making sense of algebra with realistic mathematics education

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Making Sense of Algebra with Realistic Mathematics Education. Mieke Abels Freudenthal Institute Utrecht University, Netherlands Michael Matassa and Raymond Johnson Freudenthal Institute US University of Colorado Boulder. Expectations?. - PowerPoint PPT Presentation

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Mieke AbelsFreudenthal InstituteUtrecht University, Netherlands

Michael Matassa and Raymond JohnsonFreudenthal Institute USUniversity of Colorado BoulderMaking Sense of Algebra with Realistic Mathematics Education

Ask audience about experience/knowledge with RME. Ask for keywords.1Expectations?How can RME help teachers and students to understand and meet the common core goals of Algebra?How can students develop strategies for solving systems-of-equations?What can I do with my 6th (7th, 8th) grade students?How may a learning trajectory for (systems of) equations look like?Sources?

Tug-of-war

How would a sixth grade student solve this problem?Source: Comparing Quantities (MIC)

Solving sets of linear equations (informal level, substitute)Participants start working on sixth grade problems (Informal level)Strategies: Formal? Informal? Substitution? Exchange?Or add three horses on both sides in the second picture.3Substitute and compare

This side wins

Substitute and compare

This side wins

Substitute and compare

This side winsICEBERG METAPHOR

INFORMALFORMAL

L=30y+320L=35y+295Three frogs

How would a sixth grade student solve this problem? Visualizations?Source: MiTC

from MiTC (Mathematics in the City), no number line1. Participants solve the problem as a 6th grade student. Question: If a student was to draw a picture of this situation, what would it look like?This is the informal level that sets up the possibility for students to solve more formal systems of equations. This is algebra at 6th grade!

8

Sunny1 jump = 7 stepsCal1 jump = 8 stepsLeg1352 jump = 18 steps1 jump = 9 stepsBalance scales

How can this problem be solved using informal strategies?1. Participants solve the problem themselves2. Then, look at 3 examples of student work and look for the similarities and differences between strategies.3. Share out observations (Keywords: strategy, use of drawings, symbols, equations, ) Solving sets of linear equations (Pre- formal level, substitute)The balance represents an equal sign. The total weight of the items on the left side equals the total weight of the items on the right side. Students will continue to develop their understanding of the concept of variable. They may even start to use letters as a symbol.131

Compare the student work.What are the differences, what is the same?What concepts are developed?2

ICEBERG METAPHOR

INFORMALPRE-FORMALFORMAL

MODELSL=30y+320L=35y+295

Balances are a modelEquations are more formal modelsNot a one way trajectory towards formal; Models of vs. Models forThe balance scale can become a model for 17Generate new equations

$ 6.80Generate new equations

$ 1.00are a base for a strategies to solve sets of equations: make combinations, exchange or substitute. 19Combination chart

2311$7.80$3.4022$6.80$1.0010X 2Find the differenceA Learning trajectoryFORMALFLOATING CAPACITYL=4j +5L=2j+15

Balance strategy

Equations

Compare jumps and steps3j 2 = 2j + 28 + 4x = 10 2x

Sets of Equationsy = 2x +52y = 4x 3

Generate new equationsNotebook strategySubstitute and compareTo reinvent formal and abstract mathematics, students should get the opportunity to gradually move from using informal, intuitive and concrete solving strategies through a variety of pre-formal strategies to using more formal, abstract standard solving strategies. This process is called progressive formalization (Freudenthal 1983, Treffers 1987) and is also a feature of RME.

21MIC (Mathematics in Context)http://mathincontext.eb.comUnits:Comparing QuantitiesGraphing EquationsMiTC (Mathematics in the City)http://mitcccny.orghttp://www.contextsforlearning.comUnit:The California Frog-Jumping Contest: AlgebraDME (Digital Mathematics Environment)http://dme.colorado.edu/dwo/colorado.html, or http://www.fi.uu.nl/dwo/en (login as guest)Math Secondary Education> Algebra> Lesson-Systems of equationsMore information:http://www.fisme.uu.nl/wisweb/enRME conference in Boulder.4th International RME Conference,2013sept 27-29,University of Colorado, BoulderSee for more information: www.fius.orgUtrecht Summer School on Mathematics and Science EducationAugust 19 - 30, 2013 in Utrecht, theNetherlands.http://www.utrechtsummerschool.nl/index.php?type=courses&code=H7http://www.utrechtsummerschool.nl

Mieke [email protected] [email protected] [email protected], @MathEdnetgoogle.com/+RaymondJohnson