October 2021

Engaging learners in the math classroom

By: Samantha Hirsh, Assistant Director of Curriculum & Instructional Services

Two years ago I entered a high school math class and saw desks in rows, all facing the teacher in front who was completing a problem. Students were at their desks, notebooks out, copying whatever the teacher was writing down. The teacher asked the class “Ok got it?” Some students nodded their heads, others continued to copy what they saw on the board.

“Any questions?”

No response.

“Great, next problem.”

The cycle continued through the remainder of my visit. Growing up, many of us learned math in a similar passive manner; by watching our teachers complete problem after problem, following along with a textbook, workbook, or handout, and then completing problem sets independently in the classroom and at home. How many of us understood what we were doing? Why were we doing it? Why did what we were learning matter? What was the point? When you walk into a kindergarten or a first grade classroom you see students excited with what they are learning, and what they are engaging in; however, this sense of wonder and passion for learning slowly dissipates as students matriculate through our school system, particularly in mathematics.

In 2000, The National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics, and wrote the following:

“… Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers. Alone or in groups and with access to technology, they work productively and reflectively, with the skilled guidance of their teachers.”

The picture painted by NCTM over 20 years ago is what math classrooms, particularly at the secondary level, should strive to be on a daily basis. So how do we get there? How do we empower our students to become engaged in and fully participate in mathematics? How do we make them excited to be in class, to problem solve, and struggle productively? In order to start this process, we need to look at how we approach instruction, learning, and the expression of knowledge. Admittedly, some changes are harder to make than others, but there are a few we can make immediately to embed sound pedagogical practices within our instruction and increase student learning, and engagement.

One change is to encourage and promote multiple representations and ways to approach problems. As educators, it is important for us to remember that every student is unique, and our brains are all wired differently. What works and makes sense for some may be totally foreign and confusing to others. By praising students for developing their own ideas and methods, as opposed to penalizing them for not doing something the way we wanted and/or expected them to, we are encouraging deep thinking, creating problem solvers, and valuing the process of learning over the answer.

Once, while grading tests, I heard a colleague complain about a student who completed a geometric proof in too many steps. I asked, “Did the student prove what they were supposed to?” The teacher replied, “Yes.”

I then asked, “Did they use proper mathematical theorems and postulates?” Again, the teacher replied, “Yes.”

“So why are you taking points off of their test?”

The teacher responded, “Because they didn’t do it the way I taught them, they took too many steps.”

This led to me asking, “Well, what were you assessing? The student’s knowledge and application to prove a statement, or if they could complete the proof in a number of steps you think is appropriate?”

Instead of answering me, the colleague started a new conversation with someone else.

In contrast, I observed a trigonometry class where students were practicing trigonometric proofs and identities. The teacher recognized how a student approached a problem in a way she had never thought about. After marveling at his approach, she had the student and another student simultaneously present their work to the class, highlighting the different approaches and ways of thinking, and stressing the notion that there are multiple ways of verifying and proving trigonometric identities.

Similar mathematical topics and concepts, but very different instructional approaches. In which classroom do you think students were more willing to take risks, tackle challenges, persevere through problem solving and enjoy learning?

In addition to encouraging multiple approaches, another shift one can make is to focus on teaching concepts, rather than skills, making connections between them, and highlighting the application whenever possible. When students see how mathematical concepts relate to each other and are applied in life, it is much easier for them to grasp and understand complex and confusing concepts. For example, in fifth grade when performing operations with fractions, bring in geometry; have students find the area of a rectangle or the perimeter of a shape with fractional lengths. In Algebra 1, when teaching systems of equations, provide students with real-life scenarios when first solving a system algebraically, as opposed to just giving them two equations; by connecting concepts, we are attaching an abstract idea to something concrete and building comprehension. When we ground our instruction in conceptual understanding and anchor problems using real world scenarios, students develop a stronger understanding as to what they are being asked to do and what a system of equations actually represents.

Another shift is to take a step back and facilitate students’ thinking and learning; be a guide on the side versus sage on the stage. Let your students take ownership of their learning and give them choice; by doing so, you are empowering them as learners and engaging them in the learning process. How can we accomplish this? One way is to move away from teaching lesson by lesson and move towards teaching weekly concepts. Structure your class so that students have required learning, such as a whole group mini-lesson, watching a screencast for homework with embedded assessment, or watching a pre-recorded lesson, followed by choice learning (small group instruction, short lessons through Nearpod, Peardeck, and/or the use of skill-based videos). Students then have to complete required practice and choice practice (choice in what they do, how they do it, etc.). Work is done throughout the week with the teacher conferencing and checking in with students to see where they need support. By structuring your class in this way, you are ensuring that students work on specific problems and skills you want them to master, while allowing them choice both in what and how they can learn and demonstrate their knowledge.

The learner sitting in front of us today is vastly different from the one who sat in front of us five, ten, fifteen, or twenty years ago, with changes happening faster than ever; yet many of us are still teaching the same way we did when we first started. We may have moved from blackboards to whiteboards to SMARTBoards, but in some of our classrooms our instructional and assessment practices have not adjusted to the learner in front of us. As educators we are constantly asking our students to think, learn, and grow, and we need to model this to our students through our own instructional practices.