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ABSTRACT
BORDEN, MARGARET LEAK. A Framework of Core Teaching Practices for Implementing Project-Based Learning (PBL) in the Mathematics Classroom (Under the direction of Dr. Erin Krupa).
Project-based learning (PBL) is a teaching methodology that utilizes projects to
engage students in learning the content through authentic experiences. PBL has been shown
to successfully motivate students to learn as well as reinvigorate teachers, and it can be used
to teach any discipline. However, it can be challenging to teach using PBL, and particularly
difficult for math teachers who tend to face more challenges than other disciplines when
teaching utilizing PBL. In order to make PBL more accessible to math teachers, this paper
merges several existing PBL frameworks with the various recommendations for math
education into one framework of core teaching practices for implementing PBL in a math
classroom. The framework describes what teachers should think about while planning the
project, what they need to be doing while students are working on the project, and how to
best support students when they are ready to share their work with others. The paper includes
multiple examples to help teachers visualize running a PBL in their classroom, and goes into
detail with one example to explain the different aspects of the framework. After reading this
paper, math teachers will feel empowered to be able to implement PBL in their classrooms,
non-classroom educators will be able to support their math teachers in implementing PBL,
and the myth that PBL cannot be used in math will be expelled.
© Copyright 2021 by Margaret Borden
All Rights Reserved
A Framework of Core Teaching Practices for Implementing Project-Based Learning (PBL) in the Mathematics Classroom
by Margaret Leak Borden
A thesis submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the degree of
Master’s of Science
Mathematics Education
Raleigh, North Carolina 2021
APPROVED BY:
_______________________________ _______________________________ Dr. Erin Krupa Dr. Cyndi Edgington Committee Chair _______________________________ _______________________________ Dr. Karen Hollebrands Dr. Kevin Gross
ii
DEDICATION
I would like to dedicate this paper to my daughter, Ellamarie Councill Borden,
because I want her to see that it is possible to pursue your dreams and make them your
reality, even if it takes a lot of hard work.
iii
BIOGRAPHY
Margaret Borden was born and raised in Winston-Salem, North Carolina. She moved
to Raleigh when she was accepted at NC State University in the undergraduate math
education program. She was awarded the Park Scholarship, a program which focuses on
scholarship, leadership, character, and service, and the experiences she had with her cohort
through that program inspired her to study abroad, triple major, and take on many different
leadership roles. Because she added a math degree and a communication degree to her
studies in math education, she needed to stay at NC State for a few more semesters past her
Park Scholarship allotment. She was awarded the Noyce Scholarship, a program that
supports STEM educators in their last few years in college who are interested in teaching in
underserved schools. Due to these two programs, Margaret was mentored by amazing math
education professors and math educators, who helped her to find presentation opportunities
and share her ideas with people who share her passions. She presented at various math
education conferences and led the college’s chapter of NCCTM, lighting a fire in her for
math education that extended beyond her coursework. It was through these experiences that
she discovered project-based learning (PBL), and she spent as much time as she could
learning about PBL, attending trainings on how to teach with PBL, and even presenting at the
NCCTM conference on it.
After graduating with her three degrees and a minor in Spanish, Margaret taught at
Knightdale High School of Collaborative Design (KHSCD). KHSCD was known for
utilizing innovative practices to find ways to motivate learners who did not respond well to
traditional schooling methods. They were well-versed in PBL, so it was a great fit for
Margaret since she wanted the opportunity to teach in an environment that supported PBL.
iv
She taught there for three and a half years, and found that it was exhilarating to work
alongside innovative educators, but it was also frustrating to do this work within the confines
of traditional grading practices, schedules, and other constraints.
While at KHSCD, Margaret married her husband, Peter, who was in law school at the
time. Simultaneously, Margaret decided to go to graduate school to pursue a Master’s in
Math Education, and worked on that part time while teaching full time. Peter now works as a
civil litigator in downtown Raleigh, and they moved to a home near Lake Wheeler to be
closer to his family. They decided to start their family, and learned from one of Margaret’s
mentors that there was a possibility to be funded if she finished her Master’s as a full-time
student. Thus, Margaret spent her last year at KHSCD pregnant with their daughter, who was
born in the beginning of the global Covid-19 pandemic, and then transitioned out of the
classroom the following fall to finish her Master’s as a graduate research assistant.
She spent the last year of her Master’s working on a PBI Global team with Dr. Hiller
Spires and Dr. Erin Krupa, which gave her the opportunity to work with interdisciplinary
projects at two different schools. She was able to serve as math support to the math teachers
on the team, helping them understand how their standards could be adequately addressed in
relation to the project context. She also took on a coaching role with math educators at a
charter school that is attempting to become a PBL school, helping them learn how to create
the classroom environment that could be conducive to effective PBL. All of these
experiences helped to inspire this thesis. Margaret will be pursuing her Ph.D. in Math
Education starting in the fall, and will continue working with Dr. Krupa on a different PBL
team that is specifically oriented towards high school math teachers. Margaret is very excited
v
to continue her work and looks forward to the opportunity to follow some of her own
recommendations during her doctoral pursuits.
vi
ACKNOWLEDGMENTS
I would like to acknowledge my advisor Dr. Erin Krupa for all of her support
throughout this process. Dr. Krupa received an email with my resume and saw potential for
this partnership, and I am incredibly grateful that she decided to give me this opportunity.
She has believed in my ideas every step of the way, always seeking clarification in a way that
made them stronger. I am so excited to continue working with her throughout my doctoral
studies and career.
I would also like to acknowledge the other two math education representatives on my
committee: Dr. Cyndi Edgington and Dr. Karen Hollebrands. They have both been my
mentors throughout my career at NC State, presenting alongside me, thinking of me for
opportunities, and spending time advising me. They are two of my greatest cheerleaders and I
know that I can always reach out to them when I need support, resources, or a
recommendation. I look forward to more opportunities to work with them during my doctoral
studies and through my career.
Additionally, I would like to acknowledge the statistics representative on my
committee: Dr. Kevin Gross. Dr. Gross speaks about statistics in the same way that I feel
about statistics, with enthusiasm and curiosity. It stood out to me that he took the time in
class to explain the etymology of any new terminology, sometimes to help us remember the
term better and other times to help us understand the concept around it. I am so thankful that
he was willing to serve on my committee and provide a different perspective to the feedback.
There are many other people who have been a part of my journey as a math educator
who deserve to be recognized as well. Allison McCulloch has been one of my mentors from
the beginning and has always encouraged me to find opportunities to present as well as
vii
investigate my ideas further. Teresa Pierrie has been instrumental in developing me as an
educator and sending opportunities my way to support that development. Matthew Campbell
has mentored me through the process of going back to school and provided important
insights into how to continue to be connected to the classroom even when outside of the
classroom. My teammates at KHSCD challenged my ideas and pushed me to see many
different angles in each pedagogical approach, especially Tracy Pratt and Laura Harrell who
regularly spent hours in conversation with me about different math education ideas.
Similarly, I have been inspired by conversations I’ve had with my students, mentees and my
curriculum writing team, and continually seek to learn how everyone experiences these ideas
from their perspectives. Most notably, I have appreciated my student, Kiara Bush, for her
efforts on my projects and continued work with me to share the word about PBL in math and
the joys of math in general. Additionally, I have benefitted tremendously with my work with
math teachers from Wake STEM, PECIL, and CLA who are all at different points in the
journey to teaching with PBL, and have enjoyed learning from the PBI Global team.
Finally, I would like to acknowledge my family’s support. My husband, Peter
Borden, has always been so supportive of me, encouraging me to work hard and when my
hard work still fell short. He has taken on more work around the house and with the baby
when I’ve had to work outside of daycare hours in order to meet deadlines, and listened to
me as I talked way too much about my thesis and my classes. He is my rock and my biggest
cheerleader and I am so thankful to have him by my side throughout this experience. My
parents, in-laws, grandparents, sister, and friends have also all played a huge role in helping
me get to this point. They have picked up the phone to listen to my worries, taken me to
dinners to celebrate when I was awarded funding, and tried to understand what I’m talking
viii
about when I start passionately talking about math education. I feel so blessed to have so
many people who help me believe that I am capable of doing this work and being successful
when pursuing my passion.
I have really appreciated the incredible support system that has surrounded me as I
learn and grow as a math educator. I love to learn new things and meet new people, and feel
thankful to the people that I have met through NC State and Wake County that have helped
to make that learning possible.
ix
TABLE OF CONTENTS
LIST OF TABLES ................................................................................................................ x LIST OF FIGURES.............................................................................................................. xi Chapter 1: Introduction ...................................................................................................... 1 Brief History of PBL and Mathematics .................................................................................. 1 Statement of the Problem ...................................................................................................... 4 Chapter 2: Literature Review ............................................................................................. 6 Motivation for Using PBL in the Mathematics Classroom ..................................................... 6 Barriers to PBL in the Mathematics Classroom ..................................................................... 9 Effective Mathematics Teaching Practices........................................................................... 12 Effective Project-Based Teaching Practices ......................................................................... 15 Using PBL in a Mathematics Classroom ............................................................................. 19 Chapter 3: Discussion of Theoretical Framework ........................................................... 26 Connecting the Effective Teaching Practices ....................................................................... 26
Core Practice 1: Disciplinary ................................................................................... 26 Core Practice 2: Authenticity ................................................................................... 32 Core Practice 3: Iterative ......................................................................................... 36 Core Practice 4: Collaboration ................................................................................. 40
Framework for PBL in Mathematics Classroom .................................................................. 41 Planning Phase ........................................................................................................ 42
Align to Standards ........................................................................................ 42 Working Phase ........................................................................................................ 43
Engage in Disciplinary Practices .................................................................. 43 Build Procedural Fluency ............................................................................. 47 Assess Student Progress ............................................................................... 48
Sharing Phase .......................................................................................................... 50 Find Authentic Audience .............................................................................. 50
Chapter 4: Conclusion ...................................................................................................... 52 Limitations and Recommendations ...................................................................................... 52 Conclusion .......................................................................................................................... 53
REFERENCES ................................................................................................................... 55 APPENDIX......................................................................................................................... 60
x
LIST OF TABLES
Table 3.1 Smith and Stein’s (2011) Five Practices for Orchestrating Productive Mathematical Discussions ................................................................................ 29
xi
LIST OF FIGURES
Figure 2.1 Teaching and Learning Practices. ..................................................................... 12
Figure 2.2 PBLWorks Gold Standard Design Elements (left) and Teaching Practices (right). ................................................................................ 15 Figure 2.3 The Core Practices of Project-Based Teaching.................................................. 16
Figure 3.1 Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching Framework ........................................................................................ 27
Figure 3.2 Gold Standard PBL: Design Elements (left) and Teaching Practices (right). ................................................................................ 27
Figure 3.3 NCTM Effective Mathematics Teaching Practices ............................................ 28
Figure 3.4 Teaching Practices for PBL in the Mathematics Classroom .............................. 42
1
CHAPTER 1: INTRODUCTION
It is not uncommon to hear someone say “I am not a math person,” or “I was never
very good at math.” It is a pervasive attitude shared amongst students, parents, non-math
teachers, and administrators. Even some math teachers believe that some people just aren’t
good at math. Boaler (2016) examined these fixed mathematical mindsets and learned that
these beliefs were damaging to a student’s ability to learn. However, there is hope! Her
research found that students can develop a growth mindset in mathematics, especially in a
math classroom where “mathematics is taught as a creative and open subject, all about
connections, learning, and growth, and [where] mistakes are encouraged” (Boaler, 2016,
p.20). Project-based learning (PBL) is a teaching methodology that is designed to foster this
type of learning. Although there are not a lot of examples of how PBL has been used in the
mathematics classroom, there is a large body of research on effective teaching practices in
both PBL classrooms and math classrooms. Thus, this paper will examine those practices,
outline the connections that can be drawn between them, and then recommend a framework
for utilizing PBL to support mathematical learning. To begin, the following two sections will
discuss the historical developments in education related to PBL and mathematics, and then
describe the problem still facing the effort to implement PBL in the math classroom.
Brief History of PBL and Mathematics
PBL is a relatively new term but, according to Knoll’s (1997) historical compilation
of the use of projects as a teaching methodology, the concept dates back centuries. It has its
origins in architecture and engineering education, where it was realized that learning by
lecture was not enough to become prepared to be successful as an architect or engineer
(Knoll, 1997). Just because students understand how a building should stand up on its own
2
does not mean that they are capable of building a structure that actually can stand up on its
own. Projects have since become the foundation of many other vocational disciplines, such
as medical education, that recognize the act of doing a job well cannot solely be learned in
books and lectures (Knoll, 1997). As Condliffe, et. al. (2016) points out in their literature
review of PBL, many school reforms have been based on the goal of graduating students who
are college and career ready. It stands to reason that utilizing the educational strategies that
careers employ, such as PBL, would help schools reach the goal of career readiness.
It is also not a new idea to use PBL in the K-12 learning environment. According to
Condliffe, et. al. (2016), PBL became popular to use in public education as part of the
progressive movement, coupled with the constructivist philosophy. Around the same time,
according to Klein’s (2003) history on mathematics education, there was a great debate
around which mathematical ideas are important to teach and how to teach them. The
progressive movement had sparked educational reforms in many subjects, and math teachers
were struggling to figure out how to utilize constructivist pedagogies, such as PBL, in their
practice (Condliffe, et. al., 2016). In response, the National Council for Teachers of
Mathematics (NCTM) advocated for mathematics students to spend more time problem-
solving and utilizing appropriate technology, and less time practicing tedious algorithms
(Klein, 2003; Confrey & Krupa, 2011). This period of education reform, from Klein’s (2003)
point of view, sparked a struggle between math education activists and parents, all trying to
decide if math education should be more geared towards career and citizen experiences
beyond school years or focus on preparing for academic rigor. This struggle still continues
today, but NCTM stated then and continues to advocate that mathematics education should
evolve with the needs of the current times, not rely on past ways that served needs that no
3
longer exist (NCTM, 2018). PBL, as described by PBLWorks (2019), an organization that
seeks to support educators in designing quality PBL experiences for their students, provides
structures that can support students as they solve problems and apply mathematical
procedures and concepts in an academically rigorous way. In their book on how to
implement PBL in the math classroom, Fancher and Norfar (2019) assure teachers that PBL
done well will not adversely affect academic rigor and performance on standardized tests, but
rather support deeper learning that allows students to retain more information over time.
According to Confrey & Krupa’s (2011) account of the development of the Common
Core State Standards (CCSS), a lot of work has been done over the past fifty years by a
variety of experts to support math teachers in the development of quality mathematical
curricula. There have been many iterations of content standards and process standards, with
the CCSS being the most comprehensive and widely used set to be developed (Confrey &
Krupa, 2011). The focus in CCSS on students developing conceptual understanding of each
idea aligns well with the goals of PBL, Condliffe, et. al. (2016) points out. Thus, it is
reasonable to believe that PBL would be an effective pedagogical tool for a mathematics
classroom. Additionally, studies, such as Capraro and Capraro’s (2015) longitudinal study of
the effects of PBL in secondary STEM classrooms on student learning, have shown that the
use of PBL has a positive effect on student growth, especially that of low achievers.
Condliffe, et. al. (2016) adds that many studies have shown PBL to have “positive effects on
students’ engagement, motivation, and beliefs in their own efficacy” (p. iii). Although they
go on to explain that very few empirical studies have focused on math classrooms, the ones
that did found that students who studied math in a PBL setting outperformed their
counterparts learning math in a traditional setting (Condliffe, et. al., 2016). Therefore, using
4
PBL to teach math has the potential to positively affect students from all learning
backgrounds and in all desired ways.
Statement of the Problem
Despite the many similarities between the recommended practices from PBL
advocates and mathematics education advocates, math teachers have struggled to incorporate
PBL into their instruction (Condliffe, et. al., 2016). There have even been instances of
administrators implementing schoolwide initiatives to teach using PBL, but allowing their
math teachers to choose whether or not they utilize it (S. Gibbons, personal communication,
March 12, 2021). Although much has been written on the implementation of PBL in general
classrooms, very few of the examples given are for mathematics classrooms, and of those
few examples, most are for elementary and middle school math classes (Condliffe, et. al.,
2016; Fancher & Norfar, 2019). Aslan and Reigeluth (2015) examined the challenges related
to learner-centered education, and explained that much of this difficulty comes from teachers
who do not believe that PBL can help them address deficits in prior mathematical knowledge
and meet state standards in the time they are given to teach their students. Tal, et. al. (2005)
studied science teachers in urban settings that were attempting to implement PBL, and they
found that when teachers started feeling overwhelmed by the many expectations and the lack
of resources needed to live up to the expectations, they turned back to whole class teacher-
centered instructional techniques. This phenomenon is also occurring in math classes and,
just like in Tal, et. al.’s (2005) study, it is negatively affecting student learning, contributing
to the continuation of math deficits, frustration, and dissatisfaction.
For teachers who decide to attempt to learn how to teach math using PBL, there are
very strong PBL frameworks to consult, such as the Gold Standard PBL Teaching Practices
5
and Design Elements (PBLWorks, 2019). There are also many strong math frameworks to
consult, such as the NCTM Mathematics Teaching and Learning Practices (Martin, 1998;
NCTM, 2014; NCTM, 2018). Teachers can find blog articles, videos, and example projects
on the PBLWorks (2019) website, but only a few are geared towards math teachers.
Additionally, there is one book written by math educators Fancher and Norfar (2019) that is
rich with great examples and descriptions of how to implement PBL in a math classroom
(grades 6-10), but teachers would need to purchase and read the book to access those ideas.
There is a need for a reliable framework that can be consulted when designing and
implementing PBL units in a math class, especially at the high school level. Given that math
teachers already have difficulty implementing PBL, it is unreasonable to expect them to be
willing and able to consult all of these resources to ensure that they are developing and
implementing quality projects. The goal of this paper is to research and analyze the
similarities and differences of the existing frameworks for these effective teaching practices,
and to use that research to develop a new framework specific to project-based mathematics
classrooms that accounts for those best practices.
6
CHAPTER 2: REVIEW OF LITERATURE
In this chapter, literature related to PBL and mathematics education will be reviewed
in order to provide a background for the framework. It will start by exploring the literature
related to why educators would want to use PBL in the mathematics classroom, analyzing
recommendations for mathematical learning and empirical studies of how project-based
classrooms created environments in which those recommendations could be fulfilled. Then,
the review will discuss the challenges that researchers have found when attempting to
implement PBL and similar pedagogical techniques. The following two sections will discuss
the effective teaching practices that have been defined in the literature for teaching
mathematics and teaching in a PBL classroom, respectively. The final section of the chapter
will review the math-specific PBL literature, which is mostly comprised of Fancher and
Norfar’s (2019) book.
Motivation for Using PBL in the Mathematics Classroom
NCTM “advocates for high-quality mathematics teaching and learning for each and
every student” (2017, p.1), so their research-based recommendations are widely used by
math educators and math education professionals in order to make curricula and course
decisions. In their most recent publication, Catalyzing Change in High School Mathematics
(NCTM, 2018), they recommend that students are offered mathematics courses that “do not
limit their ability to continue studying mathematics but expand their professional
opportunities, become equipped to understand and critique the world, and foster in them joy
and an appreciation for the beauty of mathematics” (NCTM, 2018, p.19). NCTM (2018)
stated in this report that they found evidence that students who do not develop a deep
understanding of mathematical content are more likely to drop mathematics courses as soon
7
as they are given the chance. Condliffe, et. al. (2016) notes that PBL seeks to create space for
deeper learning of the content, so it is a pedagogical approach that could remedy this issue. It
also is, by design, a methodology that connects students to professional opportunities, both
within discipline and across disciplines, and prepares students to understand and critique
their world, so PBL certainly would be useful for reaching the goals set by NCTM
(PBLWorks, 2019; Grossman, et. al., 2019). The final goal, fostering joy and appreciation of
mathematics, seems to be the loftiest of the goals, generating doubt among math teachers
who have worked with a wide array of student demographics (S. Graham, personal
communication, January 18, 2021). This paper will show, however, that learning how to
teach math using PBL empowers teachers to create learning experiences where students
begin to understand just how beautiful and powerful mathematics can be.
Grossman, et. al. (2019) surveyed experts in PBL and studied teachers that had been
identified as effective project-based practitioners to learn what teaching practices were
necessary to implement a good PBL unit. One of the practices that they identified was that
teachers "engaged students in disciplinary practices” (Grossman, et. al., 2019, p.45). This
means that math teachers should be ensuring that students act as mathematicians would,
practices which are outlined for teachers in the standards for mathematical practices
associated with CCSS (NCTM, 2014). As Knuth (2000) points out, in his study on how
students understand mathematical connections between graphs and equations, experts must
have knowledge that “extends beyond simple procedural competence,” (p.506), so it is not
enough for students to master a procedure involved in solving a problem. Instead, they must
be able to move flexibly between representations and to choose appropriate representations
for solving a variety of problems (Knuth, 2000). PBL is a classroom strategy that supports
8
this development of flexible understanding, since PBL is founded on authenticity in
disciplinary practices as well as real world connections (Grossman, et. al., 2019). In PBL,
students are presented with the opportunity to reason through challenging problems
(PBLWorks, 2019), author ideas, and justify their reasoning. In her work on collaborative
learning in the mathematics classroom, Langer-Osuna (2017) found that students who author
their own mathematical ideas and are able to justify their reasoning tend to form more a
robust understanding of the mathematical concepts associated with procedures that they are
expected to learn. Similarly, Herbel-Eisenmann, et. al. (2013) recommended six teacher
discourse moves, all of which foster student discourse around mathematical ideas, because
they found that when students talk about math, they are more likely to learn math. It stands to
reason that math teachers would benefit from learning how to implement PBL in their
classrooms, since PBL is capable of fostering a classroom environment in which students
engage in the learning opportunities that NCTM recommends and studies support.
Fancher and Norfar (2019) and Condliffe, et. al. (2016) both point out that as society
changes, mostly due to technological advances, the role of school, and more specifically
mathematics, changes as well. Calculators and computers can now do all of the procedural
work that used to fall on the mathematician, so now it is more important for students to be
able to utilize mathematics to make sense of a problem and create models that could help
solve the problem (Grossman, et. al., 2019). School used to prepare students to work in
factories, sit in lecture halls taking notes, and live in a society where following rules was one
of the most important aspects of being a citizen. Now, being college and career ready means
that students have developed “success skills, such as critical thinking, self-regulation, and
collaboration” (Condliffe, et. al., 2016, p.6). These skills are also described in NCTM’s
9
(2018) effective mathematical practices and are key elements of the PBL model (PBLWorks,
2019). Thus, advocates for mathematics education should turn to PBL as a primary resource
for developing curricula.
Barriers to PBL in the Mathematics Classroom
There are a number of barriers that teachers list when describing why they are
hesitant to use project-based learning approaches in their classrooms. Haag and Megowan
(2015) studied a similar teaching approach, inquiry-based learning (IBL), in science
classrooms to learn more about teacher readiness in implementing IBL. One of the most
prominent barriers they found was time. It is reasonable to conclude that math teachers
attempting to use PBL would note similar issues, in fact, Aslan and Reigeluth (2015) found
that math teachers struggled with PBL because the pace of the state standards was too fast.
DiBiase and McDonald (2015) elaborated on this barrier in their study on science teacher
attitudes towards IBL, finding that the time constraint had many facets including days in the
semester, time allotted to each class period, and time to teach the number of standards
allotted per course. Again, these issues are not unique to science courses, and certainly affect
the decisions that math teachers make each day about how they will introduce students to a
concept. In addition to concerns about having enough time with the students for PBL,
teachers also have to come to terms with the amount of time they will need to spend
“providing feedback, guiding reflective activities, and helping students consider how they
can improve their work” (Grossman, et. al., 2019, p.47). This can be especially problematic
in large class sizes, which DiBiase and McDonald (2015) and Tal, et. al. (2005) both note
when investigating teacher concerns about time. Teachers also must find time to attend
professional training to develop their PBL skills and their content knowledge (Haag &
10
Megowan, 2015), since teaching using PBL “requires expert application of knowledge and
constant adaptation to diverse contexts and students” (Grossman, et. al., 2019, p.48).
Managing all of these time constraints requires teachers to creatively redesign the way they
approach their classes, which is often too much for teachers in the midst of an already hectic
job. Thus, advocates seeking to support math teachers in implementing PBL must help them
understand how to utilize PBL as a solution for learning concerns, not an added burden.
Another prominent barrier to effectively using PBL to teach math is teacher content
knowledge. As mentioned previously, Grossman, et. al. (2019) found that effective PBL
teachers must be capable of applying expert content knowledge when developing their lesson
plans in order to create opportunities for students to engage in disciplinary practices.
Additionally, teachers must be able to collaborate with students on ideas that surface during
the lesson, regardless of whether those ideas had been thought about during the planning
phase (Grossman, et. al., 2019). Teachers that do not have a deep grasp of their content and
the disciplinary practices associated with it will have difficulty designing these opportunities
as well as recognizing and encouraging them when their students discover something that
they do not completely understand. Authenticity is also a key element of PBL (PBLWorks,
2019; Grossman, et. al., 2019), and expert mathematical knowledge is essential to being able
to find authentic connections between mathematical disciplines. Examples of these
connections include: how statistics or geometry can be used to develop algebraic ideas, how
mathematics is used in other disciplines (NCTM, 2018), how mathematics can be used to
solve a global problem (Himes, et. al., 2020), or how personal experiences in students’ lives
can influence their solution to a mathematical problem (Fancher & Norfar, 2019). It can seem
daunting to look at the long list of mathematical standards and attempt to find authentic
11
connections that can elicit sustained inquiry in a project, but Fancher and Norfar (2019)
maintain that, once a teacher starts this process, it becomes challenging to look at a standard
or a situation and not see a project opportunity.
Finally, according to DiBiase and McDonald (2015), teacher beliefs on how to best
prepare their students for standardized assessments prevents many teachers from using PBL
in the classroom. They found that there is a lot of pressure placed on teachers to prepare their
students to perform well on the standardized state assessments. This leads teachers to believe
that it is more important to develop procedural fluency and test-taking skills than it is to
develop problem solving skills or conceptual understanding. Teachers are given a long list of
standards to teach their students, most of which will be assessed by at least one question on
the test, and they believe that time spent on things like PBL will take away from the time
students need to become fluent in the procedure that will be tested (DiBiase & McDonald,
2015). The standards have been rewritten in an attempt to encourage teachers to spend more
time on problem solving, interpretation, and model building (Condliffe, et. al., 2016; Confrey
& Krupa, 2011), but the state assessments do not seem to support these adjustments in the
standards (DiBiase & McDonald, 2015). Therefore, many teachers are still resistant to
exploring innovative practices such as PBL that they believe will take away from their
students’ ability to score well on the assessment.
There will always be reasons for teachers to avoid learning how to teach using a
different tool. However, in the effort to provide quality mathematics education to each
student, it is important to learn how PBL can be utilized in the mathematics classroom and to
strive to provide support to teachers as they implement PBL in their own classrooms.
12
Effective Mathematics Teaching Practices
Over the past few decades, NCTM (Martin, 1998; NCTM, 2014; NCTM, 2018) has
been advocating for students to get the opportunity to solve problems with a variety of
appropriate strategies, communicate their mathematical reasoning, and create representations
to model and interpret phenomena in mathematics and other disciplines. They have released
short lists of teaching and learning practices for teachers to utilize, such as the “5 Process
Standards” (Martin, 1998, Figure 2.1) and the “8 Mathematical Teaching and Learning
Practices” (NCTM, 2014, Figure 2.1). This section will briefly explain these different
frameworks and then chapter three will go into greater detail about each component and how
they connect to the other frameworks.
Figure 2.1
Teaching and Learning Practices
13
Each of these lists encourage teachers to create an environment where their students
are actively making sense of problems and utilizing mathematical models to find solutions
(NCTM, 2014). NCTM (Martin, 1998) asks teachers to give students the opportunity to
communicate their reasoning with their peers, author ideas that could be used to solve the
problem, and reflect on their models and solutions in order to make adjustments.
There have been a variety of researchers seeking to help teachers understand how to
create this environment. In his investigation of how to support productive struggle in learning
math, Kapur (2013) found that teachers must begin their instruction by giving students a
challenging task. Too often, students are presented with multiple tasks that follow a
prescribed solution that they have already been taught, and they fail to develop the ability to
flexibly apply and adapt their problem-solving strategies (Knuth, 2000; Martin, 1998). For
example, as Knuth (2000) describes, students may learn about lines by looking at equations
and graphs in the context of slope-intercept form. Then, they are presented with multiple
problems, some word problems, some graphing problems, but all with the exact solution path
that they have recently been taught, using slope-intercept form. There is little flexibility in
how they might solve the problem, there is no room for creativity in solving the problem, and
the main struggles that students might face are related to misconceptions rather than the
nature of a messy problem. Heibert and Carpenter (1992), in their study of how
mathematical language supports mathematical understanding, were concerned about this
because they saw that students who were taught to simply repeat procedures that they had
learned often failed to draw the mathematical connections across representations, despite the
obviousness of those connections to their teachers. In order to address this concern, NCTM
(2014) recommends that students experience productive struggle while working on a task. In
14
response to this recommendation, Warshauer (2015) studied strategies that math teachers use
to support productive struggle. He contends that students should be allowed to experience
struggle, test their thinking, and oscillate between doubting themselves and believing in
themselves as they work through a problem and apply mathematical ideas (Warshaeur,
2015). This process does take time and requires the teacher to create an environment where
mistakes are celebrated as learning opportunities (Fancher and Norfar, 2019), but it is how
mathematicians (and other scientists) engage in a problem, so it is important for students to
experience it (Grossman, et. al., 2019). It also allows teachers to give their students agency
over their learning, and share the authority of mathematical understanding with their teacher
and peers (Langer-Osuna, 2017). When students get the opportunity to engage in this
productive struggle through a challenging task that could be solved in multiple ways, then
they are more likely to be able to make meaningful mathematical connections (NCTM,
2014).
PBL not only supports all of these suggested teaching and learning practices, but it
also provides more details for how teachers can create the environment in which students
have these opportunities. Fancher and Norfar (2019) outlined four teaching practices that
they believe are important for using PBL in the mathematics classroom: “plan lessons that
are standards-based, encourage wonder and curiosity, provide a safe environment in which
failure occurs, and give students opportunities for revision and reflection” (p.8). These
practices are exactly the same practices that NCTM has been advocating for all of these
years, which demonstrates that PBL absolutely can be used in the math classroom and could
even be one of the best pedagogical methods to support the effective mathematics teaching
practices.
15
Effective Project-Based Teaching Practices
There are two frameworks that will be used to understand what effective project-
based teaching practices look like: PBLWorks’ (2019) Gold Standard PBL (Figure 2.2) and
Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching (Figure 2.3). PBLWorks
(2019) separate their Gold Standard PBL into teaching practices and design elements, where
the teaching practices describe the culture of the classroom and the design elements describe
the project itself.
Figure 2.2
PBLWorks Gold Standard Design Elements (left) and Teaching Practices (right)
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Figure 2.3
The Core Practices of Project-Based Teaching
The environment necessary for effective PBL is very similar to the environment
described earlier for effective mathematical learning. According to the PBLWorks (2019)
Gold Standard PBL framework, teachers design and plan a project centered around a
challenging problem that is aligned to the relevant content standards and promotes
perseverant problem solving. Each aspect is important; if the problem is not challenging, then
students will not be able to sustain inquiry, and if the project is not aligned to relevant
content standards, then the time and effort will not seem worthwhile. Teachers need to build
a culture that supports student inquiry and agency, which includes teaching them how to
regularly reflect and revise (PBLWorks, 2019; Grossman, et.al., 2019). Students should be
actively exploring solutions, learning content that helps them make sense of the problem, and
17
go through an iterative process of reflection and revision until they have created the best
version of their solution possible (within the time constraint, of course).
Teachers share the authority of learning with students by engaging in student
reasoning, assessing student learning, and coaching students as they work to reach their
project goals (PBLWorks, 2019). Since teachers cannot plug their brains into the brains of
their students and download the relevant information, it is essential that students are active
participants in their own learning. It is equally important that teachers remain aware of
student learning and student goals, so they can continue to support students, hence the need
for continuous formative assessment. This collaboration between the teacher and the students
gives the students ownership of what they learn, what they need to learn, and how they plan
on learning it, with the teacher acting as a facilitator.
Finally, PBLWorks (2019) states that project-based teachers provide students with
authentic learning experiences, where they are able to engage in disciplinary practices, create
products for authentic audiences, and make sense of problems set in a real-world context.
Authenticity, according to Grossman, et. al. (2016), is a key ingredient for student motivation
since it allows students to make connections between what they are doing in math class either
to things that they care about outside of school or to why anyone in the world might care.
PBLWorks (2019) uses the phrase public product to describe the concept of an authentic
audience, because they want teachers to think about how students can present their findings
publicly. This is important for motivating students to produce quality work, as Revelle, et. al.
(2019) found after looking at the writing of second graders when asked to write for a public
librarian rather than their teacher. According to Spires, et. al. (2019), in their study of
interdisciplinary IBL, in addition to having a final public product, the authentic audience can
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be accomplished throughout the project by incorporating external experts into the feedback
process, further motivating reflection and revision. Fancher and Norfar (2019) suggest that
teachers start small, creating authentic situations for external adults to role play for students,
such as acting as clients, and then when they are comfortable with that, teachers can begin to
branch out to community members. Fortunately, communities are full of people who would
love to help students learn, such as university professors, public policy makers, school and
district leadership, and local small businessmen. Additionally, as Spires, et. al. (2017) found,
technology has made it possible for students to connect with communities worldwide, so
consulting with experts has become even easier because experts can work with students via
the internet.
Much of what has been described could simply be called great teaching, as there are
many great teachers that are creating rich, student-centered learning environments without
the use of projects (Grossman, et. al., 2019). However, as Grossman, et. al. (2019) goes on to
say, project-based teachers leverage the project context to drive sustained rigorous inquiry,
which allows students to ground their learning in a compelling context. This helps them to
make connections and retain learned concepts and procedures, as well as increases
motivation. Condliffe, et. al. (2016) makes sure to point out that motivation is a happy by-
product of PBL, and should not be the main draw or the deciding factor when choosing a
project context. The project is the central driver of the learning goals, Condliffe, et. al. (2016)
states, and should span a significant period of time in which multiple standards are addressed
in connection to the project. As Spires, et. al. (2017) points out, the project should not be
extra, but rather integrated strategically into the curriculum, serving as the conduit for
learning the desired standards. PBL does motivate students to produce quality work by
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allowing students voice and choice in their project designs and finding authentic audiences
that will experience the final product or provide helpful critiques along the way (PBLWorks,
2019). More importantly, it teaches students how to continually seek improvement while
time allows, accept feedback from teachers, peers, and external experts, and apply that
feedback productively to create a better iteration of their product or solution (PBLWorks,
2019, Grossman, et. al., 2019). PBL is great teaching, but it is also so much more, and thus
math teachers who take the time to learn how to implement PBL in their classes will
experience learning rewards beyond what they knew was possible.
Using PBL in a Mathematics Classroom
According to Condliffe, et. al. (2016), “it has been noted that math teachers have
found it [particularly] difficult to integrate PBL into their instruction” (p.iii). This is
somewhat surprising given how much similarity there is between the suggested teaching
practices for math and PBL. On the other hand, maybe this is not so surprising after all. Math
teachers have been struggling to implement the recommended teaching practices for
mathematical learning, Heyd-Metzuyanim, et. al. (2018) assert in their study of math teachers
learning from professional development around the five practices for orchestrating
mathematical discussions. Jung and Newton (2018) found, in their study of pre-service
teachers’ understanding of mathematical modeling, that this is most likely due to differing
interpretations of the practices. They assert that math teachers are likely to interpret the
practices by referencing their experiences learning math, so if they have not experienced
mathematical modeling in the way that researchers intended, or they have not seen examples
of it, then they will instead think about how they learned math when they were in school or
how they have taught it in the past (Jung & Newton, 2018). Heyd-Metzuyanim, et. al.’s
20
(2018) work confirms this claim, stating that teacher beliefs tend to change after they have
implemented a new idea and witnessed the results firsthand. Fortunately, Fancher and Norfar
(2019) provided some great examples of how to design PBL units for the math classroom,
and this paper provides more examples to help explain each aspect of the framework. Keep in
mind, however, that the research suggests that teachers will need professional development
and support that model these examples or others in order to truly believe that PBL is a good
model for teaching and that they will be capable of utilizing it (Heyd-Metzuyanim, et. al.,
2018).
Their first suggestion is to simply invert the traditional methods of mathematical
teaching. Introduce the concept with a word problem, stripped of specific numbers and rich
with potential for curiosity (Fancher & Norfar, 2019). For example, consider this textbook
problem: find the surface area and volume of a Campbell’s soup can that has a diameter of 3
in. and a height of 10 in (Fancher & Norfar, 2019). Most textbook problems include the
numbers like this for students, which allows students to identify the relevant formula that
they have learned how to use, plug in the numbers, and, voila, the problem is solved. Instead,
as Fancher and Norfar (2019) suggest, the goals of this problem can be met by having the
students act as marketing consultants to Campbell Soup. There are multiple things that
Campbell Soup wants to change about their cans: the label is tearing on the edges, so they
need students to design a label that can cover the can but leave enough room to tear, and the
soup is sloshing over the edge when the machine puts it into the can, so the students need to
figure out exactly what volume of soup to tell the machine to place into the cans. With these
problems, students might explore different size cans that Campbell Soup offers, and then
figure out the solutions to these issues for those different sizes. Adjusting textbook word
21
problems is a great strategy for beginners of PBL because math textbooks and resources are
full of word problems with contexts that are relevant to students (Fancher & Norfar, 2019).
Fancher & Norfar (2019) do want teachers to keep in mind that not all word problems are
rich enough to be adjusted this way, so there is some strategy to selecting a word problem to
turn into a project, but many problems are rich enough so this is a great place to start. The
power of stripping the numbers from the problem is that it allows students to develop a
conceptual and contextual understanding, and experience the need to know for the associated
mathematical procedure, which is something Condliffe, et. al. (2016) found to be integral to
successful PBL lessons.
As students explore the problem, Fancher and Norfar (2019) advise that teachers
strategically anticipate what students will need to know and design tasks that help them build
those skills. Similarly, Smith and Stein (2011), in their five practices of scaffolding student
learning in an inquiry-based math classroom, encourage teachers to anticipate likely student
solutions to the challenging task at hand. Each learning task experienced during the PBL
process should create curiosity around important learning goals for the project, so that
students can continue to learn the standards that are aligned (Fancher & Norfar, 2019). In the
Campbell Soup project, for example, Fancher and Norfar (2019) had a lesson planned for
students to explore various sizes and shapes of cylinders and learn how to calculate the
volume and surface area of those cylinders. They anticipated that this learning task would
help students learn about what they needed to know from the client, such as height and radius
of the cans, in order to solve the problem presented to them. Throughout the project, students
should be developing the skill of identifying areas where they need to learn more, and
seeking the resources that can teach them those skills (Fancher & Norfar, 2019; Grossman,
22
et. al., 2019). Fancher and Norfar (2019) describe how they have their students write down
things they still need to know, and how they will learn them, with ‘from the teacher’ only
allowed once per week of the project. As mentioned previously, it is essential that the project
drives the learning that occurs, and simultaneously the learning that needs to occur should
drive the project (Condliffe, et. al., 2016). Thus, the learning tasks need to be intentionally
placed so as to focus student learning after their curiosity has run wild, without stifling
student curiosity by teaching concepts prematurely. Condliffe, et. al. (2016) further explains
that the work on the project should not be peripheral to the mathematical learning goals, but
rather a reason to be learning that mathematics.
Some math teachers find it challenging to figure out what the public product or
authentic audience could be for a project. Fancher and Norfar (2019) recognize that, so they
suggest that teachers utilize the concept of a client that students must design for, which is
what they do in most of their projects. The purpose of the project is authentic by reflecting
scenarios that occur regularly in career settings, and then they bring in outside adults to play
the role of the client (Fancher & Norfar, 2019). Whenever possible, it is nice to find adults
that can play this role who actually work in a related field, such as bringing in family
members, friends, or neighbors. In the example of the Campbell Soup cans, it would be
useful for students to meet with marketing professionals or manufacturers, to understand the
importance of the labels and sloshing soup to the company. These professionals would also
be able to provide insight into potential issues with proposed student solutions, which is why
attempting to connect with related experts is so powerful. However, teachers can also do
some research around the ideas that they are presenting to provide to the ‘client’ actors so
students can still experience this type of feedback. Other PBL advocates, such as Himes, et.
23
al. (2020) in their study of interdisciplinary IBL, suggest that teachers can find local and
global experts in a related field by utilizing universities, parent communities, and local
policymakers. Again, this, as Spires, et. al. (2017) pointed out, has become especially easy to
do with the various communication technology, such as email and videoconferencing, that
exists today. Working with experts to solve part of a problem to which they have devoted
their careers creates an even more authentic level to the work students are doing (Himes, et.
al., 2020). Additionally, Himes, et. al. (2020) has found that connecting student work with an
appropriate social contribution, such as raising money for a nonprofit related to their
exploration, also creates authenticity for them. Grossman, et. al. (2019) also speaks to this
idea of making a contribution, finding that many PBL teachers figured out ways for student
products to change things at their schools or in their local communities, such as building a
community garden. These ideas, although often interdisciplinary, all have the potential to
have connections to mathematics standards with a little creativity. The more projects that
teachers complete, the more they will see connections to their standards in the world around
them, and they will become more comfortable leveraging the relationships that they make in
their communities to support student learning (Fancher & Norfar, 2019).
In addition to finding an authentic audience, teachers can create other authentic
connections to the experience, even if the public product is more of a role play or
presentation within the school community. One thing that teachers can do is learn about the
personal interests of their students and incorporate those interests into project choices
(Grossman, et. al., 2019). For example, in Mr. Smaldone’s class (personal communication,
March 10, 2021) students were given the opportunity to create art by using technology to
graph polynomial functions. In this example, students were allowed to choose what image
24
they might like to create, which gave artists the space to be more creative while other
students found images that are traceable but still representative of their interests. Another
thing that teachers can do is learn more about the associated disciplinary practices, such as
how certain mathematical standards are utilized by actual mathematicians, scientists,
economists, designers, or engineers, to name a few (Grossman, et. al., 2019). The Campbell
Soup project is an example of this, because it draws on the mathematics involved in the daily
decision-making that marketing professionals and manufacturing engineers might do on a
daily basis. Teachers could also turn to history, as Bressoud (2010) suggests in his reflection
on the history of teaching trigonometry, as well as the various mathematical disciplines
(algebra, statistics, geometry, etc.) to learn how to create an authentic learning experience. In
other words, why was the procedure or convention invented, what problem was it attempting
to solve? How can that problem be recreated for students? Additionally, how could an
algebraic concept be derived from something related in statistics or geometry? For example,
statistical regression models can be used to fit equations to a scatterplot of the data from a
data set pulled from the internet, such data about clean drinking water available to people in
different countries around the world. This would provide context to the equation in an
authentic way, since most models in the real world are not based on perfect situations. After
fitting a model, it then makes sense to learn about the different features of that model, such as
the domain and range or intervals of increasing and decreasing, because those features can be
used to communicate with an audience what the graph represents in the context of interest.
Similarly, the history of mathematics is ripe with authentic experiences for commonly used
procedures and conventions taught in math classes today, so that is another way that students
can develop the need to know that drives their learning (Bressoud, 2010). An example of this,
25
according to Bressoud (2010), is to recreate the experience of the astronomers who first
developed trigonometry as a connection between arcs and lengths in a circle in order to
understand the movement of the stars and planets in relation to the earth. Throughout history,
people made discoveries like this that required the development of new mathematical
structures, so projects made to recreate these experiences could help students understand and
appreciate the beauty of mathematics.
Math teachers can implement PBL in their classrooms and address the many state
standards that students are expected to learn in one course. In fact, PBL should not be done
unless it is addressing the mathematical learning goals of the course. There are a variety of
ways to develop PBL units for a math classroom, and once math teachers get started down
the path, they will have a hard time looking at a standard or a situation and not seeing a
project opportunity. Although it can be challenging to learn in the beginning, the rewards that
come from watching students develop deep understanding and appreciation of mathematics
make it worth the effort. In the next chapter, this paper has merged all of the discussed PBL
and mathematics frameworks into one framework that can further support math teachers on
their journey to implementing PBL in their classrooms.
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CHAPTER 3: DISCUSSION OF THEORETICAL FRAMEWORK
It is now clear that PBL should be used in math classrooms, and that math teachers
are not yet comfortable teaching using PBL with the resources that exist. The first section of
this chapter will draw explicit connections between the different frameworks that were
discussed in the two sections on effective teaching practices from the previous chapter. The
following section will describe the new framework and show how it captures the heart of
each of the previously discussed frameworks in one resource for math teachers. This section
will then explain each aspect of the new framework in detail and provide an example of how
it would look in a project.
Connecting the Effective Teaching Practices
The various frameworks discussed in the effective teaching practices sections of the
literature review have strong connections to each other. For ease of description, each
paragraph will begin with Grossman, et. al.’s (2019) Core Practices of Project-Based
Teaching framework (Figure 3.1) and draw connections accordingly.
Core Practice 1: Disciplinary
Grossman, et. al.’s (2019) framework begins with the core practice of being
disciplinary. They outline three subcategories of this practice: “elicit higher order thinking,
orient students to subject-area content, and engage students in disciplinary practices”
(Grossman, et. al., 2019, p.45). The following paragraphs will draw explicit connections
between the three parts of this core practice and the different aspects of the other frameworks
(Figures 3.2 and 3.3; Table 3.1) described in previous sections.
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Figure 3.1
Grossman, et. al.’s (2019) Core Practices of Project-Based Teaching Framework
Figure 3.2
Gold Standard PBL: Design Elements (left) and Teaching Practices (right)
28
Figure 3.3
NCTM Effective Mathematics Teaching Practices
29
Table 3.1
Smith and Stein’s (2011) Five Practices for Orchestrating Productive Mathematical
Discussions
Five Practices for Orchestrating Productive Mathematical Discussions
1. Anticipating likely student responses to challenging mathematical tasks and
questions to ask to students who produce them.
2. Monitoring students’ actual responses to the tasks (while students work on the
tasks in pairs or small groups).
3. Selecting particular students to present their mathematical work during the whole-
class discussion.
4. Sequencing the student responses that will be displayed in a specific order.
5. Connecting different students’ responses and connecting the responses to key
mathematical ideas.
Grossman, et. al.’s (2019) practice of eliciting higher order thinking is also a
mathematical teaching practice outlined by NCTM (2014, Figure 3.3), using the wording
“elicit evidence of student thinking” (p.10). It is also strongly aligned with the PBLWorks
(2019) teaching practice (Figure 3.2) of scaffolding student learning. Teachers must prepare
questions and implement tasks (NCTM, 2014, Figure 3.3) that enable students to engage in
higher order thinking, as well as practice the art of “posing purposeful questions” (NCTM,
2014, p.10, Figure 3.3) when they encounter unexpected student thinking. In order to become
more adept at this practice, Smith and Stein (2011) outlined five practices for scaffolding
student learning in an inquiry classroom. The first two are anticipating likely student
30
responses and monitoring students’ actual responses (Table 3.1), which are important in
student-centered learning because teachers must be capable of engaging with student
reasoning. Teachers who are regularly thinking about potential approaches students will take
when introduced to a challenging problem, and then actively learning about their students’
actual approaches as they work on solving a problem, are more likely to be able to pose
purposeful questions (Smith & Stein, 2011). Expecting students to “use and connect
mathematical representations” (NCTM, 2014, p.10, Figure 3.3) and providing opportunities
for students to engage with multiple connected mathematical representations also allows for
teachers to elicit higher order thinking from their students during class. Thus, the practice of
eliciting high level thinking from students is intertwined throughout the effective teaching
practice frameworks for PBL and mathematics.
The second subcategory of Grossman, et. al.’s (2019) disciplinary practice is to orient
students to subject-area content. In the PBLWorks (2019) teaching practices seen in Figure
3.2, this is called aligning to standards, and Fancher and Norfar (2019) use the words
“planning lessons that are standards-based” (p.7). In NCTM’s (2014) teaching practices in
Figure 3.3, they name this practice “establishing mathematical goals to focus learning”
(p.10). It is evident from each framework that a project must be rooted in the learning goals
of the class, or it is not considered PBL. This is important because PBL is considered to be a
pedagogical tool, where the project is central to the learning rather than simply an assessment
of the learning that has taken place, often referred to as ‘dessert projects’ (Condliffe, et. al.,
2016). Fancher and Norfar (2019) suggest that the best projects usually begin with a look at
the standards and then layer in the authenticity where they see inspiration in the world that
connects to the learning goals stated in the standards. For example, in the polynomial art
31
project described earlier, students could potentially want to use circle equations, but if the
content standards for the course do not address circle equations, then the teacher should be
aware of how to support students in creating something that does not rely on circle equations,
but rather demonstrates knowledge of the relevant polynomial standards in the course. This
might involve setting requirements for the students of the types of functions that they are
allowed to use, and providing feedback to student designs along the way when they are not
aligned with the learning goals. There are many interesting ways to use mathematics in the
world, but remember that Condliffe, et. al. (2016) cautions teachers that motivation is an
awesome side effect of PBL, not the reason to use it. PBL is only effective if it is leveraged
to teach students what they need to learn to be successful in the course.
The final subcategory of Grossman, et. al.’s (2019) disciplinary practice, engage
students in disciplinary practices, is colored purple in the diagram (Figure 3.1) because it also
provides authenticity to the project. Thus, it is connected to the PBLWorks (2019) design
element of authenticity. In mathematics, disciplinary practices include designing and testing
mathematical models in order to solve a problem, which means this category is connected to
the NCTM (2014) teaching practices of using and connecting mathematical representations
and “implementing tasks that promote reasoning and problem solving” (p.10). Additionally,
mathematicians, much like most practitioners, must be able to communicate their findings to
the mathematics community. This means the NCTM (2014) practices of eliciting evidence of
student thinking and facilitating mathematical discourse allows teachers to support student
development of mathematical communication. Since mathematicians contribute to the
mathematical body of knowledge by continually wondering about what else could be learned,
32
Fancher and Norfar’s (2019) practice of “encouraging wonder and curiosity” (p.7) is also
connected to this aspect of the disciplinary practice.
Core Practice 2: Authenticity
The next core practice in Grossman, et. al.’s (2019) framework (Figure 3.1) is
authenticity. There are two subcategories in this practice, in addition to the one previously
discussed: “support students to build personal connections to the work and support students
to make a contribution to the world” (p.46). This is a popularly discussed practice associated
with PBL, and has proven to be particularly challenging for math teachers, especially those
teaching higher level mathematics (Condliffe, et. al., 2016). PBLWorks (2019) has the exact
same core practice included in their design elements framework (Figure 3.2) because they
also believe it is one of the most important pieces of PBL.
In the math class, teachers can support students to build personal connections to the
work (Grossman, et. al., 2019) by eliciting evidence of student thinking, using and
connecting mathematical representations (NCTM, 2014) and encouraging wonder and
curiosity (Fancher & Norfar, 2019). When engaging with students about their thinking,
teachers should not only wonder about a student’s understanding of the mathematics that
they are exploring, but also about how they are making sense of the problem in context of
their previous experiences. Mathematical representations are most powerful when connected
to a context, which gives math teachers the opportunity to design projects that capitalize on
student experiences to build their models. For example, in Fancher and Norfar’s (2019)
project simulation, students were introduced to a client that runs a movie theater and needs a
new popcorn container that appears to hold lots of popcorn, but actually holds much less. In a
situation like this, students are engaging with mathematical representations that will help
33
optimize this client’s popcorn container, while simultaneously drawing upon their personal
experiences with buying and eating popcorn (Fancher & Norfar, 2019). Students will also
naturally make connections to their personal lives if teachers encourage them to follow their
curiosities as they explore the problem (Fancher & Norfar, 2019). Teachers who regularly
converse with students about their interests for social emotional purposes may also find that
they can draw upon these conversations when designing projects, and know that it will
certainly be personally connected to at least one student in the class. This practice also helps
teachers avoid situations in which they think a context will be relevant to students, such as a
quadratics roller coaster project, only to find out that none of their students care about roller
coasters and would have rather designed the animations necessary for a videogame character
to throw objects across the screen (N. Smaldone, personal communication, February 22,
2021). Since math is often seen as disconnected from personal experiences for students, it is
important for teachers to integrate relatable contexts into mathematical learning experiences.
Supporting students to make a contribution to the world (Grossman, et. al., 2019) can
seem a little daunting to teachers. This is especially true for math teachers who are often
bombarded by students and parents doubting its usefulness beyond the classroom, as
Kilpatrick and Izsak (2008) described in their historical account of algebra in school
curriculum. This is where the public aspect of the PBLWorks (2019) design element public
product (Figure 3.2) comes into play. As described before, Fancher and Norfar (2019)
usually utilize fictional clients and bring in outside adults, who preferably experience those
roles in their careers, to play those fictional clients. For teachers looking for an even more
powerful way to connect students to an authentic audience, they could find community
members with authentic needs related to the project and have students design on their behalf.
34
For example, students could meet with a local entrepreneur who is in need of a logo and
compete to design the best logo using aesthetic principles related to geometry, such as
symmetry. It can also be done by finding causes or organizations to support that are doing the
work that students are studying, like in the project done by Himes, et. al. (2020), where
students donated water to a local food bank after studying how access to clean water affects
people locally and globally. Himes, et. al. (2020) also connected students to an expert in
global sanitation through the local university, leveraging the internet and the passion of a
university professor to generate interest and knowledge amongst the students. Spires, et. al.
(2017) connected students from the United States with students from China to do a project
comparing their cultures, and one could imagine that students could learn a lot from each
other mathematically as well in a partnership like that. Additionally, NCTM’s (2014) practice
of mathematical modeling can, and should, be leveraged to support students’ ability to
communicate with an audience. An example of this might be in the case of students
proposing to build a community garden at their school. In order to get the proposal approved,
they would need to present mathematical models that help their principal know how much
the garden will cost to build, how much it will cost to run, and how they plan on funding
those costs. Maybe they will be seeking funding from an outside organization, so they will
need to take their presentation to that organization as well. These models would need to be
visually appealing and easy to understand to people who have not been studying the
mathematical ideas. They might also need to diagram how much space the garden will take
up, as well as the design of the garden plots, relying on geometrical knowledge for spatial
awareness. Students feel empowered when their work at school serves an authentic purpose
in their communities, so these opportunities have a lasting effect on student learning.
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Although not included in Grossman’s (2019) framework, the challenging problem or
question in the PBLWorks (2019) framework (Figure 3.2) is a good way to create
authenticity for students from the beginning. With a compelling question or problem,
students become naturally curious, making it easy for teachers to encourage wonder and
curiosity (Fancher & Norfar, 2019). This also aligns with NCTM’s (2014) suggestion to
implement tasks that promote reasoning and problem solving. Keep in mind, however, that
the majority of the mathematical learning tasks described by NCTM (2014), Smith and Stein
(2011), and Boaler (2016) are more useful for daily lesson plans found throughout a PBL
unit, rather than serving as the PBL unit itself. The challenging problem or question,
alternatively, is intended to overarch each lesson plan, with the tasks connecting to the goals
of the question, and should extend over a much longer period of time (Condliffe, et. al.,
2016). The daily tasks can then be used to support the sustained inquiry work throughout the
project, serving to address student need-to-knows, assess student learning, and ground
student solutions in relevant learning goals. For example, in the popcorn container project,
Fancher and Norfar (2019) had their students use clay one day to create different container
shapes and investigate the volume of those shapes and how to determine if they appeared
larger than their actual volume. This would be considered a challenging mathematical task
because it has many different solutions for students to come up with, but it is only one lesson
in the overarching project of designing a container for the client. Other necessary math
lessons in this project might include surface area of the containers for material cost purposes
and how to determine pricing options of the popcorn. It should be impossible to answer the
challenging question in just a few days, and should lend itself to the ability to continually
improve with feedback throughout the course of the project. If students are capable of finding
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a perfect solution on the first day, or uninterested in improving their initial solution, then the
question or problem is not challenging or compelling enough.
Core Practice 3: Iterative
The third core teaching practice in Grossman, et. al.’s (2019) framework is that the
process is iterative, which they explain using three subcategories of what teachers do: “track
students’ progress and provide feedback, support students to give and receive feedback, and
support students to reflect and revise” (p.47). The iterative nature of the process is very
important to PBLWorks (2019). It motivated the circular representation of the Gold Standard
PBL frameworks (Figure 3.2), and the subcategories of Grossman et. al.’s (2019) iterative
category are very similar to two of the Gold Standard design elements and one of the Gold
Standard teaching practices.
Tracking student progress and providing feedback (Grossman, et. al., 2019) is an
important teaching practice in all pedagogical methodologies, but does not always have an
iterative nature. In the PBLWorks (2019) teaching practices framework, they call this process
assessing student learning, but in their design elements framework, they use the phrase
critique and revision, taking care to include time for students to revise based on the feedback.
In Smith and Stein’s (2011) practices for supporting mathematical discourse, they refer to
tracking student progress (Grossman, et. al., 2019) as “monitoring students’ actual responses
to the task” (p.10). In other words, formative assessment is crucial to effectively teaching
PBL in the math classroom. Throughout each class, teachers need to be actively collecting
evidence of student understanding so that they can assess how students are progressing in
terms of the mathematical goals as well as in finding a solution to the challenging problem,
and then determining what tasks need to be implemented in subsequent lessons. Smith and
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Stein (2011) also provide strategies for math teachers to be able to provide feedback during
class time: “select particular students to present their mathematical work during whole-class
discussion, sequence the student responses that will be displayed in a specific order, and
connect different students’ responses [to each other] and to key mathematical ideas” (Smith
& Stein, 2011, p.10). Basically, in the process of presenting their work and making explicit
connections, students will receive feedback from their peers and their teacher and be able to
revise any misconceptions that may have occurred in the first iteration of the solving process.
Feedback loops are very important for student learning, Ferriter and Cancellieri (2017) state
in their book on creating a culture of feedback in the classroom. Students need feedback to
know if they are on the right track or have made a mistake that is affecting their process
(Ferriter & Cancellieri, 2017). However, this feedback does not have to come directly from
the teacher, as some might believe, but rather should come from a variety of places, such as
testing a model using technology like Desmos or consulting a colleague in the class. As long
as teachers are facilitating opportunities for students to receive meaningful feedback, then
students can continue to progress towards the goals of the PBL.
Additionally, according to the NCTM (2014) teaching practices, effective teachers
should be supporting productive struggle, which can only be done well if teachers are
tracking student progress and know when their students are struggling unproductively.
NCTM (2014) recommends that teachers elicit and use evidence of student thinking and pose
purposeful questions to formatively assess student progress towards the learning goals. They
also should facilitate meaningful mathematical discourse (NCTM, 2014), which can be done
using the strategies outlined by Smith and Stein (2011, Table 3.1). Mathematical discourse is
a great form of feedback for students and should be used regularly to assess student
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understanding and provide students with the opportunity to engage with each other’s
reasoning (Herbel-Eisenmann, et. al., 2013). Meaningful mathematical discourse and strong
critique and revision, especially peer critique, can only occur in a safe environment built to
support failure and iterative improvement (Fancher & Norfar, 2019). Therefore, teachers
must build and maintain the culture (PBLWorks, 2019) that allows for the PBL experiences
to occur by using the teaching practices discussed.
Supporting students to give and receive feedback (Grossman, et. al., 2019) is another
place where math teachers might feel a little more uncomfortable. There are fears that
students will give the wrong feedback to their peers, which Langer-Osuna (2017) points out
certainly does occur sometimes, such as when students who dominate socially are believed
without question. However, Langer-Osuna (2017) encourages teachers to continue to foster
meaningful mathematical discourse by intentionally positioning students to author ideas,
which Smith and Stein (2011) contend can be achieved through deliberate selecting and
sequencing. Discourse can also happen when students engage with each other’s ideas and
reasoning (Herbel-Eisenmann, et. al., 2013). Thus, according to Fancher and Norfar (2019),
math teachers must provide a safe environment in which failure occurs and “build the culture
[by] explicitly and implicitly promoting student independence and growth, open-ended
inquiry, team spirit, and attention to quality” (PBLWorks, 2019, p.1). This will allow many
recommendations to be accomplished: students will engage with each other’s thinking
(NCTM, 2014), be curious about that thinking (Fancher & Norfar, 2019), and provide and
receive feedback in a productive way (PBLWorks, 2019).
The last part of the iterative process is to support students to reflect and revise
(Grossman, et. al., 2019). The revision piece of this subcategory has been discussed at length,
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but the reflection piece has been largely ignored until now. Reflection is one of the
PBLWorks (2019) design elements because it is central to student growth. Through
reflection, students critique their own work and improve the next iteration of the model,
which is a process that mathematicians must endure in order to find the optimal model
(NCTM, 2014). According to PBLWorks (2019), this reflection also allows teachers to
“identify when [their students] need skill-building, redirection, encouragement, and
celebration” (p.1). This helps students realize that learning is a cyclical process of
production, feedback, reflection, and revision (Grossman, et. al., 2019). It is important that
students know how to reflect so that, when they receive feedback from testing their products,
as well as from their teachers and fellow students, they are able to strategize about how to
revise their product for improvement in the next iteration. Jansen (2020) wrote in detail about
this process, calling it rough-draft math, which is a term intended to remind teachers that, just
like in the process of writing, the learning process begins with ideas that are not completely
refined but become refined with time, reflection, and collaboration. She argues that
conceptual understanding and procedural fluency come from the opportunity for students to
be “imperfect but precise, unfinished and unsure” (Jansen, 2020, p.350) in their thinking, and
then allowed to reflect, “revisit and revise” (p. 351) their thoughts as they learn more and
gain a better understanding. Thus, this process also supports NCTM’s (2014) mathematical
teaching practice of “building procedural fluency from conceptual understanding” (p.10). As
students revise and test their mathematical models, they continue to practice the procedures
involved, which allows them to build the procedural fluency. If they start from a place of
conceptual understanding, then their reflections can be more efficient in developing
improvements on their models.
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Core Practice 4: Collaboration
The fourth and final core practice in Grossman, et. al.’s (2019) framework is
collaboration. They touch on two aspects of this: “support students to collaborate and support
students to make choices” (Grossman, et. al., 2019, p.47). Supporting students to collaborate
is very similar to the PBLWorks (2019) practice of supporting students to give and receive
feedback, so the connected frameworks to this subcategory have been discussed in detail.
Something to keep in mind when supporting students to collaborate is that students need
tools for “justifying whether particular mathematical ideas are reasonable or correct and press
one another for explanations” (Langer-Osuna, 2017, p. 239). This can be modeled when
facilitating meaningful mathematical discourse (NCTM, 2014) in small groups and whole
class discussions, and by implementing tasks (NCTM, 2014) that have been intentionally
selected to “require a range of strengths and strategies to solve” (Langer-Osuna, 2017, p.
240). Teachers have also utilized group roles and scripted protocols that support students
facilitating their own collaborative mathematical discourse (NSRF, 2021).
Supporting students to make choices is also strongly tied to the PBLWorks
framework. In fact, they have a student voice and choice (PBLWorks, 2019) category in their
design elements, where students are supported to make decisions throughout the project.
Notably, students and teachers are collaborating at every step of the process. In the
PBLWorks (2019) “Design and Plan” stage, teachers are designing for some degree of
student voice and choice, and sometimes are even designing and planning the project with
students. As they manage activities with students, they work with students to organize tasks
and schedules that will work well for both parties, with a few teacher-set deadlines to help
guide the decisions. Fancher and Norfar (2019) have their students keep track of what they
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know and need to know, and design most of their tasks around what their students suggest
they need. Additionally, they require students to identify how they are going to learn each
‘need-to-know,’ and only a small percentage of those resources are allowed to be ‘from the
teacher’ (Fancher & Norfar, 2019). This collaborative learning process also helps students
build procedural fluency from conceptual understanding and support productive struggle
(NCTM, 2014) because “teachers and students share in the intellectual authority of the work”
(Langer-Osuna, 2017, p. 239). Effective teaching practices place student thinking at the
center of the learning, and teachers work alongside students to help them understand how
their ideas connect to the mathematical learning goals of the day.
After analyzing these frameworks, it became clear that there are very strong
connections between effective project-based teaching practices and effective mathematical
teaching practices. Although some mathematical teaching practices are not as obviously
apparent upon first read, they are all essential in serving different components of the PBL
experience, and can be used to make sense of how to utilize the PBL frameworks in the
mathematics classroom.
Framework for PBL in the Mathematics Classroom
The connections between all of the frameworks for effective teaching practices in
math and PBL revealed a need for a single framework that fully captures everything
discussed above. Thus, based on previously mentioned research this paper presents a
framework (Figure 3.4) for implementing PBL in the mathematics classroom, which was
developed by merging the core connections discussed in the previous section. The framework
consists of three phases: planning, working, and sharing.
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Figure 3.4
Teaching Practices for PBL in the Mathematics Classroom (see the Appendix for larger view
of the model)
Planning Phase
Align to Standards. The first thing teachers should do is establish mathematical
goals and an authentic context that are aligned to course standards. It is important to start
with the standards because a project will only be able to drive the learning experience if it is
designed to support the goals of the standards (Condliffe, et. al., 2016). Fancher and Norfar
(2019) point out that some projects are best when inspired by a standard and then connected
to an authentic or relevant topic, while others are more interesting when inspired by a topic
and then connected to a standard. Either way, the project must be designed to create an
authentic learning experience in which it makes sense for students to need to learn the
mathematics in the established mathematical goals. Otherwise, Condliffe, et. al. (2016) warn,
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the project runs the risk of become tangential to the learning, possibly interesting and
motivating, but not purposeful in the grand scheme of the learning goals.
For example, if a teacher needs to teach standards related to exponential functions,
they might plan to investigate bacteria growth from drinking contaminated water and the
potential effects. They will determine what learning goals need to be met, such as students
need to be able to write an exponential function from a situation, graph an exponential
function from an equation, and explain the key features of an exponential function. Now that
the mathematical goals have been identified and the authentic context has been determined,
the teacher is ready to move to the next part of the framework.
Working Phase
Next, the project enters an iterative phase, in which three main components are
happening simultaneously: engage in disciplinary practices, build procedural fluency, and
assess student progress. This phase is the most important phase because this is when the
student learning occurs. As all teachers know, even the best laid plans can fall apart in the
hands of students, so this phase requires that teachers remain in tune with their students.
Engaging students in disciplinary practices. At this stage, teachers have a choice as
to how much planning they need to do before they introduce students to the project. They
could choose to prepare in detail how they will engage students in disciplinary practices or
choose to involve the students in collaboratively planning with them as the project
progresses. Either way, this stage begins with exploring a problem that needs solving,
collaborating both with colleagues and teacher, and then building, testing, and revising a
mathematical model of the problem in order to find a solution, just as one would do as a
practicing mathematician. This should be an iterative process, with students regularly testing
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their theories as well as seeking feedback from the teacher, external experts, and peer
colleagues. The students are active participants in the way that the project moves forward.
Their thinking should drive the conversations that occur in the classroom, and purposeful
questioning should be used to elicit curiosity that pushes student thinking towards the
mathematical goals (Fancher & Norfar, 2019). There should be many iterations of the model,
so Fancher and Norfar (2019) state that teachers must support students as they test, fail, learn
from the test, and continue to improve the model. They should also be encouraged, both by
the teacher and the nature of the problem, to continue to improve the model, even if it works
the first time. Take note, this is the first opportunity for teachers to involve an authentic
audience in the form of external experts that students are encouraged (or required) to consult.
In the bacteria example, teachers and students may start by connecting with the
school’s biology teacher and learn about different examples of bacteria so they can
investigate how those different types of bacteria grow. They may find some datasets that
measure bacteria growth over time and do a regression on that data using a technology tool
like Desmos to make sense of the growth. Maybe they have the opportunity to partner with
the science class and collect some of their own bacteria data, or establish contact with a
disease expert at a local lab who studies bacteria growth and can share some of their
expertise with the students. If none of that is possible, then the teacher might provide a
realistic, albeit fictional, dataset on bacteria growth for students to explore. Then, they will
start modeling the bacteria growth: taking a look at the graphical representation and learning
how to describe it, fitting an equation to the dataset, learning about the various components
of the equation and what they represent, and then thinking about what all of that means to
someone trying to stop bacteria from killing children. They may learn about vaccines and
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how those behave, possibly in the opposite way as an exponential decay model, and think
about how vaccines could be distributed to countries that still don’t have them. It is possible
that the teacher will even find in the process of collaborating with their students that other
mathematical standards can be addressed as well. Then they will communicate their findings
to an audience, having to practice describing the graph and what it means, as well as
important aspects that the audience should understand, which means they will reach the goal
of understanding key features of an exponential graph. This is just one way that a project like
this could go, since it is in large part up to the students, the teacher, and the people in the
community that they are able to engage in this effort.
Build Procedural Fluency. During this process, teachers should help students build
procedural fluency by eliciting student reasoning from conceptual understanding, personal
connections, and iterations of the mathematical model. Procedural fluency is developed from
practice of concepts that have been deeply understood and grounded in experience. This
experience may come from prior mathematical knowledge or non-mathematical personal
experiences, but it is essential that teachers are working to make these connections between
new information and prior experiences explicit for students. Otherwise, it will be difficult to
achieve fluency in the sense that the procedure comes naturally and can be applied flexibly.
Teachers can support their fluency development by eliciting student reasoning as they make
conceptual and personal connections (NCTM, 2014). In addition to these explicit supports,
students will be able to practice related procedures as they revise their mathematical model,
since each time they revisit the process in order to improve their model. Students should
begin to recognize that certain mathematical procedures are more efficient than others in
certain situations, and develop flexibility with applying the most efficient or effective
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procedure for a particular context (Knuth, 2000). As they test and improve their model, as
well as critique their peers’ models, they should experience the practice that they need to
become procedurally fluent. Teachers can, of course, incorporate tasks that support this
practice if they notice shortcomings in certain areas from the natural progression of the
project.
To continue with the bacteria example, students could find themselves fitting an
exponential model to a set of data, in which the teacher might ask how their model would
change if they found out that some of the records in the dataset were bad and needed to be
thrown out. Or they may learn that vaccines work as exponential decay, killing off bacteria at
the opposite exponential rate from when they grew, so the student might investigate how
long a child could live with bacteria growing in them and how quickly a vaccine could kill
the bacteria. This situation would have them creating multiple models with different starting
values of bacteria, and might lead them to wonder about timing and how that affects
distribution. In this project, different groups of students would most likely be studying
different types of bacteria, so the teacher could create practice for her students through a peer
review cycle, where they have to set up a model themselves before being able to give valid
feedback to their peers. This experience would give students lots of procedural practice,
while still serving the needs of the project.
Assess Student Progress. Throughout the project, teachers should also be assessing
student learning using formative assessment such as mathematical discourse, purposeful
questions, peer feedback, personal reflection, and a variety of learning tasks. Mathematical
discourse allows students to author ideas and engage with each other’s ideas, which often
leads to adjustments in design and improvement upon previous notions. Teachers can foster
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this discourse using a variety of strategies, such as Herbel-Eisenmann, et. al.’s (2013) six
teacher discourse moves: “waiting, inviting student participation, revoicing, asking students
to revoice, probing a student’s thinking, and creating opportunities to engage with another’s
reasoning” (p.182). The three that specifically support assessment of student understanding
are asking students to revoice, probing a student’s thinking, and creating opportunities to
engage with another’s reasoning. When teachers use these moves, they can see if other
students in the room are able to make sense of what their colleague has presented, summarize
it in a meaningful way, or explain the thought in more depth. Additionally, it supports the
teacher’s understanding of student learning across the room as well as things that still need to
be addressed.
Purposeful questioning helps teachers make sense of student reasoning as well as to
gather evidence of student learning. It can be used in whole class discussions, but is often
most effective in small group or individual settings where students feel safe to author ideas
that could be incorrect (Langer-Osuna, 2017). According to NCTM (2014), questioning
should be open-ended, with the goal of understanding student reasoning, rather than
imparting what the teacher believes the student should be saying through a series of
questions.
Peer feedback serves multiple purposes: giving students more disciplinary
experiences, giving students the opportunity to engage with each other’s reasoning and
practice skills, learning about what is missing in their own mathematical model, and showing
the teacher how well students can analyze work and provide accurate and helpful feedback.
The process of giving feedback requires students to think about and understand what their
peers did in order to assist them. The act of receiving critiques from peers requires the ability
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to reflect and decide what is beneficial and what should be ignored. Teachers should help
students reflect on all of this feedback, as the reflection process will also benefit students in
improving their models in addition to showing teachers how well students are understanding
the mathematics.
Learning tasks are also forms of formative assessment, and provide structure to
students as they explore the mathematics. This is where math teachers will feel most
comfortable, because they have been utilizing learning tasks to teach for a long time. The key
in PBL is to pace them in a way that supports student progress without giving anything away
to students before it is necessary, in contrast to the traditional pace of providing students with
the information that teachers know they will need before they experience any reason to use it.
A strategically placed learning task will help teachers to be able to focus student learning on
a mathematical goal, and limit students from spending valuable time learning ideas that do
not support the learning targets of the project. The timing is important, however, because
teaching mathematics before there is a need for it could diminish the learning that happens
during the project. Teachers should have a way to monitor student progress on the learning
task, such as written responses or utilizing the teacher dashboard on a technology tool such as
Desmos; otherwise they will have a hard time assessing what students have learned and what
they still need to learn.
In the bacteria example, students might start with an exploration of a data set
representing bacteria growth, letting technology fit an exponential model to the graph for
them, and making observations about that graph and equation. After sparking curiosity
around exponential models, the teacher could then teach a lesson on exponential equations
and what each part represents. This can be done either by examining the different equations
49
from each group’s model and identifying patterns that students notice between their contexts
and their models, or by teaching a general model with a few basic examples and having
students return to their models and analyze them with their new knowledge. Then, with the
purpose of learning how to communicate with an expert about the models, the teacher could
do a lesson on key features of an exponential graph, having students practice describing
various examples to each other. Many learning tasks during a PBL unit will look very similar
to ones that are done in a non-PBL classroom, with the main difference being that teachers in
a PBL unit should be continuously connecting learning tasks back to the challenging
question, and it should be clear to students how the learning task will serve their needs in the
project.
Assessment of student learning should be happening throughout the learning process,
in every conversation, every task, and every opportunity to engage with the students.
Traditional assessments are also appropriate to sprinkle in throughout the PBL experience,
but they should not be the only form of learning evidence collected (Fancher & Norfar,
2019). Teachers will want to include at least one summative assessment during the process as
well, and it is encouraged to do so by creating a rubric for the final product, as well as for
periodic check-in’s through the course of the project. For example, since the learning goals
were to be able to write, graph and describe an exponential function, the rubric would need to
assess whether they were able to do that. Additionally, teachers should provide intentional
opportunities for students to reflect on their work, since a lot of learning can occur when
people reflect on their own progress towards a goal (PBLWorks, 2019).
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Sharing Phase
Find Authentic Audience. The final piece of the puzzle is to find an authentic
audience that can engage in the students’ solutions. This could be integrated into multiple
phases of the project, but should at least be at the end of the project. When there is an
authentic audience, or a public product as PBLWorks (2019) calls it, producing quality work
becomes much more important to students. Just to reiterate, teachers can find many authentic
audiences in their communities, such as university experts, parents of students with specific
expertise, school leaders, or local policymakers. They can use these adults to serve as clients
or consultants in realistic situations that the teacher created (Fancher & Norfar, 2019), unless
they can find adults who have an actual need from the students, which is even better. If
teachers cannot find adults in related fields to serve in this role, Fancher and Norfar (2019)
suggest to still bring in external adults to role play for students. Fortunately, people are often
excited to be able to engage with students about their expertise, so many times all it takes is
for a teacher or student to reach out and ask. Additionally, remember to look for
opportunities for students to be able to make a contribution to their community or the world
as this can make the work even more powerful to them (Himes, et. al., 2020).
For the bacteria example, students could consult with bacterial disease experts at the
local hospital or university, learning about how bacteria grow and how vaccines work to
combat that growth. For social action, they could find a group, such as Doctors Without
Borders, that works to distribute vaccines to areas where bacteria growth causes diarrheal
deaths, and raise money or awareness about that work. They could write children’s books
teaching about how bacteria grow exponentially, and what effects that has on children around
the world, and then go to the local elementary school to read their books. In another idea,
51
they could create an infographic that informs policymakers of their discoveries, and then
meet with policymakers and advocate for them to help distribute lifesaving vaccines to these
areas. Teachers who come up with lots of exciting options like these might decide to provide
students with the opportunity to choose, which is an alternate way to fulfill the PBLWorks
(2019) recommendation of student voice and choice.
The Teaching Practices for PBL in the Math Classroom framework was developed by
merging best practices frameworks in both PBL and mathematics to become a resource for
math teachers looking to implement PBL in their classes. It is made up of three phases:
planning, working, and sharing. In the planning phase, the most important thing for teachers
to do is to identify the mathematical learning goals and an associated authentic context. The
rest of the planning can either happen during this phase or the working phase. In the working
phase, there are three main components that teachers should be doing simultaneously and
iteratively: engaging students in disciplinary practices, building procedural fluency, and
assessing student progress. Then, the final phase is the sharing phase, where teachers need to
find an authentic audience for their students, or support their students in finding their own
authentic audience. This framework was designed to be able to support math teachers who
are just beginning with PBL as well as to be a resource for math teachers to consult once they
have become comfortable teaching with PBL.
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CHAPTER 4: CONCLUSION
This chapter concludes the paper with a discussion of the limitations related to the
literature review and the recommendations for future studies that can assist in addressing
those limitations. With so much work left to do in order to understand the effects of having a
math PBL framework as a resource, the author is excited to see what future research comes
from these recommendations.
Limitations and Recommendations
There is still a lot to be learned about how to effectively implement PBL in a
mathematics classroom, and how to know if PBL is an effective learning tool for the
mathematics classroom. Condliffe, et. al. (2016) points out that there have only been a few
empirical studies done on the effectiveness of PBL for teaching math, which they say is most
likely due to the relatively few math teachers attempting to use PBL. Thus, this paper has had
to rely on studies done in other disciplines, such as science and English, as well as studies
done on effective mathematics practices that, as previously discussed, are very similar to
PBL practices. The results of the studies that have been done on PBL in math classes should
be accepted with caution, since it is possible that the students would have outperformed their
peers regardless of the method used (PBL versus traditional) because the schools were not
randomly selected (Condliffe, et. al., 2016). Similarly, there are very few studies that look at
how to prepare math teachers to be able teach using PBL, the effectiveness of the programs
that are preparing teachers to use PBL, how to support them through the process once they
begin, or the effectiveness of the coaching that is being done to support them. There have
been some studies on the effectiveness of teacher preparation for PBL, such as Himes, et.
al.’s (2019) assessment of how well teachers applied their learning after being prepared to
53
utilize the PBI Global model and Germuth and EvalWorks’ (2018) evaluation of teacher
improvement after participating in the SummerSTEM PBL teacher preparation program,
which both saw improved teaching practices and have had math teachers participate.
However, these studies have not focused specifically on math teachers, and anecdotally, it
was found in the PBI Global that actually the math teachers needed a lot more support and
were barely incorporating the project into their practice (David Hardt, personal
communication, October 16, 2020).
Analysis of the PBL frameworks and math teaching practices suggests that PBL could
be effective, and the framework proposed in this paper is designed to provide teachers,
coaches, researchers, and administrators with a tool that could be used to plan, implement
and assess a strong PBL unit in a math class. The next step for this framework would be to
assess its validity, utilizing an evaluation rubric that is designed to reveal evidence of the
framework in a classroom. This rubric should be developed and then used to test the
comprehensiveness and usability of the framework for understanding how to implement PBL
in the mathematics classroom. Once the framework has been validated, it should be used to
develop professional development for math teachers on how to utilize it, since one of the
barriers is teacher preparation (DiBiase & McDonald, 2015). The framework should also be
used in partnership with math teachers to design and implement PBL in a mathematics
classroom. This would allow researchers to analyze how effective it is in supporting math
teachers as they implement PBL, and further support professional development for its use.
Conclusion
With a brief look at education history, it is evident that PBL is a tried and true
practice and that it provides teachers with structures that can help them create the learning
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environments recommended by math education experts. Despite this evidence, math teachers
find it challenging to implement PBL and there are very few math-specific PBL resources for
interested teachers to learn from. The studies that have been done show that students who
learn via PBL demonstrate more growth in their learning than their traditional counterparts.
Additionally, students and teachers who use PBL tend to be more motivated to work hard and
produce quality work. Thus, it is important that more math-specific PBL resources be
developed and more math teachers overcome the barriers to embracing PBL as a teaching
method. This paper is a start to that effort, providing a framework that merges best practices
in math education with best practices in PBL. The framework is based on three phases,
planning, working, and sharing, which should be used as a reference guide for teachers as
they design projects and as they reflect on how well the project went. Hopefully, the
framework and examples in this paper help math teachers feel confident that they can
successfully implement PBL in their classrooms.
55
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APPENDIX
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Figure 3.4
Teaching Practices for PBL in the Mathematics Classroom
Planning Assess Student Progress: Use mathematical discourse, purposeful
questions, peer feedback, and a variety of learning tasks.
Working
Engage Students in Disciplinary Practices:
Develop tasks that support students as they collaboratively explore a problem that needs
solving, then build, test and revise a mathematical model to find a solution.
Build Procedural Fluency: Elicit student learning from conceptual
understanding, personal connections to the work, and iterations of the mathematical
model.
Find Authentic Audience: Create an opportunity for students
to share their solutions with someone beyond their class,
preferably making a contribution to their community in some way.
Sharing
Align to Standards: Establish math goals and
authentic context.