Fractions Addressing a Stumbling Block for Developmental Students
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Fractions Addressing a Stumbling Block for
Developmental Students
Wade Ellis, Jr.West Valley College (retired)
A ProblemThere 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 Solutions135
0.251.25
1.25 1352.25 135
1352.2560135 60 75
x yx x yx y
x xx
x
xy
1350.25
1.25545 44 4 4 1354 5 4 1359 4 135
4 5135 60 135 759 9
x yx x yx y
x y
x yx yx xx
x x y
A PROBLEMThere 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
5:4 5/9 4/91/9 55.5
%44.4%
OUTLINE 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
72 7a b ab
x y yx z z
2 24 16x x
x r x ry s y s
LEARNING FRACTIONS 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
LEARNING FRACTIONS (CONT’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
CROSSROADS IN MATHEMATICS 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.
BEYOND CROSSROADS 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.
JAMES STIGLER IN THE MATHAMATYC EDUCATOR 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.
TECHNOLOGY: WHAT’S A FRACTION?
TECHNOLOGY: EQUIVALENT FRACTIONS
FRACTIONS IN THE COMMON CORE 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 MATHEMATICAL PRACTICES1. 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 PROGRESSION FOR LEARNING FRACTIONS (PROF. WU)1. What is a Fraction?*2. Equivalent Fractions*3. Fractions and Unit Squares4. Creating Equivalent Fractions5. Adding & Subtracting Fractions with Common Denominators*6. Adding Fractions with Unlike Denominators7. Fractions as Division8. Mixed Numbers*9. Multiplying Whole Numbers and Fractions10. Fraction Multiplication*11. Dividing a Fraction by a Whole Number12. Division of Whole Numbers by a Fraction13. Dividing a Fraction by a Fraction14. Units Other Than Unit Squares15. Comparing Units
WHAT IS A FRACTION?
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 reflectivelyDeveloping 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
DYNAMIC INTERACTIVE TECHNOLOGY: ACTION CONSEQUENCE PRINCIPLE
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 & LEARNING
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
TECHNOLOGY AS ATOOL FOR DEVELOPING UNDERSTANDING
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
WHAT TEACHERS DO:
POSSIBLE QUESTIONS Handout
WHAT IS A RATIO?
RATIOS AND FRACTIONS
RATIO TABLE
RATIO AND SLOPE
PERCENT
QUESTIONS FOR WHAT IS A FRACTION?
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?
QUESTIONS FOR WHAT IS A FRACTION? (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.
PROBLEM 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 1/3Larger 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?
CLOSING DISCUSSION Questions Comments
REFERENCES 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.
REFERENCES 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.
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wellis@ti.com
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