Pati Ruiz is a doctoral student in Learning Technologies at Pepperdine University. She has worked as a teacher (Spanish and Computer Science), Director of Learning Technology, and is the incoming Dean of Studies at Convent of the Sacred Heart in New York City.
As a teacher studying the learning sciences in graduate school, I understood constructivist practices in theory, but I often wondered what constructivism looked like in action. Taking a constructivist perspective, Windschitl (2002) describes learning as an act of both individual interpretation and negotiation with others, where knowledge is the collection of what is constructed individually and collectively. In classrooms, a constructivist or open approach should support learners in actively constructing their own knowledge, but what does that really look like? How much time does it take? What are the challenges?
I spoke with some teachers to learn more about how they support an open approach to learning in their classrooms. This post will focus on the strategies of a middle and high school math teacher I interviewed. Future posts will focus on the work of high school Spanish and English teachers.
Math Classroom Example: Christine Trying a New Curriculum
Christine DeHaven is in her fifth year teaching middle and high school math. In her Honors Algebra 1 class at Pacific Ridge School, she always taught in a very guided or instructivist approach. For example, if the topic was lines, she would first lecture about lines, then work out one example in front of the class, and then students would do some problems on their own in class. For homework, students would complete more problems that progressed in difficulty. The class would then move on to the next topic. Christine noticed that students were just memorizing steps instead of problem solving, so she decided she needed to change her teaching approach.
Christine had learned about a new curriculum that allowed the students to learn through conferences and visits to other schools, including The Bishop’s School, Deerfield Academy, and Phillips Exeter Academy (where the curriculum was developed). Christine and a colleague decided to swap their traditional direct-instruction approach for a problem-based approach. Christine had seen the new curriculum in action, and felt that it could work at her school, too. Still, she modified the curriculum slightly for her students. Sometimes the transition to a new approach needs to be done gently. Here is what Christine’s curriculum looks like now.
First, students are assigned 8-10 homework problems per night. The goal for students is that they attempt all of the problems before class. When they arrive to class the next day, students pick a problem to solve on the board. Multiple students may put up the same problem, and everyone contributes at least one problem. After all of the problems are on the board, groups of students go to the board to present one problem at a time. If there are multiple solutions to the same problem, Christine leads a discussion about which solution is more efficient. With this new method, Christine finds that her students have more ownership of what they are learning. They apply problem solving skills to the homework and construct their own understandings through their solutions and conversations about their solutions. When they present their work and discuss the various solutions, students gain a better understanding of the concepts because they have to make a case for or against a certain way of solving a problem. Christine also encourages students’ use of graphing as a method to solve the homework problems. Students use tools like the Desmos Graphing Calculator to see a visual representation of the problem. In this way, Christine guides students to look at problems in three different ways: numerically, algebraically, and visually.
Parent education and administrator support has played an important role in the ease of adopting her new curriculum. While Christine initially received some negative feedback about her approach from parents, she felt well supported by her school administrators who are able to point concerned parents towards research and articles about the success of this approach. Open house became an opportunity for Christine to educate concerned parents–she even encouraged them to work with their children to solve the daily homework problems. Christine still attempts to engage parents by encouraging them to follow along on the course website. Many do, and often share stories of working on problems with their children. While parents were initially skeptical, many now tell Christine how much they appreciate the new approach and they have fun helping their children with their math homework. In the beginning, Christine also got negative feedback from students. But – for the most part – they have come around now that they have more practice with the approach. Something else that has helped students adjust is that the homework problems they are solving are very realistic; students can relate to them. For example, one problem, which aims to help students understand how dangerous glancing at a phone is when driving, asks students to compute how far they would drive down the highway in the time it took them to read or respond to a text message. Many of Christine’s students are learning to drive or have friends who are, so problems like these are relevant and engaging to them. (Please don’t text and drive!)
Though challenging, Christine persisted in adopting the open curriculum because she felt that it was the best approach for her students. She thinks that students have a better understanding of the concepts they have covered. For example, they understand how to factor a polynomial and aren’t just guessing and checking. She reports they are able to prove why the square root of 2 is irrational. They also have a better sense of how a graph relates to algebra, and they persist in solving problems. When solving homework problems, students don’t always know the math theories or strategies they are using, but they are developing algorithms and figuring out problems as they go. These are essential skills for mathematics. Additionally, when students don’t solve a problem the first time, they are willing to try again and again. In this way, they are developing a growth mindset and starting to see the payoffs.
This approach has been more time consuming for Christine. It’s the first time she’s seen many of these problems on the homework, so she needs to solve them all in multiple ways before going to class. She needs to think like her students and try to anticipate the problems they’ll have and the misconceptions they might bring to a problem. This means she really needs to know the content she’s teaching. It’s more prep time before class, especially in the first year, but this way she knows how to guide discussions and ask the right questions. Christine uses her expertise to help students gain a deeper understanding and make connections to content they have seen before. She’s not lecturing as much anymore, but she remains the content area expert.
This idea leads to something that might be a struggle for some. It is described by Harland (2003) like this:
“When students arrived at a position where they could function well together and drive the enquiry forward, they seldom asked for help, and the teaching team no longer had their old roles and familiar student contact. Paradoxically, we felt some sense of loss at this stage and concluded that a lot of pleasure in teaching had gone…”
For Christine, though, she simply sees her role as a teacher changing. She is now more of a facilitator who ensures that students hit certain key points. She guides students in thinking more deeply by helping them ask questions instead of giving them answers. Her connection with students is now stronger, in her opinion. Preparing for class is more involved and time-consuming and her role in the classroom is smaller. But for Christine, that’s okay. What excites her about teaching is helping students discover the math that she loves, and she’s doing that.
If you’re interested in learning more about open approaches to Mathematics education, Christine recommended the Exeter Mathematics Institute and the Mathematics Visionary Project. We would love to hear what you think and the questions you might have for Christine or other teachers.
Harland, T. (2003). Vygotsky’s zone of proximal development and problem-based learning: Linking a theoretical concept with practice through action research. Teaching in higher education, 8(2), 263-272.
Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: An analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. Review of educational research, 72(2), 131-175.
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Excellent article, Patricia! I wish this teaching approach had been available to me as a high school student. I do wonder how this method can be adapted to classrooms with a larger number of pupils. Have you found there to be an ideal number for this approach?
G Almquist – That is a great question! I don’t know of an ideal number of students for this approach, however, I do think that it can be adapted for larger class sizes in a variety of ways. If you have limited board space, you might have students put one or two problems up at a time. Alternatively, students could work through the problem set in small groups and share one or two of their solutions with the class. This way they work through the problems together, come to an agreement, present their solution(s) and still benefit from a guided discussion.
There are other strategies that could support larger or smaller class sizes, does anyone have other suggestions?