# (Not) Teaching the Way We Were Taught

Our intention is to engage our students, but many of the classroom practices we have used for years set up an environment where the learners are a passive part of the process.As a teacher educator, I have the privilege of training future mathematics teachers in methods of good classroom instruction. During one of the first days of the schoolyear, I ask my senior secondary mathematics students a simple question. What is your favorite branch of mathematics? I ask this question, even though I know what the majority of these future teachers will say. “My favorite math content is algebra.” These math majors were the students who survived and thrived in the traditional mathematics classroom.

What do I mean by traditional mathematics? Traditional math instruction is teacher-centered. The teacher is the disseminator of the information that students need to learn. While the teachers that teach within this paradigm of instruction are well meaning, students are often quiet and passive in the educational exchange.

### Making a case for a better way

to engage learners

The traditional style of mathematics instruction is teacher-centered. Our intention is to engage our students, but many of the classroom practices we have used for years set up an environment where the learners are a passive part of the process. In 2014, the National Council of Teachers of Mathematics (NCTM) published *Principles to Actions: Ensuring Mathematical Success for All* as a call to action for math teachers to rethink their practice. In this research-based book, NCTM highlights eight effective classroom practices that mathematics teachers should implement.

- 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 (NCTM, 2014)

Each of these practices add depth and richness to the classroom environment, both for the teacher and the student.

Jo Boaler, a mathematics education professor from Stanford University, is a major proponent of the growth mindset movement within mathematics teaching. In her research and the research of Carol Dweck (2007), Boaler has found that students with a growth mindset are more likely to be Students with a growth mindset believe that they can grow in the midst of challenges.successful in the long term when compared to classmates who view their mathematical ability as fixed (i.e., “I am smart.” or “I am stupid.”). Students with a growth mindset believe that they can grow in the midst of challenges. In her most recent book, *Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching*, Boaler (2016) provides six guiding questions to help move instruction from teacher-centered to an engaging, student-centered experience.

#### Can you open the task to encourage multiple methods, pathways, and representations?

Encourage learners to think of a problem from a variety of perspectives. If you are teaching 12 ÷ 3, make sure to have your learners draw this in picture form. Have your learners create a story problem where they divide a quantity of twelve items into three equal groups. When we provide our students with multiple access points for mathematics, we increase the chances that students will make conceptual connections between the representations.

#### Can you make it an inquiry task?

Boaler notes that the focus for students within an inquiry task “is not to reproduce a method but to come up with an idea” (Boaler, 2016, p.78). In an article I wrote for the *Wisconsin Teacher of Mathematics*, I showed how it is possible to find an intriguing object and turn it into an inquiry task. Look at my article entitled Curiosity and the Coin Sword for some ideas of how to create your own inquiry task (Paape, 2015).

#### Can you ask the problem before teaching the method?

Instead of showing students how…, have them try to find it on their own.This shift is possibly the easiest to make in a teacher's classroom practice. Instead of showing students how to determine the area of the shape in Figure 1, have them try to find the area on their own. Allow students to use their mathematical intuition and problem solving skills to productively struggle with the mathematics. Your learners might not figure out the exact method you had intended, but the struggle and the learning that will occur will allow for deeper connections to the concepts for your learners.

#### Can you add a visual component?

This question is an answer to one of the effective teacher practices highlighted by NCTM – using and connecting mathematical representations. If your students are learning how to determine 17 + 5, you can ask your learners to communicate that using a picture. Leaving the request as an open request for a picture will allow the students to determine what they see as the best representation. This is an intentional strategy to be somewhat ambiguous in your instructional request.

#### Can you make it low floor and high ceiling?

These kinds of tasks can vary in style and form. The idea behind the low floor, high ceiling problem is to allow learners to enter the problem solving process at a very accessible level. For instance, the problem in Figure 1 (taken from www.visualpatterns.org) is a type of low floor, high ceiling activity.

The teacher can prompt his or her learners in a number of ways for a problem like this.

- "What do you notice?"
- "How do you see the pattern growing?"
- "Describe the 100th shape."

The high ceiling aspect of this kind of problem occurs when the teacher allows for space in the learning exchange for students to develop their own ideas about what is going on mathematically. Often, the high ceiling for a problem like this occurs when the teacher asks the students to define the pattern by creating a generalized formula. This kind of prompting will often receive a variety of responses from students—some correct, some not. However, it is at this point of instruction that the "good stuff" of learning occurs.

#### Can you add the requirement to convince and reason?

When a student makes a mathematical assertion in the classroom, the learning does not end there. We need to ask our learners to defend their assertions with reasoning. One form of accomplishing this is to require justification for any mathematical statement made in the classroom.

#### Mathematics education from a Christian perspective

The ideas of existing in a community of individuals with a collective focus on growth together has been a part of our faith heritage since the very beginning. The previous six questions are a good starting point for any teacher of mathematics in creating student-centered learning experiences. I also like to make the case that this style of mathematics instruction is an application of our Christian life. Our loving God has called us to be a people of community and fellowship. While many of the instructional practices that I highlighted in this article may be new and challenging from a pedagogical perspective for some teachers, the ideas of existing in a community of individuals with a collective focus on growth together has been a part of our faith heritage since the very beginning.

Mathematics instruction needs an overhaul that puts the learner at the center of the teaching exchange. As we explore mathematics, it is important for the Christian teacher of mathematics to communicate clearly to his or her learners that the God who made the universe has also created every aspect of mathematics. With the confluence of the creativity of a loving Creator, the amazing uniqueness of mathematical exploration and learning, and the opportunities to grow together, mathematics teaching becomes a wonderfully fulfilling and exciting vocation.

Adam Paape is Associate Professor of Education and Chair of the Secondary Education Department at Concordia University Wisconsin. Dr. Paape’s instructional focus is in the area of mathematics education, mentoring future teachers of mathematics. His research interests focus on student-centered mathematics instruction, with an emphasis on implementing rich, conceptual mathematical tasks to engage all learners.

#### References

Boaler, J. (2016). *Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching.* San Francisco, CA: Jossey-Bass.

Dweck, C. (2007). *Mindsets: The new psychology of success.* New York. Random House Publishing.

National Council of Teachers of Mathematics (2014). *Principles to actions: Ensuring mathematical success for all. *Reston, VA: Author.

Paape, A. (2015) Curiosity and the coin sword. *Wisconsin Teacher of Mathematics*. 68(1), 4-6.

Reinhart, S. (2000). Never say anything a kid can say. *Mathematics Teaching in the Middle School*. 5(8), 478-483.

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