig. 4). The point of the exercise is not to ob-tain the answer b = 4n + 4 or any of its variants butrather to understand how the pattern can be visual-ized and how these different visualizations can bedescribed symbolically. If we are to foster powerfulalgebraic thinking in our students, we must encour-age a variety of well-justified generalizations of thepattern. Rather than be an end in itself, the purposeof generating rules is to develop insight into pat-terns and relationships. As Gardner (1973, p. 114)writes, “There is no more effective aid in under-standing certain algebraic identities than a good di-agram. One should, of course, know how to manip-ulate algebraic symbols to obtain proofs, but inmany cases a dull proof can be supplemented by ageometric analogue so simple and beautiful that thetruth of a theorem is almost seen at a glance.”

The SymbolicApproach, or “FruitSalad Algebra”

THE FORMAL, SYMBOLIC approach to algebra, in whichvariables are defined as letters that stand for num-bers, has been criticized as lacking meaning(Chalouh and Herscovics 1988) and has been iden-tified as the source of many difficulties faced by be-ginning algebra students (Booth 1988). OlivierFig. 2 Different ways to visualize the matchstick pattern

Pattern built of one match plus three for eachsquare, or m = 3s + 1

Pattern built of four matches for the firstsquare plus three for each subsequent square,or m = 4 + 3(s – 1)

Pattern built of two horizontal rows joined byvertical links, or m = 2s + (s + 1)

Pattern built of four matches for each square,with the overlapping match removed from allbut one of the squares, or m = 4s – (s – 1)

outline

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Approaches to Algebra

patternstructure of algebraic

expressions

compare multiple ways of seeing

compare different expressions

purpose for symbolic manipulation

nature of variable

symbolicfruit salad algebra

express generality

function multiple representations

388 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

CU R R I C U L U M M O V E M E N T S I N T H E United States and Australia, characterized bysuch documents as Curriculum and EvaluationStandards for School Mathematics (NCTM

1989) and A National Statement on Mathematics forAustralian Schools (AEC 1991), have challenged theconventional view of algebra as formal structure, ar-guing that algebra is fundamentally the study of pat-terns and relationships. Increased emphasis hasbeen given to developing an understanding of vari-ables, expressions, and equations and to presentinginformal methods of solving equations. The empha-sis on symbol manipulation and on drill and prac-tice in solving equations has decreased (NCTM1989).

Has the net effect of these changes been merelyto replace one kind of procedural knowledge withanother? This article looks at three approaches toalgebra: (1) a patterns approach, in which studentsare asked to generalize a relationship; (2) a sym-bolic approach, in which students learn to manipu-late algebraic expressions; and (3) a functions ap-proach, which emphasizes generation andinterpretation of graphs. This article examines thenature of thinking inherent in each approach andasks whether any or all of these approaches are, in

themselves, sufficient to generate powerful alge-braic reasoning.

The Patterns Approach, or “Matchstick Algebra”

THE PATTERNS APPROACH TO ALGEBRA IN THEmiddle school is typified by the matchstick patternshown in figure 1. Faced with this problem, stu-dents almost invariably describe the rule as “add 3.”Most students look at the table of values horizon-tally, observing that each time a square is added, thenumber of matches needed increases by three.Well-intentioned teachers often help students find ageneral rule from this observation, saying, for exam-ple, that if one adds 3 each time, the rule is of the

STEVE THORNTON, [email protected], is direc-tor of teacher development at the Australian MathematicsTrust, University of Canberra, Australia 2601. His inter-ests include mathematical rigor and enrichment for talent-ed students.

New Approaches to Algebra: Have We Missed the Point?

S T E P H E N J. T H O R N T O N

Fig. 1 Matchstick pattern

Examine the following pattern, complete thetable, and find a rule that shows how the num-ber of matches (m) depends on the number ofsquares (s).

Rule: m = ________

s 1 2 3 4 5 100

m 4 7 10

Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Thornton (2001)

matchstick

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Untitled

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Untitled

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Untitled

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expressions

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The Matchstick Pattern Problem is not about finding a general rule. The answer to the problem, that is, the rule itself, is unimportant. The problem is really about alternative representations. It is a visualization exercise in which different ways of looking at the pattern produce different expressions. Visualizing the pattern in different ways and writing corresponding algebraic relationships help students understand the nature of a variable and become familiar with the structure of algebraic expressions. This particular pattern can be visualized in at least four different ways.

Thornton (2001) p.389

expressions

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Writing down the number pattern in a table, an activity commonly found in textbooks and on worksheets, does not help students visualize the generality inherent in the matchstick constructions. A much more constructive approach is to ask student to build one element of the pattern physically and explain how it is put together, not in terms of numbers but in terms of its underlying physical structure. The different algebraic structures then have direct physical meaning.

Thornton (2001) p.389

NNS

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Examples from The Framework for Secondary Mathematics (UK)

http://www.standards.dcsf.gov.uk/secondary/framework/strands/881/67/17690

NNS

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different equivalent expressions

NNS

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NNS

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NNS

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NNS

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NNS

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distance

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http://www.geogebra.org/en/upload/files/arthur/distance_formula.html

Distance Formula

which is steeper

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Which is steeper?

http://www.geogebra.org/en/upload/files/arthur/steeper.html

slope

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http://www.geogebra.org/en/upload/files/arthur/slope_formula.html

Slope Formula

task

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At the end of a lesson introducing the formula for slope of straight line, this question is suggested for homework or classwork.

Find x if the slope of the line joining A(1,2) and B(x,4) is1/2.

Comment?

task

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How about this one?

Find a point B so that the slope of the line joining B and A(1,2) is 1/2.

How can student answer this?

What can they learn from this exercise?

task

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Find a point B so that the slope of the line joining B and A(1,2) is 1/2.

Can students make some sketches, tables?

consider drawing on grids:

http://web.hku.hk/~amslee/it07/Assets/F71824C5/blank_grid.html

task

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Find a point B so that the distance between B and A(1,2) is 5.

Can students make some sketches, tables?


task

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Find a point B so that the distance between B and A(1,2) is 5.

Comparing distance and slope


section

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Section Formula

http://www.geogebra.org/en/upload/files/arthur/060518a1.html

What is common among this formula and the previous two?

professional development course 08/09 advance in curriculum studies and teaching

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