6-Joy of X · George Gadanidis Page 22 01/10/01 What is Algebra? generalized arithmetic 1, 3, 5, 7, 9 ... or t n = 2n - 1 Notice the expression 2n - 1 defines all possible terms of

George Gadanidis Page 1 01/10/01

The Joy of x

When I discuss mathematics withmathematicians it is not uncommon tohear them talk about the pleasure ofdoing mathematics and the beauty thatmathematics helps them see andcreate.

The variable x is often at the heart ofmathematics.

What does xmean to you?

What is the ‘joyof x’?


Grade 4-8Curriculum

Grades 4-6

Focus shifts from exploring patternsto exploring functions, using graphs,data tables, expressions, equations,words, diagrams.

Learn that changes in one variableresult in changes in another.

Grades 7-8

Understand how the language ofalgebra can be used to generalize apattern or a relationship. (modelling)

Algebra as a problem solving tool.


The Malevolent Tyrannyof AlgebraGerald Brace (October 2000), in Education Week on the Web(http://www.edweek.org/ew/ewstory.cfm?slug=08bracey.h20)

Although algebra is all about findingvalues in equations, it has no value formost people. Its actual uselessness inmost people's lives was wonderfullyrevealed in a Washington Post articlefrom several years back. The storydescribed how parents in FairfaxCounty, Va., were rushing home fromwork, bolting down dinner, and going toschool to learn ... algebra. "They camenot for their benefit," thereporter wrote. "They hadlearned algebra yearsago and most of themhad no use for X's andY's in their current lives."


That sure gives the game away: "Mostof them had no use for X's and Y's intheir current lives." Yet, they areinflicting those useless X's and Y's onthemselves for the second time. Thistime, they're doing it so they can helptheir kids get through algebra.Apparently, it didn't occur to theseparents to ask, "If I didn't need it, whyam I suffering through it again just tohelp my kid successfully suffer throughit?"


How Teachers SeeAlgebra

“I think that we think very littleconceptually about what we’redoing. I think there’s a certainamount of just doing.”

How do you see algebra?- what is it?- how do you teach it?- how do you answer the question: “Why

do we have to learn this?”


Grades 5/6:

- patterning relationships- concept of variable

- that little curly x- kids get stumped on that

- solving simple equations- finding the mystery number

“last year when I was teaching grade 6… sharp bunch, there was hardlyanything in the textbook (on algebra), itwas just too elementary ... So I wentfrom a math book that I had from grade7 and 8 and I took the simplest pages … a – 7 = 10 and I tried to teach them… what you do to one side you do to theother side, the way I was taught algebra.


Well they went crazy, there was a brickwall, everyday they were like fighting meabout the way I was doing it, theywanted to do it in their head, I wasinsisting that they had to write it down,they had to follow by what you do on theone side … and they thought I wasinventing it, making it up.”

“I’d like to know, what am I supposed touse for resources, which was mystumbling block last year. How you canapply it to something they’d recognize,like life. How can you make themunderstand that it’s applicableelsewhere? Why they are learning it,they just don’t understand why.”


Grades 7/8:

- patterning- input/output puzzles- translate word problemsinto number sentences

- solving for unknowns- inspection- trial and error- formal process

- step-by-step; whatever youdo to one side of the equalsign, you must do to the other

- variables- the unknown values can changewithin the equation yet thesurrounding equation is always thesame

“Is there another way to teach it that’smore successful? We want a betteranswer for the kids, when they say whydo we need to know this.”


Grades 9/10:

- math that uses letters, variables,equations to carry ideas, relationships

“... that’s what we spend most of ourtime doing, showing themrelationships. Now we’re starting toshow them more algebra. We’ve sortof gone at it backwards (from howwe learned it), instead of showingthem the skills and then asking themto apply, we’re asking them to applyand then try to figure out what skillsthey were using.”

- expanding, simplifying, rearranging,mechanical

“When I took algebra in school thatwas basically what it was.”


“What I like about (algebra) is once youstart getting the hang of it then you cancount on it. It’ll work that way all thetime.”

“It’s interesting you say that because inmy grade 10 class there are 6 studentswho do absolutely nothing in class andit’s been very hard trying to findactivities they will do. They just there tosit there ... they are comfortable withknowing that they won’t pass thecourse, but as soon as we startedalgebra yesterday I noticed that theywere all working and were all tryingsteps and they weresaying “we get this,we understand” and Iwas so happy to seethese few studentsworking …”


What do StudentsLearn?Solving equations:

“As far as what you do to one side youdo to the other, (when they come tograde 9) a lot of them … don’t reallyknow why, … I had to explain that tothem. They got it now but when they firstcame they didn’t have it.”

“I find the same thing with my tens, I hadto basically teach them from scratchhow to simplify, how to subtractsomething from two sides, dividingboth sides of the equation, it wasas if they never learned it.”


Typical Misconceptions

The nature of answers

Many students assume that what isrequired is a numerical answer.

Even students who do produce thecorrect algebraic expression may notview it as a “proper” answer.

Other students appear to accept thepossibility of an algebraic answer, butthey tend to assume that at least what isrequired is a “single-term” answer.


Notation and convention in algebra

Students are confused about “n+3”being both an “instruction” and an“answer”.

Many students regard an equal sign asa unidirectional symbol preceding anumerical answer.

There is a strong tendency amongstudents to “simplify” an expression likea+b to ab.


Letters and variables

Students often translate 3a as 3 apples(as in 3 m = 3 metres) rather than 3times the number of apples.

Students assume that different lettersstand for different numerical values.

Booth, L.R. (1988). Children’s difficulties in beginning algebra. InCoxford, A.F. & A.P. Shulte, The Ideas of Algebra: K-12 (1988Yearbook). NCTM: Reston, VA. 20-32.


Solutions?

Concrete materials

Balance- show the effect of adding/subtracting

from both sides- show the effect of multiplying/dividing

both sides

Algebra tiles- operations with integers- representing 1, x, x2

- factoring


Algebra Tiles

3 – 5

= -

= -

=

= -2


x2 + 2x + 3

=

x2 + 2x - 3

=


A Sense of Algebra

The border problem

How many squares are there in theborder of a 10 by 10 grid? A 7 by 7 grid?


The border problem in algebra

b = 4e – 4b = e + e – 1 + e – 1 + e – 2b = (e x 4) - 4b = 4(e – 1)b = (e – 1) x 4b = e x 2 + e – 2 + e – 2b = e – 2 + e – 2 + e – 2 + e – 2 + 4


From Arithmetic toAlgebra

Please solve the followingmultiplication question in yourhead:

16 x 24 = ?

Compare your solution to the typicalmultiplication algorithm typically taughtand used in school:

24 x 16

144 24 384


16 x 24 = (10 + 6)24= 10 x 24 + 6 x 24= 240 + 144= 384

16 x 24 = 16(20 + 4)= 16 x 20 + 16 x 4= 320 + 64= 384

16 x 24 = (10 + 6)(20 + 4)= 10x20 + 10x4 + 6x20 + 6x4= 200 + 40 + 120 + 24= 384

16 x 24 = (20 – 4)(20 + 4)= 202 – 42

= 400 – 16= 384

16 x 24 = 16(25 – 1)= 16 x 25 - 16= 400 – 16= 384


What is Algebra?

generalized arithmetic

1, 3, 5, 7, 9 ... or tn = 2n - 1

Notice the expression 2n - 1 defines allpossible terms of the sequence, and itdoes so concisely. If we want to find anyone of the terms of the sequence, saythe sixth term, we can substitute 6 for nto get 2 ( 6 ) - 1 = 11.

variables: pattern generalizers

students: translate, generalize


a means to solve certain problems

When three is added to five times anumber, the sum is 38. Find thenumber.

variables: unknowns or constants

students: solve, simplify


a structure

Algebra as a set of methods.

Factor 3x2 + 4ax - 132a2.

variables: arbitrary marks on paper

students: manipulate, justify


a study of relationships

How does water pressure changewith depth?

variables: arguments or parameters

students: relate, graph

Usiskin, Z. (1988). Conceptions of School Algebra and Uses ofVariables. In Coxford, A.F. & A.P. Shulte, The Ideas of Algebra: K-12(1988 Yearbook). NCTM: Reston, VA. 8-19.


Area and perimeter

What is the maximum area of arectangle with fixed perimeter?


Surface area and volume

What is the relationship betweensurface area and volume of a cube, asthe side length is increased?


The Teacher as X

Marshal McLullan said:The medium is the message.

It follows that:The Teacher is the Curriculum.


The Joy of X:SinfulMathematicalPleasures

Freedom to add joy to x:

Curriculum focus is on relationships.

Teaching resources focus onlearning through exploration.

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6-Joy of X · George Gadanidis Page 22 01/10/01 What is Algebra? generalized arithmetic 1, 3, 5, 7, 9 ... or t n = 2n - 1 Notice the expression 2n - 1 defines all possible terms of