Diagnosing Mathematical Errors: Fractions and Decimals (Concepts, Equivalence, and Operations)

Chapter 4 Ashlock (2010)

Diagnosing Mathematical Diagnosing Mathematical Errors: Errors:

Fractions and DecimalsFractions and Decimals(Concepts, Equivalence, and Operations)(Concepts, Equivalence, and Operations)

College of EducationCollege of Education


Today’s Topics…Today’s Topics…• Review Chapter 3 QuizReview Chapter 3 Quiz• Review of concepts and vocabulary Review of concepts and vocabulary • DemonstrationsDemonstrations• Chapter 4 - Ashlock (2010)Chapter 4 - Ashlock (2010)

– Diagnosing Errors: Group WorkDiagnosing Errors: Group Work– Correcting Errors: Whole GroupCorrecting Errors: Whole Group


Review: Diagnosing and Review: Diagnosing and CorrectingCorrecting: Case Study Part II

Diagnosing and Correcting Mathematical Errors Diagnosing and Correcting Mathematical Errors involves four processes/steps:involves four processes/steps:

1.1. Diagnosing: collect data, analyze data, pre-diagnose, Diagnosing: collect data, analyze data, pre-diagnose, interview, and make final diagnosisinterview, and make final diagnosis

2.2. Correcting: conceptual, intermediate, procedural only, and Correcting: conceptual, intermediate, procedural only, and independent practice independent practice

3.3. Evaluating: collect data, analyze data, diagnose, and Evaluating: collect data, analyze data, diagnose, and determine effectiveness of correction strategydetermine effectiveness of correction strategy

4.4. Reflecting: instructor will provide this upon your return to Reflecting: instructor will provide this upon your return to classclass


VocabularyVocabulary

• Procedural ErrorsProcedural Errors – involve errors in skills and/or – involve errors in skills and/or step-by-step procedures (algorithms) needed to solve step-by-step procedures (algorithms) needed to solve mathematical problemsmathematical problems

• Conceptual ErrorsConceptual Errors – errors that are caused by the – errors that are caused by the misunderstanding of mathematical ideas such as place misunderstanding of mathematical ideas such as place value, meaning of operations, and number sense.value, meaning of operations, and number sense.

• Both Procedural and Conceptual ErrorsBoth Procedural and Conceptual Errors – errors – errors that involve violations of an algorithm AND a that involve violations of an algorithm AND a misunderstanding of a mathematical idea.misunderstanding of a mathematical idea.



• Algorithm Algorithm – – – ““step-by-step procedures for accomplishing a task, such as solving a step-by-step procedures for accomplishing a task, such as solving a

problem” (Ashlock, 2002, p. 256).problem” (Ashlock, 2002, p. 256).– ““a predetermined set of instructions for solving a specific problem in a a predetermined set of instructions for solving a specific problem in a

limited number of steps” (Webster, 1996, p. 34).limited number of steps” (Webster, 1996, p. 34).

• Meaning of Operations -Meaning of Operations -– Addition – Joining two or more addends together resulting in a number that Addition – Joining two or more addends together resulting in a number that

is larger than all addends. is larger than all addends. – Subtraction – Separating a smaller quantity from a larger quantity Subtraction – Separating a smaller quantity from a larger quantity

resulting in a quantity that is smaller than the minuend. resulting in a quantity that is smaller than the minuend. – Multiplication – repeated addition of a specified number or quantityMultiplication – repeated addition of a specified number or quantity– Division - the process of finding out the number of times a number (the Division - the process of finding out the number of times a number (the

divisor) is contained in another number ( the dividend)divisor) is contained in another number ( the dividend)

• Place ValuePlace Value - The understanding that the place of a digit tells - The understanding that the place of a digit tells its value AND the understanding that a number its value AND the understanding that a number can be combined and taken apart in different can be combined and taken apart in different

ways (i.e., regrouping).ways (i.e., regrouping).



• Number SenseNumber Sense - An intuition about numbers: their - An intuition about numbers: their sizesize AND how AND how reasonablereasonable a a quantity is once a number operation has occurred. quantity is once a number operation has occurred. – Use estimation as a strategy for determining whether the answer is Use estimation as a strategy for determining whether the answer is

reasonable.reasonable.– understand and represent fractions, such as ¼, 1/3, and ½ (Ashlock, 2010).understand and represent fractions, such as ¼, 1/3, and ½ (Ashlock, 2010).

• Properties of OperationsProperties of Operations• Commutative Property – the order in which two numbers are added (or Commutative Property – the order in which two numbers are added (or

multiplied) does not change the resultmultiplied) does not change the result• Associative Property – the order in which three numbers are added (or Associative Property – the order in which three numbers are added (or

multiplied) does hot change the resultmultiplied) does hot change the result• Distributive Property – adding, or multiplying, in partsDistributive Property – adding, or multiplying, in parts• Zero Property – Zero Property –

– Addition and Subtraction – the result is the non-zero number we Addition and Subtraction – the result is the non-zero number we started withstarted with

– Multiplication – the result is zeroMultiplication – the result is zero• Multiplicative Identity Property – any number multiplied by, or divided by, Multiplicative Identity Property – any number multiplied by, or divided by,

one remains unchangedone remains unchanged


Fraction ConceptsFraction Concepts– Fractional parts are equal shares or equal-sized Fractional parts are equal shares or equal-sized

portions of a whole or unit.portions of a whole or unit.– A unit can be an object or a collection of A unit can be an object or a collection of

things.things.– A unit is counted as 1.A unit is counted as 1.

• On a number line, the distance form 0 to 1 is the unit.On a number line, the distance form 0 to 1 is the unit.

– The denominator of a fraction tells how many The denominator of a fraction tells how many parts of that size are needed to make the parts of that size are needed to make the whole. For example: thirds require three parts whole. For example: thirds require three parts to make a whole.to make a whole.• The denominator is the divisor.The denominator is the divisor.

– The numerator of a fraction tells how many of The numerator of a fraction tells how many of the fractional parts are under consideration.the fractional parts are under consideration.


Equivalent Fractions…Equivalent Fractions…• Two equivalent fractions are two ways of describing the same Two equivalent fractions are two ways of describing the same

amount by using different-sized fractional parts (Van de Walle, amount by using different-sized fractional parts (Van de Walle, 2004, p. 242).2004, p. 242).– To create equivalent fractions with larger denominators, we multiply To create equivalent fractions with larger denominators, we multiply

both the numerator and the denominator by a common whole both the numerator and the denominator by a common whole number factor. number factor.

• Question: Can we use smaller parts (larger denominators) to cover exactly Question: Can we use smaller parts (larger denominators) to cover exactly what we have?what we have?

• (Activity 15.17 – Van de Walle, p. 260)(Activity 15.17 – Van de Walle, p. 260)– To create equivalent fractions in the simplest terms (lowest terms), To create equivalent fractions in the simplest terms (lowest terms),

we divide both the numerator and the denominator by a common we divide both the numerator and the denominator by a common whole number factor. whole number factor.

• Question: What are the largest parts we can use to cover exactly what we Question: What are the largest parts we can use to cover exactly what we have (Ashlock, 2006, p. 146)? have (Ashlock, 2006, p. 146)?

• Simplest terms means that the numerator and denominator have no Simplest terms means that the numerator and denominator have no common whole number factors (Van de Walle, 2004, p. 261)common whole number factors (Van de Walle, 2004, p. 261)

• ““Reduce” is no longer used because it implies that we are making a Reduce” is no longer used because it implies that we are making a fraction smaller when in fact we are only renaming the fraction, not fraction smaller when in fact we are only renaming the fraction, not changing its size (Van de Walle, 2004, p. 261) changing its size (Van de Walle, 2004, p. 261)

– The concept of equivalent fractions is based upon the multiplicative The concept of equivalent fractions is based upon the multiplicative property that says that nay number multiplied by, or divided by, 1 property that says that nay number multiplied by, or divided by, 1 remains unchanged (Van de Walle, 2004, p. 261)remains unchanged (Van de Walle, 2004, p. 261)

• ¾ x 1 = ¾ x 3/3 = 9/12¾ x 1 = ¾ x 3/3 = 9/12


Decimal ConceptsDecimal Concepts– A decimal is a number with a decimal point in it. A decimal is a number with a decimal point in it.

(6.72, 0.54, 0.019)(6.72, 0.54, 0.019)– The value of digits in a decimal is based on the our The value of digits in a decimal is based on the our

base-ten system or powers of 10.base-ten system or powers of 10.– The digits to the left of the decimal point represent The digits to the left of the decimal point represent

whole numbers whose values increase by powers of whole numbers whose values increase by powers of ten for each successive position.ten for each successive position.

– The digits to the right of the decimal point The digits to the right of the decimal point represent parts of a whole whose values decrease represent parts of a whole whose values decrease by powers of ten for each successive position.by powers of ten for each successive position.

– As with whole numbers, the place value positions to As with whole numbers, the place value positions to the right of the decimal have specific names and the right of the decimal have specific names and values.values.


Decimals ConceptsDecimals Concepts• Equivalent decimals describe the same Equivalent decimals describe the same

amount by using a different number of zeros amount by using a different number of zeros (.040 or .04 or .0400). (.040 or .04 or .0400).

• Adding zeros to the right of the last digit in a Adding zeros to the right of the last digit in a decimal does not change the value of the decimal does not change the value of the number.number.

• A repeating decimal is one in which at some A repeating decimal is one in which at some point in the notation, the sequence of digits point in the notation, the sequence of digits repeats indefinitely.repeats indefinitely.

• When comparing decimals, you progress When comparing decimals, you progress from the largest place value to the smallest from the largest place value to the smallest place value consider the digits in each place value consider the digits in each position until a determination can be made.position until a determination can be made.


DemonstrationsDemonstrations


Diagnosing ErrorsDiagnosing Errors

• Work with a group of your peers to reach a Work with a group of your peers to reach a consensus about…consensus about…– The procedural error(s)The procedural error(s)

• Ask yourselves: What exactly is this student doing to get Ask yourselves: What exactly is this student doing to get this problem wrong?this problem wrong?

– Basic FactsBasic Facts– Violations of AlgorithmViolations of Algorithm

– The conceptual error(s)The conceptual error(s)• Ask yourselves: What mathematical misunderstandings Ask yourselves: What mathematical misunderstandings

might cause a student to make this procedural error?might cause a student to make this procedural error?– Fraction ConceptsFraction Concepts

» Part-Whole RelationshipPart-Whole Relationship» Equal Parts/Fair SharesEqual Parts/Fair Shares

– Number SenseNumber Sense


GretchenGretchen


Our Conclusions:


Correction Strategies… Correction Strategies… • Correctional Strategies for Fraction Correctional Strategies for Fraction

Concepts…Concepts…– See Ashlock’s (2010) text…See Ashlock’s (2010) text…

• Gretchen’s Correction StrategiesGretchen’s Correction Strategies– See Van de Walle (2004) text.See Van de Walle (2004) text.

• Fraction ConceptsFraction Concepts – Understanding fractions build on an understanding of fair Understanding fractions build on an understanding of fair

shares – See Van de Walle (2004), pages 243-244. See also shares – See Van de Walle (2004), pages 243-244. See also Activity 15.1 and 15.2 (pp. 246-247). Activity 15.1 and 15.2 (pp. 246-247).

– Models for fractions – See Van de Walle (2004), pages 244-Models for fractions – See Van de Walle (2004), pages 244-246246

» Fraction circles, geoboard, graphing paper, pattern Fraction circles, geoboard, graphing paper, pattern blocks, paper folding, fraction tiles, number lines, blocks, paper folding, fraction tiles, number lines, folded paper strips, and set modelsfolded paper strips, and set models


CarlosCarlos


Our Conclusions:


Correction Strategies… Correction Strategies… • Correctional Strategies for Fraction Correctional Strategies for Fraction

Concepts…Concepts…– See Ashlock’s (2006) text,…See Ashlock’s (2006) text,…

• Carlos – pp. 143 -144. Carlos – pp. 143 -144.

– See Van de Walle (2004) text.See Van de Walle (2004) text.• Fraction Concepts Fraction Concepts

– Understanding fractions build on an understanding of fair Understanding fractions build on an understanding of fair shares – See Van de Walle (2004), pages 243-244. See also shares – See Van de Walle (2004), pages 243-244. See also Activity 15.1 and 15.2 (pp. 246-247). Activity 15.1 and 15.2 (pp. 246-247).

– Models for fractions – See Van de Walle (2004), pages 244-Models for fractions – See Van de Walle (2004), pages 244-246246

» Fraction circles, geoboard, graphing paper, pattern Fraction circles, geoboard, graphing paper, pattern blocks, paper folding, fraction tiles, number lines, folded blocks, paper folding, fraction tiles, number lines, folded paper strips, and set modelspaper strips, and set models


JillJill


Our Conclusions:


Correction Strategies…Correction Strategies…• Correctional Strategies for Equivalent FractionsCorrectional Strategies for Equivalent Fractions

– See Ashlock’s (2010) text…See Ashlock’s (2010) text…

– See Van de Walle’s (2004) activities… See Van de Walle’s (2004) activities… – Activity 15.4: Mixed-Number Names (p. 249) Activity 15.4: Mixed-Number Names (p. 249) – See also pages 257 – 260 See also pages 257 – 260

» Activity 15.13: Different FillersActivity 15.13: Different Fillers» Activity 15.14: Dot Paper EquivalenciesActivity 15.14: Dot Paper Equivalencies» Activity 15.15: Group the Counters, Find the NamesActivity 15.15: Group the Counters, Find the Names» Activity 15.16: Missing-Number EquivalenciesActivity 15.16: Missing-Number Equivalencies» Activity 15.17: Slicing SquaresActivity 15.17: Slicing Squares


SueSue


Our Conclusions:


Correction Strategies…Correction Strategies…

• Correctional Strategies for Equivalent Correctional Strategies for Equivalent FractionsFractions– See Ashlock’s (2010) text,…See Ashlock’s (2010) text,…– See Van de Walle’s (2004) activities… See Van de Walle’s (2004) activities…

– Activity 15.4: Mixed-Number Names (p. 249) Activity 15.4: Mixed-Number Names (p. 249) – See also pages 257 – 260 See also pages 257 – 260



TonyaTonya


Our Conclusions:


Other Correction Other Correction Strategies…Strategies…

• Correctional Strategies for Equivalent Correctional Strategies for Equivalent FractionsFractions– See Ashlock’s (2010) text,…See Ashlock’s (2010) text,…– See Van de Walle’s (2004) activities… See Van de Walle’s (2004) activities…

– Activity 15.4: Mixed-Number Names (p. 249) Activity 15.4: Mixed-Number Names (p. 249) – See also pages 257 – 260 See also pages 257 – 260



Further Information about Further Information about Fraction Concepts and Fraction Concepts and Equivalent FractionsEquivalent Fractions

• Developing Number SenseDeveloping Number Sense• Fractional-Parts Counting (Van de Walle, 2004, pp. 246-247)Fractional-Parts Counting (Van de Walle, 2004, pp. 246-247)

– Activity 15.3 p. 248 Activity 15.3 p. 248

• Fraction Number Sense (Van de Walle, 2004, pp. 251-257Fraction Number Sense (Van de Walle, 2004, pp. 251-257– Activity 15.6: Zero, One-half, or OneActivity 15.6: Zero, One-half, or One– Activity 15.7: Close FractionsActivity 15.7: Close Fractions– Activity 15.8: About How Much?Activity 15.8: About How Much?– Activity 15.9: Ordering Unit FractionsActivity 15.9: Ordering Unit Fractions– Activity 15.10: Choose, Explain, TestActivity 15.10: Choose, Explain, Test– Activity 15.11: Line ‘Em UpActivity 15.11: Line ‘Em Up


Questions…Questions…

Diagnosing Mathematical Errors: Fractions and Decimals (Concepts, Equivalence, and Operations)

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