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Publié par | Purdue University Press |
Date de parution | 15 mars 2017 |
Nombre de lectures | 0 |
EAN13 | 9781612494951 |
Langue | English |
Informations légales : prix de location à la page 0,1800€. Cette information est donnée uniquement à titre indicatif conformément à la législation en vigueur.
Exrait
Successfully Implementing Problem-Based Learning in Classrooms
Successfully Implementing Problem-Based Learning in Classrooms
Research in K–12 and Teacher Education
Edited by Thomas Brush & John W. Saye
Purdue University Press, West Lafayette, Indiana
Copyright 2017 by Purdue University. All rights reserved.
Printed in the United States of America
Cataloging-in-Publication data is on file with the Library of Congress.
Cloth ISBN: 9781557537805
ePDF ISBN: 9781612494944
ePUB ISBN: 9781612494951
Cover image Hand Drawing by Kalawin/iStock/Thinkstock
C ONTENTS
Problem-Based Learning in K–12 and Teacher Education: Introduction and Current Trends
Thomas Brush and John Saye
P ART I: P ROBLEM -B ASED L EARNING IN T EACHER E DUCATION
1 Transforming Preservice Secondary Mathematics Teachers’ Practices: Promoting Problem Solving and Sense Making,
Marilyn E. Strutchens and W. Gary Martin
2 Conexiones: Fostering Socioscientific Inquiry in Graduate Teacher Preparation,
Krista D. Glazewski, Michèle I. Shuster, Thomas Brush, and Andrea Ellis
3 An Instructional Model to Support Problem-Based Historical Inquiry: The Persistent Issues in History Network,
Thomas Brush and John Saye
4 Preservice Elementary Teachers Learning to Teach PBL Through Science-Integrated Engineering Design,
Pamela S. Lottero-Perdue
P ART II: P ROBLEM -B ASED L EARNING IN K–12 C ONTEXTS
5 Teacher as Designer: A Framework for Analysis of Mathematical Model-Eliciting Activities,
Margret A. Hjalmarson and Heidi Diefes-Dux
6 The Grand Challenge: Using a PBL Approach to Teach Cutting-Edge Science,
Peggy A. Ertmer, Sarah Schlosser, Kari Clase, and Omolola Adedokun
7 Using Technology-Enhanced Learning Environments to Support Problem-Based Historical Inquiry in Secondary School Classrooms
John Saye and Thomas Brush
8 Engaging Teachers’ Pedagogical Content Knowledge: Adopting a Nine-Step PBL Model
Karen C. Goodnough and Woei Hung
C ONCLUSION
What Is Missing; What Is Needed? Future Research Directions With PBL in K–12 and Teacher Education
Michael M. Grant and Krista D. Glazewski
Index
P ROBLEM -B ASED L EARNING IN K–12 AND T EACHER E DUCATION : I NTRODUCTION AND C URRENT T RENDS
Thomas Brush and John Saye
Welcome to this edited volume focusing on research exploring the use of problem-based learning (PBL) in preservice and inservice teacher education. Currently, there is a major debate regarding the most effective methods for providing the best educational experiences for K–12 students that will afford them with the experiences they need to succeed in the 21st century. Recent definitions of the requirements for high-quality teaching emphasize not only content and pedagogical knowledge but also the use of innovative instructional strategies to support students’ acquisition of a more flexible knowledge base as they engage in complex problem solving (Bell, 2010; Lombardi, 2007; Saye et al., 2013; U.S. Department of Education, 2010). PBL represents a widely recommended best practice that facilitates both student engagement with challenging content and students’ ability to utilize that content in a more flexible manner to support problem solving. In PBL, curriculum is anchored within authentic, ill-structured problems. Teachers guide and support students as they apply content knowledge toward problem solutions (Barrows & Kelson, 1993; Hmelo-Silver, 2004; Savery, 2015).
Current Trends of PBL in K–12 Education
Extensive research conducted over the past decade demonstrates that PBL can be an effective strategy for enhancing both student engagement with challenging content and students’ academic achievement with that content in K–12 settings (Brush et al., 2013). A number of meta-analyses focusing on the implementation of PBL in K–12 environments conclude that PBL instruction is more effective than traditional, teacher-centered instruction with regard to student achievement (Ravitz, 2009; Strobel & Barneveld, 2009; Walker & Leary, 2009). Wirkala and Kuhn (2011) explored the effectiveness of PBL with middle school social studies students and determined that students engaged in PBL instruction versus lecture-based instruction performed better on a number of outcome variables including content knowledge and argumentation. Linn and colleagues’ extensive research focusing on both the web-based science environment and “knowledge integration” framework continues to demonstrate the effectiveness of problem-based instruction when compared to typical instructional activities (Chiu & Linn, 2014; Linn & Eylon, 2011). Liu and colleagues (2014) and Pedersen and Liu’s (2002) research with Alien Rescue suggests that PBL can be an effective method for both student academic achievement in science and (perhaps more importantly) for students to transfer knowledge to both similar problems and different situations. Saye and colleagues (2013) analyzed the teaching practices of numerous middle- and high school teachers and found a strong positive correlation between teachers who engage in problem- and inquiry-based teaching practices and their students’ performance on achievement tests.
PBL has also been found to have a positive impact on a wide range of student abilities. For example, Glazewski and colleagues (2016) collaborated with a high school biology teacher on a problem-based unit focused on genetics. Results of the implementation of this unit with ninth grade students not only indicated that students had significant content knowledge gains in science (specifically genetics) but that students who were struggling with science content had significantly greater gains from pretest to posttest than their peers. Similar research demonstrating the positive impact of PBL with struggling students has been found in the areas of economics (Mergendoller, Maxwell, & Bellisimo, 2006) and scientific thinking in social science (Jewett & Kuhn, 2016).
PBL Versus Project-Based Learning
While extensive PBL research has been conducted in the areas of secondary science, social studies/history, and mathematics (e.g., Trinter, Moon, & Brighton, 2015), the research in the area of English language arts (ELA) is more limited. In addition, research focusing on PBL implementation at the elementary level is also limited. This may be due to the distinction between problem -based learning and project -based learning. Much of the literature examining inquiry-based curricular models implemented in the ELA curriculum and/or with elementary-age students tends to focus on project-based learning. This is even the case with new educational trends such as the maker movement, which tends to have students be more product/maker focused as opposed to problem focused (Halverson & Sheridan, 2014; Peppler & Bender, 2013; Smith, 2013). For example, Smith (2013) discussed an inquiry project with seventh and eighth grade ELA students in which they digitally fabricated pop-up books. The researcher specifically discussed how these types of “maker” initiatives should be considered project-based as opposed to problem-based.
From our perspective, PBL curricular models are distinct from many models proposed for project-based learning (Saunders & Rennie, 2013; Savery, 2015; Saye & Brush, 2004). In PBL, an authentic problem or central question is the overall focus of a unit; with project-based learning, the project or activity is the central focus of the unit of instruction (Savery, 2015). This sometimes can lead down a path in which the project takes over the curricular focus with little regard for the need for students to demonstrate any understanding of substantive, authentic problems. Barron and colleagues (1998) dismissed project-based learning that focuses on “doing for the sake of doing” (p. 273), or “action without appropriate reflection.” They define worthy projects as ones that integrate “doing with understanding” (p. 274). However, for Barron and colleagues, PBL is most meaningfully used when embedded in complex projects.
The Buck Institute for Education, a leading organization for development and promotion of project-based learning, has published what they refer to as “essential design elements” for any project-based curricular initiatives. They specifically state that “[t]he heart of a project … is a problem to investigate and solve, or a question to explore and answer” (Buck Institute for Education, 2015, p. 2). Similarly, Parker and colleagues (2011) used project-based learning to characterize a substantial problem-based curriculum project. Noting that project-based learning often refers to “a broad and often unspecified umbrella term for a wide range of pedagogies” (p. 538), they make clear that their use of the term project-based emphasizes activities in which students engage in deep, disciplined inquiry structured around authentic problems .
Thus, the distinction between PBL and project-based learning may become less of an issue as inquiry-based instructional models become more accepted in K–12 settings. However, the conceptualization of “problem-based learning,” “problem-based projects,” and “project-based learning” may warrant further discussion and clarification as we continue to explore the most effective methods for promoting inquiry in K–12 settings. In particular, the lack of PBL models in ELA and elementary settings may benefit from continued exploration of the commonalities between PBL and project-based learning.
The Need to Prepare Teachers to Implement PBL
The increasing evidence that PBL has a positive effect on a wide variety of student outcomes with a broad range of students is leading more K–12 schools to adopt PBL as an overarching model for their curriculum. School models such as Da Vinci Schools (2013), New Tech High (New Tech Network, 2015), and High Tech High (2014) have adopted technology-enhanced PBL approaches to their curriculum. With support from the U.S. Department of Education, Sammamish High School in Bellevue, Washington, has adopted an integrated PBL curriculum throughout their school, and the school district has plans to expand the curriculum to all schools in the district (Edutopia, 2013).
Given these trends, it seems appropriate and important for teacher educators to attempt to integrate effective PBL teaching practices into both preservice teacher education programs and professional development opportunities for practicing teachers. Unfortunately, many preservice and inservice teacher education programs still approach teaching methods with conventional practices (Feiman-Nemser, 2008; Kiggins & Cambourne, 2007), and few current and future teachers have clear conceptualization regarding effective design, development, and implementation of PBL instruction (Saye, Kohlmeier, Brush, Mitchell, & Farmer, 2009; So & Kim, 2009).
Some teacher education programs have attempted to address the need for teachers prepared to meet the instructional needs of these new school models by incorporating more PBL into their courses. A study of one program that introduced technology-enhanced PBL instruction to preservice teachers found that participating novice teachers indicated that they planned to utilize PBL in their future classrooms (Park & Ertmer, 2008). In another study, preservice teachers who had opportunities to develop collaborative PBL lessons in their methods classes demonstrated enhanced knowledge of PBL theory and practice (So & Kim, 2009). However, while more teacher education programs are recognizing the need to integrate PBL into their programs (Edwards & Hammer, 2006; Murray-Harvey & Slee, 2000), the research focused on methods to prepare both current and future teachers to successfully integrate PBL strategies in their classrooms remains limited.
The content of this volume begins to address this issue. This volume is an extension of a special issue of the Interdisciplinary Journal of Problem-based Learning (volume 8, issue 1) that focused on the topic of PBL in teacher education settings. Several of the chapters are versions of papers published in the special issue. However, this volume extends the focus of the special issue to include new chapters that focus on the integration of PBL strategies within both teacher education programs and teacher professional development in K–12 settings across a wide range of grade levels and content areas. This includes strategies to assist both K–12 teachers and teacher education faculty with implementing PBL within their teaching methods experiences as well as instructional approaches to assist preservice teachers with exploring the integration of PBL strategies into their future classrooms.
Overview of Volume
Based on the need to provide strategies for assisting both preservice and practicing teachers with implementing PBL in their classrooms, this volume is divided into two parts: “PBL Research in Teacher Education” and “PBL Research with Practicing Teachers.” Readers will find a wide range of strategies and models presented in these sections—all of which have been implemented in teacher education programs and/or K–12 classroom settings. A wide variety of grade levels are represented, including both elementary and secondary educational settings. In addition, the chapters represent a range of content areas including mathematics, science, and social studies/history. Throughout the volume, we ask “What strategies and models help prepare current and future teachers to effectively design and implement PBL teaching and learning activities?” The chapters included in this text attempt to provide insight into this question while identifying challenges and unresolved issues that invite further research on this important topic.
Part I discusses a variety of different PBL strategies that have been implemented in various preservice teacher education programs. In Chapter 1 , Strutchens and Martin provide an overview of their teacher education curriculum in secondary mathematics (including methods experiences, field experiences, and student teaching), in which they assist preservice teachers in developing research-based teaching strategies and practices to create classroom environments that foster mathematical problem solving and sense making. In Chapter 2 , Glazewski, Shuster, Brush, and Ellis present a specific model of PBL known as socioscientific inquiry and discuss how they integrated the model into a graduate-level teaching methods class in science to help prepare prospective teachers utilize this model to teach a variety of science content. In Chapter 3 , Brush and Saye describe problem-based historical inquiry, a PBL model specific to the areas of history and social studies. They discuss how they adapted this model to integrate persistent issues in history into a teacher education program and how preservice teachers applied this model as they developed PBL history units.
We have noted that there has been little empirical research investigating problem-based learning in elementary education—particularly with respect to preparing future elementary teachers to implement PBL in their classrooms. In Chapter 4 , Lottero-Perdue discusses how she inducts her preservice teachers into PBL practices for the elementary science/engineering curriculum. She integrates a model of PBL known as design problem solving (Jonassen, 2000) to provide a venue for preservice teachers to introduce advanced science and engineering concepts to elementary students.
Part II provides models and strategies for collaborating with practicing teachers on the implementation of PBL in their classrooms. In Chapter 5 , Hjalmarson and Diefes-Dux discuss a model of PBL known as model-eliciting activities and examine how middle school mathematics teachers use this model to develop tools to support student problem solving and presentation of problem solutions. In Chapter 6 , Ertmer, Schlosser, Clase, and Adedokun discuss an intensive professional development program for secondary science teachers in which the teachers applied PBL principles toward the development of a STEM unit focused on sustainable energy as well as the impact the experience had on their self-efficacy toward teaching science and their knowledge of science concepts. In Chapter 7 , Saye and Brush describe their line of research, exploring methods to support secondary social studies teachers as they integrate problem-based historical inquiry into their curriculum, and how various types of scaffolding (specifically referred to as hard scaffolding and soft scaffolding) can be used by teachers to support student inquiry. Finally, in Chapter 8 , Goodnough and Hung describe how elementary teachers used a nine-step PBL design model to develop and implement units focused on science concepts and the impact that the use of the model had on teachers’ pedagogical content knowledge and their willingness to integrate PBL practice into their future instructional activities.
Grant and Glazewski provide a summary chapter in which they discuss both common themes among the chapters as well as gaps in the current research in PBL and the need for additional exploration in specific areas (e.g., elementary education and ELA). They particularly highlight the need for more PBL research that examines sustainability of PBL curriculum efforts in both preservice and inservice teacher education, and call for longitudinal research that explores promising practices such as the ones described in this volume over a longer time period.
Technology to Support PBL Practice
While the use of digital technologies is by no means a prerequisite for the implementation of PBL in K–12 classrooms, technology is a theme throughout the various projects described in this volume. How technology is integrated within PBL can range from very specific applications to a broader suite of resources that support PBL development and implementation in multiple ways. Strutchens and Martin describe a variety of web-based tools and “apps” they have integrated into their mathematics education overall program. Glazewski and colleagues discuss how their students integrate online tools to develop their socioscientific PBL units. Saye and Brush describe a set of digital tools ( Decision Point ) specifically designed to support problem-based historical inquiry with both students and teachers. Ertmer and colleagues discuss the integration of technology as a crucial component of their professional development program with teachers as they design and develop PBL units. Thus, technology can be defined in many ways, and the various methods in which technology can be used to support PBL is diverse. However, it is important to note that in virtually all of the projects described in this volume, PBL can be effectively implemented without technology. When preparing current and future teachers to effectively implement PBL in their classrooms, the focus should always be on the pedagogical principles that make PBL an effective instructional model. Technology can definitely support the implementation of PBL, but technology is not a requirement for PBL.
Conclusion
While policymakers and K–12 school leaders continue to advocate for the adoption of more inquiry-oriented instructional models, many teacher educators have yet to fully integrate these practices into their preservice or inservice instruction. The lack of models for effectively preparing teachers to adopt a problem-based teaching practice exacerbates this problem. This volume provides multiple models for implementing problem-based learning in K–12 settings and for preparing teachers to apply these models to develop PBL instructional activities. We would like to thank the contributing authors for their scholarship and dedication in conducting exciting, innovative research that will expand our knowledge of PBL in K–12 and teacher education settings. We hope that this work will also stimulate further exploration of the continued challenges and issues that serve as barriers for successful implementation of PBL in schools.
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Parker, W., Mosborg, S., Bransford, J., Vye, N., Wilkerson, J., & Abbot, R. (2011). Rethinking advanced high school coursework: Tackling the depth/breadth tension in the AP US Government and Politics course. Journal of Curriculum Studies, 43 (4), 533–559.
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Saye, J. W., & Brush, T. (2004). Promoting civic competence through problem-based history learning experiments. In G. E. Hamot, J. J. Patrick, & R. S. Leming (Eds.), Civic learning in teacher education (Vol. 3, pp. 123–145). Bloomington, IN: The Social Studies Development Center.
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1
T RANSFORMING P RESERVICE S ECONDARY M ATHEMATICS T EACHERS’ P RACTICES : P ROMOTING P ROBLEM S OLVING AND S ENSE M AKING
Marilyn E. Strutchens and W. Gary Martin
Introduction
Since the 1970s, there have been calls for changes in mathematics instruction emphasizing problem solving, culminating with the recommendation by the National Council of Teachers of Mathematics (NCTM) in 1980 that “problem solving be the focus of school mathematics” (p. 1). Throughout the 1980s, there was considerable research on how to promote students’ problem solving in mathematics (cf. Charles & Lester, 1984; Goldin & McClintock, 1984; Schoenfeld, 1985). Near the end of the decade, a distinction was drawn between a focus on problem solving as an end of instruction (i.e., “teaching about problem solving”) and a focus on problem solving as a means of instruction (i.e., “teaching via problem solving”) (Schroeder & Lester, 1989, p. 32). This process view of problem solving promotes students’ development of a relational understanding of mathematics (Skemp, 1976/2006) in which students understand both mathematical procedures and the reasoning behind those procedures.
In the NCTM’s first set of national standards, both perspectives were valued; the first standard for each of its three grade bands was “Mathematics as Problem Solving,” which called for students to “use problem-solving approaches to investigate and understand mathematics content” as well as “to develop and apply strategies to solve a wide variety of problems” (1989, p. 23). The subsequent Professional Standards for Teaching Mathematics (NCTM, 1991) described ways of supporting students’ development of problem solving through the use of worthwhile tasks, classroom discourse, and an effective learning environment. This primary focus on problem solving in mathematics education was maintained through multiple “NCTM standards” documents produced over the following two decades (NCTM, 1995, 2000, 2006, 2009). As stated in Focus in High School Mathematics: “In the three decades since the 1980 publication of An Agenda for Action , NCTM has consistently advocated a coherent prekindergarten through grade 12 mathematics curriculum focused on mathematical problem solving” (NCTM, 2009, p. xi). Focus in High School Mathematics further framed problem solving in terms of “reasoning and sense making”—that is, reasoning is “the process of drawing conclusions on the basis of evidence or stated assumptions,” while sense making requires “developing understanding of a situation, context, or concept by connecting it with existing knowledge” (p. 4).
In 2010, the National Governor’s Association (NGA) and the Council of Chief State School Officers (CCSSO) developed the Common Core State Standards for Mathematics , which more than 43 states and territories adopted as their state course of study. In addition to promoting a more “coherent and focused” mathematics curriculum (p. 3), the document includes a required emphasis on problem solving and mathematical sense making in the Standards for Mathematical Practice , process and proficiencies that students should develop across the grades. These practice standards require that students develop the ability to make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning (NGA & CCSSO, 2010).
While the Common Core set both content and practice standards for K–12 mathematics, the document did not provide guidance for how schools and teachers might support students in achieving those standards. In response, the NCTM released Principles to Actions: Ensuring Mathematical Success for All in 2014. This document describes “essential elements” of school mathematics programs that will support student attainment of the Common Core , along with research-informed “mathematical teaching practices” to support students’ mathematical learning, which is explicitly described to include problem solving and sense making. This latest standards-genre document from the NCTM is quickly gaining support as a primary source in framing discussions around mathematical teaching and learning.
In this chapter, we discuss how we work to transform the mathematical teaching practices of preservice secondary mathematics teachers to develop an equitable, inquiry-based approach to teaching in a manner that will help them to create classroom environments that foster mathematical problem solving and sense making. We discuss a range of pedagogical strategies and describe how these strategies might be introduced in methods courses and reinforced during teacher candidates’ clinical experiences, including early field experiences and student teaching.
Instructional Practices to Promote Problem Solving and Sense Making
In this section, we describe research-based instructional practices that promote problem solving and sense making. These are the targets for our preparation of secondary mathematics teachers. We begin by describing tenets related to teaching and learning mathematics, consider the importance of the classroom environment, and, finally, discuss specific mathematics teaching practices described by the NCTM in 2014 to promote students’ mathematics learning of problem solving and sense making.
Foundational Tenets for Teaching and Learning Mathematics
As teacher candidates begin to conceptualize how to teach mathematics in a meaningful way, they need to understand some tenets that should undergird their development of lesson plans and how they enact those lessons with students.
Relational versus instrumental understanding . Skemp (1976/2006) defined relational understanding as “knowing what to do and why” and instrumental understanding “as rules without reasons” (p. 89). Preservice secondary mathematics teachers need to know the difference between the two kinds of understanding because most preservice secondary mathematics teachers have largely experienced learning mathematics in an instrumental way. Realizing the differences between the two types of understanding, experiencing learning mathematics in a relational manner, and seeing the results of students developing relational understanding of concepts are important experiences for secondary mathematics preservice teachers.
Placing preservice secondary mathematics teachers in situations in which they develop a relational understanding of a concept helps them to see the difference between relational understanding and instrumental understanding. For example, allowing preservice secondary mathematics teachers to work with mystery pouches and coins can help them understand how to solve equations in a meaningful way. Preservice teachers are given a picture such as the one in Figure 1.1 .
Figure 1.1 Task promoting meaningful use of equations. (From Moving Straight Ahead: Linear Relationships by G. Lappan, J. T. Fey, W. M. Fitzgerald, & S. N. Friel, 2014, Upper Saddle River, NJ: Prentice Hall. Reprinted with permission.)
The pouches in the figure are related to the variables in an equation, and the coins represent the constants. The equation represented by the picture is 3 x + 3 = 2 x + 12. In thinking about the picture, the students know that each of the pouches contains the same number of coins. In balancing the equation, they know that the two pouches on each side are equal and three of the coins on the right side are the same number as three of the coins on the left side. They then know that each pouch has to contain nine coins in order for the equation to be true because they are left with one unmatched pouch on the left side of the equation. These problem types enable preservice teachers to understand that variables represent unknown quantities, which will make the equation true.
In addition to asking preservice teachers to solve these types of problems, they are asked to reflect on their thinking and to think about how helping students to develop a relational understanding of concepts and skills such as this will enable them to reason and make sense of mathematics. In addition, showing preservice teachers videos of students solving problems and presenting their solutions in a relational manner helps to confirm why it is important to teach mathematics in a relational manner.
Furthermore, Pesek and Kirshner (2000) posited that in order “to balance their professional obligation to teach for understanding against administrators’ push for higher standardized test scores, mathematics teachers sometimes adopt a two-track strategy: teach part of the time for meaning (relational learning) and part of the time for recall and procedural-skill development (instrumental learning)” (p. 524). Moreover, they specifically addressed the problems that might occur when rote-skill development (instrumental understanding) occurs prior to teaching for relational understanding. In addition, they found that students who were taught area and perimeter via instrumental instruction before they received relational instruction “achieved no more, and most probably less, conceptual understanding than students exposed only to the relational unit” (pp. 537–538). Pesek and Kirshner also found that students who learned area and perimeter as a set of how-to rules referred to formulas, operations, and fixed procedures to solve problems, whereas students whose initial experiences were relational used conceptual and flexible methods to develop solutions. Finally, the authors shared that extensive time spent on routine exercises to consolidate rote or procedural knowledge does not lead to the kind of learning that takes place through the time given to students’ intuitive and sense-making capabilities.
Equity . Another tenet of inquiry-based teaching is equity. Understanding that all students need the opportunity to reach their full mathematics potential is crucial for secondary mathematics preservice teachers. Preservice secondary mathematics teachers need to understand what it means to achieve equity in a mathematics classroom, examine barriers related to student engagement and achievement, develop equitable pedagogical strategies, and examine their beliefs about students from different race/ethnicity, socioeconomic status, gender, ability, and sociolinguistics groups, and confront the negative beliefs (NCTM, 2014; Strutchens, 2000).
Preservice teachers need to think about the meaning of equity from different perspectives. They should understand that “equity does not mean that every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students” (NCTM, 2000, p. 12). Equity also means “being unable to predict students’ mathematics achievement and participation based solely upon characteristics such as race, class, ethnicity, sex, beliefs, and proficiency in the dominant language” (Gutiérrez, 2007, p. 41). Teachers should also think of equity as a bidirectional exchange—which is different from equity as primarily benefitting growth of students and student groups that have historically been denied equal access, opportunity, and outcomes in mathematics to a reciprocal approach where teachers are learning from the students and what they bring from their cultural backgrounds (Civil, 2007). In addition, the concept of equity includes “the equitable distribution of material and human resources, intellectually challenging curricula, educational experiences that build on students’ cultures, languages, home experiences, and identities; and pedagogies that prepare students to engage in critical thought and democratic participation in society” (Lipman, 2004, p. 3). In addition, according to a joint position statement of the National Council of Supervisors of Mathematics and TODOS: Mathematics for All (2016), “a social justice stance requires a systemic approach that includes fair and equitable teaching practices, high expectations for all students, access to rich, rigorous, and relevant mathematics, and strong family/community relationships to promote positive mathematics learning and achievement. Equally important, a social justice stance interrogates and challenges the roles power, privilege, and oppression play in the current unjust system of mathematics education—and in society as a whole” (p. 1).
These conceptions of equity are made concrete to preservice teachers when they actually experience equitable pedagogy as students in their methods classes, and then practice implementing equitable pedagogy in their field experiences. According to Banks and Banks (1995), equitable pedagogy focuses on the “identification and use of effective instructional techniques and methods as well as the context in which they are used,” “challenges teachers to use teaching strategies that facilitate the learning process,” and “provides a basis for addressing critical aspects of schooling and for transforming curricula and schools” (p. 153). Teachers may use mathematics autobiographies and other means to get to know their students and foster positive mathematics identities in students. Teachers may ask students to engage in social justice activities, such as studying statistics related to racial profiling and determining whether injustices have occurred, and then suggesting what steps should be taken next (Gutstein, 2003). Teachers may ask students to develop and explore mathematical problems related to situations that they have observed in their community, such as comparing convenience store prices to those in a major grocery store and discussing which store has the better deals and why (Tate, 1995). Teachers may ask students to write and share reports on the contributions of various cultures to the advancement of mathematics (NCTM, 1991). These strategies, along with the NCTM’s (2014) mathematics teaching practices discussed in a later section, will enable teachers to foster reasoning and sense-making skills in students from a variety of backgrounds.
Mathematical habits of mind . Mathematical habits of mind are ways of thinking that are indigenous to mathematics (Cuoco, Goldenberg, & Mark, 2010). In the middle grades, students develop habits of mind beyond what they learn in elementary school, such as abstracting regularity from calculations and articulating a generalization using mathematical language, which are both related to algebraic thinking (Mark, Cuoco, Goldenberg, & Sword, 2010). At the high school level, students develop general habits of mind and habits of mind that are more subject specific (Cuoco, Goldenberg, & Mark, 2010). Some of the habits of mind discussed by Cuoco, Goldenberg, and Mark (2010) are: performing thought experiments; finding, articulating, and explaining patterns; creating and using representations; articulating generality in precise language; generalizing from examples; expecting mathematics to make sense; modeling geometric phenomena with continuous functions; seeking regularity in repeated calculations; and chunking—changing variables to hide complexity. In addition to the aforementioned habits of mind, the NCTM (2009) contended that students should develop reasoning habits such as analyzing a problem and implementing a strategy, which are described as “productive way[s] of thinking that become common in the processes of mathematical inquiry and sense making” (p. 9).
The mathematical habits of mind and the reasoning habits are in alignment with the Standards for Mathematical Practice of the CCSS (CCSSO & NGA, 2010). The Standards for Mathematical Practice were built on the NCTM’s (2000) process standards of problem solving, reasoning and proof, communication, representation, and connections, as well as the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up (Kilpatrick, Swafford, & Findell, 2001).
Throughout their program of study, preservice secondary mathematics teachers should experience tasks and problems that help them to develop the mathematical habits of mind, reasoning habits, and the skills and proficiencies embedded in the Standards for Mathematical Practice. These reasoning and sense-making skills are paramount for teachers who will be facilitating an inquiry-based classroom.
Curriculum coherence . It is necessary for preservice secondary mathematics teacher candidates to understand the importance of using a set of curriculum materials that are well aligned with their course of study and that are problem based with the Standards for Mathematical Practice or other reasoning and sense-making skills embedded. It is also important that preservice secondary mathematics teacher candidates are able to examine curricular materials to determine which ones are more in alignment with the problem-based approach to teaching. They should also be provided an opportunity to look across grades to determine how the curriculum grows and how topics are connected. Examining learning trajectories (Confrey, Maloney, & Corley, 2014; Daro, Mosher, & Corcoran, 2011) is also helpful.
Assessment . Assessment in an inquiry-based classroom should be in alignment with the type of mathematics teaching that is occurring. Preservice teachers need the opportunity to examine and experience a variety of tasks that lead to more information about what students know and can do. Equitable assessments include open-ended, open-middle, and a small number of closed tasks. Preservice secondary mathematics teachers should learn about exit slips, writing prompts, projects, portfolios, and other measures of student mathematical understanding. They should also learn about and use rubrics in evaluating student work. Preservice secondary mathematics teachers should learn the difference between formative assessment and summative assessment, and the affordances of each. Formative assessment is particularly important in the inquiry-based classroom. Formative assessment is any assessment for which the first priority is promoting students’ learning. Furthermore, formative assessment aids in the learning process when it provides feedback that teachers and students can use to modify the teaching and learning activities in which they are engaged (Black, Harrison, Lee, Marshall, & Wiliam, 2004).
Classroom Environment
The classroom environment for a problem-based inquiry classroom is one of the main determinants of successful implementation. Students need to know that teachers care about and value their opinions. How teachers interact with students around problem solving will impact students’ mathematics identities. Creating a classroom environment in which students are empowered because they are able to see themselves as capable of participating in and being doers of mathematics is important (Solomon, 2007). Solomon asserted that when students identify themselves as participatory and doers of mathematics, they make positive connections and are motivated to achieve at high levels.
Cooperative learning groups . One important component of a problem-based inquiry classroom is the opportunity for students to work in cooperative learning groups. Cooperative learning groups have been shown to improve academic achievement, improve behavior and attendance, increase self-confidence and motivation, and increase liking of school and classmates. In a cooperative group, members discuss their approaches to solving a math problem, explain their reasoning, and defend their work. However, cooperative learning involves more than placing students into groups. Students need to be taught how to work effectively in groups.
Complex instruction (CI) is a form of cooperative learning where students are assigned open-ended, interdependent group tasks and serve as academic and linguistic resources for one another. Their teacher addresses status issues to ensure that all students have the opportunity to be heard and to participate well (Cohen, Lotan, Scarloss, & Arellano, 1999). In CI, teachers use cooperative group work to teach at a high academic level in diverse classrooms. One goal of this approach is to let students make mistakes, figure out why things did not work, and then try to think of new ways to solve the problems they have encountered. The teacher should be very involved in the following activities: (1) observe students, (2) listen to the discussions in the groups at a proper distance, (3) ask questions to groups who are stuck, (4) provide positive feedback to individuals and groups, (5) watch for who is participating and who is not, and the reasons for this, and (6) remind the groups about rules, roles, and norms (Kujansivu & Rosell, 2000).
Boaler (2006) found that when CI is implemented well, an added benefit for students and teachers occurs—relational equity. Relational equity is the respectful relationships that students develop through a collaborative problem-solving approach in which students work together and learn to appreciate the diverse insights, methods, and perspectives that different students offer in the collective solving of problems (Boaler, 2006, 2008; Boaler & Staples, 2008). Preservice teachers need to be engaged in cooperative group work as learners of mathematics and facilitators of student learners. Figure 1.2 provides a list of rules and roles that can be shared with preservice teachers to help them facilitate cooperative learning.
Figure 1.2 Rules and roles for effective cooperative learning.
Differentiated instruction . Differentiation consists of the efforts of teachers to respond to variance among learners in the classroom. It is an approach designed to improve classroom learning for all students. At least four classroom elements can be differentiated based on student readiness, interest, or learning profile: (1) content: what the student needs to learn or how the student will get access to the information; (2) process: activities in which the student engages in order to make sense of or master the content; (3) products: culminating projects that ask the student to rehearse, apply, and extend what he or she has learned in a unit; and (4) learning environment: the way in which the classroom works and feels (Tomlinson, 2000). Furthermore, Small (2009) listed three elements that are essential to effectively differentiating instruction:
1. The focus of instruction must be on the big ideas being taught to ensure that they are addressed across all levels.
2. Pre-assessment is essential to determine what needs different students have.
3. There must be some aspect of choice for students, whether in content, process, or product. (p. 4)
Differentiated instruction in the problem-based classroom can be implemented through using multiple entry-level tasks and tasks designed to meet students at their levels of mathematics understanding. Moreover, flexible grouping is also a hallmark of effective differentiated instruction. It is important for preservice secondary mathematics teachers to understand what it means to provide effective differentiated instruction.
Classroom discourse . As first defined by the NCTM in 1989, “[t]he discourse of a classroom—the ways of representing, thinking, talking, agreeing and disagreeing—is central to what students learn about mathematics” (p. 34). The teacher is envisioned as an orchestrator or facilitator of rich exchanges in which students share their ideas about mathematics, consider the ideas of others, and question the teacher and others. Such discourse effectively promotes students’ engagement in mathematical problem solving and sense making. Moreover, it helps to surface the shift from everyday use of language to more mathematical and precise use of language that is essential for effective communication about mathematics (Moschkovich, 2007). Preservice secondary mathematics teachers need to understand the importance of classroom discourse in promoting student learning, as discussed further in the following section.
Mathematics Teaching Practices
In 2014, the NCTM proposed a set of eight mathematics teaching practices, which is quickly gaining currency as a useful “framework for strengthening the teaching and learning of mathematics” (p. 9), including work with preservice mathematics teachers. These practices are grounded in research that has been conducted over the past four decades on how to promote students’ ability to build relational understanding of mathematics through engagement with challenging mathematics problems. Each of these practices is briefly described in the following sections.
Establish mathematics goals to focus learning . Hiebert and colleagues (2007, as cited in NCTM, 2014) stated: “[F]ormulating clear, explicit learning goals sets the stage for everything else.” Those goals should not be established in isolation but should rather be situated within a progression of mathematics learning. The goals established will guide decisions about the tasks that are selected, the directions that classroom discourse should take, and assessments of student learning.
Implement tasks that promote reasoning and problem solving . This practice states that “effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies” (NCTM, 2014, p. 10). The selection of tasks should be in alignment with the goals that are set for students. Moreover, tasks should be selected that allow many students to approach them but provide challenge for all. Consider, for example, the following task:
A slab of soap on one pan of a scale balances three-quarters of a slab of soap of equal weight and a three-quarters-pound weight on the other pan. How much does the slab of soap weigh? (Adapted from Kordemsky & Parry, 1992, as cited in NCTM, 2009.)
Students might solve the problem using a variety of methods, including:
• Writing an algebraic equation and solving it: x = 3/4 x + 3/4.
• Reasoning arithmetically: Remove three-quarters of a slab from both sides of the scale. That means one-quarter of a slab measures three-quarters of a pound. Since there are four quarters in a pound, the weight must be 4 x 3/4, or three pounds.
• Guess and check: Let’s guess that it’s a pound. Then the weight on the left pan will be 3/4 + 3/4, or 1.5 pounds. That’s more than one pound, so let’s try a larger amount. If it weighs two pounds, then the left side will be 3/4(2) + 3/4, or 9/4. That’s still too much, so let’s try three pounds. 3/4(3) + 3/4 = 3. So it weighs three pounds.
Note that students can approach this problem at a variety of levels, which may increase student engagement in the problem and provide students with an opportunity to learn from and with one another.
Use and connect mathematical representations . Mathematical representations “embody critical features of mathematical constructs and actions” (NCTM, 2014, p. 24). Representations can be physical, visual, verbal, or symbolic. The use of representations, and connecting among representations, is essential to developing deep mathematical understanding and effective problem solving (NCTM, 2014). For example, students might explore what happens when one multiplies one more than a number by one less than that same number by making a drawing, making a chart, or using a symbolic representation, as shown in Figure 1.3 . These representations become both a tool for problem solving and for students to communicate their reasoning with others.
Figure 1.3 Representations of multiplying one more than a number by one less than that same number.
Facilitate meaningful mathematical discourse . As stated in the previous section, mathematical discourse is essential for developing mathematical problem solving and sense making. In particular, Smith and Stein (2011) describe five practices for effectively orchestrating classroom discourse:
1. Anticipating student responses prior to the lesson
2. Monitoring students’ work on and engagement with the tasks
3. Selecting particular students to present their mathematical work
4. Sequencing students’ responses in a specific order for discussion
5. Connecting different students’ responses and connecting the responses to key mathematical ideas
These practices provide a mechanism through which teachers can move from initial discourse about a mathematical task that may be occurring in pairs or small groups to discourse of the full mathematics classroom that is aimed at allowing students to share problem-solving methods and the sense they are making of mathematical problems with the goal of developing common mathematical understandings.
Pose purposeful questions . A teacher’s use of effective questioning is essential in promoting students’ mathematical reasoning and sense making (NCTM, 2014), as such questions encourage students to share their thinking and reflect on the thinking of others. According to the NCTM (2014), two critical issues must be addressed. First, rather than relying on questions that primarily count on the recall of facts, teachers must ask effective questions that encourage students to discuss mathematics, make connections among ideas, and justify the validity of their work. Second, teachers must use effective patterns of questioning. A funneling pattern, in which a series of questions is used to guide students to a desired outcome, is all too frequently implemented with minimal attention to divergent perspectives, whether or not they may be productive (Herbel-Eisenmann & Breyfogle, 2005). Alternatively, a teacher may use a pattern of questioning that focuses on student thinking. In this pattern, the outcome of the discussion is guided by the students’ thoughts, and alternative perspectives are encouraged. The teacher’s goal is to encourage students to clearly communicate their thinking and to reflect on what other students say. This latter pattern is foundational to developing deep discourse that supports mathematical learning and problem solving.
Build procedural fluency from conceptual understanding . Procedural fluency requires more than recall of facts and procedures. As stated by Martin (2009): “Students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results” (p. 165). Such procedural fluency is built on a solid base of relational understanding that is developed through engagement with meaningful problem situations. A “rush to fluency” is, however, not productive, as it may cause mathematics anxiety and decrease interest in the study of mathematics (NCTM, 2014, p. 42). On the other hand, development of procedural fluency is essential, as it forms a solid base for students’ expanding problem solving and sense making, allowing them to focus on the more challenging aspects of a problem while off-loading its more trivial aspects.
Support productive struggle in learning mathematics . As students face a challenging mathematics problem, they may feel that they are unable to solve it if they do not readily see its solution. Teachers must resist the inclination to become overly helpful in the face of student frustration and encourage students to persist in attempting to solve the problem. This corresponds with the first standard for mathematical practice in the Common Core , which states that students must “[m]ake sense of problems and persevere in solving them” (CCSSO & NGA, 2010, p. 6). All too often, mathematics students have a fixed mind-set in which they consider mathematical ability to be an innate characteristic of a person that cannot be changed (Dweck, 2008). Building a growth mind-set in which students recognize that mathematical ability can be improved through hard work and persistence requires teachers to expect their students to persist. They praise effort over perceived intelligence and provide feedback on students’ progress in becoming more proficient (NCTM, 2014).
Elicit and use evidence of student thinking . “Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning” (NCTM, 2014, p. 10). This mathematics teaching practice is essentially formative assessment. An example of an effective way to elicit students’ thinking is a strategy called “My Favorite No” (Teaching Channel, n.d.). Students are given an entry task/warm-up, which serves as a review problem. Students are asked to write their solutions on index cards. Next, the teacher collects the cards, and then chooses the “best” wrong answer to share with the class, usually a misconception that the teacher wants to highlight so that students will learn from the solution. The teacher rewrites the solution so that students cannot tell who solved the problem in that manner. The teacher displays the solution to the problem so that the students can analyze the solution to determine its strengths and weaknesses. Next, the teacher asks the students to tell her what they think she likes about the thinking displayed in the solution. After discussing what is right about the solution, the class talks about where the error occurred. This discussion helps the teacher to assess the thinking of multiple students and provides the class with immediate feedback on the problem they just solved. This strategy leads to growth in students’ ability to recognize their weaknesses and identify exactly where they need to improve.
Practices to Foster Preservice Teachers’ Facility With Effective Instruction
In this section, we discuss experiences integrated into our secondary methods program that can help preservice secondary mathematics teachers gain familiarity and facility with instructional practices that promote and support mathematical problem solving and sense making. In our program, students are required to take three methods courses:
• CTSE 5040, Technology and Applications in Secondary Mathematics. This course focuses on the use of technology that supports mathematics teaching and learning.
• CTMD 4010, Teaching Mathematics in the Middle School. This course focuses on teaching middle school mathematics and the middle school mathematics curriculum.
• CTSE 4030, Curriculum and Teaching in Secondary Mathematics. This course focuses on teaching high school mathematics and the high school mathematics curriculum.
Throughout these methods classes, students are exposed to the importance of mathematical problem solving as the core of school mathematics by careful analysis of the Standards for Mathematical Practice, which are presented as required content that they will need to address as mathematics teachers. We also emphasize the importance of “teaching via problem solving” as a way to both meet the practice standards and make sense of mathematics. A variety of readings (NCTM, 2009, 2014; cf. Schroeder & Lester, 1989; Skemp, 1976/2006), vignettes, videos, and in-class “laboratory experiences” are used to illustrate these points, as well as an extensive microteaching experience in CTSE 4030. These approaches are described in more detail in the following sections.
“Laboratory Experiences” With Problem Solving
A major obstacle in preparing preservice secondary mathematics teachers to teach mathematical problem solving and sense making is that they themselves may not have developed proficiency in these areas. Moreover, they frequently lack the relational understanding of core mathematical topics in the secondary mathematics curriculum needed to promote their students’ learning. Volumes such as The Mathematical Preparation of Teachers II (Conference Board of the Mathematical Sciences, 2012) recommend rethinking the mathematics courses preservice teachers take to integrate attention to mathematical practices and processes, including changing how mathematics courses are taught and adding mathematics courses specifically designed for mathematics teachers. While implementing these recommendations is a long-term goal for our program, in the short term we compensate by including “laboratory experiences” within the methods courses where students explore mathematics concepts using problem solving. Particular focus is placed on building their mathematical habits of mind, especially the Standards for Mathematical Practice.
These experiences generally incorporate the use of appropriate tools and technology that are useful in making sense of mathematics and problem solving. Physical manipulatives (or their virtual, online counterparts) can be useful in helping students visualize mathematical relationships. While emphasis is often placed on using manipulatives with younger children, they can also be very useful with secondary mathematics students (NCTM, 2014). Technology also provides a useful context for mathematical problem solving, particularly the use of “mathematical action technologies” (Dick & Hollebrands, 2011), which “perform mathematical tasks and/or respond to the user’s actions in mathematically defined ways” (p. xii). As argued in NCTM (2014), given the pervasiveness of technology in students’ lives, it should be an inherent part of doing mathematics in classrooms.
CTSE 5040, experiences with technology as a learner . A major focus of CTSE 5040 is developing the preservice teachers’ own mathematical problem solving and sense making. Technology provides a context in which they can build their experiences with problem solving, taking them out of their comfort zone. They explore a wide range of mathematical action technologies that can be accessed via tablet devices (e.g., iPads) or laptops. Applications used include spreadsheets (e.g., Microsoft Excel or Google Sheets), dynamic geometry software (e.g., Geogebra or Geometer’s Sketchpad), computer algebra systems (e.g., WolframAlpha or TI-Nspire), statistical analysis software (e.g., Fathom or Core Math Tools), and software designed to explore specific mathematical concepts (e.g., the NCTM’s Illuminations website or the Shodor Interactivate website).
Students are generally given “labs” in which they use particular mathematical tools in order to solve mathematical problems. A sample activity from the first day of class follows. Students are directed to a web-based app, pictured in Figure 1.4 , and are asked to complete the following directions:
1. What does changing the black slider do?
2. Consider the purple slider:
a. How does changing its value change the line? The equation?
b. What happens when the number is large? A negative number? Zero?
3. Consider the green slider:
a. How does changing its value change the line? The equation?
b. What happens when the number is large? A negative number? Zero?
4. In the equation y = mx + b , summarize what m tells you and what b tells you.
Figure 1.4 Web-based app exploring slope-intercept form at http://x.co/mwjU .
The preservice teachers typically work on lab activities in pairs so they can discuss their thinking in a safe environment. The course instructor then facilitates full-class discourse around their solutions, keeping their focus on their own mathematical solutions rather than discussing how secondary students might solve the problems. Thus, the instructor is essentially modeling instruction designed to promote mathematical problem solving without directly addressing instruction. Only after the discussion of their mathematical problem solving and sense making is complete does the discussion turn to a reflection on the activity from the perspective of a preservice teacher, including the methods the instructor used to facilitate instruction, the approaches secondary students might use to solve the problem, and how the activity or technological tool might be used to promote mathematical problem solving and sense making in the secondary classroom.
Throughout the course, the preservice teachers are given open-ended projects to complete, in which they solve extended mathematics problems that may take several days to complete. For example, following several labs focused on dynamic geometry, students were given their choice of several problems to pursue, such as the following (adapted from Dick & Hollebrands, 2011):
Two brothers are part of an expedition that discovers a new island. From the sky, they notice that the island is shaped like an irregular convex quadrilateral. They are not sure of the actual dimensions of the island, but they want to determine a way that they can fairly divide the island between the two of them. The older brother suggests that they divide the land along one diagonal, and each would take one triangular part. He states that this is a fair method since it creates two triangles, and each will receive one triangle.
1. Is this method fair? Why or why not?
2. Devise another method that is more fair.
3. Can you come up with other approaches that will be more advantageous?
Note that this task is amenable to multiple solution paths and is likely to engage multiple mathematical habits of mind. Use of dynamic geometry software, such as Geogebra or Geometer’s Sketchpad, is very helpful in supporting preservice teachers’ exploration of the problem. Figure 1.5 shows a possible exploration of the problem in the NCTM’s Core Math Tools. Students can investigate various subdivisions of the quadrilateral to see which meet the conditions of the problem.
Figure 1.5 Exploration of the “island problem” using NCTM’s Core Math Tools.
In the project write-up, the preservice teachers are asked to discuss their own mathematical problem solving; in a second section, they are asked to discuss how that problem might be incorporated into a mathematics classroom.
CTMD 4010, experiences with manipulatives as a learner . In this course, preservice teachers use a variety of physical models or manipulatives to solve problems in order to develop a relational understanding of mathematical concepts and skills. They solve problems and reflect on their problem-solving processes and think about how students in the middle grades would solve the same type of problem and what the students would learn from solving the problem. A sample problem with a variety of solutions is shown in Figure 1.6 .
Figure 1.6 “Lab” problem along with possible solutions.
The “sharing the money” problem allows students to see multiple ways to solve the problem and that a fraction is relative to the whole. They also may use Cuisenaire rods, fraction circles, or counters to solve the problem.
CTSE 4030, experiences as a learner of high school mathematics . Throughout CTSE 4030, preservice teachers are asked to solve challenging problems that focus on the development of mathematical habits of mind within key areas of the high school mathematics curriculum. Given that CTSE 5040 is a co-requisite for this course, the preservice teachers are expected to integrate the use of technological tools into their problem-solving activities, as appropriate. For example, they use spreadsheets to model the concentration of a drug in a person’s blood for given dosages and filtration rates by the kidneys (NCTM, 2000). This problem both builds their mathematical habits of mind and familiarizes them with an area of the high school curriculum that they may not have experienced, functions that are defined recursively.
A number of problems incorporate use of physical manipulatives; for example, algebra tiles are used to explore operations with polynomials. While the preservice teachers may be able to carry out the algebraic procedures, they often do not possess a relational understanding of why those procedures work. For example, they explore the algebraic technique of “completing the square”; using algebra tiles makes it evident that they are literally completing a square, as shown in Figure 1.7 (NCTM, 2009).
Figure 1.7 Visual representation of completing the square.
The preservice teachers are generally asked to work on the problems in small groups, with a subsequent full-group discussion led by the instructor, modeling instructional approaches that promote the development of mathematical problem solving and sense making. While the initial focus of the discussion is on developing their mathematical habits of mind and knowledge of the curriculum, subsequent discussion considers the problems from the perspective of a high school mathematics teacher, including common ways students may think about the problem and instructional approaches that may help students develop productive ways of thinking.
Case Studies
In order for preservice teachers to fully understand the instructional practices that promote mathematical problem solving and sense making, such as the mathematical teaching practices (NCTM, 2014), they need to see depictions of such instruction within the context of high school classrooms. While they participate in school classrooms as a part of their field experiences, those experiences are not common across the preservice teachers within a given course. To provide a common base for discussion and reflection, we use classroom cases—either textual vignettes or videotapes. The prototypical approach used is to begin by having the preservice teachers participate as a class in the activity depicted in the case and to subsequently analyze its potential for promoting student learning. Consider, for example, the “Pay It Forward” task (Smith, 2014), presented to the CTSE 4030 class:
In the movie “Pay It Forward,” a student, Trevor, comes up with an idea that he thought could change the world. He decides to do a good deed for three people and then each of the three people would do a good deed for three more people and so on. He believed that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage. (p. 1)
The vignette or videotape is then shown to the class, including appropriate reflection questions. In the case of the “Pay It Forward” task, a case titled Exploring Exponential Relationships: The Case of Ms. Culver (Smith, 2014), which details a teacher’s attempt to teach the task, was presented to the class along with the following reflection questions:
• How does this classroom compare to the typical high school classroom?
• Which of the Standards for Mathematical Practice did you observe?
• What did the teacher do to support students’ use of the Standards for Mathematical Practice?
As is typical with our use of vignettes or videotapes, the preservice teachers first discussed the vignette in small groups, and then shared their responses with the class as a whole.
Lesson Plan Format
Preservice teachers write lesson plans for each course that are designed for problem-based classrooms. The lesson plans contain three major phases that are centered around worthwhile tasks that support what students should know and be able to do: launch, explore, and summarize. A sample lesson plan is provided in the Appendix.
Launch . In the first phase, the teacher launches the problem with the whole class. This involves helping students understand the problem setting, the mathematical context, and the challenge. The following questions can help the teacher prepare for the launch:
1. What are students expected to do?
2. What do the students need to know to understand the context of story and the challenge of the problem?
3. What difficulties can I foresee for students?
4. How can I keep from giving away too much of the problem?
The launch phase is also the time when the teacher introduces new ideas, clarifies definitions, reviews old concepts, and connects the problem to previous experiences of the students. It is critical that, while giving students a clear picture of what is expected, the teacher leaves the potential of the task intact. He or she must be careful not to tell too much and lower the challenge of the task to something routine or to cut off the rich array of strategies that may evolve from an open launch of the problem.
Explore . In the explore phase, students work individually, in pairs, in small groups, or occasionally as a whole class to solve the problem. As they work, they gather data, share ideas, look for patterns, make conjectures, and develop problem-solving strategies. The teacher’s role during this phase is to move about the classroom, observing individual performance and encouraging on-task behavior. The teacher helps students persevere in their work by asking appropriate questions and providing confirmation and redirection where needed. For students who are interested in and capable of deeper investigation, the teacher may provide extra challenges related to the problem. The following questions can help the teacher prepare for the explore phase:
• How will I organize the students to explore this problem? (Individuals? Pairs? Groups? Whole class?)
• What materials will students need?
• How should students record and report their work?
• What different strategies can I anticipate they might use?
• What questions can I ask to encourage student conversation, thinking, and learning?
• What questions can I ask to focus their thinking if they become frustrated or off-task?
• What questions can I ask to challenge students if the initial question is “answered”?
Summarize . The summarize phase of instruction begins when most students have gathered sufficient data or made sufficient progress toward solving the problem. In this phase, students discuss their solutions as well as the strategies they used to approach the problem, organize the data, and find the solution. During the discussion, the teacher helps students enhance their understanding of the mathematics in the problem and guides them in refining their strategies into efficient, effective problem-solving techniques.
Although the summary discussion is led by the teacher, students play a significant role. Ideally, they should pose conjectures, question one another, offer alternatives, provide reasons, refine their strategies and conjectures, and make connections. As a result of the discussion, students should become more skillful at using the ideas and techniques that come out of the experience with the problem.
During the summarize phase, content goals of the problem, investigation, and unit can be addressed, allowing the teacher to assess the degree to which students are developing their mathematical knowledge. At this time, teachers can make additional instructional decisions that will enable all students to reach the mathematical goals of the activities. The following questions can help the teacher prepare for the summary:
• How can I help the students make sense of and appreciate the variety of methods that may be used?
• How can I orchestrate the discussion so students summarize their thinking about the problem?
• What concepts or strategies need to be emphasized?
• What ideas do not need closure at this time?
• What definitions or strategies do we need to generalize?
• What connections and extensions can be made?
• What new questions might arise and how do I handle them?
• What will I do to follow up, practice, or apply the ideas after the summary?
Microteaching
In addition to lab experiences focusing on particular areas of mathematics, CTSE 4030 incorporates an extended microteaching experience, in which the preservice teachers teach lessons to their peers. This creates a simplified environment in which the participants can feel safe in trying out various instructional methods without the complexities of running a classroom. They can also receive detailed feedback from their peers, a highly valued feature of the microteaching experience (Benton-Kupper, 2001). To maximize the benefits of the experience, lessons are drawn from an exemplary high school textbook series, The Interactive Mathematics Project (IMP). This approach has a number of benefits:
• Many students have not experienced instruction that is focused on the development of mathematical problem solving and sense making. While the lab experiences within the methods courses give them a taste of what this might look like, working through a complete unit designed to promote mathematical habits of mind gives them a better sense of how such instruction might play out over time in the classroom.
• Rather than having students pick potentially unrelated topics to teach, using a full unit provides a “coherent sequencing of core mathematical ideas” that “pose[s] problems that promote conceptual understanding, problem solving, and reasoning and are drawn from contexts in everyday life and other subjects” (NCTM, 2014, p. 72).
• Use of exemplary curriculum materials has the potential to enhance teacher learning if curriculum enactment is addressed in the supporting materials (Ball & Cohen, 1996). The IMP materials include information on student thinking, useful questions to ask, and tips on organizing the activity that may inform preservice teachers’ thinking about the task being proposed.
• Having an exemplary curriculum to draw on simplifies aspects of the planning process, such as task selection, allowing the preservice teachers to focus on their instructional approaches. The IMP materials also provide useful guidance to inform lesson planning.
The instructor launches the microteaching experience by teaching the first several days of the unit; this allows the instructor to establish effective classroom procedures and norms for participation and the preservice teachers to become familiar with the general approach used in the curriculum materials. As routines are established, members of the class begin taking turns teaching. In contrast to many microteaching experiences, the preservice teachers were asked to prepare a full lesson and, both for the sake of time and to increase their sense of safety in carrying out the lesson, they are allowed to plan and teach their lesson in pairs.
One of the important norms established in the initial days of the experience is that the participants can only draw on the prerequisite mathematical knowledge that a high school mathematics student studying that unit would have; this is characterized as wearing their “student hats.” Following the lesson, a transition occurs; participants put on their “teacher hats,” which gives them permission to discuss the mathematics from the perspective of a teacher, how students might respond to the lesson, and the efficacy of the instructional approaches used. Note that this debriefing session begins during the initial days of the unit, while the course instructor is teaching, in order to establish norms for participation. The class collaboratively develops “ground rules” for the debriefing; typically, rules address the need for constructive criticism, maintaining a respectful tone, and including positive observations. The goal is to create a safe environment in which preservice teachers can exchange perspectives about critical aspects of instruction that promote mathematical problem solving.
Field Experiences That Foster Preservice Teachers’ Facility With Effective Instruction
Each of the methods courses includes an early field experience, in which preservice teachers are required to spend time working in mathematics classrooms, as follows:
• CTSE 5040, Technology and Applications in Secondary Mathematics. Six three-hour observations are required in a middle or high school mathematics class that incorporates extensive use of technology, assigned in pairs. Preservice teachers are asked to inventory available technology and teach a sample activity focusing on the use of mathematical action technology (Dick & Hollebrands, 2011).
• CTMD 4010, Teaching Mathematics in the Middle School. Seven three-hour sessions are required in a middle school class where the preservice teachers teach in small groups and assist the cooperating teacher with lessons.
• CTSE 4030, Curriculum and Teaching in Secondary Mathematics. Twelve three-hour observations are required in a high school mathematics class, assigned in pairs. This includes three lessons that are designed and taught by the pair.
In addition, preservice teachers complete a semester-long, full-time internship in a middle or high school mathematics class. These experiences are critical in helping students develop their ability to support mathematical problem solving and sense making as they plan lessons, implement them, and then reflect on the success of their lessons and how they might be improved.
Middle school Pi Day . As a part of CTMD 4010, preservice secondary mathematics teachers write lesson plans that focus on a nonroutine problem-solving activity that may require the use of manipulatives to solve. The activities usually involve pi or circles in some way. The preservice teachers create tri-boards to simulate a fair-like atmosphere. The preservice teachers facilitate this activity at a middle school during the school’s Pi Day, typically held around March 14 (or 3/14), since 3.14 is a common approximation for pi. Preservice teachers may end up facilitating the same activity three or four times during the day. Thus, they see a variety of responses and thought processes. They reflect on the experience in the form of a written report. This report describes the conditions under which the activity took place, students’ thought processes, and examples of students’ solution strategies. The preservice teachers enjoy the opportunity to teach multiple groups of students. Their lessons are based on the Common Core and NCTM’s (2014) Principles to Actions .
Middle school small-group lessons . During their practicum experience for CTMD 4010 (the middle school methods class), preservice teachers are required to teach lessons to small mixed-ability groups of students in order to understand students’ thought processes. The preservice teachers are required to write at least one lesson and teach their peers’ lessons as well. They are required to reflect on how well the lessons were taught and how well the student understood the content. The preservice teachers assist their mentor when they are not teaching by tutoring students. By working with a small group, they are able to ask more questions and focus more on teaching, rather than on having to deal with classroom management issues. Ideally, the preservice teachers will be able to work with the same group of students throughout the semester so that they can see how their students have grown mathematically. Each of the mentor teachers from the school works with two to four preservice teachers, and each of the preservice teachers teach the same lesson to the groups of students within the same class period. Preservice teachers are then able to debrief at the end of the lessons about how well the students understood the lessons and the strengths and weaknesses of the lessons. Oftentimes, a university supervisor (usually the professor of the course) is able to observe the preservice teachers’ lessons in their small groups and provide feedback.
The preservice teachers are encouraged to use Connected Mathematics (CMP) textbooks as resources to write their lessons. CMP is a problem-centered curriculum promoting an inquiry-based teaching-learning classroom environment. For example, in a lesson related to volume and surface area, the preservice teachers used the following problem:
ABC Toy Company is planning to market a set of children’s alphabet blocks. Each block is a cube with 1-inch edges, so each block has a volume of 1 cubic inch. The company wants to arrange 24 blocks in the shape of a rectangular prism and then package them in a box that exactly fits the prism.
1. Find all the ways 24 cubes can be arranged into a rectangular prism. Make a sketch of each arrangement you find, and give its dimensions and surface area. It may help to organize your findings into a table.
2. Which of your arrangements requires the box made with the least material? Which requires the box made with the most material?
Students use 1-inch cubes to create and find the surface area and volume of different rectangular solids that can be made with 24 cubes. Students see that the volume is always 24 cubic inches, but the surface area changes with the shape. This problem solidifies student understanding of volume and surface area, and it is connected to an authentic problem. The preservice teachers ask students questions and provide scaffolding for the students as they complete the tasks.
The lessons must cover objectives that are included in the Alabama Course of Study and emphasize problem solving and critical thinking. The preservice teachers spend three hours in the schools for six sessions for three weeks at two days per week. The lessons are graded and approved by the professor of the course. The preservice teachers teach at least four different lessons to students.
Full-group instruction . The focus of the field experiences in CTSE 4030 (the high school methods class) is on full-group instruction using problems that promote problem solving and sense making. Students are assigned in pairs to a high school mathematics classroom to conduct 12 three-hour visits. The trajectory of experiences over the course of the semester builds from observing for the first two to three visits to familiarize themselves with the context, to assisting with small-group instruction or segments of full-class discussion over the next four to five visits, to actually planning and teaching three lessons. The first lesson is co-taught by the pair in order to create a nonthreatening experience; each member of the pair takes the lead for one of the remaining lessons, with the other member assisting. Prior to teaching each lesson, they prepare a lesson plan following the format for the program, which must be approved by both their course instructor and their cooperating teacher. Lessons must promote mathematical problem solving and sense making, including attention to the mathematical teaching practices (NCTM, 2014). Following each lesson, both members of the pair prepare a reflection on the lesson. For the individually taught lessons, the lead preservice teacher must prepare a professional work sample in which they provide detailed commentary on the lesson—including the decisions made in planning the lesson, how the lesson was implemented, and support for student learning with a focus on problem-based instruction. In addition, the students keep a journal in which they reflect on their experiences and how they can better support students’ problem solving and sense making.
Internship
The internship is an extension of the total program experience. Preservice teachers continue learning content and pedagogical skills during their internship. The internship lasts for 15 weeks. The preservice teachers gradually take on teaching and end up teaching for 20 full days and gradually give the courses back. Three facets of this experience are designed to help preservice teachers increase their ability to support mathematical problem solving and sense making: focus on the craft of teaching, continuous reflection on practice, and collaboration.
Craft of teaching . The craft of teaching is the ability to design lessons that involve important mathematical ideas, to design tasks that help students access those ideas, and then to successfully teach the lesson, which entails effectively launching the lesson, facilitating student engagement with the task, orchestrating meaningful mathematical discussions, and helping to make explicit the mathematical understanding students are constructing (Peterson & Leatham, 2009). Designing tasks that are accessible to students requires consideration of how students think mathematically. This focus on the craft of teaching brings together many aspects of their experiences in the methods courses related to supporting mathematical problem solving and sense making.
Reflecting on practice . During the internship, preservice teachers are required to reflect on their daily interactions with students and teachers. They keep a daily journal, and they have other reflections that are specific to lessons, which they teach. We ask them to reflect on the learning environment, the type of questions they or their cooperating teachers ask, student learning, and so forth. This consistent focus on reflection supports our goal of helping them to become facilitators of learning in a problem-based setting.
Collaboration with teachers . It is important that preservice teachers learn how to communicate effectively with their cooperating teacher, administrators, and others. We have found that placing students in pairs during their internship experience helps them to become more reflective, collaborative, and focused on student learning. Whether placed in pairs or as singletons, this focus on collaboration with their peers, their cooperating teacher, and others in the educational environment sets a foundation for a career-long trajectory focusing on promoting mathematics problem solving and sense making (NCTM, 2014).
Conclusion
Given that most preservice secondary mathematics teachers have been largely taught in an instrumental way, they typically have limited relational understanding of mathematics and a narrow view of what it means to teach mathematics, with little awareness of mathematical problem solving and sense making. Moreover, they are generally the “success stories” of traditional mathematics instruction; that is, they have been recognized as doing well in mathematics and thus may not recognize the need to change. Helping them to reconsider their views of what it means to be an effective mathematics teacher may be challenging, requiring them to face their limitations in what they had always perceived to be an area of strength. To help them build knowledge, dispositions, and skills to support the development of mathematical problem solving in each student requires a long-term, multisemester, multidimensional approach. In this chapter, we have outlined some of the approaches we have used across our program to ensure that our graduates are well prepared to integrate mathematical problem solving and sense making into instructional activities they design for their future classrooms.
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APPENDIX Sample Lesson Plan: S-Pattern Task
Audience: Algebra II With Trigonometry students
Content Objectives
• Quadratic functions can model relationship between figures in a sequence.
• Two quadratic functions are equivalent if each can be rewritten using the distributive property and combining like terms.
• Second-order differences of quadratic functions are constant.
Alabama Course of Study (Algebra II With Trigonometry)
• #12. Interpret expressions that represent a quantity in terms of its context. 1 [A-SSE1]
• #12a. Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a]
• #21. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]
Behavioral Objectives
The students will
• observe patterns in a set of figures,
• sketch and describe figures based on the patterns they observe,
• write a function rule for a pattern that is growing quadratically and relate the rule to the diagram of figures, and
• explain why the relationship between the figure number and the number of tiles is not linear.
Prerequisites
• The students should be familiar with the differences between linear and quadratic functions.
Materials
• “S-Pattern Task” handout
• Calculators (if desired)
• Square tiles
Procedure
1. Warm-up (small groups)
Warm-up on the screen—figures from “S-Pattern Task.”
Directions: In your small groups, talk about the figures below. Write down at least two patterns that you notice in the set of figures.
• Have you noticed any patterns yet?
• What do you notice about the figures?
• How many tiles are in the first figure?
• What about the second figure?
• What about the fifth figure?
• How are the figures growing?
• What is changing?
• What is staying the same?
• Can you think of any other patterns?
Transition : All right, everyone! Let’s talk about the warm-up.
2. Discussion of warm-up and launch of today’s investigation (full group)
Call on several groups for the patterns they identified. Continue asking until no groups have additional patterns to add.
• What was one pattern you identified?
• Did anyone else find this pattern?
• Did anyone notice a different pattern?
• What other patterns did you notice?
Say, “For the rest of today’s class, we will be investigating this pattern further.” Pass out the task sheet.
• Before we get started, what do you notice about problem 1? (It’s the warm-up.)
Read through the task together.
• What kinds of things are you trying to find when solving the problems in this task?
• Does anyone have any questions about the activity?
Stress that they will have to explain how their function relates to the diagram.
Transition : Let’s get to work. Focus on #2–4. If you finish early, move on to #5 and #6.
3. Investigation of “S-Pattern Task” (small groups)
Circulate and monitor groups as they work on “S-Pattern Task.”
• How did you know how to construct the sixth figure?
• Did you use any of the patterns you identified in the warm-up?
• What changes between the sixth and seventh figures?
• How can you determine a function rule for the total number of tiles in any figure?
• What should your input values be? (Figure number)
• What should your output values be? (Total number of tiles)
• What patterns did you see in the figure that relate to the number of tiles?
For different formulas, have students relate them to the picture.
• What does each term represent in your picture?
• Can you simplify your expression?
• Can you relate your simplified expression to the picture?
If they rely on a table:
• How did you get the values in your table?
• What pattern did you find in the table?
• How can you represent the pattern you found algebraically?
• How does this relate to the explorations we did with tables in the Meaningful Math textbooks?
• What was the term for what you looked at? (Second differences)
Transition : Okay, everyone. Let’s share what we found!
4. Discussion of “S-Pattern Task” (full group)
Call on a group to present #2.
• Does anyone have any questions for this group?
• Can someone else explain what they did?
• How does this relate to the patterns we talked about in the warm-up?
Call on a group to present their function.
• Can someone else explain what they did?
• How do you know that their solution is correct?
• Will your function tell us how many square tiles are in the 100th figure?
• How does your function relate to the pictures?
• What does each term in the function represent in the picture?
Call on another group to present their function.