Teaching Secondary Mathematics

Teaching Secondary

Mathematics

What do students think?

Some students believe that:

Algebraic letters are abbreviations for words or things

Algebra is a sort of shorthand

Algebraic letters stand for a secret code.

“The meaning of letters in algebra”

• Students often have difficulty with algebra because of misconceptions in various areas.


• If I add 3 to x + 4 I get 7.

“The meaning of letters in algebra” Examples of Misconceptions

Reason : Ignoring completely the presence of letters. l

• 8 m and 8m are the same.


Reason : Not distinguishing between letters used as units of measure and as variables..

• Shirts cost s rupees each and pants cost p rupees a pair. If I buy 3 shirts and 2 pairs of pants, what does 3s + 2p represent?


Reason : Treating letters as objects.

• c = 3 because c is the 3rd letter of the alphabet. y = 4 because in the previous questions y was 4.


Reason : Believing there are rules used to determine whichnumber a letter stands for..

Teaching for understanding

• What can you say about p if p + q = 12 and p is a natural number greater than q? p = 7


Reason : Thinking that letters always have one specific value

• a + b cannot equal a + c.


Reason : Thinking that different letters always representdifferent numbers.

• 6x = 13, then x = 2


Reason : Thinking that letters can only stand for natural numbers..

• a + b = ab• 2x + 3 = 5x

Misconceptions aboutNotation

Reason : Combining letters and numbers incorrectly becausethey think that operation symbols answercannot be part of an

• The area of this rectangle is 3 x x + 4 or x + 4 x 3.


Reason : Neglecting to use brackets when needed

• –6x = 12 x=12/6 x= 2


Reason : Disregarding signs when manipulating expressions.

Copying other symbol systems letters are often used as:

• Abbreviations for words in everyday life

• Teaching – fruit salad algebra

Students may have developed these misconceptions from:


Anything wrong with this reasoning?


Confusions about the meanings of letters: Chacolates ChcoC

Students were asked the following question:

Write an equation which describes the situation » “6 Chacolates cost 12 Rupees”

Correct equation, written by nearly all students,

» 6d = 12, but what does it mean?

Instructions to students:

After you have written the equation, say what quantity each of the numerals and pronumerals represents.


Student 6 d 12

Anna* amount of Chacolates

Chacolates cost

Ben numeral pronumeral numeral

Cath amount of Chacolates

cost per doughnut

Total price

Dan six Chacolates cost Rs.12

Ellie number of Chacolates

Chacolates price

Interpreting student work 6d=12

* by far most common response

“6 Chacolates cost 12 Rupees”.

Student 6 d 12

Anna* amount of Chacolates

Chacolates cost

Ben numeral pronumeral numeral

Cath amount of Chacolates

cost per doughnut

Total price

Dan six Chacolates cost Rs.12

Ellie number of Chacolates

Chacolates price

* by far most common response

Only one

correct!

“6 Chacolates cost 12 Rupees”


Fran wrote this incorrect equation

2 d = 12

2 = cost of each doughnut

d = number of Chacolates

12 = overall cost

Unusual incorrect response, but Fran is one of the few students who thought carefully about what she really meant.

“6 Chacolates cost 12 Rupees”


Famous problem

• At a university there are 6 students for every professor. Let S be the number of students, P be the number of professors,and write an equation.

• Letter as object misconception again

6S = P

“The meaning of letters in algebra 4.25”

Misconceptions about what a letter stands for in

algebra affect formulating equations

• Students need to understand that:

– A letter stands for one quantity

– The meaning is fixed through the problem


Illustration 2- Diagnostic item

Write an equation that describes the following situation.

Use b to stand for the number of blue pencils and r to

stand

for the number of red pencils.

I bought some red pencils and some blue pencils and spent

a

total of 90 cents. The blue pencils cost 10 cents each and

the

red pencils cost 6 cents each.


Explain the following student answers

• 10b + 6r = 90 (correct)

• b+r = 90

• 6b+5r = 90

• b=3, r = 10

I bought some red pencils and some blue pencils and spent a total of 90 cents. The blue pencils cost 10 cents each and the red pencils cost 6 cents each. Write an equation to describe this situation.

share Rs.47, but Surya gets Rs.5 more than

Jothika.

How much do they each get?

Diagnosing students’ thinking


Main ideas: how many letters in my name

• Letter stands for number – unknown to audience – possibly can be found by audience

• Reinforces simple substituting, basic syntax, etc

• Students may make harder equation than teacher expects

– creativity, diversity

• Some are equations and some identities: some equations can belong to one person, some to more than one person, and some to everyone

• Equation solving by guess-check-improve.


How many letters in my Name?

Let a = the number of letters in my first name.

Let b = the number of letters in my family name.

For Lini Marandri, a = 4, b = 8

Sample equation: a + b

= 4 + 8

= 12

For Thy Vo a = 3, b = 2

Sample equation: a + b

= 3 + 2

= 5

.


Try it at your table:

• Make up 3 equations for your name

• Try to include the variety of equations which

students might write (correct and incorrect)

• Pool your equations and think about what

different

equations will reveal about students’ thinking.


Sample Equations

Lini Marandri, a=4, b = 8 4+a = b a+a = b ba=84

Thy Vo, a = 3, b = 2 a+b = 5 ab = 1.5 a+a=a 2

Robert Menzies, a=6, b = 7

b-a=1 a-a = b-b b-a+1 = 0

John Curtin, a=4, b=6 ab = 24 2 a + b + a + 1 = 19

Place value confusion is

common with beginners

Some will be identities – true for everyone!

Probably a bracketing

error b – (a+1) = 0

Lini Marandri, a=4, b = 8

4+a = b a+a = b ba=84

Thy Vo, a = 3, b = 2 a+b = 5 ab = 1.5 a+a=a 2

Robert Menzies, a=6, b = 7

b-a=1 a-a = b-b b-a+1 = 0

John Curtin, a=4, b=6 ab = 24 2 a + b + a + 1 = 19

No need to stay with linear equations

These equations can be easily solved by guess-check, because there are only a few numbers to try.

Sample Equations

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jUf;fKiw mikg;g[

• 200 igfspd; tpiy 20Qgha; vd;gij 200 igfs; = Q/20 vd vGJtJ jtW

jUf;fKiw mikg;g[

• xU bgl;o ghjp jpwe;jpUg;gJk;. ghjp |oapUg;gJk;

xnu bghUs;gLk;/ Mdhy; ,ij rkd;ghlhf vGjpdhy; jpwe;jpUg;gJ = |oapUg;gJ vd bghUs;gLk;

jUf;fKiw mikg;g[

• 4 kP x 3 kP = 12 r/kP vd vGJtJ jtW (4 x 3 ) r/ kP = 12 r/kP vdf; Fwpg;gp ntz;Lk;

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Teaching Secondary Mathematics

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