Teaching Secondary Mathematics

1
Teaching Secondary Mathematics
5
Module 5

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

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Outline of Module 5

• The learner at the centre
• The Meaning of Letters in Algebra 5.25
• Assessment

3
Putting the learner at the centre by
Assessment for Learning
4
The meaning of letters in algebra 4.25

• This indicator of progress provides some ways to
  determine students 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 standingfor numbers, not objects

5
The meaning of letters in algebra 4.25

• 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.

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The meaning of letters in algebra 4.25

• 2(3 apples 4 bananas )
• 2 3 apples 24 bananas
• 6 apples and 8 bananas
• Using pro numerals
• 2 ( 3a 4b)
• 2 3a 24b
• 6a 8b

Illustration 1 Algebraic letters do not stand
for things
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The meaning of letters in algebra 4.25
Students may have developed these misconceptions
from

• 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

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The meaning of letters in algebra 4.25
Anything wrong with this reasoning?
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Confusions about the meanings of letters
doughnuts
The meaning of letters in algebra 4.25

• 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.

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Interpreting student work 6d12
6 doughnuts cost 12 dollars.
by far most common response
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Interpreting student work 6d12
6 doughnuts cost 12 dollars
Only one correct!
by far most common response
12
Interpreting student work 6d12
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.
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Famous problem
The meaning of letters in algebra 4.25

• 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

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The meaning of letters in algebra 4.25

• 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?)

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The meaning of letters in algebra 4.25
Make 7 bicycles and 28 tricycles, for maximum
profit - correct
The possible numbers of bicycles and tricycles
are in the shaded region
Number of bicycles (B)
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The meaning of letters in algebra 4.25

• 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
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The meaning of letters in algebra 4.25
Make 7 bicycles and 28 tricycles, for maximum
profit - correct
wrong line
The possible numbers of bicycles and tricycles
are in the shaded region
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Misconceptions about what a letter stands for in
algebra affect formulating equations
The meaning of letters in algebra 4.25

• Students need to understand that
• A letter stands for one quantity
• The meaning is fixed through the problem
• x doesnt just stand for what is being sought at
  the time

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Illustration 2- Diagnostic item
The meaning of letters in algebra 4.25

• 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.

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Explain the following student answers

• 10b 6r 90 (correct)
• br 90
• 6b5r 90
• b3, 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.
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Characteristics of students thinking
The meaning of letters in algebra 4.25

• 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 dont 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.

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Using algebra to solve problems
The meaning of letters in algebra 4.25

• 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.

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The meaning of letters in algebra 4.25
Diagnosing students thinking

• MARK AND JAN
• Mark and Jan share 47, but Mark gets 5 more
  than Jan.
• How much do they each get?

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The meaning of letters in algebra 4.25

• 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

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The meaning of letters in algebra 4.25

• Wylie (June Year 11)
• Algebraic solution do same to both sides
• x (x 5 ) 47
• 2 ? x 5 47
• 2 ? x 42
• x 21

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The meaning of letters in algebra 4.25
How has Les used x?

• 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 xs.
• I What are the two xs?
• L Unknownsthey 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.

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Les refers to x as meaning several different
things, he informally tracks thinking with
algebra
The meaning of letters in algebra 4.25

• 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 xs.
• I What are the two xs?
• L Unknownsthey 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)

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How has Joel used x?
The meaning of letters in algebra 4.25

• Joel writes x (for Jans amount)
• Then writes x 5 (for Mark )
• Then x 5 47
• I Points to x 5 47. What does this say?
• J (its) 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.

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Joel multiple and shifting referents for x
The meaning of letters in algebra 4.25

• Joel writes x (for Jans amount)
• Then writes x 5 for Mark
• Then x 5 47
• I Points to x 5 47. What does this say?
• J (its) 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 Jans amount

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How has Tim used x?
The meaning of letters in algebra 4.25

• Tim writes x 5 for Marks amount
• Then writes x x 5, saying the x after the
  equal sign is Jans x
• T (Pointing to first x in x5 x) Thats
  Marks x.
• I And why do we add 5 to it?
• T Because Mark has 5 more dollars than Jan. No,
  thats not right, it should be Jans x plus 5
  equals Marks x.
• I Could you write an equation to say that Mark
  and Jan have 47in total ? You dont have to
  work out the answer first.
• T x divided by a half equals x (writes x ? 1/2
  x)

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Tim uses x as a general label for all unknown
quantities.
The meaning of letters in algebra 4.25

• Tim writes x 5 for Marks amount
• Then writes x x 5, saying the x after the
  equal sign is Jans x
• T (pointing to first x in x5 x) Thats
  Marks x.I And why do we add 5 to it?
• T Because Mark has 5 more dollars than Jan. No,
  thats not right, it should be Jans x plus
  5 equals Marks x.
• I Could you write an equation to say that Mark
  and Jan have 47 in total ? You dont have
  to work out the answer first.
• T x divided by a half equals x (writes x ?
  1/2 x)

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Can you see what is bothering Leonie?
The meaning of letters in algebra 4.25

• Leonie writes (x 5) y 47, and cannot
  progress beyond this point
• Leonie explains that
• (x5) is the money that Mark has
• this says it is 5 more than Jans money
• y is the money that Jan has
• Leonie believes that the equation (x5) x
  47 is wrong.

Why?
Why?
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The meaning of letters in algebra 4.25

• Explanation of Leonies 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
  students worklong after introductory lessons on
  algebra.

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The meaning of letters in algebra 4.25

• 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.

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How many letters in my Name?
The meaning of letters in algebra 4.25

• 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
• .

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Try it at your table
The meaning of letters in algebra 4.25

• 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.

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Sample Equations
Probably a bracketing error b (a1) 0
Place value confusion is common with beginners
Some will be identities true for everyone!
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Sample Equations
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.
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The meaning of letters in algebra 4.25

• 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.

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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.

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End of Module 5

• This is the last slide of the module
• Further questions
• studentlearning_at_edumail.vic.gov.au
• Subject field- Teaching Secondary Mathematics