teaching secondary mathematics

Teaching Secondary Mathematics

Understanding students’ mathematical thinking:Focus on algebra and the meaning of letters.

Module 5: 55

Outline of Module 5

1. The learner at the centre

2. The Meaning of Letters in Algebra: 5.25

3. Assessment

Assessment for Learning

Putting the learner at the centre by;

“The meaning of letters in algebra 4.25”

• This indicator of progress provides some ways to determine

student’s misconceptions behind all algebra

• 2 illustrations and 4 teaching activities from the teaching

strategies will:

– provide items for teachers to use to diagnose how

students

are thinking about algebraic letters

– offer suggestions for introducing algebra with letters

standing

for numbers, not objects

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.


2×(3 apples + 4 bananas )

2× 3 apples + 2×4 bananas

6 apples and 8 bananas

Using pro numerals

2× ( 3a + 4b)

2× 3a + 2×4b

6a +8b

Illustration 1: Algebraic letters do not stand for things


Copying other symbol systems letters are often used as:

• Abbreviations for words in everyday life

• Teaching – fruit salad algebra

• False analogies e.g. with codes

Students may have developed these misconceptions from:


Anything wrong with this reasoning?


Confusions about the meanings of letters: doughnuts

Students were asked the following question:

Write an equation which describes the situation » “6 doughnuts cost 12 dollars”

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 doughnuts

doughnuts cost

Ben numeral pronumeral numeral

Cath amount of doughnuts

cost per doughnut

Total price

Dan six doughnuts cost $12

Ellie number of doughnuts

doughnuts price

Interpreting student work 6d=12

* by far most common response

“6 doughnuts cost 12 dollars”.

Student 6 d 12

Anna* amount of doughnuts

doughnuts cost

Ben numeral pronumeral numeral

Cath amount of doughnuts

cost per doughnut

Total price

Dan six doughnuts cost $12

Ellie number of doughnuts

doughnuts price

* by far most common response

Only one

correct!

“6 doughnuts cost 12 dollars”


Fran wrote this incorrect equation

2 d = 12

2 = cost of each doughnut

d = number of doughnuts

12 = overall cost

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

“6 doughnuts cost 12 dollars”


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


• A factory makes bicycles and tricycles, using the same

wheels

• Supplier provides no more than 100 wheels per day

• Their customer requires at least 4 tricycles for every

bicycle.

• Profit is $300 for either a bicycle or a tricycle.

• Aim is to maximise profit

(How many of each should the factory make?)


Number of bicycles (B)

-2 2 4 6 8 10 12 14 16 18 20

3

6

9

12

15

18

2124

27

3033

36

39

Num

ber

of

tric

ycl

es

(T)


The possible numbers of bicycles and

tricycles are in the shaded region

Make 7 bicycles and 28 tricycles, for maximum

profit - correct

Supplier provides no more than 100 wheels per day

Their customer requires at least 4 tricycles for every bicycle.

B = number of bicycles made per day

T = number of tricycles made per day

Number of wheels less than 100: 2B + 3T 100

4 tricycles for each bicycle: Is it 4T B or 4B T?

Writing 4T B or 4T B is one of the most common errors


The possible numbers of bicycles and tricycles

are in the shaded region

Make 7 bicycles and 28 tricycles, for maximum

profit - correct

wrong line

Number of bicycles (B)

-2 2 4 6 8 10 12 14 16 18 20

3

6

9

12

15

18

21

24

27

30

33

36

39

Num

ber

of

tric

ycl

es

(T)


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

– x doesn’t just stand for what is being sought at the time


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.

Characteristics of students’ thinking

• Students are experiencing problems with the meaning of letters.

• The equation that they write may look like what we write, but the meaning is not the same!

• Students have achieved success without using algebra. They don’t understand algebra is helpful - often do the problem firstby logical arithmetic reasoning and then dress up as algebra by sprinkling letters around.

• Research shows similar observations around the globe, but todifferent extent.


Using algebra to solve problems

• Algebra provides a very different method of solving problems,

it is not just a new language

• Students may experience difficulties when making the transitionfrom arithmetical to algebraic thinking in

– having different idea of the unknown (transient vs fixed)

– Believing that an equation only describes the information in a question

– algebraic solving proceeds by transforming one equation into another: very different way of thinking.


MARK AND JAN

Mark and Jan share $47, but Mark gets $5 more than Jan.

How much do they each get?

Diagnosing students’ thinking


Brenda (Year 9) Uses logical arithmetic reasoning – letters added at the end

• 47 / 2 = 23.5 - 2.5 = x

• 47 / 2 = 23.5 + 2.5 = y

Wylie (October Year 10) Uses logical arithmetic reasoning & writing answer as a “formula”

• y = (47-5) / 2 + 5 = 42/2 + 5 = 26

• x = (47-5) / 2 = 42/2 = 21

Other students wrote:

• y = (T - D) / 2 + D , x = (T - D) / 2

• Guess and check (Year 9): 15 + 32 = 47, 16 + 31 = 47, …., 21 + 26 = 47


Wylie (June Year 11)

Algebraic solution “do same to both sides”x + (x + 5 ) = 472 x + 5 = 472 x = 42x = 21


Les begins by writing 5 + x = 47

L: x is what is left out of $47 if you take 5 off it.

I: What might the x be?

L: Say she gets $22 and he gets $27. They are sharing two x’s.

I: What are the two x’s?

L: Unknowns…they are two different numbers, 22 and 27.

I: So what is this x? (pointing to 5 + x = 47)

L: That was what was left over from $47, so its $42.

How has Les used x?


Les refers to x as meaning several different things, he informally tracks thinking with ‘algebra’

Les begins by writing 5 + x = 47

L: x is what is left out of $47 if you take 5 off it.

I: What might the x be?

L: Say she gets $22 and he gets $27. They are sharing two x’s.

I: What are the two x’s?

L: Unknowns…they are two different numbers, 22 and 27.

I: So what is this x? (pointing to 5 + x = 47)

L: That was what was left over from $47, so its $42.

(3 different meanings for x simultaneously)


How has Joel used x?

• Joel writes x (for Jan’s amount)

• Then writes x + 5 (for Mark )

• Then x + 5 = 47

• I: Points to x + 5 = 47. What does this say?

• J: (it’s) the amount they both get. The amount that Jan gets.

I just like to keep the three of them, 47 dollars, x and 5

dollars and make something out of them.


Joel: multiple and shifting referents for x

• Joel writes x (for Jan’s amount)

• Then writes x + 5 for Mark

• Then x + 5 = 47

• I: Points to x + 5 = 47. What does this say?

• J: (it’s) the amount they both get. The amount that Jan gets.

I just like to keep the three of them, 47 dollars, x and 5 dollars

and make something out of them.

• x as “the amount they both get” ($42) and as well as Jan’s amount


How has Tim used x?

Tim writes x + 5 for Mark’s amount

Then writes x = x + 5, saying the x after the equal sign is “Jan’s x”

T: (Pointing to first x in x+5 = x) That’s Mark’s x.

I: And why do we add 5 to it?

T: Because Mark has 5 more dollars than Jan. No, that’s not right, it should be Jan’s x plus 5 equals Mark’s x.

I: Could you write an equation to say that Mark and Jan have $47in total ? You don’t have to work out the answer first.

T: x divided by a half equals x (writes x 1/2 = x)


Tim: uses x as a general label for all unknown quantities.

Tim writes x + 5 for Mark’s amount

Then writes x = x + 5, saying the x after the equal sign is “Jan’s x”

T: (pointing to first x in x+5 = x) That’s Mark’s x.I: And why do we add 5 to it?

T: Because Mark has 5 more dollars than Jan. No, that’s not right, it should be Jan’s x plus 5 equals Mark’s x.

I: Could you write an equation to say that Mark and Jan have $47 in total ? You don’t have to work out the answer first.

T: x divided by a half equals x (writes x 1/2 = x)


Can you see what is bothering Leonie?

Leonie writes

(x + 5) + y = 47, and cannot progress beyond this point

Leonie explains that

• (x+5) is the money that Mark has

• this says it is $5 more than Jan’s money

• y is the money that Jan has

• Leonie believes that the equation

(x+5) + x = 47 is wrong. Why?

Why?


Explanation of Leonie’s thinking

• Leonie knows that the numerical amounts of x and y are the

same, but she uses the separate letters because she

believes Mark and Jan

have physically different money.

• For Leonie, x represents the actual money, not the amount

of money.

• Another instance of “letter as object” evident in student’s

work

long after introductory lessons on algebra.


Summary

• Uncertainties and misconceptions about the meanings ofletters lie behind many difficulties with algebra

– writing expressions

– formulating equations.

• Examine students’ work closely to identify their difficulties, and then address them.

• Use teaching strategies that emphasise that algebraic letters stand for numbers, and that there is a specific meaning for a letter throughout one problem.


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

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.


Assessment

• Assessment practices are an integral part of teaching and learning (PoLT Principle 5)

• Teachers are encouraged to use evidence from assessment toinform planning and teaching.

• These Continuum items described in the module are intended

as diagnostic assessment.

• Teachers should aim to understand what their students are trying to say when they try to write algebra. Teachers can then design instruction whichmakes sense to students and hence changes their thinking more effectively.

End of Module 5

• This is the last slide of the module• Further questions…•

[email protected]• Subject field- Teaching Secondary

Mathematics

teaching secondary mathematics

Documents

meaning of letters

meanings of letters

things algebra

number of students

symbol systems letters

students misconceptions

incorrect equation

dollarscorrect equation