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Equivalence and Equa-on Solving with Mul-ple Tools: Toward a Local Instruc-onal Theory
Nicole L. Fonger, Ph.D. Research Associate, North Carolina State University
Friday Ins-tute for Educa-onal Innova-on
PME-‐NA 2013 Chicago, IL
November 14-‐17
Focus
• Purpose: Conjecture, test, and revise an instruc-onal theory for students’ representa-onal fluency in solving equa-ons with CAS and paper-‐and-‐pencil.
• Research Ques-on: What processes of learning and means of support seem to facilitate students’ change in RF in a combined CAS and paper and pencil environment?
• Themes: Representa=onal fluency, equivalence and equa=on solving, mul=ple tools
BACKGROUND & MOTIVATION
Star-ng Points
• Representa-ons—fluency is an indicator of conceptual understanding (Na-onal Research Council, 2001; Kieran, 2007)
• Tools—the coordinated use of tools can support co-‐development of technique and theory (Kieran & Drijvers, 2006)
• Mathema-cs—Understanding “=” ma_ers for students’ abili-es to solve equa-ons (Knuth, Stephens, McNeil, Alibali, 2006)
Fluency
• Students demonstrate shortcomings in all or some aspects of representa-onal fluency in solving problems involving linear equa-ons (e.g. ,Bieda & Nathan, 2009; Huntley, Marcus, Kahan, & Miller, 2007; Moschkovich, Schoenfeld, Arcavi, 1993).
• For non-‐rou-ne tasks, students demonstrate difficul-es in coordina-ng informa-on across graphs and symbols, hence have superficial connec-ons (Huntley et al., 2007; Knuth, 2000).
Mul-ple Tools
• Cul-va-ng opportuni-es to learn with mul-ple tools and methods is valued (NCTM, 2000;
CCSSI, 2010), and needed for beginning algebra (e.g., Heid & Blume, 2008; Kieran & Yerushalmy, 2004).
• More research is needed to specify the roles of representa-ons and a balance between CAS and paper-‐and-‐pencil tools (e.g., Kieran & Saldnaha, 2008).
Mathema-cs
• We expect students to jus-fy solving linear equa-ons by reasoning about equivalence of equa-ons and to use appropriate tools (CCSSI, 2010).
• Students need to understand the equal sign as expressing a rela-on to be successful in symbolic manipula-on of equa-ons (Knuth, Stephens, McNeil, & Alibali, 2006).
Mo-va-on
• Linking research and prac-ce is a priority (Arbaugh, Herbel-‐Eisenmann, Ramirez, Knuth, Kranendonk, Quander, 2010)
• Cul-va-ng opportuni-es to learn with mul-ple tools and methods is valued (NCTM, 2000; CCSSI, 2010),
and needed for beginning algebra (Heid & Blume, 2008; Kieran & Yerushalmy, 2004).
• More research is needed to specify the roles of representa-ons and a balance between CAS and paper-‐and-‐pencil tools (Kieran & Saldnaha, 2008).
CONCEPTUAL & THEORETICAL FRAMEWORKS
Conceptual Frame on Learning
• Interpre-ve lens on classroom ac-vity (Cobb & Yackel, 1996)
• Individual cogni-on and ac-vity emerge as classroom prac-ces are nego-ated
Psychological Social Individual cogni-on and ac-vity
Classroom mathema-cal prac-ces
Theore-cal Frame on Ac-vity
• Coordinated tool use (Ar-gue, 2002; Kieran & Drijvers, 2006)
• Organizes task design for specific content by intent of tool use and theore-cal backing based on learning
Tasks Technique Theory
What—the mathema-cal ac-vity guided by learning goal
How—accomplish tasks with a tool, approach, representa-on
Why—disciplinary discourse that underpins a sequence of goal-‐directed tasks
A DESIGN RESEARCH STUDY
Phases of Design Experiment
I. Prepare for experiment (Cobb, 2000; Simon, 1995);
design tasks based on progression (Kieran & Drijvers, 2006; Kieran & Sfard, 1999)
II. Conduct collabora-ve teaching experiment (Cobb, 2000); engage in ongoing analyses, thought experiments (Gravemeijer & Cobb, 2006)
III. Retrospec-ve analyses of case studies, classroom condi-ons, revised theory (Cobb et al., 2003; Gravemeijer & Cobb, 2006; Stake, 1995)
(Gravemeijer & Cobb, 2006)
Design to Link Research and Prac-ce
Gravemeijer and Cobb (2006, p. 28): Reflexive rela-on between theory and prac-ce
Mathema-cal Learning Goals
I. Develop representa-onal fluency with linear expressions and equa-ons.
II. Use representa-ons of linear expressions and equa-ons to solve problems.
III. Understand the meaning of the equal sign as a rela-onship between expressions.
IV. Understand solving equa-ons as a process of reasoning and explain that reasoning.
(CCSSI, 2010; Knuth, Stevens, McNeil, Alibali, 2006)
Assump-ons
• A more adaptable understanding is possible with the rule of four—create, interpret, connect (e.g., Huntley et al., 2007; Janvier, 1987).
• Students need to come to understand the meaning of “=” as an equivalence rela-on (Knuth et al., 2006).
• Coordina-ng equivalence of expressions and solving equa-ons is significant to build meaning with mul-ple tools (Chazan & Yerushalmy, 2003; Kieran & Drivers, 2006).
Collabora-ve Teaching Experiment
• All ac-vi-es were created in close communica-on and considera-on of district planning guides.
• The teacher taught all lessons; the researcher was a par-cipant observer (field notes, technical assistant).
• Daily debriefing sessions/thought experiments about alignment between enactment and goals.
• Revision and crea-on of subsequent ac-vi-es were based experimenta-on and reflec-on.
Retrospec-ve Analyses
• The data analysis method resembled a constant compara-ve method (Glaser & Strauss, 1967).
• Conjectures about the instruc-onal theory from ongoing analysis were confirmed/refuted; then tested against the next episode.
• Process of confirming and refu-ng conjectures was repeated un-l all teaching episodes were analyzed in chronological order.
RESULTS
Processes of learning and means of suppor=ng learning—Gravemeijer & Cobb (2006)
Emerging Local Instruc-onal Theory
• Task. An ac-vity sequence was constructed from empirical studies on students’ understanding of the equal sign and representa-onal fluency.
• Technique. An ac-vity structure guided the coordinated use of techniques with both paper-‐and-‐pencil and computer algebra systems.
• Theory. A learning progression was conjectured based on literature, revised for a mul--‐representa-onal lens, then tested and revised throughout the teaching experiment.
Chapter Enacted Revised
1 Multiple Representations of Equivalent Expressions
The “Cartesian Connection” in Graphs, Symbols, Tables, and Words
Equivalent Expressions and Non-Equivalent Expressions in Graphs, Tables, Symbols, and Words
2
Equations are Equivalence Relations that are Sometimes, Always, or Never True
Equations are Equivalence Relations that are Sometimes, Always, or Never True
3
Solving Linear Equations with Multiple Representations
Identifying Solution Sets of Linear Equations in Graphs, Tables, Symbols, and Words
Equivalent Equations Have the Same Solution Sets
Revised Sequence of Ac-vi-es (Tasks)
Suppor-ng Representa-onal Fluency
• Transla-on involves crea-on and interpreta-on (Janvier, 1987)
An Ac-vity Structure
• Ac-on-‐consequence principle (Dick & Hollebrands, 2011) • Ac-on on objects (Moschkovich, Schoenfeld, Arcavi, 1993)
• Reflec-on, CAS Check, and reconciling CAS and paper-‐and-‐pencil (Kieran & Saldanha, 2008)
A Learning Progression
• Descrip-on of students’ progress from informal understandings to more sophis-cated “big ideas” based on research on student learning (Confrey, Maloney, Nguyen, Mojica, & Myers, 2009)
• Star-ng points – Linking arithme-c and algebra – Equivalence of expressions
• Ending points – Representa-onal fluency in solving equa-ons – Linking expressions, equa-ons, and func-ons
Kieran & Drijvers (2006)
Expression1 = Expression2
Interpret CAS Inscrip-ons “No cause its [sic] a word” – Annie “[Y]es because if you put the numbers and variable together it makes sense” – Bryon “I think it does because the CAS says the two equa-ons are equil [sic]” – Carlos
Technique & Theory
D1*: Solu=ons to equa=ons can be determined by equality of expressions.
• “Students must understand the equal sign as expressing a rela-on in order to make sense of the transforma-ons performed on such an equa-on” (Knuth, et al., 2006, p. 229).
• Ms. L: “If two lines cross how many solu-ons do you have? [gestures an X with hands]” Student: “One” Ms. L: “You’re going to have no solu-ons when they’re parallel [gestures | | with hands and points to graph of 5x+7+x=6x]… what was the third case?” Student: “Always” Ms. L: “Always … infinite solu-ons [points to graph of -‐1(x+4)=-‐x-‐4].”
“Solve each equa-on for the variable. Show your work. Check your solu-on.
–r – 5 = – (r + 5).” Ms. L: Looks like he used the distribuKve property. […] And he ended up with 0 = 0. What does zero equals zero mean? [interrup-on and inaudible student response] Ms. L: Infinite solu-ons. Why? Why is it infinite solu-ons? Cause it means the same thing on either side of the equal sign. Look back at this step right here. What do you see about the expression on the lez and the expression on the right? Ethan: They're the same. Ms. L: They're exactly the same. So you have equivalent expressions. If your expressions on either side of the equals sign are exactly the same that means you have the same line, you have exactly the same line, so infinite solu-ons on number 2.
Kieran & Sfard (1999); Davis (2005)
f(x)=ax+b vs. g(x)=cx+d to Solve
Iden-fying Solu-ons: -‐7x+13=4x-‐9 Ms. L: What happened to those lines? Katrina: They overlap. Ms. L: They cross don't they? They sure are intersec-ng. […] And if I look at my graph can I figure out what value of x they're going to cross at? Student: Yeah. Ms. L: About where? Abila: Nega-ve one and two. Ms. L: So x = 2, y = –1. What if I look at the table? Davon: You can check. Ms. L: That's the way we do it, too. So if I go to my table, Control T, go to your table, and look where my y-‐values are the same, OK, that's the only place where the y-‐values are the same. Right here [gestures to table] x = 2, y = –1. That's the only place. So one solu-on.
technique
CONCLUSIONS & DISCUSSION
Design Research: Method and Evalua-on of Outcomes
• Revision and tes-ng over -me in several cycles
• Design principles for coordina-ng mul-ple tools (predict-‐act-‐reflect-‐reconcile) more stable than progression of learning (remains a conjecture).
Improvements to Future Design
• Integra-on of task-‐techniques-‐theory (e.g., see Stephan & Cobb, 2013)
• Too large of conceptual shizs proposed without adequate -me to coordinate perspec-ves—concep-ons of algebra and mul-ple representa-ons.
• Shiz focus of thought experiments from reflec-on on theory in prac-ce to imagining hypothe-cal learning trajectories (Simon, 1995).
Expanding Instruc-onal Theory
“An instruc=onal theory is more than learning goals and a means of support for students, it also includes a means of support for the instructor to determine the alignment/efficacy of the goal outcomes and student supports (i.e., a theory of evidence or assessment).” (Reviewer comment)
Ongoing Research
• Adaptability in mul-ple contexts – pre-‐service secondary teachers – middle grades students
• Usability and Prac-cality – design principles – learning theory – integra-on of task-‐technique-‐theory
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