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Solving Equations: Students’ Solving Equations: Students’ Algebraic ThinkingAlgebraic Thinking

Chris LinsellChris Linsell

Supported by TLRISupported by TLRI

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OutlineOutline

IntroductionIntroductionConceptual DifficultiesConceptual DifficultiesThe PhD StudyThe PhD StudyThe TLRI projectThe TLRI projectSome early conclusionsSome early conclusionsWhere to next?Where to next?

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IntroductionIntroduction• Differences in perception between maths Differences in perception between maths

teachers and studentsteachers and students• One. ..fact must astonish us, or rather would One. ..fact must astonish us, or rather would

astonish us if we were not too much accustomed astonish us if we were not too much accustomed to it. How does it happen that there are people to it. How does it happen that there are people who do not understand mathematics? If the who do not understand mathematics? If the science invokes only the rules of logic, those science invokes only the rules of logic, those accepted by all well-formed minds. ..how does it accepted by all well-formed minds. ..how does it happen that there are so many people who are happen that there are so many people who are entirely impervious to it?entirely impervious to it? (Poincare, 1908) (Poincare, 1908)

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Hewitt 2001Hewitt 2001

• A girl in the class was looking at the following:A girl in the class was looking at the following:• Example l: x+3=9, x=6 Example l: x+3=9, x=6 • Example 2: x+7=10, x=3Example 2: x+7=10, x=3• I asked her whether she understood what was I asked her whether she understood what was

written on the board and she replied “I don't written on the board and she replied “I don't understand, I just copy it down.” understand, I just copy it down.”

• I then asked “What is the 'ex' on the board?” I then asked “What is the 'ex' on the board?” • To which she replied “What 'ex'? That's a times.”To which she replied “What 'ex'? That's a times.”

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Algebraic Thinking?Algebraic Thinking?

• Is it appropriate to talk about algebraic thinking?Is it appropriate to talk about algebraic thinking?• We no longer talk about simply arithmetic We no longer talk about simply arithmetic

thinking, but specify the strategies that students thinking, but specify the strategies that students use in a number of domains.use in a number of domains.

• Suggested domains of algebra Suggested domains of algebra

- Generalised number- Generalised number- Patterns and relationshipsPatterns and relationships- Specific unknownsSpecific unknowns

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Specific UnknownsSpecific Unknowns

• How do students solve equations?How do students solve equations?• What are the prerequisite skills and What are the prerequisite skills and

knowledge?knowledge?• How can we find out?How can we find out?• Observational studiesObservational studies• Diagnostic interviewsDiagnostic interviews

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Some Conceptual DifficultiesSome Conceptual Difficulties

• Arithmetic structureArithmetic structure• Inverse operationsInverse operations• Notation and conventionNotation and convention• Unknown / generalised number / variable / Unknown / generalised number / variable /

parameterparameter• Operating on unknownsOperating on unknowns• Binary / unary operationsBinary / unary operations• IntegersIntegers

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Process / object duality and Process / object duality and proceptsprocepts

• Counting→natural→subtracting→negativeCounting→natural→subtracting→negative→roots→imaginary and complex→roots→imaginary and complex

• Adding→sumsAdding→sums• Evaluating expressions→equationsEvaluating expressions→equations• Procepts, pivotal symbolismProcepts, pivotal symbolism

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The = signThe = sign

• 3+5=3+5= • compute nowcompute now • is equivalent tois equivalent to• Process/object againProcess/object again • 3+5=8+2=103+5=8+2=10• lack of appreciation of the structure of lack of appreciation of the structure of

mathematical statements mathematical statements • 3x+154=475 and 3x=475+154 3x+154=475 and 3x=475+154

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NumeracyNumeracy• x+3=7 can be solved by advanced counters by x+3=7 can be solved by advanced counters by

guess and check, but can be solved much more guess and check, but can be solved much more easily by part/whole thinkers able to visualise 7 as easily by part/whole thinkers able to visualise 7 as 3+43+4

• 2x+3=11 requires an understanding of numbers 2x+3=11 requires an understanding of numbers beyond simple additive part/whole or multiplicative beyond simple additive part/whole or multiplicative part/whole thinkingpart/whole thinking

• 2(x+3)+2x=22 requires an understanding of the use 2(x+3)+2x=22 requires an understanding of the use of brackets, the hierarchy of operations, and an of brackets, the hierarchy of operations, and an understanding of the commutative, associative and understanding of the commutative, associative and distributive lawsdistributive laws

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The PhD StudyThe PhD Study

• QualitativeQualitative• 4 Year 9 students studied for 27 lessons4 Year 9 students studied for 27 lessons• Activity based approachActivity based approach• Data – videotapes of lessons, SRIs, Data – videotapes of lessons, SRIs,

student’s work, field notesstudent’s work, field notes• Transcription and analysisTranscription and analysis

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Process/object and procedureProcess/object and procedure

• Superficial similarity of process and procedureSuperficial similarity of process and procedure• Analogy of two-digit subtractionAnalogy of two-digit subtraction• Getting the right answerGetting the right answer• Clues from errorsClues from errors• Clues from notationClues from notation• Learning – Learning – process 2/4process 2/4

object 0/4 object 0/4

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Prerequisite NumeracyPrerequisite Numeracy

• Numeracy as a predictor of learningNumeracy as a predictor of learning• Arithmetic structureArithmetic structure• Inverse operationsInverse operations• Operating on integersOperating on integers• Binary/ unary viewBinary/ unary view

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Strategic ThinkingStrategic Thinking

• Multiplicative part / wholeMultiplicative part / whole• Expressions as objectsExpressions as objects• Value of Value of guess and checkguess and check• Constraints on choice of partsConstraints on choice of parts• Unknown partsUnknown parts• Algebraic part / whole thinkingAlgebraic part / whole thinking• Equations as objectsEquations as objects• Time spent on stagesTime spent on stages

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The TLRI ProjectThe TLRI Project

• How do students solve equations?How do students solve equations?• What diagnostic questioning is appropriate What diagnostic questioning is appropriate

for eliciting the knowledge and strategies for eliciting the knowledge and strategies used by students?used by students?

• Action Research ParadigmAction Research Paradigm

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Structure of Diagnostic Structure of Diagnostic AssessmentAssessment

Separated into knowledge and strategy Separated into knowledge and strategy componentscomponents

Strategy assessed by interview, need for Strategy assessed by interview, need for supplementary questionssupplementary questions

Knowledge assessed by written test, Knowledge assessed by written test, guided by literature and our own guided by literature and our own experienceexperience

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StrategyStrategy

A focus on HOW students solve equationsA focus on HOW students solve equationsA series of increasingly complex A series of increasingly complex

questions, students solve and explain questions, students solve and explain strategiesstrategies

Teachers classify the strategies used by Teachers classify the strategies used by studentsstudents

Parallel questions – in context and Parallel questions – in context and abstractabstract

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Strategy – Sample QuestionsStrategy – Sample Questions

3n – 8 = 193n – 8 = 19 Here are 3 packets of beans, that started Here are 3 packets of beans, that started

with the same number in each packet, but with the same number in each packet, but a mouse ate 5 of the beans. There are a mouse ate 5 of the beans. There are now 16 beans left in the packets, like this now 16 beans left in the packets, like this pile over here. How many were in each pile over here. How many were in each packet at the beginning? packet at the beginning?

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Strategy ClassificationStrategy Classification

0.         Unable to answer question0.         Unable to answer question 1a.       Known basic facts1a.       Known basic facts 1b.       Counting techniques1b.       Counting techniques 1c.        Inverse operation1c.        Inverse operation 2.         Guess and check2.         Guess and check 3a.       Cover up3a.       Cover up 3b.       Working backwards3b.       Working backwards 3c.3c. Working backwards then known factsWorking backwards then known facts 3d. Working backwards then guess and check3d. Working backwards then guess and check 4.         Formal operations / Equation as Object4.         Formal operations / Equation as Object 5. Use a Diagram5. Use a Diagram

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Knowledge – Areas InvestigatedKnowledge – Areas Investigated

Using symbols and letters to represent an unknown Manipulating symbols/unknowns (lack of closure) Forming expressions with unknowns/symbols in

them Understanding of the equals sign Operations on integers Understanding of arithmetic structure Understanding of inverse operations

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Knowledge – Sample QuestionsKnowledge – Sample Questions

What answer do you get for each of the following?

1) 5 + 6 × 10

2) 8 × ( 7 – 5 + 3 )

3) 6+12

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4) 18 -12 ÷ 6

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Applying the Strategy AnalysisApplying the Strategy Analysis

Watch the videos, try to classify the kind of Watch the videos, try to classify the kind of strategies used by the students in each strategies used by the students in each case.case.

In groups discuss the strategies used. In groups discuss the strategies used. Although we do not have a hierarchy yet, Although we do not have a hierarchy yet, what would be the next teaching step for what would be the next teaching step for these students?these students?

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3) n+46 = 1133) n+46 = 113 AndrewAndrew(n=67)(n=67)

Video removedVideo removed

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3) n+46 = 1133) n+46 = 113 JackJack(n=67)(n=67)

Video removedVideo removed

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3) n+46 = 1133) n+46 = 113 OliverOliver(n=67)(n=67)

Video removedVideo removed

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3) n+46 = 1133) n+46 = 113 OscarOscar(n=67)(n=67)

Video removedVideo removed

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3) n+46 = 1133) n+46 = 113 SeanSean(n=67)(n=67)

Video removedVideo removed

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5) 4n+9=375) 4n+9=37 AndrewAndrew(n=7)(n=7)

Video removedVideo removed

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6) 3n-8=196) 3n-8=19 AndrewAndrew(n=9)(n=9)

Video removedVideo removed

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5) 4n+9=375) 4n+9=37 Nicola Nicola (n=7)(n=7)

<20 seconds to answer<20 seconds to answer

77How did you work that out?How did you work that out?

I timesed 4 by 7 and added 9I timesed 4 by 7 and added 9Did you guess seven?Did you guess seven?

I took a close estimateI took a close estimateSo you tried seven and it was right?So you tried seven and it was right?

YesYes

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6) 3n-8=196) 3n-8=19 NicolaNicola(n=9)(n=9)

Takes longer than previous question, but still less than <30 secTakes longer than previous question, but still less than <30 sec

99How did you work that out?How did you work that out?

I just took a close estimate. I just took a close estimate. Was 9 the first number you tried?Was 9 the first number you tried?

No I tried 8. I went 24 – 8No I tried 8. I went 24 – 8So you went 3x8 is 24 minus 8 is …So you went 3x8 is 24 minus 8 is …

Not 19!Not 19!

So then you tried 9?So then you tried 9?

And 9 was rightAnd 9 was right

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5) 4n+9=375) 4n+9=37 SusanSusan(n=7)(n=7)

Approx 40 sec to answerApprox 40 sec to answer

77How did you work that out?How did you work that out?

Because 9 minus 37 is 28 Because 9 minus 37 is 28

and 28 divided by 4 is 7and 28 divided by 4 is 7

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6) 3n-8=196) 3n-8=19 SusanSusan(n=9)(n=9)

After 4 minutes of trying prompted and replies:After 4 minutes of trying prompted and replies:

Something like 3 times 4 Something like 3 times 4 remainder 1 or something.remainder 1 or something.

Because 3x4 is 12 but 8-19 Because 3x4 is 12 but 8-19 equals 11equals 11

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Some early conclusionsSome early conclusions

Clear parallels with NDP approachClear parallels with NDP approach Numeracy teaching is having an impact Numeracy teaching is having an impact Instead of looking at how hard equations are to Instead of looking at how hard equations are to

solve and whether students get them right, it is solve and whether students get them right, it is more useful to look at what strategies the more useful to look at what strategies the students usestudents use

The strategies we have identified are consistent The strategies we have identified are consistent with the research literature but extend itwith the research literature but extend it

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More early conclusionsMore early conclusions

Contexts are a help, not a hindrance!Contexts are a help, not a hindrance!At some point contexts stop being helpfulAt some point contexts stop being helpful““Working backwards” is less Working backwards” is less

homogeneous than has been previously homogeneous than has been previously suggestedsuggested

Formal operations, which treat equations Formal operations, which treat equations as objects, are clearly difficult for studentsas objects, are clearly difficult for students

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What next?What next?

Guess and CheckGuess and Check and and Cover UpCover Up are worth are worth teaching!teaching!

We want to develop an Algebra We want to develop an Algebra Framework similar to the Number Framework similar to the Number FrameworkFramework

The next step is to gather lots of data, The next step is to gather lots of data, using the diagnostic interview we have using the diagnostic interview we have developeddeveloped

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20072007

• Find interested teachersFind interested teachers• Teachers must have numeracy data on Teachers must have numeracy data on

their studentstheir students• Train teachers to administer diagnostic Train teachers to administer diagnostic

tooltool• Teachers and research team to gather Teachers and research team to gather

data on 800 – 1000 studentsdata on 800 – 1000 students• Enter results into databaseEnter results into database

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2008 – Data Analysis2008 – Data Analysis

• Factor Analysis, Rasch AnalysisFactor Analysis, Rasch Analysis• What is relative difficulty of test items?What is relative difficulty of test items?• What is relative difficulty of strategies?What is relative difficulty of strategies?• What prerequisite knowledge is required What prerequisite knowledge is required

for each strategy?for each strategy?• What stage of numeracy is required for What stage of numeracy is required for

each strategy?each strategy?• What is the impact of context?What is the impact of context?