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Dr Vicki Steinle University of Melbourne

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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 “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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MAV Conference 2004 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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MAV Conference 2004 Task given to pre-service teachers: Create a word problem which requires the addition of decimal numbers.

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MAV Conference 2004 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?

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MAV Conference 2004 Do you like this problem? What if they scored one more behind in the second quarter? =? Is this 5.10 or 6.0? Is 5.10=5.1? Is a decimal number a pair of numbers?

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MAV Conference 2004 Details of Longitudinal Study Conducted with Prof Kaye Stacey Over 3 thousand students involved Year 4 to Year Melbourne schools (Thank you!) Almost tests completed (1995 to 1999) The diagnostic test used was the Decimal Comparison Test (described later)

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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 How can we diagnose decimal misconceptions? Consider these items: Circle the larger number Circle the larger number There are three common patterns of responses

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MAV Conference 2004 Pattern Pattern x Pattern x

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MAV Conference 2004 Pattern Pattern x Pattern 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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MAV Conference 2004 More examples Pattern 1Pattern 2Pattern 3 Apparent Expert (A) Longer-is-Larger (L) Shorter-is-Larger (S) X X X X X X X X X X

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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 Some thinking behind A behaviour Money thinking: 4.3 is less than 4.65 as $4.30 is less than $4.65 But I don’t know what to do with 4.65 and 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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MAV Conference 2004 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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MAV Conference 2004 Prevalence of misconceptions a) Prevalence of expertise*b) Prevalence of L behaviour

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MAV Conference 2004 Prevalence of misconceptions c) Prevalence of S behaviour d) Prevalence of other

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MAV Conference 2004 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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MAV Conference 2004 Two students in study Yr4 Sem2 Yr5 Sem1 Yr5 Sem2 Yr6 Sem1 Yr6 Sem2 Yr7 Sem LLAAL LASSS *Both students regressed

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MAV Conference 2004 Good news! It is possible to remove misconceptions

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MAV Conference We need “good” models to represent decimal numbers: LAB (Linear Arithmetic Blocks) Number Expanders Number Slides 2. Choose your language carefully 3. Make sure that students with misconceptions know that they have something to learn.

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MAV Conference 2004 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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MAV Conference 2004 Introducing LAB to students Start with the one, then discuss chopping into 10 pieces

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MAV Conference 2004 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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MAV Conference 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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MAV Conference 2004 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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MAV Conference 2004 Components or building blocks Entire object Literacy26 Letters a,b,c,……x,y,z Words Cat, dog, banana Numeracy10 Digits 0,1,2,3,4,5,6,7,8,9 Numbers 324, 2.5, 7/8

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MAV Conference /8

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MAV Conference 2004 Multiplication 10 x 3.1 should be (No) A number slide can help here

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MAV Conference 2004 Number Slide showing 3.1 multiplied (and then divided) by 10 and 100

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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 Number Density But is a long way from 0.3 isn’t it? No! is between 0.3 and 0.4 It is also between 0.31 and 0.32 How 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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MAV Conference 2004 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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MAV Conference 2004 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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MAV Conference 2004 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

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