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# Problem-based Learning in Math Takes a Student-centered Approach

*By Fanny Sosenke, Upper Division Math Teacher, Avenues New York*

Starting in 8th grade math and continuing into the upper grades, the math curriculum is centered around having students learn mathematics by solving and discussing a well- organized set of problems that leads students into mastering content in a student-centered way. These problems are carefully chosen so that content and skills are connected and build on each other. This approach is different from the one in a traditional math class, where a teacher is the “owner” of the knowledge and his or her task is to pass that knowledge to the students. Learning by solving problems and discussing solutions results in students owning their learning, which makes mathematics meaningful to them.

When I first started teaching this course in September, we worked together to build a unique culture in the classroom where the feeling was, “we are all in this together.” This student-centered and collaborative approach to learning mathematics has been shown to have lasting results.

The problems we tackle are open-ended and can be solved in a variety of ways. They are problems that will lead students to learn a specific algebra skill, problems that strengthen students’ number and visual sense. There is great emphasis on proportional reasoning and patterns – two crucial component of mathematical thinking.

Here are some examples of the problems students explore:

“On a road map of Uganda, the scale is 1 : 1, 500, 000. The distance on the map from Kampala to Ft. Portal is 17 cm. What is the real-world distance in km between these two cities?” (proportional reasoning/open ended)

“Davis says that adding a two-digit number to the two-digit number formed by reversing the digits of the original number results in a sum of 65. Avery says that’s impossible. Is it impossible?“ (number sense/open ended)

“You have seen that multiplication distributes over addition. Does multiplication distribute over subtraction? Does multiplication distribute over multiplication? Does multiplication distribute over division? Use examples to illustrate your answers.”

“Jess has just finished telling Lee about learning a wonderful new algebra trick: 3 + 5x can be simplified very neatly to just 8x, because a + bx is the same as (a + b)x. Now Lee has to break some bad news to Jess. What is it?“ (exploring algebra skills)

Gemma, one of my students, offers her point of view of the class in the following narrative: “This class is taught where you go home and try your homework and try to understand and interpret it, and put as much as you can down on the paper (it doesn't always have to be the right answer). Then when you get to class the next day you get taught how you do it using the strategies of your classmates or your teacher, which I like much better than just being told what to do and how to do it.” According to Jack: “This year, math was definitely different from the years before but in a good way. It allows students to share their homework in an open and non-judgmental space, giving them time to help and support any students who don’t fully understand the work and get them back on track as quick as possible with the help of your peers always available at any time during class.”

One of the new and crucial topics in the year was linear functions, one of the foundations of Algebra I. This topic was new to students and it took them longer to make connections. This is explained again by Jack and Gemma: “At first I a had a really hard time with linear functions, memorizing the different forms of equations of lines (y=mx+b, y-y1=m(x-x1)) and connecting them with problems. Then in the second day of learning it in class I just connected them, I can't really explain how, but someone was presenting and it just clicked.” And, “There was one problem that I really struggled with about linear equations and it took me around a week to understand it but then during presentations a student explained it one way which just made everything seem like it made so much sense and that was through the help of all of my peers pitching in and doing their job to ensure that the class understood the concepts.”

If you visit an 8th grade math classroom, it may look chaotic at first. You might see some students writing solutions on the board and others working on a puzzle or a study guide. It may seem as if nobody is in charge. That’s because we are all in charge, in an organic way. Students call on each other to offer alternative solutions or to ask clarifying questions. Students disagree with each other and try hard to prove their points. And my role as a teacher is to guide students into their learning by asking them clarifying questions, making them expand and deepen their thinking, rephrasing their thoughts using more accurate mathematical terms and developing assessments that help them evaluate their learning.

If you are reading this and you are in 7th grade, here is some advice from an 8th grader: “Just do your homework; always be open to asking questions if you don’t understand a concept; and take risks. You have nothing to worry about because you will always have your peers around you at any time there to give you the help you need.”