early algebra_GB

Algebra and Representation in the Middle GradesGail BurrillMichigan State Universityburrill@msu.ed : 

Algebra and Representation in the Middle GradesGail BurrillMichigan State Universityburrill@msu.ed

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A Beam of Length 5 Roodhardt,

Draw a beam of length 6 : 

Draw a beam of length 6 Using your drawing, find a general rule for the number of rods you would need for a beam of any length.

Math/Science Lessons : 

Math/Science Lessons Weiss, et al 2003b

What is algebra? : 

What is algebra? Structured system that supports reasoning with symbols Language of mathematics Study of functions Generalized arithmetic

Algebra in Realistic Mathematics : 

Algebra in Realistic Mathematics Reasoning with symbols that are temporarily removed from the context in which they originated Roodhardt

Algebra : 

Algebra Understand patterns, relations, and functions Represent and analyze mathematical situations and structures using algebraic symbols Use mathematical models to represent and understand quantitative relationships Analyze change in various contexts NCTM, 2000

Represent and analyze situations and structures using symbols (6-8) : 

Represent and analyze situations and structures using symbols (6-8) develop an initial conceptual understanding of different uses of variables; explore relationships between symbolic expressions and graphs of lines, the meaning of intercept and slope; use symbolic algebra to represent situations and to solve problems; recognize and generate equivalent forms for simple algebraic expressions and solve linear equations NCTM, 2000

Visual representation as a tool… : 

Visual representation as a tool… 40% of the teachers are males. There are 18 more female teachers than male teachers. How many teachers are there altogether? 0% 40% 60% 100% 18 60%-40%=20% 20% ->18 100% -> Singapore, Primary Mathematics 6A, p. 66

A very special snake has one red ring on its skin. Each year it sheds its skin, and the red ring is replaced by a red ring, a black ring, and a red ring.After 4 years, how many red rings does the snake have?How many black rings? : 

A very special snake has one red ring on its skin. Each year it sheds its skin, and the red ring is replaced by a red ring, a black ring, and a red ring.After 4 years, how many red rings does the snake have?How many black rings? Roodhardt

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R RBR RBRBRBR ……..

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Can the snake have 64 red rings? If so, when? If a snake has 250 rings, does it below to this family of snakes? Why or why not? Find a way to tell how many rings the snake will have for any number of years.

Research indicates that students : 

Research indicates that students need to see that they are learning something useful and relevant. abstract important concepts and are flexible in representing and using their knowledge when a subject is taught in multiple contexts with examples that apply what is being taught (NRC, 1999)

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PCMI 2004

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PCMI, 2004

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PCMI, 2004

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PCMI, 2004

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one fourth of the remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with? : 

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one fourth of the remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with? NAEP

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Ferrini-Mundy, 2005

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Ferrini-Mundy, 2005

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Ferrini-Mundy, 2005

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Ferrini-Mundy, 2005

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Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one fourth of the remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with? A. 36 B. 48 C. 60 D. 96 PROM/SE ITEM, GRADES 6,7,8 Grade 6 35.2% Grade 7 43.0% Grade 8 43.6%

Research indicates that students : 

Research indicates that students learn by building on or transferring knowledge from previous experiences. They misinterpret new knowledge because they fail to make the connections with what they had previously learned (NRC, 1999).. “...fail to connect everyday knowledge to subjects taught in school” ( NRC, p. 57,1999).

Research on Functions : 

Research on Functions Teaching issues Students accept different answers to same problem rather than reject a procedure they feel is correct or explore why the difference (Sfard, 1994) Form has consequences for learning (y = mx + b vs y = b + x(m); point slope form-y=y1+ m(x-x1) (Confrey & Smith, 1994)

Difficulties : 

Difficulties Noticing the domain and range (Markovitis et al, 1998) Decoding the graphic notation itself(Evan, 1990; Confrey & Smith,1994; Markovktis et al, 1998) Role of units (Confrey & Smith, 1994) Focusing on process of finding solution rather than result(Evan, 1994) Constructing meaning for algebraic terms (Monk, 1989) Making connections between graphical and symbolic representation (Evan, 1990;Markovitis et al, 1998) Can identify from graph; harder to do from symbols(Confrey & Smith, 1994; Evan, 1990)

Algebra in Realistic Mathematics : 

Algebra in Realistic Mathematics Representing and reasoning with symbols that are temporarily removed from the context in which they originated where each representation reveals more information about the story.

References : 

References Confery, J. & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics. 26: 135-164. Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics. 21: 521-544. PROM/SE (2004). Michigan State University Mathematics Science Partnership. Markovitis, Z., Eylon, B. S., & Bruckheimer, M. (1988). Difficulties students have with the function of concept, the ideas of algebra, K-12 (PP. 43-60). Reston, VA: National Council of Teachers of mathematics. Monk, G. S. (1989). A Framework for Describing Student Understanding of Functions: Department of Mathematics. Paper presented a the annual meeting of the American Educational Research Association

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National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston VA: Author National Research Council. (1999). How People Learn: Bain, mind, experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press. Park City Mathematics Institute Secondary School Teachers Program. (2004). Lesson Study Working Group. Roodhardt, A., Kindt, M., Burrill, G., & Spence, M. (1997). Patterns and Symbols. From Mathematics in Context. Directed by Romberg, T. & de Lange, J. Chicago IL: Encyclopedia Britannica Roodhardt, A. Private conversation related to the development of the Math in Context algebra strand. Singapore Ministry of Education. (2000). Primary Mathematics 6A. (2000). Curriculum Planning & Development Division. Ministry of Education. Singapore: Federal Publications.

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Sfard, A., & Linchevski, L. (1994). Between Arithmetic and Algebra: In the search of a missing link. The case of equations and inequalities. Rendicondi del Seminario Matematico, 52 (3), 279-307.. Wijers, M., Roodhardt, A., van Reeuwijk, M., Dekker, T., Burrill, G, Cole, B., & Pligge, M. (2006) Building formulas. Directed by Romberg, T. & de Lange, J. Chicago IL: Encyclopedia Britannica. Weiss, I. R.; Pasley, J. D.; Smith. S.. Banilower, E. R.; & Heck, D. J. (2003). Looking Inside the Classroom: A Study of K-12 Mathematics and Science Education in the United States. Chapel Hill, NC: Horizon Research, Inc. horizon-research.com