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