Post on 17-Dec-2015
Fraction Applets for Developmental
Mathematics Students
Wade Ellis
West Valley College (retired)wellis@ti.com
Fraction Applets for Developmental Mathematics Students
Developmental Mathematics students have problems learning algebra based in part on their misconceptions about fractions and fraction operations. This presentation will demonstrate the instructional use of fraction and proportional reasoning applets along with inquiry questions that enhance and deepen student understanding of fractions sufficient to improve student performance in algebra.
OutlineIntroduction
Procedures and Understanding
Research Basis for an Approach to Fractions
Examples of Applets (Lua and Nspire)
Basic Ratio Concepts Require Fractions
Research Basis for Applet & Activity Development
Examples of Activities for Applets
Comment and Questions
Learning FractionsIf 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
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
James Stigler: UCLA Psychology Dept.in May 2011 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.
James Stigler: UCLA Psychology Dept.Author of The Learning Gap
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.
Fractions According to Prof. Wu
A fraction is a point on the number line
Unit fractions are emphasized• (1/b is a unit fraction, b is a positive integer)
Common denominators
Improper fractions presented long before mixed numbers
Fraction Applets
What is a Fraction?
Creating Equivalent Fractions
Adding and Subtracting Fractions withCommon Denominators
Fractions and Unit Squares
Adding Fractions with Unlike Denominators
Division of a Fraction by a Fraction*
Basic Concepts
1. Ratio as a ratio relationship between two quantities
2. Ratio and rate
– ratio as a relationship
– rate as a fraction with units
3. Unit rate b/a associated with a ratio a:b with a ≠ 0
4. Equivalent ratios
5. Percent of a quantity as a rate per 100
6. Constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions
Ratio of Quantities a:b
Part to Whole Part to Part
Fraction
Number
Point on the Number Line
Length Area
Percent Unit RateProportion
Rate y kxa c
b d
What do we gain?Ratios are much more than just a different notation for fractions; ratios communicate a relationship between quantities
The story emphasizes paired quantities over and over.
Paired quantities leads naturally to graphs and to proportional relationships
The constant of proportionality relates to both the graphical idea of slope, the physical idea of rate, geometrical notion of scaling
Emphasis on variety of strategies to solve ratio/proportion problems (ratio tables, double number lines, graphs, …)
Algebraic Use of Fractions and Ratios
Students use fractions and ratios in algebra when they study similar figures, slopes of lines, solving linear equations and proportional reasoning problems (and later when they study sine, cosine, tangent, and other trigonometric ratios in high school).
Evidence from many sources suggests students often do not understand fundamental mathematical concepts.
Our hypothesis: Consider another approach rather than continuing
what has been unsuccessful for many Use interactive dynamic technology to support the
development of understanding, especially of “tough to teach/tough to learn” fundamental concepts.
Building Concepts
Burrill, Dick, & Ellis, 2013
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
As a tool for doing mathematics - a servant role to perform computations, make graphs, …As a tool for developing or deepening understanding of important mathematical concepts
The Role of Technology
Dick & Burrill, 2009
Principles for effectively integrating interactive dynamic technologies in the classroom:
The Action-Consequence Principle
The Questioning Principle
The Reflection Principle
Burrill, Dick, & Ellis, 2013
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
Focus of an Activity
On fundamental concepts
One or two ideas per activity
Follow a learning trajectory supported by research
Recognize student misconceptions/difficulties
Ratio Activity Examples
What is a Ratio?
Ratio Tables
Building a Table of Ratios
Connecting Ratios to Equations
Variables and Expressions
Teacher Notes
The Mathematical Focus of the Activity
Objectives of the Activity
About the Applet
Sample Questions
A Ratio Problem
7. Suppose the ratio was 5 to 3. If there were a total of 120 circles and squares, how many squares would there be? Explain how you found your answer.
The Learning TrajectoryAlgebra
FractionsRatios
Rates
Coordinate Axes
by x
a
y mx b
3rd Grade
5th Grade
7th Grade
What you teach.
How you teach it.
1. What is a Fraction?
2. Equivalent Fractions
3. Fractions and Unit Squares
4. Creating Equivalent Fractions
5. Adding & Subtracting
Fractions with Common Dens.
6. Adding Fractions with Unlike Denominators
7. Fractions as Division
8. Mixed Numbers
Building Concepts: Fractions9. Multiplying Whole Nos.
and Fractions10. Fraction Multiplication11. 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
Building Concepts: Ratios1. Introduction to Ratios
2. Introduction to Rates
3. Building a Table of Ratios
4. Ratio Tables
5. Comparing Ratios
6. Connecting Ratios and Fractions
7. Double Number Line
8. Connecting Ratio to Rate of Change
9. Adding Ratios10. Proportions11. Proportional
Relationships 12. Solving Proportions
13. Ratio and Scaling14. Ratio and Similarity
Questions and Comments
wellis@ti.comWu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79
www.education.ti.com
Understand ratio concepts and use ratio reasoning to solve problems.1. Understand the concept of a ratio and use ratio language to describea ratio relationship between two quantities.
CCSSM, 2010
What is a ratio?
a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compareratios.
CCSSM, 2010
Ratio Tables/ Connection to Graphs
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.c. Represent proportional relationships by equations.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
CCSSM, 2010
Proportions
Connection to GeometryThey can apply a scale factor that relates lengths in two different figures, or they can consider the ratio of two lengths within one figure, find a multiplicative relationship between those lengths, and apply that relationship to the ratio of the corresponding lengths in the other figure.
When working with areas, students should be aware that areas do not scale by the same factor that relates lengths.
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Empson, S., & Knudsen, J. (2003). Building on children’s thinking to develop proportional reasoning. Texas Mathematics Teacher, 2, 16–21.
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Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing.
Lane, D. M., & Peres, S. C. (2006). Interactive Simulations in the Teaching of Statistics: Promise and Pitfalls. In A. Rossman and B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.National Assessment for Educational Progress (2013). Released Item. Grade 8. National Center for Educational Statistics.
National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web 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. www.nap.edu.
Program for International Assessment (PISA). 2012 Release items. Organization for Economic Co-operation and Development. http://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf
Progressions for the Common Core State Standards in Mathematics (2011). 6-7, Ratio and Proportional Reasoning.
Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students, Educational Studies in Mathematics, 43(3), 271-292.
Wu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79
James Zull, ( 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.