Project-Based Learning in the Math Classroom

By Chris Fancher and Telannia Norfar

Project-Based Learning in the Math Classroom (grades 6 - 10) explains how to keep inquiry at the heart of mathematics teaching and helps teachers build students' abilities to be true mathematicians. This book outlines basic teaching strategies, such as questioning and exploration of concepts. It also provides advanced strategies for teachers who are already implementing inquiry-based methods. Project-Based Learning in the Math Classroom includes practical advice about strategies the authors have used in their own classrooms, and each chapter features strategies that can be implemented immediately. Teaching in a project-based environment means using great teaching practices. The authors impart strategies that assist teachers in planning standards-based lessons, encouraging wonder and curiosity, providing a safe environment where failure occurs, and giving students opportunities for revision and reflection.

• Language: English
• Publisher: Sourcebooks
• Release date: Apr 30, 2019
• ISBN: 9781646320851

Chris Fancher

Chris Fancher is an eighth-grade teacher at a public charter school in Texas.

Book preview

Authors

Introduction

As we work with math teachers around the world, we find that there are some common fears and anxieties about teaching mathematics using inquiry practices, such as project-based learning (PBL). Almost all of these fears come from the idea that mathematical skills, such as learning multiplication tables, must be mastered through repeated practice. The question we most often hear is: How can you help students learn these skills and do PBL?

If we do nothing else in this book, we hope to show that you can successfully implement PBL in the math classroom. We will show you how to incorporate math practice into the scaffolding of your project. We will provide guidance on how to transform your classroom into one of discovery rather than show-and-tell only. We believe that, as teachers master the classroom management of discovery, they will find time to implement PBL and the practices they find successful.

Teaching in a project-based environment means using great teaching practices. Many teachers may feel overwhelmed by all of the research-based practices out there. However, we like to keep it simple. We see great teaching practices as strategies that involve:

1. planning lessons that are standards-based,

2. encouraging wonder and curiosity,

3. providing a safe environment in which failure occurs, and

4. giving students opportunity for revision and reflection.

When considering the main components of a PBL-based classroom, we see that great teachers are doing most of these practices already. Often, the missing components include a problem or issue that will interest students and lead to student inquiry and questions, an opportunity for students to direct their learning, and an opportunity to demonstrate their learning to an audience.

In this book, we provide in-depth tasks and describe a project-based approach to teaching. The book includes details about how to create a PBL unit, as well as models that you can implement immediately. The book aligns with leading research available regarding inquiry-based learning methods. If you have chosen to read this book, you are either a great teacher already, or you desire to be a great teacher. Thank you for your desire to perfect your art form.

SECTION I

Understanding Project-Based Learning in a Math Classroom

CHAPTER 1

PBL Is Modeling Mathematics

In mathematics, the art of proposing a question must be held of higher value than solving it.

—Georg Cantor

In our technologically-advanced world, the ability to model mathematics is a necessary skill to participate in society. According to the National Research Council (NRC, 2001), people will be called on more and more to evaluate the relevance and validity of calculations done by sophisticated machines… All young Americans must learn to think mathematically, and they must think mathematically to learn (p. 16). Given its necessity, defining and recognizing how to develop mathematical understanding is important. This chapter provides a rationale for changing the structure of the mathematics classroom and a methodology to enable students to reason mathematically in our ever-advancing world.

A Look at the Standards

As stated in the Common Core State Standards (CCSS), mathematical understanding is shown through justifying a statement or mathematical rule (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010). The writers explained how mathematical understanding and procedural skills are equally important, and both are accessible using mathematical tasks of sufficient richness. Mathematical understanding and procedural skills are what all educators desire to develop in their students. The key to developing both conceptual understanding and procedural fluency is for students to engage in rich mathematical tasks. Mathematics curriculum in the United States has long had memorization of processes and procedures as a dominant component. The CCSS and other state standards strike a balance between conceptual understanding and procedural fluency. In addition, the CCSS highlight a consistent skill set to be developed from kindergarten through 12th grade.

The CCSS define thinking mathematically to learn as mathematical practices (NGA & CCSSO, 2010). These are the habits of mind students should practice with all mathematical content. According to the CCSS, the eight mathematical practices developed were based upon work by the National Council of Teachers of Mathematics (NCTM) and the NRC (2001). The Standards for Mathematical Practice are:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Both entities (NCTM and NRC) have developed concepts based upon more than 30 years of research on the teaching and learning of mathematics. Although the eight practices are desired by educators and required by many standards, most educators find the real challenge to be creating the proper mathematical environment in their classrooms. Modeling mathematics is the fourth practice of the CCSS; however, it is the gateway to the other seven practices. It is how educators can create the proper mathematical environment.

According to the CCSS, modeling mathematics means that "students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace [emphasis added] (NGA & CCSSO, 2010). Modeling can range from simple to complex situations. In the early grades, modeling might be as simple as writing an addition equation to describe a situation. In the middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community, and in high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another." The CCSS explain how students can simplify a complicated situation and analyze data in multiple representations. A mathematically proficient person can interpret the information and determine whether it is accurately represented.

The CCSS further explain that modeling mathematically involves making assumptions, approximations, simplifying complex situations, reflecting on results, and modifying results as necessary (NGA & CCSSO, 2010). Assumptions, approximations, simplifications, reflections, and modifications are all skills that are embedded within other practices. As students work toward solutions to real-world problems, they will make sense of problems and persevere in finding solutions. They will use reasoning, construct arguments, and use tools to ensure precision—all while examining structure and applying reasoning.

Not all content standards lend themselves to modeling, but often the wording of a particular standard will show when modeling would be appropriate. For instance, in Algebra I, the standards might state that students will use functions to represent and model problems, as well as analyze or interpret relationships. The CCSS often use specific wording and notation in the content standards that identify which content should have a real-world focus (NGA & CCSSO, 2010). For example, high school content standards that should include modeling with mathematics have an asterisk. In the 6–8 standards, the CCSS use real-world or model in the language of the standards. A grade 6 standard related to geometry asks students to find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. Students are then to apply these techniques in the context of solving real-world and mathematical problems.

At the high school level, modeling is not only a practice in the CCSS, but also embedded in the content (NGA & CCSSO, 2010). There is a complete section within the high school domain that further explains the purpose of modeling as

the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods.

To further assist with understanding the content, the domain provides examples of empirical situations. They include:

–Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed.

–Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.

–Relating population statistics to individual predictions.

As the list shows, modeling mathematics is more than a word problem. Textbook word problems often deter students from the mathematical practices because the problems follow a predictable format. Modeling mathematics is more complex. It is the application of mathematics in the context of the current world. The word current is emphasized because application of mathematics changes as technology advances. What is done this year may not be done 5, 3, or even 2 years from now. For example, construction managers used to calculate the number of beams needed to frame a building manually. This is now completed through software systems, such as AutoCAD. The skill used to have an emphasis on precision for getting the calculation correct. Now the emphasis is on estimating the reasonableness of the computer’s calculation.

A Look at the Research

In addition to meeting the demands of the standards, modeling mathematics is supported by the current research on teaching and learning mathematics. In 2010, Mid-continent Research for Education and Learning (McREL) published the third edition of What We Know About Mathematics Teaching and Learning, which outlined the latest findings in relation to equity, teaching, assessment, curriculum, instructional technology, and learning in regard to mathematics instruction. McREL noted that, in order to create an equitable classroom, teachers must engage all students in higher order thinking skills and help students make connections among related mathematics concepts across other disciplines and with everyday experiences (p. 3). Conceptions of higher order thinking skills vary, but these skills often include synthesizing, analyzing, reasoning, comprehending, applying, and evaluating.

To build equitable classrooms, McREL (2010) emphasized that we must identify elements that address the needs of a diverse population, such as English language learners (ELLs) and students with disabilities. The following are elements that have proven effective for diverse mathematics classrooms:

–a meaningful math curriculum (contexts that give facts meaning and complex problems);

–emphasis on interactive endeavors that promote divergent thinking (construct knowledge with peers);

–diversified instructional strategies;

–assessment that is varied, ongoing, and within instruction (performance, portfolio, projects, complex problems); and

–focused lesson planning (based on what students need to learn and what they already know; p. 5 ).

This identification of elements led to a detailed explanation of the teaching strategies and curriculum necessary to cultivate mathematical thinking in all students, which include mathematically rich environments conducive to investigations and community settings in which teachers carefully select problems, materials and grouping practices; provide opportunity for mathematics discourse and use assessment practices designed to provoke and support student thinking (McREL, 2010, p. 19). McREL also cited the need for a standards-based curriculum that is built upon the central topics as identified in the Principles and Standards for School Mathematics (NCTM, 2000), and contains rigor, focus, coherence, reading, and writing.

McREL (2010) stressed the need for students to make connections across the mathematics curriculum and to connect to the outside world, noting that When mathematics is taught in rich and realistic contexts, more students are able to build deep understanding. Students who learn mathematics through complex problems and projects outperform other students whose learning is more compartmentalized and abstract (p. 64). McREL also suggested integrating the various branches of mathematics, allowing students to struggle and teachers to do more than a casual observation to find ways to connect concepts. McREL reported, Real life is a rich source of mathematics problems. Learning is highly interactive as students explore problems, formulate ideas, and check those ideas with peers and with their teacher through discussion and collaboration (p. 66).

In 2014, NCTM published its latest findings of mathematics learning in Principles to Actions: Ensuring Mathematical Success for All. As the preface stated, the publication was created to describe the conditions, structures, and policies that must exist for all students to learn. One of the main sections of the book explained eight research-informed teaching practices that support the learning of mathematics by all students. The eight practices are:

–Establish mathematics goals to focus learning.

–Implement tasks that promote reasoning and problem solving.

–Use and connect mathematical representations.

–Facilitate meaningful mathematical discourse.

–Pose purposeful questions.

–Build procedural fluency from conceptual understanding.

–Support productive struggle in learning mathematics..

–Elicit and use evidence of student thinking.

Most of the practices discuss the need for students to be involved in meaningful tasks (modeling mathematics). NCTM (2014) wrote, For students to learn mathematics with understanding, they must have opportunities to engage on a regular basis with tasks that focus on reasoning and problem solving and make possible multiple entry points and varied solution strategies (p. 23). Whether it is called a task, modeling, problem or project, the need for students to learn with real-world context is clear and a necessary change to the mathematics classroom.

NCTM (2014) showed a need to look at a constructivist pedagogy and development of social skills within the mathematics classroom. Teachers, especially at the secondary level, may struggle with how to have students construct their own understanding through collaboration. This is due to what NCTM called unproductive beliefs:

Teachers’ beliefs influence the decisions that they make about the manner in which they teach mathematics. It is important to note that these beliefs should not be viewed as good or bad. Instead, beliefs should be understood as