Project Narrative - Florida Atlantic University
Project Narrative
1. Significance
Purpose of the project
The overarching goal of this R&D, proof-of-concept project, titled Collaborative Problem Solving Impact: Students’ Learning to Co-Think Algebra/Geometry (CPSI), is twofold: (i) co-develop, with teachers, problem-based materials for integrating central concepts in algebra and geometry, and (ii) study the impact on students’ achievements and dispositions of incorporating these materials into existing Algebra-I Geometry courses. That is, this project will address a national burning issue of gaps in students’ achievements in solving non-routine, mathematical word problems, in ways that are consistent with Common Core State Standards and measured by end-of-course Florida state exams. 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, the CPSI project will (i) develop 36 clusters of integrated, algebra/geometry model problems, (ii) pilot 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-I is taught before geometry; thus, integrated problems for the former course will be developed to fit with students’ understandings prior to taking the latter. 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 problem cluster illustrating the twofold purpose of the CPSI 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 (tentative) 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 solution in Appendix B.) |
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|Word Problem: An agriculture company uses square-shaped lots to study how a sprayed fertilizer impacts plants. It takes 1 gallon of |
|fertilizer to spray 1 squre yard (Yrd2) of soil. The company encloses each square-shaped lot with a fence (2-yard high). To date, the |
|company used lots with a side of 4 yards. Thus, it needed 16 yards of fence and 16 gallons of fertilizer for each field. 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: |
|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? |
|[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 lines in your GeoGebra represent the length of the fence? What does the slope of the straight |
|line in your GeoGebra graph mean? |
|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) perfect hexagons. |
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|Figure A: Square-shaped fields Figure B: Hexagon-shaped fields |
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The problem cluster above helps to elucidate the fourfold concept, which the CPSI 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-I and Geometry courses. In contrast, the CPSI project will develop problems for each course which 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 translations across representations (e.g., Lesh, Middleton, Caylor, and Gupta, 2008). The CRA model enables every student to participate productively in the intended learning processes and achieve expected outcomes. Fourth, the project will focus on mathematics teaching that employs low-tech (e.g., wooden cubes) as well as high-tech (e.g., GeoGebra) methods. That is, the CPSI project will provide a proof of the feasibility and advantages of integrating the content and methods of central mathematical ideas via genuine collaboration among pertinent problem solvers (students, teachers, and researchers).
The CPSI project will produce three main outcomes. First, a set of 18 problem clusters per each of the two courses (Algebra-I and Geometry) will be produced. 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) 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 probelm 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 research findings about the impact of these materials on student learning. Consistent with the project’s tenet of integration, these findings will couple the study of changes in students’ achievements with changes in their dispositions toward mathematical 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-solving approach. All in all, the focus of this project will be to make algebra and geometry problem solving a central theme in the curriculum and to make all high school students 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), and 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. Via a collaboration with local teachers, the CPSI 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 the local 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 impact on student 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 National Burning Issue
The CPSI project is significant because it addresses a national burning issue—the massive proportion of American students who lack the mathematical understanding and 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’ (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 the world’s most advanced countries. 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 CPSI 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-I 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 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-I and Geometry classes of 12 teachers in both counties, giving priority to socio-economic disadvantage schools.
The CPSI 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-I and Geometry. This effort will revolve around an extensive infusion 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, because this domain is 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). 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 indicated 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 (except in cases for which the Algebra-I intended learning would require geometrical concepts that students do not yet have, in which case co-thinking will revolve around other topics).
Along with the focus on problem solving in algebra and geometry, the project 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. (The latter two points are discussed below.)
1.2. Nature of Problem Clusters to be Developed
The CPSI 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 which one’s puzzlements play in both triggering and regulating a search for resolution that, in turn, brings about abstraction and generalization of a new idea. Similarly, Polya’s (1945/2004) work articulated the process of interpreting and solving a problem. In a nutshell, he asserted that solving a problem begins with understanding the problem (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 mental and physical 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). 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 these patterns (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 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 CPSI 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 employers 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 CPSI 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, the use of high-tech tools, such as dynamic software for geometry or algebra (e.g., GSP, Matlab, Excel), have found their way into mathematics classrooms around the country. However, 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 reflection on activity-effect relationship constructivist framework, the innovative (free) GeoGebra software and its suitability for cooperative learning, and a curriculum development strategy that centers on genuine partnership with classroom teachers and school experts. The following discussion elaborates each of these four 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 and sustained via concrete manipulation of objects, actions 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 that 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-shaped 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 ray 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 and analyze the behavior of a much larger set of points that lie on each graph.
Whereas mathematics educators seem to respect 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 which 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 her extant (assimilatory) conceptions (e.g., she knows squares, fencing and spraying, measuring sides, calculating perimeter and area). The learner’s assimilatory conceptions set her situation and goal—a desired, anticipated state to guide her 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 and effects not noticed previously (e.g., the graph for perimeter is a straight line 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 and graph 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, accepted for publication).
1.3.3 High-Tech Tool to Co-Think: GeoGebra in Cooperative Learning Situations
An imperative feature of the CPSI project significance is the intended wide use of the innovative, open-source, multi-language, dynamic GeoGebra software (). 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 back-and-forth shifts between real-world problems the software as a representational tool).
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 exemplary 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 constitute 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 occur. 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 CPSI 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 CPSI 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, leading to highly synergized improvements of both the software and its use in service of co-thinking throughout the Algebra-I and Geometry courses.
The significance of the CPSI project is augmented by ensuring that each of the 18 Algebra-I clusters and 18 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 hinders 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 initiate the 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), and by definition promote 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) 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 exemplary 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. Tzur & Lambert (in press) further theorized the link between those two types of zones, postulating that the prompt-dependent participatory stage can be regarded as a cognitive correlate of ZPD. 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.
1.3.4 R&D Partnership with Teachers
A major challenge to every project that develops new materials for teaching mathematics to students is the recruitment of and collaboration with classroom teachers who will 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 spirit of partnership with teachers that 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 CPSI 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 CPSI 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 3 Algebra-I and 3 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 also 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 impact of the revised materials on students’ learning and dispositions.
One “Professional Development 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 6 classroom teachers from each district mentioned above. We also expect to employ at least 5 additional 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 hundreds of teachers can adopt (and adapt) the materials based on their peers’ guidance, and thousands of students can benefit from learning Algebra-I and Geometry via proficient problem solving.
2. Research Plan
Research and Development Aims and Plan
The overarching purpose of the research component of the CPSI project is to address the problem: How can students’ achievements and dispositions in Algebra-I and Geometry courses be improved through continual engagement in solving non-routine, innovative problems that coordinate central concepts from both mathematical domains? To this end, we will:
1) Partner with 12 high school teachers, 2 district math curriculum specialists, and 10 coaches (expert teachers) in two districts to develop approximately 36 problem clusters (18 per course), each consisting of puzzlers, real-life, and content-specific problems.
2) Collaborate with and document those teachers’ implementation of the clusters in their classrooms during the first academic year of the project, and revise the materials continually as well as retrospectively in succeeding years.
3) Collaborate with and document those teachers’ implementation of the revised clusters in their classrooms while studying the impact of using them on students’ achievements and dispositions, then revise and finalize the materials.
Accordingly, the development process will be iterative in nature and done entirely in partnership with classroom teachers (12), coaches (10), and district math coordinators (2). The first phase (year 1 of the project) will consist of scrutinizing available curricula and the Common Core State Standards and corresponding Florida Benchmarks, identifying core concepts for each course (Algebra-I, Geometry) and corresponding concepts from the other domain, and plausible places in the curriculum for interweaving problem clusters into the existing curriculum. This scrutiny will lead to the creation of approximately 18 problem clusters for each course by designated teams comprised of half the school district peers (one team for Algebra-I, the other for Geometry). The second phase (year 2 of the project) will consist of pairing up the 12 teachers with investigators and coaches, piloting the implementation of problem clusters ("alpha" version) in one classroom of each teacher, documenting the implementation process (video recording, field notes), debriefing about the implementation immediately after the lessons, and beginning to revise the clusters as time permits. The third phase (ongoing during year 2 of the project) will consist of revising all 18 clusters for each course (producing the "beta" version). The fourth phase (year 3 of the project) will consist of implementing the beta version of problem clusters in half of the classes of each teacher, while studying the impact of using those clusters on students’ achievements (end-of-course scores) and dispositions (self-report survey consisting of Likert scale and open-ended items). As control, the same instruments will be administered to students in the other half of the participating teachers’ classrooms, as well as all students’ in their (non-participating) colleagues’ classrooms. Similar processes to those used in the second phase of observing, documenting, debriefing, and beginning the final revision of problem clusters will take place during the fourth phase. Finally, the fifth phase (ongoing during year 3 of the project) will consist of fine-tuning revising and finalizing the problem clusters, completing data collection and analysis, disseminating findings and materials, and preparing a model for future teacher workshops.
This iterative development process, which includes a research study, will enable to address three research questions:
1) To what extent does the implementation of co-think algebra/geometry problem clusters, as an integral part of the existing curriculum, impact Algebra-I and Geometry students’ achievements on Florida end-of-course exams as compared to peers in control classes?
2) To what extent does this implementation impact participating students’ dispositions toward mathematical problem solving as compared to peers in control classes?
3) To what extent do the participating students’ achievements and dispositions show tendency toward closing achievement and disposition gaps among social sub-groups (gender, race/ethnicity, ELL)?
2.1 Theoretical Underpinnings of our Study
An overarching constructivist conceptual framework will guide the CPSI research and development effort. This framework will consist of the four aforementioned components: The Concrete-Representational-Abstract (CRA) model of problem solving; 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 approach. Each of these components has been articulated in Section 1.3 above. Here, it suffices to depict briefly the theoretical underpinnings of constructivism:
1) Mathematical knowing does not exist independently 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 (von Glasersfeld, 1995).
2) Consequently, coming to know mathematics entails an active 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. 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 (Piaget, 1985; Simon et al., 2004; Tzur & Simon, 2004; von Glasersfeld, 1995).
3) Teaching mathematics commences with the premise that one person’s knowing 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 different 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).
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, indicating that a critical facet of the problem involves students’ inability to solve mathematical problems (coupled with negative dispositions). Starting in the 1980s, this state of affairs has generated substantial reform efforts, heralded 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. Cockcroft (1982) also advocates 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 reinforcing 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 becomes the main road to successful learning and career choices. 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 impact on students’ achievements and dispositions.
The CPSI project is resubmitted in order to address this challenge. Reviewers of the first 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 pilot and test problem-clusters in regular classes, before attempting to train a large number of teachers for using those materials and conduct a large-scale study of their impact. Consequently, the context and rationale for the CPSI project amount to focusing on the first step—develop problem-clusters with teachers, and pilot study their implementation and impact on students in this proposed 3-year effort. A future, large-scale professional development program for mathematics teachers can be proposed as a follow-up project.
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 students, respectively, with more than 71,000 and 35,000 students 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 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 of interweaving CRA-based problem clusters into the regular Algebra-I and Geometry courses have on achievements and dispositions of all students (particularly those from underrepresented groups).
Located at the heart of this region, Florida Atlantic University (FAU) and its mathematics, engineering, and education schools are strategically situated to conduct the proposed R&D 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 (), 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 exemplary problem cluster, GeoGebra allows learners to operate algebraically and geometrically in realistic problem situations, 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.
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 intervention will consist of unique and innovative set of materials and professional development workshops for teachers, targeting problem solving in algebra and geometry.
The proposed intervention has been outlined in Section 2 above. Once 36 problem clusters have been created and revised, the 12 teachers will interweave them into their regular Algebra-I 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 about two clusters during a 3-week period. (Some problems may be implemented to initiate a unit, others during its evolution, and still others toward the end of a unit.)
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. At the core of this rationale is (i) the premise of genuine partnership with teachers in developing, implementing, revising, and studying the impact of the problem clusters and (ii) the first phase in which existing curricula and the Common Core State Standards and corresponding Florida Benchmarks will be scrutinized.
2.2.2 Practical importance
The proposed intervention will improve all students’ achievements and dispositions, particularly those from underrepresented groups, 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 is likely to also lay the groundwork for affordable, easily implemented workshops for teachers and consequent problem- and technology-based learning experiences for students.
2.2.3 Rationale Justifying the Importance of the Proposed Research
Whereas the previously described overarching rational for this project seems reasonable, and the problem clusters to be developed are potentially powerful, the ultimate test is quite simple: Does implementation of the materials make a difference in students’ progress as gauged by publicly accepted measures (e.g., Florida’s standard end-of-course exam)? Accordingly, the justification for the proposed research is straightforward—it will address the critical linkage between using materials informed by constructivist ideas, in a cooperative-learning mode, with student achievements and dispositions. Addressing the three research questions listed in Section 1.4 will provide valuable testimony 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 to provide empirically grounded proof-of-concept. The following section delineates the CPSI project research and evaluation plans.
2.3 Methodological Requirements (Research & Project Evaluantion Plans)
2.3.1 Sample
Teachers:
Due to the pilot nature of this R&D project, and the focus on studying impact on students, only a small sample of teachers will be used. This sample is purposeful, as we will recruit teachers who are interested to participate and commit to stay with the project for the full 3 years; we will also select and work in schools where the student population fits the project’s focus on underrepresented sub-groups. Further, the teacher sample will be comprised of 6 teachers from each school district, 3 who teach Algebra-I and 3 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 pilot study will not attempt to examine variables pertaining to the teachers. (This will change when a future, large-scale project is proposed.) Although teachers are not studied directly, IRB-approved Consent forms will be signed to ensure due processes.
Students:
In both school districts, students are assigned to the Algebra-I and Geometry classes 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 non-random assignment of teachers to classes and the need to select those who teach particular levels—low and middle (quite often, teachers who are assigned to teach those levels are less experienced).
In the experimental group (EG), students who will participate in the study will be those who take an Algebra-I 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, the EG will consist of 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 implementation of the alpha version, while the teacher learns how to use the materials. In Year-3, students in two participating teachers’ classrooms will benefit from implementation of the beta version, which the teachers are expected to use with facility. Each year, the two classes from which students will be pulled into the EG will be chosen randomly among all classes that the participating teacher is assigned to teach. This method, again, contributes to making the research design as close as possible to fully experimental. As class size in Algebra-I and Geometry courses in the two districts are typically about 25 students per class, the EG will consist of approximately 600 students in each of the three years (300 in Algebra-I and 300 in Geometry).
The control group (CG) will consist of two sub-groups. The first sub-group (CG1) will comprise of two, randomly selected classes taught by each of the participating teachers (in the last 2 years of the project), in which the project materials will NOT be used. The purpose for creating CG1 is to control for all variables, including the teacher. The assumption is that the participating teacher, in spite of knowing about the possibility to teach with problem clusters, will continue teaching the curriculum in essentially the same way as she or he has done it in the past. To avoid the teacher’s awareness (and potential bias) of the specific classes that were selected into CG1, students in all the teachers’ classes will take the pre- and post-intervention survey; the random selection of the two classes to be pulled into CG1 will be done only after the post-intervention survey is administered. The approximate number of students in CG1, in each year of the project, will be the same as in EG.
The second sub-group (CG2) will be created only during the year when the beta version will be taught by participating teachers. CG2 will comprise of all students in two randomly selected classes taught by another teacher in the participating teacher’s school building. This will enable controlling for students’ demographics, because within the same school, through random assignment of students to classes (per their level), these variables are not expected to impact students’ achievements and dispositions beyond chance. The approximate number of students in CG2, in the single year that this sub-group will be pulled, will be 600 (300 in Algebra-I and 300 in Geometry). The table below summarizes the sample.
| |EG |CG1 |CG2 |
|No intervention |Alg-300; Geo-300 | | |
|Alpha version |Alg-300; Geo-300 |Alg-300; Geo-300 | |
|Beta version |Alg-300; Geo-300 |Alg-300; Geo-300 |Alg-300; Geo-300 |
2.3.2 Hypotheses
To address the three research questions of the proposed study, several hypotheses will be tested (see table below). For abbreviation purposes, the “>” sign is used to indicate that students in one group outperform students in the other group (statistically significant difference set out to p < 0.5). For example, if the hypothesis is that students in the EG that was taught with the alpha version will outperform their counterparts in the no-intervention EG on the end-of-course exam, the hypothesis is expressed as: EG(alpha) > EG(no). The table below summarizes all the hypotheses to be tested. Note: For the disposition survey, “outperform” (also symbolized as “>”) means that, on the average, students’ responses were more favorable to mathematical problem solving than their counterparts’ responses. It should be recalled that disposition surveys will be administered to each group at the beginning (pre) and end (post) of the course.
|Achievements |Dispositions |
|EG(alpha)>EG(no); |EG(no, post)=EG(no, pre); |
|EG(beta)>EG(alpha) |EG(alpha, post) > EG(alpha, pre); |
|EG(alpha)>CG1(a); |EG(beta, post) > EG(beta, pre); |
|EG(beta)>CG1(b); |EG(alpha, post) > CG1(a, post); |
|CG1 = CG2 for all group pairings |EG(beta, post) > CG1(b, post); |
|EG(alpha)>CG2 |EG(beta, post) > CG2(b, post); |
|EG(beta)>CG2 (most important) |CG1(b, post) = CG2(b, post); |
2.3.3 Data Collection
Achievements:
The instruments for collecting data about sampled students’ academic achievements will be the Florida end-of-course exams for Algebra-I and Geometry. These are standard (reliable and valid) instruments, administered by the state in May or June of each academic year. Data from those exams are reported to the schools and to the districts’ central offices. They will be obtained and delivered to the FAU team members by the Math Curriculum Specialist in each district.
Dispositions:
The instrument for collecting data about sample students’ dispositions toward mathematics, its learning, and mathematical problem solving will be determined during the first 3 months of the project. Preferably, an existing, validated instrument will be identified and obtained. However, due to the specific nature of survey items the team may elect to include, it is likely that an existing instrument will be identified, adjusted, and tested for its validity and reliability. The project evaluator (Dr. Tzur) has developed and used such instruments, and experienced the adjustment process they may require. He will guide the team in selecting and adjusting those instruments.
Once the disposition survey is finalized, it will be administered to the students during the first two weeks of their course—soon after signed consents were obtained and prior to any use of problem clusters (pre). A team member (but not the class’s teacher) will administer the survey and collect the responses during a regular class period. A similar procedure will take place to administer the post-intervention survey after students have taken the end-of-course exam but prior to being informed about their scores. This way, the disposition measure will include their experience of taking the exam without being biased by how they scored on it.
2.3.4 Data Analysis
In general, studies of teaching-learning processes in school systems are considered to yield "nested" data—students are nested within classrooms, which are nested within schools, which are nested within districts. Analysis of nested data typically requires hierarchical analytic methods (e.g., HLM). However, due to the small-scale, pilot nature of the project, to the sufficient control for students’ variables, and to the exclusion of collecting and analyzing teacher data, such methods are not needed for the proposed study. On the other hand, the large number of students in each of the study groups and the random assignment to classes allow for the assumption of normality of data distributions. These assumptions underlie the following analytic plan, in which the significance level of all statistical tests is set to p < .05.
Achievements:
Because all achievement scores for students will be obtained from a single event (end-of-course exam), which consists of variables measured on an interval scale, and compared between groups, the ANOVA statistics will be used. Students’ achievements in the three groups (EG, CG1, and CG2) will be compared, as well as those of social sub-groups within those groups (pending the size of N for sub-groups).
Dispositions:
Analysis of between-group differences will be tested via the ANOVA statistics.
Analysis of within-group differences, that is, changes from pre- to post-intervention, will be done via a paired t-test statistics. It should be noted that this test is considered to require variables that are measured on an interval scale. However, when the sample size is large, a t-test has also been considered appropriate for variables measured on an ordinal scale, such as responses on a 5-point Likert scale (Agresti & Finlay, 1997, p. 232).
2.3.5 Evaluation Plan
Project evaluation will serve formative and summative functions (Nevo, 1983; Stake, 1995), by identifying, producing, and using 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. As a mathematics education expert, 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. Both evaluation types will use statistical and qualitative methods, to increase trustworthiness of measurement through triangulation (Lincoln & Guba, 1985).
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, because the time needed to detect change extends 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 are problem clusters developed, what role do teachers play, 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 focused-group interview, focusing on questions that help link |
| |particular materials developed in the observed session with the specific team actions taken toward such development. |
| |c) Artifact collection of materials produced by the team during all three years. |
|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. |
Three 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 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 12 teachers at the beginning and end of the three years. Following the end-of-year survey, the evaluator will observe classes of 4-6 teachers and then interview those teachers.
2.4 Summary Time-Line of Activities
|Year |Work Plan / Outcomes |
|1 |Phase 1 |
| |Select group of Common Core State Standards and corresponding Florida Benchmarks for cluster problems. |
| |Identify core concepts for Algebra and Geometry |
| |Develop the 36 cluster problems (alpha version) with CRA model and GeoGebra implementation similar to the model cluster problem |
| |in the proposal, but with more detail. |
| |Indicate for each how to approach--the pedagogy. |
| |Faculty works with district coordinators, professional development assistants (classroom teachers and coaches/expert teachers) |
| |on the mathematical concepts and how to present. |
| |Validate collectively that material is ready for trial in a classroom. |
| |Collect student baseline data on end-of-course exams |
| | |
| |END OF YEAR RESULTS |
| |36 cluster problems with supporting material (alpha version). |
| |Formative assessment of effectiveness and what to do to improve. |
|2 | Phases 2 and 3 |
| |Pair up teachers with investigators and coaches |
| |Pilot alpha version of cluster problems in one classroom of each teacher |
| |Document the implementation process |
| |Debriefing about the implementation |
| |Revise all cluster problems and develop beta version |
| |Continue collection of data on end-of-course exams |
| | |
| |END OF YEAR RESULTS |
| |Beta version of 36 cluster problems with supporting material. |
| |Teacher experience on using the alpha version of cluster problems |
| |Assessment of effectiveness and what do to do improve. |
|3 |Phases 4 and 5 |
| |Implementation of beta version in half of the classes of each teacher |
| |Study the impact of using those clusters on students’ achievement and dispositions |
| |As control, the same instruments will be administered to students in the other half of the participating teachers’ classrooms, |
| |As control also, give instruments to all students’ in their (non-participating) colleagues’ classrooms. |
| |Observe, document, debrief, and begin the final revision of problem clusters |
| |Fine-tune revision and finalize the problem clusters |
| |Complete data collection and analysis |
| |Disseminate findings and materials |
| |Prepare a model for future teacher workshops. |
| | |
| |END OF YEAR RESULTS AND END OF GRANT |
| |36 well refined and tested cluster problems with evaluations in each district |
| |Plan for integration into the full curriculum in the districts. |
| |Disseminate findings and materials |
| |Prepare a model for future teacher workshops |
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
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, Department of Mathematical Sciences, FAU
Ms. Escuder is an instructor in the Department of Mathematical Sciences at FAU. She taught a wide range of middle and high school mathematics courses for 18 years, integrating leading technology tools in all of her classes. She is currently the project manager of Dr. Heinz-Otto Peitgen’s NSF-funded Math and Science Partnership project with Broward County Public Schools. Through her work with the MSP project, she has mentored middle and high school teachers in Broward County, instructing in mathematical content and pedagogical methods, developing course materials, and evaluating participating teachers. She has extensive experience in running high-quality innovative teacher development programs in school districts.
Ms. Escuder is the chair of the Florida Chapter of the International GeoGebra Institute, a non-profit organization which works with independent regionally-based GeoGebra chapters around the world. The goals of the chapter are to train and support teachers in the use of GeoGebra in mathematics teaching and learning, to develop and share workshop resources and materials, and to conduct research in teaching and learning mathematics with the use of technology. She is also an active participant in the mathematics councils of the community, state, and nation.
Ms. Escuder has a Master in Science in Teaching Mathematics from FAU and is currently a doctoral student in the Curriculum and Instruction Program in the College of Education at FAU.
Co-Principal Investigator Joseph Furner, Department of Math-Education, FAU
Dr. Furner is an Associate Professor of Mathematics Education at FAU on the Jupiter (Florida) campus. His scholarly research interests are related to math anxiety, the implementation of the new national and state standards, English language issues as they relate to mathematics instruction, the use of technology in mathematics instruction, math manipulatives, family mathematics, and children's literature in the teaching of mathematics. He is the author of more than 50 published refereed articles/book chapters in mathematics education. Dr. Furner has worked as an educator in New York, Florida, Mexico, and Colombia.
Dr. Furner has a B.S. in Mathematics Education from the State University of New York at Oneonta and an M.A. and Ph.D. in Curriculum and Instruction-Mathematics Education from the University of Alabama.
Co-Principal Investigator Lee Klingler, Department of Mathematical Sciences, FAU
Dr. Klingler is Professor and Chair of the Department of Mathematical Sciences at FAU. In the 2008-2009 academic year he implemented an overhaul of the College Algebra curriculum as course coordinator, and in the 2009-2010 academic year coordinated and updated the curriculum of the Math for Liberal Arts 1 and 2 courses. He has served on FAU's Secondary Teacher Education Coordinating Committee, the Engineering-Mathematics Liaison Committee, FAU's Core Curriculum Committee, and as the mathematics department’s Master Teacher. He has worked with Dr. Heinz-Otto Peitgen's MSP project developing a Master of Science in Teaching Mathematics degree program for middle school teachers.
Dr. Klingler is an active researcher in algebra with special interest in module theory, commutative rings, systems over rings, and integral representations of finite groups.
Dr. Klingler has a B.S. in Mathematics and Philosophy from Lebanon Valley College and a Ph.D. in Mathematics from the University of Wisconsin-Madison.
Co-Principal Investigator Daniel Raviv, College of Engineering, FAU
Dr. Raviv is a professor in the Electrical Engineering Department at FAU. In 2007 he became a special assistant to the provost and led a group of mathematics instructors to develop a comprehensive report on improving students’ success in College Algebra. Initial implementation of some recommendations from that report have resulted in a doubling of students’ success. He has been a visiting professor at Johns Hopkins University and at the University of Maryland, where he taught and conducted research in innovative thinking. Over the past 16 years, Dr. Raviv has been teaching creative, inventive, and innovative thinking using new methodologies and hands-on discovery-based activities. He shared his experience with College and high school students at FAU, Johns Hopkins University, and the University of Maryland. Jointly with the University of Maryland, he is developing a national program titled "Ideation to Innovation" (I2I).
Dr. Raviv developed a fundamentally different approach to teaching “out-of-the-box” problem solving. For his unique contributions he received the prestigious Distinguished Teacher of the Year Award, the Faculty Talon Award, the University Researcher of the Year AEA Abacus Award, and the President’s Leadership Award. He has published in the areas of vision-based driverless cars, green innovation, and innovative thinking. Dr. Raviv is a co-holder of a Guinness World Record.
Dr. Raviv has a B.Sc. and M.Sc. from the Technion, Israel Institute of Technology, and a Ph.D. from Case Western Reserve University.
Evaluator/Consultant Gail Wisan, University Director of Assessment, Institutional Effectiveness and Analysis, FAU
Dr. Gail Wisan is currently the University Director of Assessment at FAU in the Office of Institutional Effectiveness and Analysis. She has over ten years of experience in assessment and institutional research. She has supervised multiple professional and administrative staff and coordinated mandated state, federal, and accreditation reporting. She has also developed decision-support data management tools (e.g., Dashboard Indicators) related to institutional effectiveness. She has taught research methodology, research design, statistics and other courses in both a face-to-face setting (e.g., George Washington University) and online (UMUC) and is well versed in quasi- and experimental designs and evaluation research. She has also worked as a SAS statistical data analyst for a large educational research contractor.
Dr. Wisan's academic research has provided an excellent background for her work supporting the Academic Division in implementing student learning outcomes assessment and institutional effectiveness. She has published articles and presented papers at national and regional meetings on various topics, such as student learning outcomes assessment and comparing outcomes in different learning environments (face-to-face vs. online classrooms).
Dr. Wisan has a B.A. in Sociology from Hunter College, a B.S. in Information Systems Management from UMUC, and an M.A. and Ph.D. in Sociology (social research methodology emphasis) from the University of Illinois, Champaign-Urbana.
External Evaluator/Consultant, Ron Tzur, Professor of Mathematics Education, UCD
Dr. Tzur is Professor of Mathematics Education at the University of Colorado, Denver, where he also serves as the Faculty Chair of the Doctoral and Research Program. After teaching mathematics (K-16) for over 15 years, Dr. Tzur completed his graduate work in mathematics education. He became a mathematics educator whose research has combined qualitative and quantitative methods for studying children’s number and fraction knowledge, mathematics teacher development, and the impact of assessment on student achievements and dispositions. He has gained a national and international reputation from his extensive publications and presentations. He has served as co-PI and PI on several large NSF-funded projects, including the ongoing, 5-year/$3M Nurturing Multiplicative Reasoning in Students with Learning Disabilities. He has developed a comprehensive framework that addresses how (i) cognitive change occurs and can be assessed as learners progress from not knowing to knowing a particular mathematical idea (reflection on AER), (ii) teaching can promote such progress, (iii) teachers develop ways of thinking about and teaching conducive to such conceptual learning, and (iv) this model of learning can be linked to research about the brain. Recently, he conducted and completed the evaluation of a 5-year, $2.5M project funded through the Department of Education Jacob Javits grant program (Project Bright IDEA in North Carolina).
Dr. Tzur has a B.S. from Haifa University, an M.S. from the Technion, Israel, and a Ph.D. from the University of Georgia in Athens.
Professional Development Specialist, District Coordinator, James Chinn, Broward County School District
James Duke Chinn is the Secondary Mathematics Curriculum Supervisor for the School Board of Broward County, which is the sixth largest school district in the nation. He works with secondary mathematics classroom teachers and administrators to establish and maintain high mathematics standards across the county. With a standards based curriculum, he provides a multitude of professional development opportunities for secondary mathematics teachers, department instructional leaders and secondary mathematics administrators.
Mr. Chinn has worked with mathematics professors at FAU for the past 15 years, developing and delivering high quality workshops for teachers taking classes at FAU. Currently he also works as an adjunct as part of an NSF-sponsored grant with the Department of Mathematical Sciences at FAU.
Professional Development Specialist, District Coordinator, Diana Snider, Palm Beach County School District
Diana Snider is the Secondary Mathematics Program Planner for the School District of Palm Beach County, which is the thirteenth largest school district in the nation. With over 100 public, charter and alternative middle and high schools, she works with secondary mathematics classroom teachers and administrators to establish and maintain high mathematics standards across the county. With a standards based curriculum, she provides a multitude of professional development opportunities for secondary mathematics teachers, department instructional leaders and secondary mathematics administrators.
Ms. Snider also teaches at Palm Beach Atlantic University in the MacArthur School of Continuing Education. She teaches the single required mathematics course for non-traditional students earning an associate or bachelor degree. The course, Finite Mathematics, offers a survey of a variety of topics in mathematics applications.
4. Resources
Florida Atlantic University and the School Board of Broward County, the nation sixth largest and fully accredited school district, have enjoyed a successful working relationship for more than 15 years, with NSF-funded Teacher Enhancement and Math and Science Partnership grants. Thus, the partnership team has extensive experience in running high-quality, innovative teacher development programs with the local school district. Some of the key successful features of the past collaboration will also be implemented in this new project, specifically:
• Supporting hands-on workshops and follow-up meetings based on a "teachers-teaching-teachers" model, and the evolution of a highly motivated community of teacher leaders.
• Design of all materials and instruction to correlate with relevant classroom content and pedagogy as well as state and nation standards.
• Opportunities for teachers to experiment with their newly acquired knowledge and teaching methods, and immediate feedback provided to the participants and the program staff.
• Enrichment of all materials, content and pedagogy instruction by science and interdisciplinary applications and technology integration.
The integration of the School District of Palm Beach County, the thirteenth largest school district in the nation, brings an exciting new dimension to the proposed project. One of the Co-PIs (Ana Escuder) was the project manager of the NSF MSP grant, and most of the researchers have extensive experience working with teachers. The existing institutional cooperation and the dedicated team of researchers, administrators, and teachers provide a strong foundation for the success of the new project.
Budget
|Category |Year 1 |Year 2 |Year 3 |
|Faculty--PI and Co-PIs |$187,272 |$192,890 |$198,677 |
|Professional Development Specialists/District Coordinators |$21,920 |$22,578 |$22,578 |
|Professional Development Assistants (Trainers of teachers) |$42,500 |$42,500 |$42,500 |
|Consultants-Developers/Classroom teachers |$9,000 |$9,000 |$9,000 |
|Student assistants |$18,200 |$18,200 |$10,400 |
|External evaluator/Math-Ed Consultant |$25,520 |$26,796 |$28,136 |
|FAU internal evaluator |$10,000 |$10,000 |$10,000 |
|FAU personnel travel |$16,000 |$16,000 |$16,000 |
|External evaluator travel |$4,000 |$4,000 |$4,000 |
|Part-time budget coordinator |$11,391 |$11,391 |$11,391 |
|Overhead |$153,651 |$145,319 |$14,902 |
|Total |$499,454 |$498,673 |$498,660 |
Budget Justification
Faculty (Investigators)—Create sample problems and identify how students understand, reformulate, and visually represent problems. We want a broad spectrum of application areas and have faculty that represents engineering and mathematics. Professional development materials need to be created as modules that can be used in the classroom that emphasize pedagogy and how to use (and modify) the examples. A database of cluster problems will be created. The budget is based on salary and benefits for each faculty member. Each will receive 25% of their base salary during the academic year for a course release each semester, and approximately 1-month of summer salary, along with a 3% salary increase per year.
Professional Development Specialists (District Coordinators)—Work with the faculty on the initial cluster problems and on reformulations of them that will ultimately lead to modules in professional development to assure materials are correlated to the state standards and the districts’ math curriculum. They need to recruit and select with the Investigators the Professional Development Assistants who will work in their schools and be the first to apply the techniques in their classrooms. They will need to supervise with the Faculty gathering of data for evaluation. (Our focus is 9-12 but certification in Florida is for 6-12. Hence, we will have some teachers not necessarily in 9-12.)
Professional Development Assistants/Teachers & Coaches--Work with the Professional Development Specialists and Investigators on creation of the initial problems for teachers to use in the classroom. We expect 6 teaches and 5 coaches/expert teachers in each district. We include at least three algebra and three geometry teachers from each of the two districts. They will provide the first level for formative analysis for the evaluators. These teachers will be active members of the team in creating the problems for co-thinking algebra/geometry. They will be the first to validate the experiment in their classrooms with students. They will be the first to yield feedback on the efficacy of the material developed, suggest modifications, and be involved in the modification process. They will also be involved in modifying and refining the problems. The coaches/expert teachers will also be involved in developing professional development workshops.
Program Assistant--Will work with the PIs and Co-PIs on documentation and coordination of developed material. They will be involved in data collection and entry for evaluation. Manage budgets and contracts for faculty and professional development staff, as well as manage consent forms.
Evaluation—Budgeting substantial amount to ensure we have support for evaluation of results on an on-going basis. On-going results will guide subsequent activities. The External Evaluator will ensure appropriate Math-Ed rigor through the evaluation process. He will work with the FAU internal evaluator on ensuring meaningful data are collected and analyzed. Fees for our internal evaluator are $10,000 per year. Fees for our external evaluator will begin with $25,520 for the first year, and include increases of 5% per year.
Travel—For all the faculty, representing engineering, mathematics, and evaluation, we should be attending relevant seminars as well as granting agency meetings in Washington. Estimate budget as submitted. Approximately $14,000 per year will be allocated for the faculty, and $3,900 will be allocated for the external evaluator.
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