Fractions Addressing a Stumbling Block for Developmental Students Wade Ellis, Jr. West Valley College (retired)

A Problem There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class?

Possible Solutions

A P ROBLEM There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class? 100% 135 1%1.35 5:45/94/9 1/9 55.5% 44.4%

O UTLINE Setting the stage AMATYC Crossroad and Beyond Crossroads MathAMATYC Educator What is a Fraction? and Equivalent Fractions Using Technology ― Action/Consequence Principle Questions that Advance Student Learning A Progression for Learning Fractions Ratios and Proportions & Percents Comments and Suggestions

1988 NCTM Yearbook on Algebra Common Mistakes in Algebra (Marquis, 1988) 10 of 22 were related to fractions

L EARNING F RACTIONS If you are training someone to be a retail clerk, and you believe that that person will never need to know much more math than a retail clerk knows, then you can teach fractions using standard algorithms for doing common fraction problems. But, if you think that the person you are teaching might need to know more advanced mathematics later, then you should teach fractions in a different way. Jim Pellegrino Distinguished Professor of Cognitive Psychology at the University of Illinois at Chicago

L EARNING F RACTIONS (C ONT ’ D ) In math, you can teach arithmetic by simply teaching the most efficient arithmetical algorithms or you can teach it in a way that greatly facilitates the learning of algebra – so you understand the idea of equivalence..., not just what you need to do to execute procedures.... Research shows what kids understand and what they don’t understand depends very much on how we teach the material. Jim Pellegrino

C ROSSROADS IN M ATHEMATICS First, technology can be used to aid in the understanding of mathematical principles. Second, students will use technology naturally and routinely as a tool to aid in the solution of realistic problems.

B EYOND C ROSSROADS Inquiry. Effective mathematics instruction should require students to be active participants. Students learn through investigation. Advances in neuroscience confirm that students’ active involvement in learning mathematics is important in the process of building understanding and modifying the structure of the mind.

J AMES S TIGLER IN THE M ATH AMATYC E DUCATOR Students who have failed...[might succeed] if we can first convince them that mathematics makes sense key concepts in the mathematics curriculum... included comparisons of fractions, placement of fractions on the number line, operations with fractions/decimals/percents, ratio, the ability to correctly remember and execute procedures... is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas. Finally, when students are able to provide conceptual understanding, they also produce correct answers.

T ECHNOLOGY : W HAT ’ S A F RACTION ?

T ECHNOLOGY : E QUIVALENT FRACTIONS

F RACTIONS IN THE C OMMON C ORE Grade 3 Develop understanding of fractions as numbers. Grade 4 Extend understanding of fraction equivalence and ordering. Build fractions from unit fractions. Understand decimal notation for fractions, and compare decimal fractions. Grade 5 Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division. Grade 6 Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

CCSS M ATHEMATICAL P RACTICES 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

A P ROGRESSION FOR L EARNING F RACTIONS (P ROF. W U ) 1.What is a Fraction?* 2.Equivalent Fractions* 3.Fractions and Unit Squares 4.Creating Equivalent Fractions 5.Adding & Subtracting Fractions with Common Denominators* 6.Adding Fractions with Unlike Denominators 7.Fractions as Division 8.Mixed Numbers* 9.Multiplying Whole Numbers and Fractions 10.Fraction Multiplication* 11.Dividing a Fraction by a Whole Number 12.Division of Whole Numbers by a Fraction 13.Dividing a Fraction by a Fraction 14.Units Other Than Unit Squares 15.Comparing Units

W HAT IS A F RACTION ? Teacher Guidance Document I. The Mathematical Focus References Common Core Standards Covered II. About the File III. Possible Objectives IV. Sample Questions

Engaging in a concrete experience Observing reflectively Developing an abstract conceptualization based upon the reflection Actively experimenting/testing based upon the abstraction People learn by Zull, 2002

Conceptual Knowledge: Makes connections visible, enables reasoning about the mathematics, less susceptible to common errors, less prone to forgetting. Procedural Knowledge: strengthens and develops understanding allows students to concentrate on relationships rather than just on working out results NRC, 1999; 2001

Take an action on a mathematical object Observe the mathematical consequences and Reflect on the mathematical implication of those consequences Conceptual Understanding

Action Consequence Principle Interactive Dynamic Technology

D YNAMIC I NTERACTIVE T ECHNOLOGY : A CTION C ONSEQUENCE P RINCIPLE Students take an action on a mathematical object, observe the consequences of that action, and reflect on the mathematical implications of those consequences Burrill & Dick, 2008

A/C DOCUMENTS & L EARNING Take an action on a mathematical object Observe the consequences Reflect on the mathematical implications Engage in concrete experience Observe reflectively Develop abstract conceptualization Experiment and test concepts

T ECHNOLOGY AS A T OOL FOR D EVELOPING U NDERSTANDING Key is asking good questions Predict consequence in advance of action (what would happen if…?) Consider action that would produce a given consequence (what would make … happen?) Conjecturing/Testing/Generalization (When…?) Justification (Why…?)

The only reasons to ask questions is to: (Black et al., 2004) Probe to uncover students’ thinking discover misconceptions that exist Push to advance students’ thinking make connections justify or prove their thinking W HAT TEACHERS DO :

P OSSIBLE Q UESTIONS Handout

W HAT IS A R ATIO ?

R ATIOS AND F RACTIONS

R ATIO T ABLE

R ATIO AND S LOPE

P ERCENT

Q UESTIONS FOR W HAT I S A F RACTION ? Describe where three fifths will be. How will three fifths differ from seven fifths? Explain your thinking, then check your answer using the tns file. Where will 4/8 be? b) 0/8? c) Is eleven eighths closer to one or to two? How do you know? If the number of 1/5’s is larger than the 5, what can you say about the size of the fraction? Explain. Suppose the unit fraction was 1/5 and the numerator was between 11 and 14. Where is the fraction? If the unit fraction were 1/6, where would fractions with a numerator between 25 and 29 be?

Q UESTIONS FOR W HAT I S A F RACTION ? ( CONT ’ D ) How many copies of ½ are in 2? Use the file to make a conjecture about whether the following sentences are correct. a) 0 is a fraction. b) A whole number cannot be a fraction. c) A fraction can have many names.

P ROBLEM At a dance, 2/3 of the girls dance with 3/5 of the boys. What proportion of the students are dancing?

A constant way to think: k/p is k copies of 1/p - the length of the concatenation of k segments each of which has length 1/p. Behavior similar to whole numbers: k/3 is a multiple of 1/3 Larger fraction is to the right on the number line Connection of whole number to fractions. One number has many names and none more important than another. No difference between proper and improper fractions What does fraction as a point on a number line buy us?

C LOSING D ISCUSSION Questions Comments

R EFERENCES Burrill, G. & Dick, T. (2008). What state assessments tell us about student achievement in algebra. Paper presented at NCTM 2008 Research Presession Dick, T. & Burrill, G. (2009). Technology and teaching and learning mathematics at the secondary level: Implications for teacher preparation and development. Presentation at the Association of Mathematics Teacher Educators, Orlando FL. National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press Zull, J. ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Association for Supervision and Curriculum Development, Alexandria, Virginia.

R EFERENCES What Does it Really Mean to be College and Work Ready? : The Mathematics Required of First Year Community College Students, National Center on Education and the Economy, 2013.,