Math Awareness: The Importance of Inquisitiveness and Student Struggle

Math can be a daunting subject for some students, but having the right teacher can make a world of difference. The best teachers draw connections to real-world applications, making the subject approachable and engaging. We asked two math education instructors to talk about methods for teaching math to a variety of learners.

Katie McFarland is a special education teacher at Pentucket Regional School District in West Newbury, Massachusetts, and has taught seventh-grade mathematics. Emily Reed of Troutdale, Oregon, previously taught math for Grades 3-5 and has created professional development sessions about integrating technology in instruction.

What qualities make a good math teacher?

KM: A good math teacher is patient and positive and extremely knowledgeable of the concepts being taught. Math anxiety is one of the most challenging things math teachers have to overcome in the classroom, and trying to break down that mindset takes time and a lot of positive reinforcement.

ER: A good math teacher knows the importance to being inquisitive, the importance of multiple approaches, differentiated instruction, and reflective thought. A good math instructor also knows the importance of student struggle and is always looking for new ways to engage students in the learning process.

How do you help students experience success in math? How do you define success in a math class?

KM: Success is measured by any type of growth in math. Students who move their learning to the next level, no matter where they start, are a success. Students who are able to conceptualize learning and utilize it in real-world applications are successful. Success is also defined by students who are able to take learning from mistakes to keep from making them again because the learning is now realized.

ER: It’s important to engage students in problem-solving. It is through the problem-solving process that students begin to develop a relational understanding of content. Success is found when students are able to take the content they have learned in the classroom and apply it to their everyday lives.

Mistakes are a key component to the learning process. By working though mistakes, students are engaging in productive struggle, and they recognize that mistakes are not the end-all. They can learn from their errors. This is one way students begin to develop mathematical resolve.

Describe any innovative math projects you have been involved in developing.

ER: I was lucky enough to work with Dr. Margaret Niess in a program that aimed to get more educators passionate about midlevel mathematics. Throughout this program, we shared ideas and strategies. We learned how to differentiate instruction in the math classroom and how to integrate technology into instruction to encourage student engagement. We then shared these ideas and helped other educators develop them in their math instruction. It was such a pleasure to have the opportunity to work with such an innovative mathematics educator.

KM: I worked as part of a collaborative team to refine and create math projects that involve real-world scenarios. Although application in the real world would be ideal — it hasn’t happened yet — but we have designed and planned for it.

Some of these projects involve:

• Defining and creating a garden with three-dimensional art pieces
• Creating a memorial space (utilizing geometry) to honor a special person or event
• Creating linear artwork
• Understanding the impact of an individual’s carbon footprint by looking at the output of town solar panels
• Planning a trip where students have to budget for everything from travel expenses, hotel, food, and souvenirs
• Reviewing marketed games for probability, and writing letters to game manufacturers based on the fairness of the games

What is an example of a curriculum project you developed that generated high motivation and engagement among your students?

KM: A favorite project that I have helped develop is the Math Memorial project. My current school district is very big on personal meaning and having students connect personally with what they are doing to build motivation and connection.

Students were asked to choose an event or person to memorialize, as well as pick a space to put it. There were criteria regarding how big or small the memorial could be, and it had to contain a three-dimensional figure (stone, bench, reflecting pond), but otherwise, it was open. Students had to think about the importance of memorializing someone or an event, and explain why they chose the space and design. Then they had to determine a materials list and cost for materials based on their design. They really connected and jumped into learning.

ER: In the classroom, I was a part of a technology grant. I wholeheartedly believe in the benefits of integrating technology into math instruction. Through a variety of virtual activities and applications, students were highly motivated and demonstrated that they were mastering the content. Students no longer dreaded math. They were engaged and active participants in the learning process.

I also developed podcasts for students to review if they were absent, if they needed reminders, or if parents needed support to help their child. Additionally, students utilize games, virtual field trips, and developed their own podcast. The student-developed podcasts is probably one of my favorite activities. Students not only had to explain their thought process, but they engaged in reflective thinking. It was a very telling project-based assessment.

How would you individualize instruction for students?

ER: To individualize instruction, it’s important to know your students. One must know their starting point, as well as what motivates each student. Once a base understanding of each student is developed, true differentiated instruction can be integrated. Tasks can be tiered based on the different levels of learners and modified for any special needs. Additionally, lessons can be developed that speak to a variety of the multiple intelligences.

KM: My goal is to give students exactly what they need, when they need it. Whether it be guided notes, visuals, manipulatives, calculators, or video support materials, there is no one thing that I do — it really depends on what is needed.

With this said, there is one assignment I give that is individual to each student. Students have to write in a journal each week. They have to choose three different problems where they made a mistake. It can be from classwork, homework, or assessments. They have to write the problem out, show the work of how they did it incorrectly, the work of how to solve it correctly, and write a reflection that explains what their mistake was, why they think they made the mistake, and how they can keep from making the mistake again. This helps students to identify consistent mistakes they make, and hopefully build better habits to keep from making them.

How do you get students interested in the problem-solving process?

KM: I ask a question, typically open-ended, and let them decide how to answer it. I let them know that I don’t care about how they get to an answer, but instead I care that they can explain how they get there.

ER: To integrate a problem-based approach, it’s vital to develop rich and relevant problems that draw students in to the task. This generates interest, and by making content interesting, students become motivated to engage in the problem-solving process.

Teaching through a problem-based approach takes much time, practice, and patience not only on the teachers’ part, but on the part of the students as well. We live in a time of instant gratification. Students want to be able to punch numbers into a formula and find the answer. However, critical thinking is essential. Not all problems are solved within a day, and that is all right. In their book, John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams suggest, “Classrooms where students are making sense of mathematics do not happen by accident – they happen because the teacher establishes practices and expectations that encourage risk taking, reasoning, sharing, and so on.” This is the type of classroom I work to develop.

It’s important to remember that with any new program or approach there will be transition time. We cannot integrate new methods and expect students or teachers to excel right away. These things take time. Additionally, parents may also struggle with the approach because it is different from what they’re used to. The most important thing is to teach students how to become problem solvers. No matter where their path will take them: college, the workforce, or trade school, students will need problem-solving skills in all walks of life.

How would you challenge a slow learner and an advanced learner within the same class?

ER: To challenge a slow learner and an advanced learner, differentiation is key. We must know our students and then develop tasks that not only meet their needs, but also stretch them at a level they are ready for. To accomplish this, I would use a tiered task model. The complexity of the problems may not be the same for each student, but students are still learning the same content.

KM: Involve them actively in the learning process. For a “slow” learner, I would challenge them to achieve to a given benchmark. A benchmark is something that I believe the student can achieve, and I work to encourage the student to get there, and celebrate with them when they do. Then, we set a new benchmark. I would do the same for an advanced learner, raise the bar, and let them work to achieve.

These two students could be working on similar concepts, just at different levels. Differentiated instruction is the essence of a successful mathematics classroom. It’s easier to do at some times than others — but my goal is to always keep students engaged and learning.

Describe the management strategies and techniques you use to maintain an effective classroom environment.

ER: To maintain an effective classroom environment, it’s important to get students to buy into the learning process and drive instruction. We want students to develop the skills to question and engage in reflective thinking. There are many different strategies that help in this process. The most important strategy is developing a positive classroom climate that is conducive for learning.

My greatest tools are the relationships I build with my students. Once I know my students and what motivates them, I’m better able to develop lessons that speak to them. These lessons are engaging, rich with motivating tasks, student-to-student mathematical discourse, reflective thinking, and problem-solving. Additionally, students build confidence each time they are successful, especially when I demonstrate that I believe in their abilities.

KM: The No. 1 rule in my classroom is “no stress.” Mistakes are going to happen. I expect them and celebrate them because that is where the real learning takes place. I also have a “no nonsense” classroom, where my expectation is that we are all respectful of each other. When one is speaking, everyone else is listening, where hard work and participation is an expectation, not an option, and where students who work hard find success, and all measures of success are celebrated.

What is the most effective way to communicate with students?

ER: To effectively communicate with students, it’s key to get them to become active members of the learning process. Setting aside time for individual and team meetings illustrates to students that they have ownership and a voice in class.

KM: I am an open communicator with students. I always try to start with what they are doing “right,” before I discuss what they are doing “wrong,” so we can build off the positive. I find that talking to students and being honest with them helps me get them where they need to go. I love questions. I often turn the question back to students and encourage them to take risks answering. So if a student says “I don’t understand how to do this.” My first question back is, “What part don’t you understand?” Then I begin my explanation with, “You tell me where to start, and if you are wrong or get stuck, I’ll guide you back on track.”

How do you evaluate student math progress other than testing?

KM: I am constantly evaluating: homework completion and accuracy, exit tickets, or student check-ins.

ER: Rick Stiggins, an expert in assessment, says students are over-tested and under-assessed. As educators, we are constantly making observations and evaluating students’ understanding of content. While summative assessments are an important and necessary form of assessment to determine students’ understanding at a particular point, formative assessments also have their place in instruction.

Formative assessments drive instruction and help educators take a closer look at student understanding. Formative assessments help educators determine if they need to approach content in a different manner or if there are any misconceptions. Think of it as being the difference between a checkup with your doctor versus an autopsy. Formative assessments can range from observations, discussions, self-assessments, journals, exit slips, and a wide variety of other informal tools that allow educators to design and modify instruction.

What do you believe is the most useful part of instruction in your course?

KM: For [my master’s program], it would be the discussion of how to reach all learners, and things that could impact learners, and the importance of assessment to guide and differentiate instruction.

For my middle-schoolers, I would say that the consistent practice of concepts, real-world application, and guided notes to support learning.

ER: I believe the most useful part of Transformative Mathematics in the Differentiated Classroom is the process of taking a unit plan and transforming it to better meet our ever-changing students’ needs. Throughout the course, we are not only learning strategies and techniques, but we are taking these ideas and integrating them into our lesson plans.

Prospective students of the MEd Curriculum and Instruction: Mathematics program are able to see how the content we are discussing has the ability to truly impact their classroom students’ understanding of content and possibly develop a passion for mathematics.

What advice would you give a student considering the MEd in Curriculum and Instruction: Mathematics?

KM: Go for it!! Math is my happy place, so anyone looking to teach math is fabulous to me. Understanding things beyond the mathematics itself — the outside factors that impact learning — is also important to consider and learn.

ER: You can do it! Many people have this idea that “they are not good at math” or “math is not for them.” This is not true. Math is such a big part of our everyday lives. It’s all around us. For example, the purpose of algebra is not to just prepare students for high school, but to teach them how to solve for an unknown. The use of logic is something we do on a daily basis. Therefore, it’s vital to help students understand that mathematics is not a scary subject. It is useful.

Throughout the MEd in Curriculum and Instruction: Mathematics program, students will look at math through many different lenses. Prospective teachers will gain a better understanding of how students learn math and how to better involve them in the learning process.

Why is continuing education important for math teachers?

KM: Math is constantly changing, so continued education is essential in supporting all learners. In our technological savvy world, it is so important to be up on what’s available to students, and how to integrate it in the classroom.

You don’t have to “do it all” in regard to technology. But if you can incorporate some of it into the classroom, the students will benefit — as long as it is done to support the learning, instead of replace it, or having it done in isolation without connection.

ER: Continuing education is very important for math teachers. There are new and innovative ideas being developed constantly. As educators, we modify lessons regularly to better meet our student’s needs. The traditional approach to education is no longer working. We cannot continue to teach in a one-size-fits-all mentality, or simply present information. Each student is an individual and each student does not always understand content in the same manner. This reminds me of the old saying “you cannot fit a round peg into a square hole.”

We must make modifications to tasks in an effort to ensure each student is exposed to multiple representations of content, is challenged at the appropriate level, and most importantly is successful. Our students are ever-changing, and our teaching methods should be as well. It’s through continuing education that we continue to evolve in our teaching craft.

What kinds of graduate programs help math teachers?

ER: Any program that encourages students to think outside the box and develop innovative techniques benefits math teachers. These programs can range from mathematics content courses, technology, education, engineering, art, and beyond. These courses provide educators with opportunities to look at math through a different paradigm.

KM: Programs that have a balance of content and pedagogy are fabulous. [My school] does a fabulous job in balancing these two areas, as well as considering all other factors in education that are important for MEd students to understand:

• Building relationships with parents and the school community
• Teaching differentiated instruction
• Working with students with differing needs
• Using research to support and guide teaching practice