Project NarrativePurpose of the project
The overarching goal of this Development and Innovation, proof-of-concept project, titled Collaborative Problem Solving: Teaching Students to Co-Think Algebra/Geometry (CPS), is twofold: (i) co-develop, with teachers, problem-based materials for integrating central concepts in algebra and geometry into existing Algebra 1 and Geometry courses, and (ii) study how teachers learn to incorporate these materials and students’ dispositions toward these materials. That is, this project will address a burning national issue of gaps in students’ learning to solve non-routine, mathematical word problems, in ways that are consistent with Common Core State Standards (CCSS) and Florida benchmarks. Relying on close collaboration with teachers in Broward County and Palm Beach County School Districts (Florida), while drawing on a Backward Design approach (Wiggins & McTighe, 2005), on the cutting-edge, open-source GeoGebra software (Hohenwarter & Preiner, 2007), and on contemporary cognitive-change frameworks (Weick, 2001), the CPS project will (i) develop 16-20 clusters of integrated, algebra/geometry model problems, (ii) test implementation of these clusters in those teachers’ regular courses, and (iii) examine students’ learning via solving such problems. It should be noted that, typically, Algebra 1 is taught before Geometry; thus, integrated problems for Algebra 1 will be developed to fit with students’ intuitive understanding of geometry from elementary and middle school. Using GeoGebra as a technological platform for integrating innovative thinking in algebra and geometry is a distinctive feature of this project, because the developer of this software (Markus Hohenwarter) introduced the software into the US while in a post-doctoral position at Florida Atlantic University (FAU), and one of his students and collaborators (Ana Escuder) is a member of the project’s team.
We begin with an example of a possible problem cluster illustrating the twofold purpose of the CPS project and the nature of materials to be developed, after which we elaborate on key goals and outcomes of the project. This example (see Box 1) targets integrated learning of the central concepts of linear function (algebra benchmark) and relationships between dimension changes and the perimeter and area of common figures (geometry benchmark). This cluster includes a “puzzler”, a realistic word problem, and a GeoGebra representation of how dynamic changes of the side of a square and scale factor vary with the perimeter and area of the square.
Box 1: A linear function + dimension change problem cluster
Missing Square Puzzler: If you re-arrange the pieces of the upper triangle to form the lower triangle, a square goes missing. Can you explain? Please note that the colored pieces in both pictures are identical. (See hints in Appendix B.)
Word Problem: An agriculture company uses square lots to study how a sprayed fertilizer impacts plants. It takes 1 gallon of fertilizer to spray 1 square yard (yd2) of soil. The company encloses each square lot with a fence (2-yard high). To date, the company used lots with a side of 4 yards. Recently, however, the company decided to use two different sizes—a mid-size lot (side = 8 yards) and a large-size lot (side =10 yards). Solve the following problems:
1. How much fence and how much fertilizer does the company need for each new lot? What if they decided to increase the side of a lot to 12 yards? To 15 yards?
2. [Once problem #1 was solved outside the computer] Use GeoGebra to show/explain your work and solutions if the side of the lot changes to any other size. What do the x-values on your graph mean for the fence? For the fertilizer? What do the y-values mean for the fence and for the fertilizer? Which of the two graphs in your GeoGebra represent the length of the fence? Is it a linear function? If so, why? What does the slope of the straight line in your GeoGebra graph mean? Why aren’t the red dots in a straight line?
3. Use GeoGebra to discuss how the quantities of fence and fertilizer will change if the company changes the shape of the lots to (a) equilateral triangles or (b) regular hexagons.
Figure A: Square-shaped fields Figure B: Hexagon-shaped fields
The problem cluster above helps to elucidate the fourfold concept, which the CPS sets out to prove. First, as the phrase "co-think algebra/geometry" implies, the materials draw on a non-separable view of these two domains, which are typically taught separately in Algebra 1 and Geometry courses. In contrast, the CPS project will develop problems for each course that require students’ coordination and use of central concepts from both domains. Second, problem clusters will integrate puzzlers—problems that "puzzle" one’s thinking (brainteasers)—to foster an innovative, out-of-the-box (and often counter-intuitive) mindset to problem solving and to co-thinking algebra/geometry. Third, the project will attempt to further a two-facet notion of collaboration: among students as a way to augment their learning, and among classroom teachers and researchers as a way to establish 21st century practices. The model that will underlie each problem cluster emphasizes a progression from Concrete, through Representational, to Abstract (CRA) ways of operating and expressing solutions to problems. This model is consistent with recent approaches to learning via translating problem solutions across representations (e.g., Lesh, Middleton, Caylor, and Gupta , 2008). The CRA model enables students to participate productively in the intended learning processes. Fourth, the project will focus on mathematics teaching that is content-based for the presentation of central mathematical ideas. Collaboration and student learning will be facilitated via low-tech puzzlers (e.g. wooden cubes) as well as high-tech representations (e.g. GeoGebra).
The CPS project will produce three main outcomes. First, a set of 8-10 problem clusters per each of the two courses (Algebra 1 and Geometry) will be produced. The clusters will be created via an iterative development process. That is, each problem-cluster will be revised several times based on feedback from its use. Results and teachers provide feedback for (a) improving new versions of already tried problem clusters and (b) guiding and improving the creation or modification of other problem clusters not yet tried. Each cluster will consist of puzzlers, content-based problems, and real-world problems that link key concepts from both domains. Teachers will play a central role in the development process to ascertain that problem clusters are suitable, simultaneously, for (i) all students’ needs and backgrounds, (ii) use within existing curricula, and (iii) aligning current courses with Common Core State Standards and corresponding Florida Benchmarks. For each course, the project will provide student and teacher materials that enrich and empower an entire year of study. As the example of a problem cluster shows, the proposed project is consistent with the NCTM (1989, 2000, 2006) recommendations to develop students’ competence in solving unfamiliar problem situations, in gathering, organizing, interpreting, and communicating information, in formulating conjectures, in analyzing problems, in discovering patterns, in taking risks and experimenting with novel ideas, in transferring skills and strategies to new situations, and in developing curiosity, confidence and open-mindedness. Second, the project will provide preliminary research findings about how these materials promote student dispositions toward and learning through problem solving. Third, the project will provide a model of how to develop and implement innovative problem solving approaches to fit within regular school courses of study. That is, it will lay the groundwork for future large-scale professional development efforts for mathematics teachers that will lead to widespread dissemination of the integrated, algebra/geometry problem-based approach. All in all, this project focuses on making algebra and geometry problem solving a central theme in the curriculum so that high school students become stronger, more proficient problem solvers.
1. Significance of the project
Too many students in the US fail to develop robust mathematical understandings required for fully and productively functioning in a 21st century, innovation-driven, technological society (PCAST, 2010). Within mathematics, this failure is particularly evident in the domains of algebra and geometry. A mathematical aptitude that seems in serious jeopardy is solving non-routine, realistic word problems, a skill which requires making sense of relationships within a given situation, mindfully translating and expressing these relationships in proper mathematical ways, competently using standard or non-standard methods for finding a solution (answer and justification), and making sense of the solution while situating it back within the context of the given problem (Ferrucci and Carter, 2003). Via collaboration with local teachers, the CPS project will address these issues of national importance by developing problem-based learning materials and experiences that will be incorporated into the regular courses of study in large school districts. The nature of these problems and the pedagogical approach that underlies their development and implementation constitute a unique and promising strategy for overturning the current state of affairs for all students. The extent to which the promise is fulfilled will be studied empirically to both guide the curriculum development process and provide evidence for its promise to change teachers’ practices and students’ learning. Consequently, the project will provide a basis for future professional development of teachers that will expand adoption and impact of these materials and experiences. The following discussion elaborates on each of the points in this logical chain.
1.1. Mathematical Problem Solving: A Burning National Issue
The CPS project addresses a burning national issue—the massive proportion of American students who lack the mathematical understanding and problem solving competence needed to cope in today’s increasingly technological society. International comparisons, such as the TIMSS (Schmidt, 1998; TIMSS Video Mathematics Research Group, 2003), provided chilling evidence that US students are losing ground when compared to their counterparts in other countries. Likewise, the Programme for International Student Assessment’s (PISA) (OECD, 2010) report on mathematics achievements in 65 countries showed that US students’ average score (487) on this problem-based proficiency measure was below the average, far behind China (600), Singapore (562), Korea (546), and Japan (529). Moreover, US high achievers in problem solving (those scoring in the top 10 percent) were significantly outperformed by their counterparts in countries that lead the world in mathematics. According to Schmidt (1998) and the TIMSS Video Group (2003), unlike Asian teachers who use problem solving as the heart of their pedagogy, most US teachers still teach in a very traditional way—lecture based attempts to transmit facts and procedures while submitting students to memorization, drill, and practice of routine exercises.
This dismal state of affairs is echoed in national reports (NAEP, 2011). The National Center for Education Statistics (NAEP) (2009) results in mathematics for 12th graders—which include a substantial portion of realistic word problems in number properties and operations, measurement, geometry, data analysis, and algebra—reports that when considering 12th graders’ performance at or above proficiency level the national average was 25% (Florida – 19%). That is, while the critical and most challenging domain of problem solving is supposedly addressed in most mathematics curricula of each and every grade level (K-12), too many students are still lacking competence and confidence in it (NCTM, 1989, 2000, 2006; Burns, 1998).
The CPS project will take place in Broward County and Palm Beach County School Districts, the 6th and 13th largest in the country (respectively). Approximately 60% of the students in Broward County and 55% in Palm Beach County are eligible for Title I programs. At the end of the second semester of 2010, Palm Beach County reported that 48% of their students enrolled in Algebra 1 and 33% of the students enrolled in Geometry received a grade of D or F. In Broward County, 10th graders’ average scores on the annual Florida Comprehensive Assessment Test (FCAT, Spring 2011) were 51% and 45%, respectively. Broward County has more than 900 in-service high school course mathematics teachers and 263,000 students; Palm Beach County has more than 400 teachers and 172,000 students. This project will work with students in the Algebra 1 and Geometry classes of 8 teachers, 4 in each county, giving priority to socio-economic disadvantage schools.
The CPS project is significant in its concentrated effort to improve the teaching and learning of two courses that constitute the core of high school mathematics, Algebra 1 and Geometry. This effort will revolve around an extensive use of problem solving as a prominent means for and outcome of student learning. The project emphasizes every student’s learning to solve problems in algebra and geometry, because these domains are necessary not only for any future STEM study but also for productive partaking in STEM-related careers (NCTM, 2000, 2006; The National Academies, 2007). As the recent report by the President’s Council of Advisors on Science and Technology (2010) stated, “too many American students conclude early in their education that STEM subjects are boring, too difficult, or unwelcoming, leaving them ill-prepared to meet the challenges that will face their generation, their country, and the world” (p. vi). In particular, algebra has become a notorious gatekeeper to STEM studies and professions (Rech & Harrington, 2000), to the extent it is regarded as the civil rights of 21st century citizens (Moses & Cobb, 2001). As emphasized by the co-think maxim, the team embraces a perspective on the needed integration of algebra and geometry and will make sure that each problem cluster will foster such co-thinking.
Along with the focus on problem solving in algebra and geometry, the project’s significance must be understood in terms of its innovative approach to: (i) the development and implementation of problem-based learning materials as a genuine partnership among teachers, district experts, and teacher educators and researchers; (ii) the nature of problem clusters to be developed; and (iii) the pedagogical tenets that underlie the sought after changes.
1.2. Nature of Problem Clusters to be Developed
The CPS project draws on the growingly accepted wisdom about the central role that problem solving can and should play in students’ learning of mathematics. The grand theories of Dewey (1902, 1933) and Piaget (1971, 1985) delineated the role that one’s puzzlements play in both triggering and regulating a search for problem resolution that, in turn, brings about abstraction and generalization of a new idea. Similarly, Polya (1945/2004) articulated the process of interpreting and solving a problem. In a nutshell, he asserted that solving a problem begins with understanding it (including identifying what is given and what is unknown), proceeds to devising a plan for figuring out the requested unknown and carrying out this plan (including adjustments to meet unforeseen constraints), and finally looking back to the problem to assess whether the solution actually fits with the problem and makes sense in the larger scheme of things.
Learning a novel concept through these processes occurs as the problem solver explicates, operates, and reflects on obvious and implicit relationships among givens and unknowns. The need to continually translate between concrete and abstract representations of such relationships supports cognitive change (Lesh et al., 2008). For example, in the problem cluster above, making sense of the shape (square), dimensions (size of side), and properties (perimeter, area) yields a dynamic geometrical diagram. By operating on the diagram (e.g., changing the size of the side) and observing the behavior of the graph in comparison with one’s initial anticipation (e.g., linear), the solver notices and distinguishes between straight and curve lines produced as representation of the co-variation between the variables. Then, translating to an algebraic representation (equation, function) of the relationship provides a link and reason for distinguishing what is a linear function of the side as dependent variable (e.g., perimeter) from what is not. By reflecting across different instances of this relationship, a new algebraic concept—linear function—is likely to emerge for various geometrical properties. This concept can and should be linked to other contexts in which the invariant ratio between quantities underlies a linear relationship (e.g., constant price per product), and endows the slope with a meaning for that ratio. Schoenfeld (1985) emphasized that in this way solving problems is a means to introduce and explore new, fundamental ideas. Taplin (2008) argued that such a problem solving approach would augment teachers’ view of themselves as competent problem solvers, who can develop various strategies to deal with change in their classrooms (Taplin & Chan, 2001). Simply put, pedagogy shifts from "teaching problem solving" (after learning and mastery) to "teaching via problem solving".
To augment students’ learning via problem solving, Reed and Smith (2005) stressed using a variety of materials and strategies to solve problems. Drawing on Montessori’s focus on understanding children’s thinking, they suggested that variation of problem types and difficulty levels, and discussions of multiple solutions to those problems, provide teachers with a window into each student's understanding. In turn, the teacher's understanding of the students' mathematical thinking is used to select, adjust, and follow-up on problems that seem conducive to the students’ progress. According to Reed’s (2007) classification of problem types, the cluster above seems to belong in the "Inducing Structure" category. The questions in the cluster require identifying relationships among the components, fitting the relationship into a pattern, and testing and changing conjectures about this pattern (e.g., changing an initial anticipation that the graph for area will be linear). To identify the relationships, students need to develop and apply four skills: encoding, inferring, mapping, and applying. Identifying underlying structural relationships and representing them in abstract forms (e.g., a graph, an equation) was proposed as an effective strategy to help students not only understand mathematical concepts but also retain information in long-term memory and become competent in transferring and applying the knowledge in novel situations. Similar findings, about the role that teaching through variation of problems and solutions serves in students’ learning, have recently been reported in studies of Chinese mathematics teaching (Gu, Huang, & Marton, 2006; Jin & Tzur, 2011). There, variation of problems, and materials and processes used for solving them, serves as a central pedagogical tool for shaping students’ (and teachers’) learning environment, gauging what students understand, and providing suitable challenges to perturb and promote their thinking.
An important question concerning teaching mathematics via problem solving is how problems are to be selected and structured. The CPS project team will answer this question by using a Backward Design approach (Wiggins and McTighe, 2005). This approach draws on the observation that, too often, teachers and curriculum designers begin with favored activities and lessons. Instead, backward design begins with the articulation of learning goals—understandings, competencies, and skills expected of students. Six facets of understanding are distinguished in this approach: explaining, interpreting, applying, developing a perspective, empathizing, and self-knowing. These facets should constitute a vision of lasting comprehension of "big-ideas." For each learning goal, the designer then selects instruments to obtain evidence for student progress. McTighe and Thomas (2003) stressed the multiplicity of data sources needed for analyzing and assessing student growth (e.g., performance tasks, tests, homework, self-assessment). Finally, planning of learning experiences and instructional methods takes place. This approach differs from traditional planning that mainly attempts to "cover" materials. For example, the problem cluster above was generated by integrating the key understandings of function (starting with linear) as an expression of co-variation (algebra) and the impact of changes in dimensions on properties of 2-D figures (geometry). These key understandings were identified via scrutiny of reform-oriented curricula, of Common Core State Standards and corresponding Florida Benchmarks, and discussions with mathematicians and teachers about understandings that are critically needed and too often missing in high school graduates. Then, oral, written, and bodily manifestations of initial states of knowledge (e.g., students’ expectation to see a straight line produced for the area of a square) and desired states of new understandings were identified (e.g., hand gestures of curve, or a written explanation of the constant change that underlies changes in perimeter). Finally, a commencing puzzler, a GeoGebra applet, a real-world problem (fencing and spraying lots), and follow-up prompts (e.g., “What if …?”) were created.
This unique, purposeful clustering of puzzlers, real-life, and algebra/geometry content-specific problems, which begins from the intended concepts and proceeds to assessment and teaching methods, heightens the significance of the CPS project. Separately, each of these types of problems has been addressed. For example, Movshovitz-Hadar and Webb (1998) provided ample examples of puzzlers that can instigate curiosity and learning. Other researchers have advocated the use of puzzlers and games in teaching mathematics because solving them contributes to students’ motivation and learning (Hill, et. al. 2003, Rao, et. al. 2006, Levitin 2005). Through solving puzzlers and games, students process mathematical ideas that can be linked to various contents. A recent study (Deslauriers, 2011) demonstrated that, in an interactive class that employed puzzlers and brainteasers, 71% of the students have productively participated in the learning process as compared to 41% of their counterparts in a non-interactive class. The increased level of engagement seemed supportive of students’ grasp of complex concepts. Working with puzzlers, a component of each problem cluster, promotes students’ learning because it enables them to explore mathematical concepts and develop abstract reasoning while engaged in hands-on, visual, curiosity-enhancing activities. Moreover, principles that can be demonstrated using puzzlers and games include dealing with constraints, intuition and counter-intuition, and visual and verbal thinking that help promote an inquisitive mindset in students.
Apart from the use of puzzlers, in the past two decades real-life problems have become commonplace in reform-oriented curricula such as Connected Mathematics (Lappan et al., 2002) and Core-Plus (Coxford et al., 1998). Also, high-tech tools, such as dynamic software for geometry or algebra (e.g., GSP, Matlab, Excel) have been incorporated into mathematics classrooms around the country. GeoGebra has emphasized integrating geometry and algebra into a single, free-software package. Mindful integration of the three problem types (puzzlers, real-world, content-based) and solution processes (low-tech, high-tech), to support teachers’ work and students’ learning, seems direly needed. The pedagogical tenets to accomplish this are elaborated in the next sub-section.
1.3. Project Pedagogical Tenets
The pedagogical approach that will guide development and implementation of the problem clusters achieves significance from its unique synthesis of frameworks of mathematics learning and teaching. The project will draw on the Concrete-Representational-Abstract (CRA), evidence-based model of teaching mathematics, the corresponding reflection on activity-effect relationship constructivist framework, the innovative (free) GeoGebra software and its suitability for cooperative learning, puzzlers, and a curriculum development strategy that centers on genuine partnership with classroom teachers and school experts. The following discussion elaborates each of these five points.
1.3.1 The Concrete-Representational-Abstract (CRA) Model
The CRA model draws on Dewey’s (1933) and Piaget’s (1971, 1985) constructivist theories. Dewey asserted that to help students develop understandings of abstract concepts teachers should commence learning of those concepts by solving problems in concrete contexts. He emphasized that manipulation of objects and reflection on ways in which a puzzling aspect of a problem situation is being addressed provide the human mind with "raw materials" needed to meaningfully grasp adults’ highly structured ideas. In Dewey’s (1902) words: “Hence the need of reinstating into experience the subject-matter of the studies, or branches of learning. It must be restored to the experience from which it was abstracted. It needs to be psychologized; turned over, translated into the immediate and individual experiencing within which it has its origin and significance” (p. 29). Likewise, Piaget contended that the construction of new schemes through transformation of existing ones is an active mental process. He stressed that mental activity suitable to the construction of intended, new schemes is often triggered through a cognitive perturbation (puzzlement) and sustained via activities on concrete and/or mental objects—activities that the mental system then interiorizes and coordinates. Scholars who drew on these giants’ works, such as von Glasersfeld (1995), Steffe (1991), Thompson (1985, 1991), Pirie and Kieren (1992), and Lesh et al. (2008), have all maintained the key role played by actions on concrete objects, coordination of those actions, and continual shifts between expressions of those actions, play in the reflective process of abstracting a new mathematical idea. This constructivist premise, of the need to organize learning experiences that proceed from concrete to abstract, has become commonplace in the mathematics education community (NCTM, 1989, 2000, 2006). In recent years, researchers in the learning sciences further supported this premise (Bransford et al., 1999). Witzel (2005) contended that, when applied to teaching mathematics, the CRA model is an evidence-based instructional practice that consistently engenders successful learning and progress of students at all achievement levels (low, medium, and high) and for all grade bands (elementary, middle, and secondary).
The model problem cluster above illustrates the nature of CRA-based design of problems. To solve the real-world problem of fencing and spraying square lots, students can be given cardboard sheets and asked to draw a 1cm X 1cm grid on them, then cut out squares of the size given in the problem (e.g., using 4 cm in place of 4 yards), surround the edges with a piece of paper (to measure all four sides), and find the perimeter and area of the given lot. They can repeat this concrete, low-tech experience for a few more lots, and record their measurements in a 3-column table (side, perimeter, area). By reflecting on and coordinating their actions of producing the squares and the measures, patterns and conjectures about them can be noticed and analyzed. Next, the few points students have produced concretely can be charted on a graph, to be followed by a discussion of ways to extrapolate from these points to an entire line in the first quadrant, including reasoning as to the shape of the graph (straight or curve line) and why, for this real problem, it should not include the origin and points in the third quadrant. Next, the graph and table of values can help students to write the equation of each situation (perimeter, area), which would naturally lead to using GeoGebra as a tool for co-thinking the relationship between the evolving models of their work, from concrete to representational to abstract, hence to conceptualizing and consolidating the intended concepts. With the help of dynamic sliders in the GeoGebra file, students can easily change the values given in the problem, and thus conjecture about (“what if …?”) and analyze the behavior of a generalized set of points that lie on each graph.
Whereas mathematics educators seem to espouse the CRA model, it has not been widely implemented in US schools. In part, this is explained by Cooney’s (1999) contention that teachers are substantially influenced by their own experience as students in traditional, non-CRA classrooms. Another reason seems to be the lack of materials that are both organized in the CRA model and fit within the already packed, test-driven curricula and school culture. To foster students’ development of the dispositions and competencies required of a 21st century problem solver, teachers must become such solvers themselves and have the proper tools for the task at hand. At the concrete level, puzzlers provide such tools, enabling students to learn in a fun and rewarding way while playing “low-tech” games that promote group cooperation, discussion, and reflection. At the representational level, various media (e.g., GeoGebra) encourage the representation of data and expression of ideas, which in turn facilitate abstraction and support transfer of knowledge to novel situations. In short, providing teachers with CRA-based practices and learning materials will make them more likely to help their students become expert solvers, by constructing a rich variety of mental schemes that consist of strategic knowledge (conceptual and procedural) for solving problems.
1.3.2 Refection on Activity-Effect Relationship
Building on the aforementioned constructivist works, Tzur, Simon, and their colleagues (Simon et al., 2004; Simon & Tzur, 2004; Tzur & Simon, 2004; Tzur, 2007) have recently proposed a comprehensive framework that articulates a mechanism of cognitive change in learning a new mathematical conception, along with a corresponding account of mathematics teaching. The mechanism of reflection on activity-effect relationship (AER), postulated to underlie abstraction of a new conception, is the core of their framework. This mechanism commences with a learner’s assimilation of problem situations into extant (assimilatory) conceptions (e.g., he/she knows squares, fencing and spraying, measuring sides, calculating perimeter and area). The learner’s assimilatory conceptions set the situation and goal—a desired, anticipated state to guide the problem solver’s activity (e.g., produce conjectures about graphs for different perimeters and areas, relative to the side). The learner’s situation and goal then call up, and regulate from within the mental system (Piaget, 1985), execution of a pertinent activity sequence (e.g., calculate values of sides and corresponding perimeters and areas, organize these values in a table, chart them as points on a graph, link the points). While running the activity sequence, a learner may notice gaps between its actual effects and the anticipated result, as well as effects not noticed previously (e.g., the graph for perimeter is a straight line, as anticipated, whereas the graph for area is not). Through reflection on and reasoning about solutions to similar problems (e.g., changing the sides, changing from square to hexagon), the learner abstracts a new invariant—a relationship between an activity and its anticipated and justified effects (e.g., co-variation of the side and perimeter is constant, so all points end up on the same straight line, and the slope represents the multiplicative constant of the number of sides). The ensuing regularity (invariant AER) involves a reorganization of the situation that brought forth the activity in the first place, that is, the learner's previous assimilatory conceptions. For instruction, the crucial implication of this mechanism is that it clearly distinguishes between the teacher’s goals for what students need to learn (e.g., linear function and its link to side-perimeter co-variation) and the student’s own goal in the activity (e.g., produce, graph, and make sense of a set of points).
Accordingly, the AER framework defines conception as the abstract relationship between an activity and its effects, implying that an activity is a constituent of a conception (e.g., producing and charting points, motivated by a conjecture of the shape of the graph, becomes part of a co-thinking linear function). This view is contrasted with the view of activity as a catalyst to the learning process or a way to motivate learners, to which von Glasersfeld (1995) referred as "trivial constructivism". The view of conception and learning mechanism defined by the AER framework is consistent with and draws on recent studies on brain and learning (Bransford et al., 1999; Tzur, 2010; Tzur, 2011).
1.3.3 High-Tech Tool to Co-Think: GeoGebra in Cooperative Learning Situations
An important feature of the CPS project significance is the intended wide use of the innovative, open-source, multi-language, dynamic GeoGebra software ( www.geogebra.org ). As the diagram in Box 1 (problem clusters) shows, in this platform students can work together to quickly and easily produce various geometrical figures and algebraic expressions (e.g., value tables, graphs, equations). Those figures and expressions are linked in the software so students can act on any of them and observe changes in the other. Our experience of introducing the software to teachers and students indicate that they (i) swiftly become facile with the software, (ii) enjoy the explorative nature of problem solving processes, (iii) work cooperatively to solve and pose problems in it, and (iv) learn mathematical ideas through this work (largely through the geometric visualization of the real-world problem and the algebraic, abstract representation).
To the best of our knowledge, GeoGebra was developed based on a constructivist theory of learning (Hohenwarter, 2006) but independently of the reflection on AER framework. However, the above description of that framework, including specific allusions to the sample problem cluster, indicates that the software can become an essential tool for engendering students’ construction of concepts as explained by the framework. In the hands of dedicated and well-informed teachers, such a tool will support the two key aspects of the reflection on AER mechanism, namely, that learning entails a transformation of one’s anticipation of the effect of an activity, and that it occurs through comparisons between the anticipated and actual effect (e.g., anticipated a straight line for the graph of area but found a curve) as well as across situations in which the new anticipation proved valid (e.g., the graph for perimeter is linear for any polygon). That is, students’ actions in, reflections on, and coordination of actions afforded by GeoGebra seem to support the very mental comparisons postulated by the framework to foster construction of a new, intended concept. Such continual coordination of algebraic and geometrical mental actions, which underlie a student’s (and teacher’s) use of GeoGebra as a representational tool, is precisely the reason we expect co-thinking concepts from both domains to develop. Accordingly, guided and independent co-explorations of problem clusters will draw heavily on GeoGebra as the "representation" in the CRA model.
An added value to the significance of the CPS project is found in the close ties between the developer of GeoGebra and personnel at FAU. Based on his doctoral dissertation, Dr. Hohenwarter continued developing GeoGebra while at FAU, working with classroom teachers and implementing changes suggested by them. He also worked closely with team members of the CPS project. In particular, Ana Escuder was his student and became an expert in extensive use of the software to promote mathematics learning (and teaching) at FAU and the school districts where the project will be conducted. These ties will enable continual exchanges between the team and Dr. Hohenwarter (now a professor in Austria), leading to highly synergized improvements of both the software and its use in service of co-thinking throughout the Algebra 1 and Geometry courses.
The significance of the CPS project is augmented by ensuring that each of the 8-10 Algebra 1 clusters and 8-10 Geometry clusters will be developed to suit cooperative learning. Cooperative learning approaches have been advocated in general (Johnson & Johnson, 1983) and in mathematics education in particular (NCTM 1989, 2000; Davidson, 1990). One reason this approach is conducive to learning via problem solving is the support given to students whose inability to adequately read and comprehend a word problem may hinder their productive participation in the process. Benko (1999) asserted that cooperatively using ample illustrations for problems, which are read in learning groups, enhances students’ attitudes toward solving word problems—willingness to embark on this challenging process and persistence in completing it. Seen through the reflection on AER lens, another reason that the cooperative approach is conducive to learning via mathematical problem solving is the continual cognitive support provided by exchanges of ideas among students in a group. Every student in a group assimilates these exchanges into her own evolving anticipation of AER. In turn, this assimilation prompts comparison to others’ anticipations (e.g., I thought the graph for area would be a straight line and you said it would not—let’s figure this out), which by definition promotes comparisons across mental runs of the activity sequence as well as renegotiation of the goal that regulates one’s own mental activity. Tzur (2008, 2010, 2011) stressed that these cross-situational comparisons are necessary for transition to the robust, transfer-enabling (anticipatory) stage but are not occurring automatically in the brain. A cooperative learning group repeatedly provides cross-situational comparisons and thus enhances transition from none, to participatory, to anticipatory stage of the novel, intended concept.
As the sample cluster in this proposal shows, the GeoGebra platform provides substantial support to cooperative learning. A students’ group approaches problems in a cluster as a shared task. They have to negotiate and renegotiate sequences of actions, potential operations in the software to create algebraic or geometrical objects, and manipulate them to test specific conjectures (anticipations of effects to their GeoGebra actions), and thus engender continual coordination of their mental actions. Key to the support that the GeoGebra platform gives to solving problems cooperatively are the multiple ways in which every member of the group may use it to work on a problem (e.g., you created a slider for the side; let me try to use the same slider for a hexagon; no, we should have different sliders for each figure; why? because we can better control the graph). Such negotiations are the core of critical learning processes of socio-mathematical norms and practices in classroom environments that emphasize social interaction (Cobb, Yackel, and Wood, 1992; Davidson, 1990; Yackel et al., 1990).
Consequently, employing GeoGebra in service of solving CRA problem clusters seems highly conducive to cooperative learning experiences, which in turn is highly supportive of reflecting on one’s and others’ actions to abstract taken-as-shared ideas. Steffe & Tzur (1994) have discussed the theoretical underpinnings of such pedagogy, suggesting that it promotes corresponding, socially rooted zones of proximal development (ZPD) (Vygotsky, 1978) and cognitively rooted zones of potential construction via cooperatively facing and resolving perturbations that are bearable to the students. Cooperative learning through co-exploring CRA-based problems in GeoGebra seems highly supportive of group members’ providing and assimilating prompts that enable one’s work (and learning) at a level not accessible to him or her independently.
Although “computer-supported cooperative work” is a generic term, it is well accepted that it deals with computer systems (e.g., GeoGebra) that help to support collaborative goal-oriented problem solving activities. It deals with sharing knowledge, cooperation between individuals that share their work, and adaptation to technology. Ackerman (2000) discussed the socio-technical gap in “computer-supported cooperative work,” i.e., the “divide between what we know we must support socially and what we can support technically.” There are several different elements in “computer-supported cooperative work.” Baecker (2000) discussed a four-quadrant related matrix that presents possibilities for collaboration in time (same or different) and space (same or different).
(a) “same time same space,” e.g., face to face interaction;
(b) “same time different space,” e.g., real-time remote interaction;
(c) “different time same space,” e.g., work that is accomplished by the same or different group(s) in the same room but at different time slots; and
(d) “different time different space,” e.g., off line e-mail communication.
In this proposal we are focusing mostly on (a) face-to-face computer-supported interaction. “Student’s learning of mathematics is enhanced in a learning environment that is built as a community of people collaborating to make sense of mathematics ideas” (NCTM, 1991, p. 58). According to Duarte, Young, and DeFranco (2000) technology enhances the learning environment by providing opportunities for students to investigate ideas, verify their thinking, construct graphs and diagrams, and discuss their ideas with peers and adults. Sheets and Heid (1990) believe that even if teachers do not plan for group work, technology fosters the development of collaboration in small groups as a result of the public character of the computer screen, the need for interaction with computer programs, and the need for discussion when students share computers. There is growing interest in the research community to explore creativity and leadership in “computer-supported cooperative work,” (Farooq 2008; Olson 2008; Thompson 2006). Mullins (2011) has studied collaborative learning in student computer supported activities in mathematics. For practicing procedures, collaboration may result in reduction of skill capability. However, for conceptual material, there was a positive impact in conceptual knowledge acquisition by collaboration.
1.3.4 The Role of Puzzlers in Concept-Based Learning
Giguette (2003), Rajaravivarma (2005) and, more recently, Luxton-Reilly and Denny (2009), among many others, have shown how games can be used to teach introductory concepts with successful results. The use of puzzles in early stages in the computer science undergraduate curriculum is also illustrated by the “Using puzzles in IT” introductory course described by Cha, Kwon, and Lee (2007). They asserted that such a course “is not only helpful to introductory students but also to non-major students.” They also indicated that “After the course of using puzzles in IT, students comment that it is more easy and interesting than other courses” (p. 140 from 135-140) Hill and colleagues (2003) demonstrated how puzzles and games could address different learning styles. Recently, Cicirello (2009) described an exercise called Collective Bin Packing -- an adaptation of the well-known combinatorial optimization problem known as bin packing -- and showed that it can be used to successfully engage and allow students to learn by discovering their own solutions rather than being told what to do (p. 182-186)
The work by David Ginat at Tel-Aviv University provided another eloquent source of support for the ideas presented in this proposal. Ginat (2007) wrote: “Mathematical games arouse enthusiasm and challenge. They usually involve clear and simple rules, with physical, visual, or numerical entities, which raise motivation and intuition” (p. 32). He emphasized that game playing does not take away the rigor with which algorithms and associated topics must be treated. In fact, it reinforces the notion that rigor is necessary! Ginat reported additional successful outcomes, such as increased student enthusiasm, successful pair interactions, free flow of richly creative ideas, and increased problem solving ability among students.
Consider the following example problem, which can be solved by an individual, or collaboratively by a group of students with or without the use of a computer. “Take the integers from 1 to 25 (inclusive) and arrange them in a row such that each pair of adjacent numbers sums to 2^i or 5^i, for some positive integer i.” For example, consider the following arrangement:
1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
The numbers 12 and 13 are adjacent, and they sum to 25, which is 5^2. Also, 3 and 5 are adjacent, and they sum to 8, which is 2^3. However, 1 and 2 are adjacent, and they sum to 3, which is not 2^i or 5^i for any positive integer i. So, the above arrangement is not correct. A discussion of this example is included in Appendix B . The important part of solving this problem is the multiple directions for approaching it, open discussions, divergent thinking, and thus how a solution may be achieved.
1.3.5 Partnership with Teachers
A major challenge to every project that develops new materials for teaching mathematics to students is the extent to which classroom teachers actually endorse and implement the project’s products (Zaslavsky, 2008). The final reason we bring forth in support of the significance of this project is the likelihood for such implementation to happen due to the spirit of partnership with teachers, which has characterized our work with school districts in the region where FAU lives and functions. For more than 15 years, the Department of Mathematical Sciences at FAU has had a close working relationship with the (adjacent) Broward County School District, through a series of NSF-funded mathematics education projects of Dr. Heinz-Otto Peitgen and Dr. Richard Voss. Most recently, they are completing the 5-year Math and Science Partnership Institute titled Standards Mapped Graduate Education and Mentoring for Middle Grade Math Teachers, in which one of the co-PIs on this CPS proposal (Ana Escuder) has been the project manager. As part of these projects, FAU has partnered with classroom teachers, expert teachers who coach them, and district mathematics coordinators, while carrying out tasks conducive to their mission and professional development. This CPS project will rest on and expand the existing personal and professional relationships and networks that we have built during those years.
We are therefore confident in our ability to recruit teachers as genuine partners in the project, to work with the team continuously to co-develop the problem clusters. We will have two Algebra 1 and two Geometry teachers in each of the two school districts. We will attempt to recruit them in pairs, an algebra teacher and a geometry teacher from the same school building. They will be working with us from the initial stages of development, to provide their insights as to (i) the central concepts for which problem clusters should be developed (based on curricula they use as well as Common Core State Standards and corresponding Florida Benchmarks), (ii) how and where in the existing courses of study the problem clusters will fit, (iii) which problems could be most helpful to their students, and (iv) how to organize the materials to be as teacher-friendly as possible. We will partner with them in implementing the materials, debriefing about the ways teaching-learning processes were promoted, suggesting revisions, and collecting and analyzing data about the promise of the revised materials for promoting students’ learning and dispositions.
One Mathematics Curriculum Specialist (district coordinator) in each school district has already been working with us in conceptualizing and writing this proposal, and they are excited to support the innovative, challenging, and promising work on which we all wish to embark (see Letters of Support in Appendix C). These district coordinators will be instrumental in recruiting the 4 classroom teachers from each district mentioned above. We also expect to employ two expert teachers (coaches) from each district to work as “Professional Development Assistants” on the project, helping to create the problem clusters and working on the modification process. Certainly this partnership between FAU and the local school districts will pave the way to future scale-up of the project, so that eventually hundreds of teachers can adopt (and adapt) the materials based on their peers’ guidance, and thousands of students can benefit from learning Algebra 1 and Geometry via proficient problem solving.
2. Research Plan
2.1 Theoretical Underpinnings of our Study
The Piagetian cooperative social constructivist conceptual framework will guide the CPS research and development effort. This framework will consist of the aforementioned components: The Concrete-Representational-Abstract (CRA) model of instruction within the problem solving process; the GeoGebra dynamic tool for co-thinking problems in both domains; the reflection on activity-effect relationship framework for both learning and teaching a new mathematical concept; and the cooperative learning problem solving approach to instruction. Each of these components has been articulated in Section 1.3 above. Here, it suffices to summarize briefly the theoretical underpinnings of constructivism:
(1) Mathematical knowing does not exist independently and outside of humans’ minds; rather, it is afforded and constrained by one’s (mental) activities. People achieve high degrees of shared understandings based on having compatible anticipations and cooperative interactions with others through compatible learning experiences (von Glasersfeld, 1995). It is critical that students know the mathematics content from the Common Core State Standards and then be able to apply this knowledge to solve sound word problems in a CRA, cooperative manner.
(2) Consequently, mathematics learning entails an interactive process of constructing new (to the learner) anticipations—coordinated, justified mental actions and their meanings for the person—via continual interactions in one’s social and physical milieu where students have learned this while using worked examples. These anticipations are held in continual check against newly noticed effects of mental activities and adjusted to fit one’s experiential reality, which always includes social exchanges while solving problems cooperatively (Cobb, 1994; Piaget, 1985; Simon et al., 2004; Tzur & Simon, 2004; von Glasersfeld, 1995).
(3) Teaching mathematics commences with the premise that one person’s knowing (e.g., teacher) cannot be directly transmitted to and passively received by another person, nor does it amount to fostering memorization and mastery of facts and procedures. Rather, it requires indirect orientation of students’ thought processes, via engaging them in problem solving situations that trigger particular goals and activities (mental operations that may be part of physical actions) toward those goals, orienting students’ noticing and linking effects of those activities, including possible changes in the original anticipations, and orienting students’ reflection onto things that change and things that are anticipated to remain the same across situations (Steffe, 1991; Simon & Tzur, 2004; Tzur, 2008).
(4) Students’ productive engagement in the learning process is a crucial variable in their learning, so they both apply themselves and advance toward the intended concepts. To support such engagement, an inquisitive and risk-taking mindset is needed, including a willingness to bring forth intuitive thoughts that may turn out to be wrong, and a healthy disposition toward making and correcting mistakes as part of the learning process. Thus, teachers need to create a learning environment in which students feel safe to think, share, and critique, and are eager to explore new ideas (NCTM, 2000; Marzano, 2011). A cooperative constructivist approach to doing problem solving enables students to gain confidence, to extend understanding, to see others’ views and approaches, and to lessen anxiety of doing the problem solving alone (Cobb, 1994).
2.2 Context and Rationale for the Proposed Intervention
The context and rationale for the proposed intervention—material development and research—is the aforementioned dismal state of affairs of US students’ mathematical achievements and dispositions, as measured by both international (Schmidt, 1998; OECD, 2010) and national (NAEP, 2011) studies. This state of affairs has been discussed in the Purpose and Significance sections of the proposal, which explain that a critical facet of the problem involves students’ inability to solve mathematical problems (coupled with negative dispositions toward such activities). Starting in the 1980s, this state of affairs has generated substantial reform efforts, as published by the NCTM (1989, 1991, 2000, 2006) and supported by federal, state, and local educational agencies. While the problems are widespread, particular subgroups (e.g., females, African American, Hispanic, and ELL students) have been disproportionally found at the underachieving end of the spectrum (Stiff, 1990; Tate, 1994, 1997).
Mathematics is an essential discipline because of the vital role it plays in individuals’ lives and the society at large. Resnick (1987) recommended a problem solving approach to appreciate the practical use of mathematics. Likewise, Cockcroft (1982) advocated problem solving as a means of developing mathematical thinking, and as a tool for daily living, saying that problem solving ability lies “at the heart of mathematics” (p. 73), because it is the means by which mathematics can be applied to make sense and function in a variety of unfamiliar situations. Problem solving is a vehicle for teaching and promoting students’ mathematical knowledge, and can help to meet everyday challenges and to enhance logical reasoning. In the 21st century, individuals can no longer function optimally in society by just knowing the rules to follow for obtaining a correct answer (Taplin, 2008). Rather, as pointed out by Lester, et al. (1994), the emphasis must shift from teaching problem solving to teaching via problem solving.
A decade ago, a major national undertaking commenced under the No Child Left Behind Act (US Department of Education, 2002). This law attempted to rectify the situation by creating measures for students’ success, teachers’ quality, and school district effectiveness (to the extent of closing schools that fail the target growth in students’ achievements). In turn, the NCLB law brought forth a wave of pleas for generating nationally accepted standards for mathematical understandings and mastery, as well as measures (benchmarks, end-of-course state exams) to assure that those standards are met. These led to the recent publication of Common Core State Standards, which were adopted (and adapted) by most states, Florida included. But all these reform efforts have yet to yield substantial change in the way mathematics is taught and learned, so that solving non-routine problems will become the main road to successful learning and career pathways. To this vital end, problems that can be interwoven into the regular courses of study and which promote students’ understandings of central mathematical concepts must be created, tried out, revised, and tested for promise in promoting students’ learning and dispositions.
The CPS proposal has been revised and resubmitted in order to address this challenge. Reviewers of the first and second submissions of this proposal pointed out a number of weaknesses, and these criticisms were extremely helpful in revising the proposal. (Responses to those are detailed later in the narrative.) However, the most critical realization on the part of the team was the need to field test problem clusters in regular classes, before attempting to train a large number of teachers for using those materials and conducting a large-scale study of their impact. Consequently, the context and rationale for the CPS project amount to focusing on a critical first step—developing problem clusters with teachers, and study their implementation and promise in promoting students’ learning through this proposed 3-year effort.
As explained above, the project will be conducted in the context of Broward County and Palm Beach County school districts. These are two of the largest school districts in the country, with 263,000 and 172,000 total students, respectively, of which more than 71,000 and 35,000 students are enrolled in high school mathematics classes, respectively. Approximately 60% of the students in Broward County and 55% of the students in Palm Beach County qualify for Title I programs. These figures indicate that, if properly selected, schools and teachers who participate in the project can help to substantiate the impact that interweaving CRA-based problem clusters into the regular Algebra 1 and Geometry courses has on student learning and dispositions—particularly those from underrepresented groups as well as students interested in the STEM fields.
Located at the heart of this region, with its main campus on the border between these two counties, Florida Atlantic University (FAU) and its mathematics, engineering, and education schools are strategically situated to conduct the proposed project. The FAU team is not only solidly networked in those districts, but also brings cross-disciplinary expertise for generating and interweaving innovative approaches and tools into mathematics classrooms. In particular, the FAU team has developed expertise in teaching teachers how to use GeoGebra ( www.geogebra.org ). This is a new, interactive mathematics learning technology which has gained growing international recognition since its official release in 2006, because of its open source status, international developers, and a cross-disciplinary user-base of mathematicians, mathematics educators, and classroom teachers (J. Hohenwarter & M. Hohenwarter, 2009; Hohenwarter & Preiner, 2007). The rationale for using such a software platform is that mathematics teaching-learning technologies are reshaping the representational dimension of mathematics education and providing the world community with easy and free access to powerful mathematical processes and tools (Kaput et al., 2002). As shown in the sample problem cluster, GeoGebra allows learners to operate algebraically and geometrically in realistic problem situations, and invent and experiment with personally meaningful models while using multiple representations and tools to construct increasingly abstract mathematical ideas. GeoGebra is web-friendly and freely available to the international community (with multiple languages), and, most importantly, it is highly supportive of instigating the reflection on AER mechanism of cognitive via individual and social interactions.
Sparks (2011) asserted that as the STEM fields become more important for our students to study in the USA, schools and teachers need to do more to promote confidence in mathematical problem solving so that our students are confident to study areas related to STEM. Attending to methods that are effective at boosting such confidence are important. Geist (2010) felt that negative dispositions toward mathematics and problem solving are serious obstacles for young people in all levels of schooling today. In his paper, the literature is reviewed and critically assessed in regards to the roots of negative dispositions and their detrimental effect on "at-risk" populations, such as special education, low socioeconomic status, ESOL, and females. He asserted that a confidence-boosting curriculum is critical for students’ learning and use of mathematics. Willis (2010), a mathematics teacher and neurologist, in her book, Learning to Love Mathematics, provided over 50 research-based strategies teachers can use right away in any grade level to decrease negative attitudes about mathematics and problem solving, increase willingness to make mistakes, and relate mathematics to students' interests and goals. These strategies support students’ understanding and can help build foundational skills in mathematics and other subjects, including long-term memory of academic concepts and problem solving. Willis felt that classroom interventions that can help students include the following: changing students' mathematical intelligences by incorporating relaxation techniques, humor, visuals, and stories; eliminating stress and increase motivation to learn mathematics by using problem solving strategies, estimation, and achievable challenges; and differentiating strategies to fit with diverse students' skill levels via scaffolds, flexible grouping, and multisensory input.
The CPS project will promote student confidence by emphasizing the strengthening of problem solving in our students and by promoting students’ use of processes of modeling, representing, and then solving them from concrete to an abstract stages via innovative technology (GeoGebra). Our goal is to develop a set of problems conducive to all stages of the CRA approach, and making them available online for teachers. Overall, the project emphasizes problem solving, cooperative learning, the CRA model for learning mathematics, and the use and implementation of technology in the learning of mathematics. Our goal is to improve our students’ ability and confidence to do mathematics and to solve a variety of applied problems. In light of work from Kirschner, Sweller, & Clark (2006), educators need to be careful and exert minimal guidance when instructing students. While using a constructivist inquiry and problem-based approach, it is important that the problem solving process consist of appropriate selection of problems and making sure that students have prior knowledge needed to successfully solve them collaboratively. This project will use ongoing and sustained guidance and assessment of student performance based on Marzano’s (2011) assessment research, which is being implemented in Florida school districts and, in particular, in both partnering districts. Marzano (2011) contended that educators and leaders should engage teachers in focused practice, helping them develop specific classroom strategies, and integrating technology as an instructional strategy for student success. By co-developing materials with teachers and our use of technology (e.g.,GeoGebra), we can provide ample guidance for adoption and implementation.
2.2.1 Intervention, Theory of Change, and Theoretical/Empirical Rationale
As the nation is adopting the Common Core State Standards and the state of Florida is implementing end-of-course exams for algebra and geometry, mathematics teachers need help to get their students ready to perform based on these new requirements. The proposed project will consist of developing unique and innovative materials and professional development supports for teachers, targeting problem solving in algebra and geometry.
The proposed work is outlined in Section 2.3 . As the 16-20 problem clusters are being created and revised, the 8 teachers will interweave them into their regular Algebra 1 and Geometry courses. As the examples of problems in this proposal indicate, students’ work on each problem cluster may span 2-3 lessons. To make the interweaving reasonable for both teachers and students, and support rather than interfere with the courses during the first seven months of an academic year (prior to the spring standard exam), these numbers entail implementing one cluster about every 3-4 weeks. Some problem clusters may be implemented to initiate a unit, others during its development, and still others toward the end of a unit. (See Appendix A. Model of Change.)
The theoretical rationale for the intervention has been articulated in the Purpose and Significance sections of this proposal. The practical rationale lies in the development strategy outlined in Section 1.4. The core of this rationale is the premise of genuine partnership with teachers in developing, implementing, revising, and studying the impact of the problem clusters along with the mapping of the Common Core State Standards and corresponding Florida Benchmarks into the existing curriculum.
2.2.2 Practical importance
The proposed intervention promises to improve all students’ achievements and dispositions by fostering their understanding, integration, and mastery of central concepts in algebra and geometry. Aligned to the Common Core State Standards and corresponding Florida Benchmarks, the materials will engage students in learning the intended concepts through solving non-routine, challenging problems tailored to their developmental levels and present context. Moreover, to ensure fidelity of implementing these materials, teachers will be genuine partners in the development of materials and pedagogical approaches needed to teach such problem solving (via the CRA approach and the use of GeoGebra). It is this approach that constitutes the innovative nature of the proposed project, which will lay the groundwork for the development of workshops for training large numbers of teachers.
2.2.3 Rationale Justifying the Importance of the Proposed Research
Whereas the previously described overarching rationale for this project is clear, and the problem clusters to be developed are potentially powerful, the ultimate test is how developed materials are accepted and used by teachers and their promising effect on students. Accordingly, the proposed research will address processes of developing problem solving clusters and infusing them into existing courses of study, and the impact these materials make on student dispositions. Addressing the five research questions listed in Section 2.3.1 below will provide valuable information about the potential to improve high school students’ capacity to solve problems, and for designing learning materials and creating teaching-learning processes that make substantial use of an open-source technological tool (GeoGebra). Simply put, the rationale for the proposed research is that it will provide empirically grounded proof-of-concept. The following section delineates the CPS project research and evaluation plans.
2.3 Methodological Requirements - Research Plans
2.3.1 Research and Development Aims and Plan
The overarching purpose of the research component of the CPS project is to address the problem: How can teaching/learning experiences of solving non-routine, innovative problems be designed, implemented, and refined to coordinate central concepts of Algebra 1 and Geometry? To this end, we will:
(4) Partner with 8 high school course teachers, 2 district math curriculum specialists, and 4 coaches (expert teachers) in two districts to develop 16-20 problem clusters (8-10 per course), each consisting of puzzlers, real-life, and content-specific problems.
(5) Collaborate with and study those teachers’ implementation of the clusters in their classrooms during the first academic year of the project, and revise the materials in succeeding years.
(6) Collaborate with and study those teachers’ implementation of the revised clusters while studying the promise of using them on students’ successes/difficulties and dispositions toward problem solving, then revise and finalize the materials.
Accordingly, the development process will be iterative in nature and done entirely in partnership with classroom teachers, coaches, and district math coordinators. We will give a high level overview of our development process and elaborate below. The Year-1 Summer phase will consist of scrutinizing available curricula, the Common Core State Standards, and corresponding Florida Benchmarks. The goal is to identify the big ideas for each course (Algebra 1, Geometry) and corresponding concepts between the two domains, and plausible places in the existing curriculum into which to interweave problem clusters. This scrutiny will ground the creation of problem clusters for each course. These problem clusters will be sequenced for projected insertion into the curriculum. The Year-1 Fall Semester phase will consist of developing initial versions of these clusters that will be deployed in the Year-1 Spring Semester phase. We will pair up the 8 teachers with investigators and coaches, piloting the implementation of initial problem clusters in one classroom of each teacher, studying the implementation process (video recording, field notes), and debriefing about the implementation immediately after the lessons (individual teachers, focused groups, student surveys). Further, we will complete the initial pass on the remaining clusters, iterating by feeding back any information that we have gathered to date on desirable cluster properties with supporting material. The next phase takes place during the Year-2 Summer and consists of developing the next pass for clusters for each course. The Year-2 Fall Semester phase will see the field-testing of the clusters yet to be tried in the classroom, as well as refinement of the clusters to be implemented (for the second time) in the Year-2 Spring Semester phase. Further, we will continue classroom observations and formative and diagnostic assessments. The Year-2 Spring Semester phase will consist of implementation of the second pass of the spring semester clusters in the classroom and gathering appropriate information as indicated above. Year-3 Summer will see iteration on results to refine the clusters and supporting material and produce improved of clusters. The Year-3 Fall Semester will consist of implementing these improved clusters in half of the classes of each teacher, while studying the promise of using those clusters on student success in solving the problems and on their dispositions toward problem solving (focused-group interviews, self-report survey consisting of a Likert-type scale and open-ended items). The same instruments will be administered to students in the participating teachers’ classrooms before implementation begins (i.e., Year 1). Similar processes to those used in the second phase of observing, documenting, debriefing, and beginning the final revision of problem clusters will take place throughout year-3. Finally, the Year-3 Spring Semester phase will consist of analyzing any additional desirable enhancements to the clusters and possibly fine-tuning revisions and finalizing the problem clusters, completing data collection and analysis, and disseminating findings/materials. The above description is captured in the project time-line (see Appendix A).
The rationale for developing about 8-10 problem clusters for each course is rooted in the need to infuse them into an existing curriculum. We estimate that each cluster will require about 2-3 lessons. There are about 180 days of school (hence, lessons) per academic year. A substantial number of those days (estimated at 30) is devoted by teachers to test preparation and other non-mathematical activities, which leaves about 150 lessons per year (or 30 weeks). Developing about 8-10 clusters per course thus entails infusing a cluster into the existing curriculum about once every 3-4 weeks.
This iterative development process, which includes a research study (see table in section 2.3.2), will enable us to address the following research questions:
(6) How and to what extent do teachers believe the co-think algebra/geometry problem clusters align with existing curricula and with accepted benchmarks (Common Core, Florida)?
(7) How do teachers develop knowledge and practices needed for effectively infusing problem clusters into their daily teaching of existing curricula? What supports are needed (and provided) for such teacher learning?
(8) To what extent does each problem cluster seem to work in terms of (a) teachers’ use and (b) students’ engagement in learning through problem solving? What criteria inform determining a problem cluster’s value (low/high) and, when low, how are problem clusters being refined and re-tested?
(9) What successes and difficulties do students experience when solving the problem clusters? How do student solutions inform revision of problem clusters and implementation?
(10) To what extent does teaching with problem clusters impact participating students’ dispositions toward mathematical problem solving?
2.3.2 Field Study – Participants
Teachers:
Due to the pilot nature of this project, only a small sample of teachers will be recruited. We will recruit teachers who are interested to participate and commit to stay with the project for the full 3 years. Further, the teacher pool will be comprised of 4 teachers from each school district, two who teach Algebra 1 and two who teach Geometry. There will be no attempt to control for teacher demographics (e.g., years of teaching experience, gender, highest academic degree, etc.), because this study will not attempt to examine variables pertaining to the teachers. (These questions are left to a possible future project, based on the results from this pilot development project.) IRB-approved Consents will be signed to ensure due processes with human subjects.
Students:
The scope of this study, with its focus on developing, field testing, and refining worthy materials, precludes a careful study of how the co-thinking geometry-algebra problem clusters improve learning of these mathematical contents as compared to current curricula. A major reason for not being able to do this evaluation is that measuring such impact requires first to have all materials ready. Only then can one develop measures that are sensitive and relevant to students who study only geometry or only algebra (in regular classes) as well as to students who studied contents through the combined (CPS) curriculum.
As reviewers pointed out, existing examinations (e.g., Florida’s end-of-year) are not sensitive to such deviations in what and how is being taught. We thus focused this revised proposal, our proposed project, to serve as a first, necessary phase in the longer agenda toward examining the impact of the materials developed on student learning. That is, in this project we will develop and study implementation of problem clusters, making a preliminary assessment of how useful they are in coordinating algebra and geometry concepts. As a possible follow-up project, we can then compare the impact of the co-thinking curriculum with standard programs. Certainly, the scope of a 3-year project does not permit in-depth look into students’ learning and outcomes, let alone comparison among programs, because the new program must first be ready for effective implementation – the very goal of our project. We will, however, begin a study of the promise on students’ dispositions toward and competencies developed through solving the co-thinking problem clusters in this project.
Accordingly, students who will participate in this project will be those who are assigned to the Algebra 1 and Geometry classes of the 8 participating teachers. In both school districts, students are assigned in a stratified method. First, their entry level (low, middle, advanced) is determined. Then, if more than one section per level is needed, they are essentially assigned to their class randomly. This method will support a quasi-experimental design. True experimental design cannot be supported due to the inability to assign teachers to treatment vs. a comparison group and the intent to select those who teach particular levels—low and middle. (Quite often, teachers who are assigned to teach those levels are less experienced.) Such a selection will enable examining the teaching of higher-level thinking processes—coordinated problem solving in algebra and geometry—to the lower achieving student populations.
Students who will participate in the study will be those who take an Algebra 1 or Geometry course with a participating teacher. Typically, teachers in Broward and Palm Beach districts are expected to teach 4 to 6 different classes (periods) per day. In each of the three years of the project, participants will include all students in two classes of the participating teacher for whom IRB-approved Parental Consent and Student Assent will be signed. In Year-1, data on those students will be obtained prior to any use of the project materials, thus creating a baseline for comparison. In Year-2, students in two classrooms of the participating teachers will benefit from improved versions of the clusters and the teacher will learn how to use the materials and help identify any shortcomings. In Year-3, students in two participating teachers’ classrooms will benefit from implementation of the iterated versions. We expect by this iteration of our clusters and supporting material, the teachers will be able to use the material with facility. Each year, the two classes from which students will be pulled into the experimental groups will be chosen randomly among all classes that the participating teacher is assigned to teach. As class size in Algebra 1 and Geometry courses in the two districts are typically about 25 students per class, approximately 400 students will participate in each of the three years (200 in Algebra 1 and 200 in Geometry).
2.3.3 Field Study – Data Collection and Analysis
In the table below, we specify methods and instruments to be used for addressing each of the research questions. It should be noted that the specified methods and instruments constitute the iterative nature of the material development. Research data provide feedback used for (a) improving new versions of already field-tested problem clusters and (b) guiding and improving the creation or modification of other problem clusters not yet tried or piloted.
Question
Instruments/Methods and Analysis
1. How and to what extent do teachers believe co-think algebra/geometry problem clusters align with existing curricula and with accepted benchmarks (Common Core, Florida)?
(1a) Responses (8 teachers & 4 coaches, 5-point Likert-type scale) about the extent to which each problem cluster (i) meets the standards and (ii) fits their curriculum; (1b) responses to open-ended questions about which standards/units are addressed by each problem cluster. In each case, the polarity of sentiment will be examined using a test due to Whitney (1978), or in the case of zero variance, due to Cooper (1976), with appropriate adjustment for multiple hypothesis testing.
2. How do teachers develop knowledge and practices needed for effectively infusing problem clusters into their daily teaching of existing curricula? What supports are needed (and provided) for such teacher learning?
(2a) Three Account of Practice (Simon & Tzur, 1999) sets of video recorded observations/interviews with each teacher (focus on content and pedagogical knowledge in classroom practice) when s/he implements a problem cluster in Year-2 and Year-3 (start, middle, and end of year); (2b) Systematic documentation (video) of all coaching sessions provided with teachers, of their requests for additional support, and of how these requests are addressed.
3. To what extent does each problem cluster seem to work in terms of (a) teachers’ use and (b) students’ engagement in learning through problem solving? What criteria inform determining a problem cluster’s value (low/high) and, when low, how are problem clusters being refined and re-tested?
(3a) Written survey, filled by each teacher about each problem cluster upon its completion, followed by focus-group sessions with Algebra-1 and with Geometry teachers; (3b) Video recording and analysis of team sessions in which the teachers’ input is considered for refinement of that problem cluster.
4. What successes and difficulties do students experience when solving the problem clusters? How do student solutions inform revision of problem clusters and implementation?
(4a) Using same data (video) from the Account of Practice sets to identify students difficulties and teacher responses; (4b) Focus-group sessions (video recorded) with 4 students (2 high and 2 low) from two classes (randomly selected) in which a cluster was used; (4c) Video recording and analysis of team sessions in which students’ difficulties are discussed and used to inform revision.
5. To what extent does teaching with problem clusters impact students’ dispositions toward mathematical problem solving?
(5a) A math problem solving disposition survey (5-point Likert –type scale) adapted from an existing, validated instrument administered to participating students at the beginning (pre) and end (post) of every year; ANOVA for one between group and one within group (pre to post) independent variable will be used.
2.3.4 Pilot Study – Participants
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2.4 Methodological Requirements - Evaluation Plan
In addition to the research plan above, project evaluations will serve formative and summative functions (Nevo, 1983; Stake, 1995) to inform and improve the project’s functioning. To this end, the evaluators will identify, produce, and use information about two aspects: (i) project activities and (ii) accomplishment of project goals (i.e., an outcome measurement as suggested by United Way of America, 1996). Formative ongoing evaluation will proceed from project start to completion to inform the team about project progress and deficiencies. Written briefs will be provided with the project team twice a year. The evaluator will advise the project team about desired changes, including aspects of the materials and experiences that can be improved. Summative evaluation will occur during the final months of implementation, through collecting the final data sets and completing analysis of the project as a whole. Starting with an executive summary, the final report will focus on lessons learned from the project, along with responses to key questions that arise during material development and implementation.
Educational projects should evaluate not only goal accomplishment but also project activities (Nevo, 1983), because there is no one-to-one correspondence between those activities and goal accomplishment. Project activities may be of high quality whereas some goals are not fully met (e.g., time needed to detect change may extend beyond the project’s tenure). Thus, evaluation will examine the nature and quality of project activities. Three questions will guide evaluation of project activities: (1) How does the team develop problem, what do teachers contribute to this process, and what reasoning processes inform this work? (2) What are project team’s reasons for adjusting specific project activities? (3) How does the project team address unforeseen hurdles? To address these questions, the following methods will be used. (Observations and interviews will be videotaped.)
Q
Method
1
a) Observations of team sessions of task development (two per year).
b) One-hour individual interviews with team members followed by a focus-group interview, concentrating on questions that help link particular materials developed in the observed session with the specific team actions taken toward such development. Member check will be used to increase credibility and validity of the study and to ensure that the summaries reflect their views, feelings, and experiences.
c) Artifact collection of materials produced by the team during all three years and comparison of the materials to the Common Core State Standards (and the Florida Benchmarks.)
2
d) Same as above, all analyzed qualitatively as text that conjoins justifications about intended students’ learning and suitability of problem clusters.
3
e) Hurdle-focused, open-ended interviews with relevant project personnel.
f) Analysis of team communication (e.g., e-mail) about such hurdles through their resolution.
Five main questions for evaluating project outcomes will be identical to the research questions of the project and addressed via the same data collection methods and analysis described above (Section 2.3). An additional sixth question will address teachers’ impression of changes in students’ level of engagement in the class. To this end, the evaluator will develop a survey to be filled out by each of the 8 teachers at the beginning and end of the three years. Following the end-of-year survey, the evaluator will observe classes of 4 teachers and then interview those teachers.
3. Experiences of the Principal Investigator, Co-Principal Investigators, and collaborative team
Principal Investigator: Roger M. Goldwyn, Department of Mathematical Sciences and
Director of the Math Learning Center at FAU
Responsibilities: Project coordinator and mathematics applications
Level of effort: Equivalent of 1 course per semester during the academic year (course release) and 1 month during the summer
Dr. Goldwyn is Research Professor of Mathematics and Director of the Math Learning Center at FAU. He is course coordinator for the courses Calculus for Engineers 1 and 2 as well as Engineering Mathematics 1 and 2. He has coordinated the undergraduate courses Trigonometry and Pre-calculus Algebra--both prerequisites for Calculus for Engineers 1. As chair of the Engineering-Mathematics Liaison Committee, he was instrumental in the adoption of the ALEKS pretest for placement/assessment/remediation of our introductory mathematics courses. He has advised other colleges and universities in South Florida on our approach to placement, assessment, and remediation. He led a Faculty Learning Community on coordination of multi-section courses at FAU, and specific recommendations are being implemented this fall.
Dr. Goldwyn led a number of application areas while at the IBM Thomas J. Watson Research Center, including biomedical data processing, relational database applications, expert systems, and speech recognition. He was the IBM Worldwide Development Manager for IBM’s speech recognition activities. His mathematical research interests include systems theory, biomedical data processing, complex data analysis, transform theory, perturbation theory, and speech recognition and processing.
Dr. Goldwyn has a B.A., a B.S. in Electrical Engineering, and a M.S. from Rice University and an A.M. and Ph.D. in Applied Mathematics from Harvard University.
Co-Principal Investigator, Ana Escuder, Curriculum Development with GeoGebra and Curriculum Integration, Department of Mathematical Sciences, FAU
Responsibilities: Incorporating GeoGebra for CPSI with integration into curriculum; school districts-FAU liaison
Level of effort: Equivalent of 1 course per semester during the academic year (course release) and 1 month during