Dr Vicki SteinleUniversity of Melbourne

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Outline

• What are misconceptions?• Characteristics of misconceptions• Why do they occur?• Brief details of longitudinal study• How we can diagnose decimal

misconceptions?• How prevalent are they?• What can we do to remove them?

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Various terms in use in research literature

• misconceptions• alternative conceptions, • pre-conceptions, • conceptual primitives, • alternative frameworks, • systematic errors, and • naïve theories.

Confrey (1990)

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Characteristics of misconceptions

• self-evident (one doesn’t feel the need to prove them),

• coercive (one is compelled to use them in an initial response) and

• widespread among both naïve learners and more academically able students.

Graeber and Johnson (1991)

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Why do they occur?

“In learning certain key concepts in the curriculum, students were transforming in an active way what was told to them and those transformations often led to serious misconceptions.”

Confrey (1990)

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“Errorful rules, …. are intrinsic to all learning- at least as a temporary phenomenon- because they are a natural result of children’s efforts to interpret what they are told and to go beyond the cases actually presented. ….Errorful rules, then, cannot be avoided in instruction.”

Resnick et al (1989)

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It is helpful for teachers to know• that misconceptions do exist, • that they do not signal

recalcitrance, ignorance, or the inability to learn;

• how such errors and misconceptions …..can be exposed; and

• that simple telling does not eradicate students’ misconceptions or “bugs”

Graeber and Johnson (1991)

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Task given to pre-service teachers:

Create a word problem which requires the addition of

decimal numbers.

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One student’s story:

Carlton scored 3 goals and 4 behinds in the first quarter of the game and then 2 goals and 5 behinds in the second quarter.

What had they scored by half-time? 3.4+2.5 5.9

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Do you like this problem? What if they scored one

more behind in the second quarter? 3.4 +2.6=?

Is this 5.10 or 6.0? Is 5.10=5.1?Is a decimal number a pair

of numbers?

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Details of Longitudinal Study

Conducted with Prof Kaye Stacey• Over 3 thousand students involved • Year 4 to Year 10• 12 Melbourne schools (Thank you!)• Almost 10 000 tests completed (1995 to

1999)• The diagnostic test used was the

Decimal Comparison Test (described later)

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• Nearly 60% of the one thousand students tested in primary school were tracked to secondary school

• More than 600 students completed 5, 6 or 7 tests during this study.

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This very large dataset has provided information about which decimal misconceptions are more prevalent at which year levels and which misconceptions are hardest to leave.

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How can we diagnose decimal misconceptions?

Consider these items:Circle the larger number 4.8

4.75Circle the larger number 4.65

4.3There are three common patterns

of responses

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Pattern 1 4.8 4.75 4.65 4.3

Pattern 2 4.8 4.75 x 4.65 4.3

Pattern 3 4.8 4.75 4.65 4.3 x

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Pattern 1 4.8 4.75 4.65 4.3

Pattern 2 4.8 4.75 x 4.65 4.3

Pattern 3 4.8 4.75 4.65 4.3 x

Correct choices in each case

Decimal with more digits has been chosen in each case

Decimal with fewer digits has been chosen in each case

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More examples

Pattern 1 Pattern 2 Pattern 3

Apparent Expert (A)

Longer-is-Larger (L)

Shorter-is-Larger (S)

4.8 4.63 X

0.5 0.36 X

0.8 0.75 X

0.37 0.216 X

3.92 3.4813

X

5.736 5.62 X

0.75 0.5 X

0.426 0.3 X

2.8325

2.516 X

7.942 7.63 X

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You want proof?

“A reader with a healthy quota of scepticism should have some alarm bells ringing by now!”

Proof that students do really exhibit A, L and S behaviours!

• in 1997, 3531 tests were completed• remove exactly 1200 with no errors

(30/30)• examine the remaining 2331 tests for the

response patterns on these 10 items

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0 1 2 3 4 5

0123450

100

200

300

400

500

600

Score on Type 2Score on Type 1

S

L

A

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Some thinking behind L behaviour

Decimal point ignored thinking:4.8 is less than 4.63 as 48 is less than 463

Whole number thinking:4.8 is less than 4.63 as 8 is less than 63

Column overflow thinking:4.8 is less than 4.63 as 8 tenths is less

than 63 tenths

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Some thinking behind S behaviour

Reciprocal thinking:4.65 is less than 4.3 as 1/65 is less than 1/3

Negative thinking:4.65 is less than 4.3 as -65 is less than -3

Denominator focussed thinking:4.65 is less than 4.3 as hundredths are less

than tenths (based on the correct idea 1/100 < 1/10)

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Some thinking behind A behaviour

Money thinking:4.3 is less than 4.65 as $4.30 is less than

$4.65But I don’t know what to do with 4.65 and

4.653. Does the 3 at the end make it bigger or smaller?

There is nothing to compare the 3 with…I’ve run out of digits in 4.65……

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Brekke (1996) makes this comment regarding the context of money:

“Teachers regularly claim that their pupils manage to solve arithmetic problems involving decimals correctly if money is introduced as a context to such problems. Thus they fail to see that the children do not understand decimal numbers in such cases,…. it is possible to continue to work as if the numbers are whole, and change one hundred pence to one pound if necessary. It is doubtful whether a continued reference to money will be helpful, when it comes to developing understanding of decimal numbers.”

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Prevalence of misconceptions

0%

10%

20%

30%

40%

50%

60%

70%

80%

Yr 4 Yr 5 Yr 6 Yr 7 Yr 8 Yr 9 Yr 10

0%

10%

20%

30%

40%

50%

60%

70%

80%

Yr 4 Yr 5 Yr 6 Yr 7 Yr 8 Yr 9 Yr 10

a) Prevalence of expertise*

b) Prevalence of L behaviour

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Prevalence of misconceptions

0%

5%

10%

15%

20%

25%

30%

Yr 4 Yr 5 Yr 6 Yr 7 Yr 8 Yr 9 Yr 10

0%

5%

10%

15%

20%

25%

30%

Yr 4 Yr 5 Yr 6 Yr 7 Yr 8 Yr 9 Yr 10

c) Prevalence of S behaviour

d) Prevalence of other

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Trends in L and S

• Lots of students start in L and some persist for several years. Once they leave, they tend not to return.

• S behaviour is more complex: different students move in and out at different times. About 1 in 3 students will exhibit S behaviour at some time in primary school and about 1 in 4 in secondary school. Students in S in Year 8 have high persistence (about 40% retest as S).

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Two students in study

Yr4Sem

2

Yr5Sem

1

Yr5Sem

2

Yr6Sem

1

Yr6Sem

2

Yr7Sem

1

410504041

L L A A L

210403026

L A S S S

*Both students regressed

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Good news! It is possible to remove

misconceptions

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1. We need “good” models to represent decimal numbers:

• LAB (Linear Arithmetic Blocks)• Number Expanders• Number Slides 2. Choose your language carefully3. Make sure that students with

misconceptions know that they have something to learn.

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LAB• Made from lengths of

pipe/tubing & washers• Ones, tenths, hundredths &

thousandths• Is the only model which helps

students to build a number line

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Introducing LAB to students

Start with the one, then discuss chopping into 10 pieces

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Which students would predict 0.13 was larger?Round 0.13 to the nearest tenth

Comparing 2 tenths (0.2) with 13 hundredths (0.13)

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2 tenths (0.2)2 tenths + 6 hundredths (0.26)3 tenths (0.3)

Comparing 0.2, 0.3 and 0.26

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Language

Parents and teachers would be rather concerned if a child did not understand the difference between a letter and a word.

Do we feel the same way about digits and numbers?

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Components or building blocks

Entire object

Literacy 26 Lettersa,b,c,……x,y,z

WordsCat, dog, banana

Numeracy

10 Digits0,1,2,3,4,5,6,7,8

,9

Numbers324, 2.5, 7/8

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320 330325

2 32.5

0 17/8

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Multiplication

10 x 3.1 should be 30.10 (No)

A number slide can help here

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Number Slide showing 3.1 multiplied (and then divided) by 10 and 100

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Features of the Number Slide

When a number is multiplied by 10, the digits move into the next biggest column (rather than the usual rules of add a zero or move the decimal point.)

Similarly, when a number is divided by ten, the digits move into the next smallest column.

Notice that the decimal point is fixed (between the ones and tenths columns) and that it is the digits that move to other columns.

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Number Between Game

• Choose any two numbers as endpoints and then ask for a number that fits between them.

• Choose one of the two subintervals and repeat.

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Number Density

But 0.3157 is a long way from 0.3 isn’t it?No!0.3157 is between 0.3 and 0.4It is also between 0.31 and 0.32How can students use approximation to

check the reasonableness of their calculations?

Furthermore, how can students make sense of the rules for rounding?

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Rounding

Be aware that always rounding the result of a calculation to two decimal places can reinforce the belief that decimals form a discrete system and that there are no numbers between 4.31 and 4.32, for example.

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Check your texts & worksheets

• Can a student with a misconception (e.g. whole number thinking) answer correctly?

• They will if all the decimals have the same number of decimal places!

• Example: Order these jumps from shortest to longest: 1.34m, 1.87m, 1.24m, 1.62m

• You need ragged decimals!

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Resources

• Fraction materials based on length (in preference to circles and squares which are based on area)

• Number Sense (McIntosh, Reys & Reys)• Teaching and Learning about Decimals

(Steinle, Stacey & Chambers, 2002) online sample at

http://extranet.edfac.unimelb.edu.au/DSME/decimals/