Research and Development in the Teaching and Learning of Algebra

Lisandro Peña Auditorium - Architecture

Topic Study Group 11 brings together researchers and developers who investigate and develop theoretical accounts of the teaching and learning of algebra. The group especially welcomes empirically grounded contributions that focus on (a) learning and teaching in diverse classrooms settings from around the world; (b) the evolution of algebraic reasoning from elementary through university schooling.

**Topics of interest include: **

-Classroom processes that enhance the learning of algebra.

-The role of problem situations and technological tools (and the interrelationship between the two) in the teaching and learning of algebra.

-Similarities and differences between algebra curricula and how this influences teaching and learning in the classroom.

-Professional development for teaching and learning algebra.

Organizing team composition

DISCUSSION SESSIONS-Overview

How can I prepare for the discussion sessions?

SESSION I- Tuesday: Contexts for Learning Algebra

SESSION II-Wednesday: Early Algebra

SESSION III-Friday: Syntax and Semantics

SESSION IV-Saturday: Special discussion of Puig, Rojano, & Filloy

Papers and discussion documents

Rosamund Sutherland (United Kingdom)

ros.sutherland@bristol.ac.uk

2067 Massachusetts Ave. Cambridge, MA

Claudia L. Oliveira Groenwald (Brazil)

Within the framework set out above we would like to consider the following questions:

Can algebraic notation and reasoning take on a ‘life of its own’, by the end of secondary schooling, if we introduce young learners to algebra as a means for summarizing observed patterns and making empirical generalizations?

What does research tell us about (the early introduction of) algebraic notation?

How does one shift from using algebra (a) to express mathematical relationships (especially with regard to extra-mathematical phenomena) and (b)to derive additional expressions?

Deeper shared understanding of the Topic Questions, new methodologies (approaches and methods and how they shape the work) as well as emerging knowledge/understanding

Exemplars of research-grounded and research-generative practice

Post-ICME report/publication: a report of the working of the group that will include a subset of accepted papers, as well as overview comments and discussion written by the organizers.

DISCUSSION SESSIONS-Overview

DISCUSSION SESSIONS OVERVIEW

Topic Study Group 11 has a total time of 5 hours to meet during the Congress. The programme and schedules are shown below:

Session Focus Issues Readings

I—Tuesday June 8 12-13 PM

Contexts for learning algebra

* Can a) Observing and formulizing rules about patterns b) Making empirical generalizations and c) Thinking about relations among variable quantities provide solid foundations algebra?

* Or are these likely to lead instruction the wrong direction?

* What empirical evidence supports your claim(s)?

Dekker, Wijers, van den Zwaart, Spek; Egodawatte; Zhang

II—Wednesday June 9 12-13:30 PM

## What does research tell us about the early introduction of algebraic notation and the syntactic rules of algebra?

## Does the equals sign in arithmetic mean something fundamentally different from in algebra?

## Is this necessary or an artifact of limited instruction?

Carraher, Schliemann, & Brizuela; Schliemann, Goodrow,Caddle, Porter & Carraher; Cusi & Malara; Nogeira de Lima; Maranhao

III—Friday June 11 12:30-13:30 PM

How do students shift from using algebra to express extra-mathematical relationships to derive additional statements?

Huang; Martinez; Campos & DeSouza; Groenwald & Becher

IV—Saturday June 12 12-13:30 PM Special discussion: Puig, Rojano, & Filloy Within this session Rosamund Sutherland will lead a conversation with Teresa Rojano, Eugenio Filloy and Luis Puig about their recently published book “Educational Algebra”. The focus will be on the question: What is the role of algebraic notation in the development of algebraic understanding? Rojano, Filloy and Puig will also be asked to comment on the main issues that have been raised in the previous sessions of the working group.

Tarp; Linsell; Vieira da Silva

How can I prepare for the discussion sessions?

1-If your name is listed next to one of the discussion sessions, you should: (a) try to prepare some thoughts on how your paper is related to the issues that will be the focus of discussion for that day; (b) Prepare no more than 3 Powerpoint slides to make your points; and © read the papers of the other authors who are listed for that day.

2-If your name is not listed for the session then: (a) read the papers of those whose names ARE listed and (b) think about the relations of those papers (and your own understanding of the research literature) to the issues of the day.

**Session** Focus Issues Readings

I—- Tuesday: Contexts for Learning Algebra

Time Focus Issues Readings

I—Tuesday June 8 12-13 PM

Contexts for learning algebra

# Can (a) Observing and formulizing rules about patterns (b) Making empirical generalizations and (c ) Thinking about relations among variable quantities pave the way for algebra?

# Or are these likely to lead instruction the wrong direction?

# What empirical evidence supports your claim(s)?

Zwaart; Egodawatte; Zhang

Dekker, Truus; Wijers, Monica; van der Zwaart, Pieter; Spek, Wim Learning Lines: From arithmetic to algebra, from elementary towards secondary education In the Netherlands a discussion is going on about the role of arithmatical and algebraic skills in the ongoing learning lin form primary education to acadamic education and to vocational education.

In this line especially the transition from arithmetics to algebra gives interesting possibilities for mantaining and expanding the arithmecal skill in the learning of algebra. The ReAL-project investigated these posibilties and constructed three (of many possible) learning lines from arithmetics to algebra. These learning lines are described in overview in this paper.

The learning lines are based on gaps that were found in the most commonly used schoolbooks for mathematics. The gaps were translated into changes for improving the learning of algebra and by doing that giving a constructive role to the maintenance of arithmetic skills in secondary education. This translation resulted in the learning lines mentioned above.

Zhang, Guang-xiang On Pattern Intuition and Symbol Intuition in Teaching and Learning Algebra Abstract Geometric intuition is important in mathematical thinking. However, we rely on intuition even when we deal with algebraic problems in which no diagrams are present. Geometric thinking relies heavily on diagrams, and geometric discovery and reasoning arise naturally from intuitional observations. This would appear not to be the case for algebraic operations, which are relatively abstract. Nevertheless, if we investigate the details, we may also find that intuition plays a major role in algebraic reasoning and furthermore that it is an important task for teachers to introduce algebraic concepts and algorithms in intuitionistic ways. Here we put forth the concept of pattern intuition as it applies to algebra and discuss teaching approaches based on pattern intuition. We also provide examples to show the value of pattern intuition in mathematical reasoning.

SESSION II-Wednesday: Early Algebra

Time Focus Issues Readings

II--Wednesday June 9 12-13:30 PM|^.

What does research tell us about the early introduction of algebraic notation and the syntactic rules of algebra?

Does the equals sign in arithmetic mean something fundamentally different from in algebra?

Is this necessary or an artifact of limited instruction? |^.

Carraher, Schliemann, & Brizuela; Schliemann, Goodrow, Caddle, Porter & Carraher; Cusi & Malara; Nogueira de Lima|

Carraher, David; Schliemann, Analúcia D.; Brizuela, Bárbara M. Algebra in Early Mathematics: a Longitudinal Intervention We report on partial results of a study aimed at introducing algebra in elementary school. We describe the main findings of the longitudinal study for a first cohort of students, from grades 3 to 5, taught by the researchers, and for a second cohort, grades 3 to 4, taught by their regular classroom teachers. The quantitative analysis of the data shows that, throughout the grades, students in the experimental group performed significantly better than the control group on written assessments developed by the research team and on standard measures.

Schliemann, Analúcia D.; Goodrow, A.; Caddle, Mary; Porter, Megan; Carraher, David From Functions to Equations in Elementary School We report on a study aimed at introducing algebra, including equation solving, from grades 3 to 5. We describe how students learn to work with functions and express relationships between quantities in narratives, tables, graphs, and equations, and how they come to understand and work with equations as the setting equal of two functions. Equations were challenging but within reach of participants

Cusi, Annalisa & Malara, Nicolina Games of interpretation, anticipating thought and coordination between verbal and algebraic register: key-aspects in the analysis of students’ proofs in elementary number theory This work is part of a wide-ranging long-term project aimed at fostering students’ acquisition of symbol sense through teaching experiments on proof in elementary number theory (ENT). Our aim is to analyze the use and the role of algebraic language in the development of such proofs. In this paper we present the analysis and classification of students’ behaviour in facing the proof of a conjecture while working in small groups. The analysis of students’ protocols was made by reference to the following interpretative-keys: the application of specific conceptual frames, the games of interpretation between different frames, anticipatory thoughts, the use of conversions and treatments and coordination between different registers of representation. Our analysis highlights the incidence of anticipatory thoughts and of the flexibility in the coordination between different frames and different registers of representation in the development of proof in ENT. Moreover our work testifies the effectiveness of the analysis of students’ discussions during small group activities as a methodological instrument to highlight all these aspects.

Nogueira de Lima, Rosana Procedural embodiment & quadratic equations In this paper, we present the results of a research study involving 77 14-15 year-old students working with quadratic equations. Data were analysed in the light of a theoretical framework that considers three different worlds of mathematics (Tall, 2004; Lima, 2007). Evidence shows that students give to equations and the solving methods they use meanings related to procedural embodiments (Lima & Tall, 2008), taking symbols as physical entities that can be moved around, “putting them on the other side” of the equation, with the additional magic of, for example, “changing signs”, or “transforming the exponent in a square root” in the case of quadratics. Procedural embodiments may be effective in some cases, but it is necessary to relate them to their underlying mathematical concepts, in order to make it possible for students to relate the meanings they give to equations to the algebraic principles they should be connected.

SESSION III-Friday: Syntax and Semantics

Time Focus Issues Readings

III—Friday June 11 12:30-13:30 PM

How do students shift from using algebra to express extra-mathematical relationships to derive additional statements?

Huang, Cai, & Ye; Martinez; Campos & Giusti De Souza; Groenwald & Becher; Ribeiro

Huang, Rongjin; Cai, Jinfa; Ye, Lijun Mathematical tasks implementation in the U.S. and Chinese classrooms This paper investigates the features of classroom instruction in the U.S. and China through examining the cognitive demands of mathematical tasks and the strategies used to implement the tasks.

Based on find-grained analysis of 10 consecutive lessons in each of the two countries, we came to the following conclusions: the U.S. and Chinese teachers tried their best to implement high level cognitive demands in the classrooms through effectively demonstrating high level performance, appropriate soliciting and use of students’ answers, and appropriately organized exploratory activities; the Chinese teacher seems to be successful at maintaining a high level when implementing the tasks while the U.S. teacher had some difficulties in doing so.

Martinez, Mara Integrating Algebra and Proof in High School:

The Case of the Calendar Sequence

|Previous studies (Friendlander & Hershkowitz, 1997; Harel & Sowder, 1998; Healy & Hoyles, 2000) set the stage for the need to conduct more research in the integration of algebra and proof in the high school curriculum. Given the evidence provided by Barallobres’ (2004) work, my hypothesis is that an integrated approach can provide meaning to students’ learning of both algebra and proof. The goal of this paper is to report on the challenges that students (9th and 10th graders in a public school in Massachusetts, USA) faced in their work with variables and equivalent expressions when engaged in producing and proving conjectures, and how these challenges were overcome. Students worked on problems from the Calendar Sequence, a didactical sequence (Brousseau, 1997) of twenty problems that was engineered (Artigue, 1988, 1994; Douady, 1997) to promote a specific algebraic functioning: the production of new knowledge about a system (Chevallard, 1985, 1989) such as the calendar.

Groenwald, Claudia Lisete Oliveira & Becher, Ednei Luis The characteristics of the algebraic thinking of school students using first degree equations This work presents the results of a pilot experiment implementation; with high school students of a public school in Rio Grande do Sul, a Brazilian state, using the SCOMAX system (Student Concept Map Explore), and the first degree equation content. The SCOMAX is a very intelligent system developed by The Educational Technologies group of La Laguna University in agreement with the Curricular Studies of Mathematics Education group of Universidade Luterana do Brasil, in Canoas-Brazil, which aims to model the students’ knowledge, in any knowledge area. The objective of this work was to investigate the characteristics of the algebraic thinking, developed by the Elementary School, in first degree equations, with High School students.

Alessandro Ribeiro The multimeanings of equation The present work discusses different meanings that can be given to the notion of equation in

teaching and learning of Mathematics and their potentialities in the construction of mathematical knowledge. It is a theoretical work, as it summarizes and amplifies the arguments presented by the author’s doctorate thesis (Ribeiro, 2007). Grounded in the epistemological study developed in such thesis, it is presented, as can be seen from history of Mathematics facts analysis, different ways of understanding equation. Besides, Research studies on the subject have also been analyzed, showing other ways of explaining the notion of equation. The author validates such results with Semiotic Representation Registers Theory (Duval, 1993, 2003). Afterwards, it is presented and discussed possibilities and potentialities that multimeanings of equation approach can bring to Algebra teaching and learning processes. Among the final conclusions it is pointed out the importance of discussing Elementary Education subjects in teacher training courses without doing it as a simple review of mathematics contents. It is also discussed the relevance that the articulation of multimeanings of equation can provide to teacher and students´ conception enlargement on the notion of equation.

SESSION IV-Saturday: Special discussion of Puig, Rojano, & Filloy

Time Focus Issues Readings

IV—Saturday June 12 12-13:30 PM Special discussion: Puig, Rojano, & Filloy Within this session Rosamund Sutherland will lead a conversation with Teresa Rojano, Eugenio Filloy and Luis Puig about their recently published book “Educational Algebra”. The focus will be on the question: What is the role of algebraic notation in the development of algebraic understanding? Rojano, Filloy and Puig will also be asked to comment on the main issues that have been raised in the previous sessions of the working group.

Tarp; Linsell; Vieira da Silva

Tarp, Allan Pastoral algebra deconstructed Presenting its choices as nature makes modern algebra pastoral, suppressing its natural alternatives. Seeing algebra as pattern seeking violates the original Arabic meaning, reuniting. Insisting that fractions can be added and equations solved in only one way violates the natural way of adding fractions and solving equations. Anti-pastoral grounded research identifying alternatives to choices presented as nature uncovers the natural alternatives by bringing algebra back to its roots, describing the nature of rearranging multiplicity through bundling & stacking.

Linsell, Chris Exploring Connections Between Numeracy and Algebraic Thinking:

Some Current Directions in New Zealand At present there are initiatives to extend the New Zealand Numeracy Development

Projects from number into algebra. This paper describes an approach to linking

numeracy with students’ strategies for solving linear equations. Preliminary data from diagnostic interviews with 500 Year 7 to Year 10 students suggests that there is a hierarchy of sophistication of strategies. The most sophisticated strategy that a student is able to use is associated with the stage of numeracy of the student.

Vieira da Silva, Magda Influence of intuition and analytical thinking on graphic representaton of problem situations This work aims to report the results of a research carried out with beginners from the Bachelor’s degree (BA) course in Mathematics to verify the most common errors in the graphic representation of problem situations. Different activities were applied to 23 students and each of them involved: 1) graphic representations and intuitive thinking; 2) graphic representation, exploring intuitive and analytical thinking; 3) graphic representation, exploring intuitive and analytical thinking through more complex situations.

In general, intuition and analytical thinking were observed for this study. 33.09% of correct answers were obtained from the overall average, in percentage, among the activities. Such figure is considered very low given the fact it refers to real problem situations. It can be noticed that most learners have difficulty in perceiving what is being proposed and, many times, they perceive it, their intuitive thinking is equivocated, preventing them from getting to a correct analytical thinking.

Ibarra Lidia, Formeliano Blanca, Alurralde Florencia, Silvia Baspiñeiro, Graciela Méndez, Mirta Velásquez Successions stand always, but what about generalization?

Papers and discussion documents

Martinez-Integrating Algebra and Proof in High School: The Case of the Calendar Sequence (1.00 MB)

Cusi_Malara-Games of interpretation, anticipating thought and coordination between verbal and algebraic register:of students’ proofs in elementary number theory (233.00 KB)

Linsell-Exploring Connections Between Numeracy and Algebraic Thinking:Some Current Directions in New Zealand (295.00 KB)

Groenwald_Becher-ALGEBRAIC THINKING OF STUDENTS USING FIRST DEGREE EQUATIONS (1.00 MB)

Zhang-On Pattern Intuition and Symbol Intuition in Teaching and Learning Algebra (629.00 KB)

Carraher-Algebra in Early Mathematics: a Longitudinal Intervention (451.00 KB)

Schliemann_Goodrow_Caddle_Porter:From Functions to Equations in Elementary School (222.00 KB)

Lima-PROCEDURAL EMBODIMENT AND QUADRATIC EQUATIONS (849.00 KB)

Huang-Cai-Ye MATHEMATICAL TASKS IMPLEMENTATION IN THE U.S. AND CHINESE CLASSROOMS (125.00 KB)

Tarp-Pastoral Algebra Deconstructed (348.00 KB)

DeSousa_Campos-INEQUATIONS RESOLUTION: A FUNCTIONAL GRAPHIC APPROACH (434.00 KB)

Maranhao-OPERATIONS WITH WHOLE NUMBERS: TEACHERS’ UNDERSTANDING (128.00 KB)

Zwaart-Learning Lines: From arithmetic to algebra, from elementary towards secondary education (357.00 KB)

Silva-INFLUENCE OF INTUITION AND ANALYTICAL THINKING ON GRAPHIC REPRESENTATION OF PROBLEM SITUATIONS (347.00 KB)

Egodawatte-Algebraic procedures used by 14 to 15 year old Sri Lankan students (278.00 KB)

Ribeiro-The multimeanings of equations (205.00 KB)