If you were to walk into most math classrooms, it would not be unusual to see students taking notes at their desks. They would be neatly arranged in rows facing a teacher who would be explaining content on the board. After the explanation, the students would go about familiarizing themselves with the concepts using drill exercises and working independently. Now, with technology, the only difference is perhaps an LCD project and a tablet for the instructor to assist in their explanation. I believe that Mathematic educators are at risk of falling into this traditional classroom setting for many reasons. For me, this is typically the way I approach many math lessons. The main reason for adopting this routine is because of the convenience and efficiency offered by this teaching approach; but after taking one of my old lesson on polynomial and approaching the graphing concepts using a constructivist mindset, I found the process to be innovative and exciting.

My typical lesson plan consists of templated notes that students fill out while following along with my explanation. I usually start off with definitions and then explain the reasoning and mathematical processes used to solve several common math problems that they would encounter in their homework or tests. The students end the lesson working on the assigned textbook questions individually or in pairs. For some topics, like graphing polynomials, I include a discovery activity to be completed in pairs and accompanied with technology. In my graphing polynomial lesson, I have students complete the discovery activity before going over the math problems as a class. The discovery activity for this particular topic requires students to manually enter a variety of polynomials into a Ti-83 graphing calculator in order to find patterns. Aside from instructions to use the graphics calculators, I do not give them guided instruction and students determine how they want to go about discovering these patterns. Some students randomly graph polynomials while other students systematically change one co-efficient/exponent at a time. Most students find the task too daunting and would not even attempt to graph anything. Because of the low success rate, I end up telling the students the pattern at the end of the discovery activity; as a result, I have adapted the activity so that I provide the students with a variety of graphs and their corresponding equations. Students use my examples to find the patterns of polynomials with increasing success. Unfortunately, this reduces the interactivity students have with technology and the flexibility students have to experiment with the polynomial functions. Although more students are able to notice the patterns and perform recognition tasks (such as matching the equation with the graph), I still find that students have difficulty applying the concepts in more varied, complicated scenarios (like graphing or finding the equation when given the characteristics), and recall tasks (like sketching the graph from merely looking at the equation and vise versa). Despite the discovery activity, I still find that the abstract nature of the topic and lack of real world application causes students to feel disassociated and disinterested with the topic.

Following a constructivist model design, I attempted to address all the weaknesses of a traditional lesson using the phases outlined by Driver-Oldham (1986). I started with the orientation phase by stating the real world application of polynomial functions. Students can decide to further investigate the different applications that interest them using the links that I have provided on the ‘Introduction’ page. This freedom gives an individualized approach and helps students realize that the topic has many useful applications. Students are also asked to share their reasoning behind learning polynomials which personalizes the learning experience and helps each student contribute to a learning culture specific to their class. By sharing their opinion freely on the subject and without judgment, students can relate to each other in order to establish a safe and cooperative learning community.

In an attempt to further motivate the students, I have the students designing a rollercoaster using authentic physics principles instead of completing a set of traditionally assigned textbook questions. Students are asked to share their prior knowledge on rollercoaster design on the ‘orientation’ page and then share what they have learned about roller coaster designs after exploring virtual simulations on the ‘elicitation’ page. This situated learning experience will engage and motivate students to investigate the topic.

In a traditional setting, I usually forgo the introduction, orientation, and elicitation phases. I start immediately with what I call the ‘pre-structuring’ phase. In the ‘pre-structuring’ phase, I assess the students’ prior knowledge on polynomials. This process usually consists of explaining some previous graphing examples on the board and reviewing some old vocabulary. For a constructivist classroom, it is more appropriate to have students take an online test that would provide immediate feedback on the specific areas they need to work on. This gives each student the unique guidance that is needed to correct their misconceptions before building new knowledge structures; the immediate feedback offered by this approach has been reported to be the single most powerful influence affecting student achievement (Hattie, 1987).

In the restructuring phase, I provide students with a dynamic and virtual simulation. This is where students can explore and investigate the relationships between graphs and their functions interactively and easily. Compared to the static pictures in textbooks, this computerised environment offers students the opportunity to immediately explore and observe the effects of changing each exponent and coefficient on the graph of a polynomial. According to Vygotsky (1976), the ability to play offers the cognitive support and framework essential to developing higher order mental processes so that students can move toward the abstract and graphical applications of functions. However, this exploration needs to be accompanied by proper guidelines in order to provide a learning task that offers the optimal level of challenge for students. It is believed that by going through the simulations under the guidance of a more knowledgeable individual, the “Zone of Proximal Development” (which is defined by Vygotsky as the disparity between the actual developmental level and the potential developmental level) will be reduced. For this reason, guiding questions have been provided alongside the app to help students construct their conceptual understanding. Unlike my previous discovery activities which leaves students to their own accord. This allows the teacher to identify the important variables they must pay attention to while running the simulation. This guidance helps the weaker students through their construction of knowledge, and helps the stronger students reduce cognitive overload. For the weaker students, they are given additional support through collaboration with peers as students are then required to post their observation and graphs on a cloud graphic organizer. This gives students a chance to clarify and exchange ideas with each other while making their learning visible in a learning community. Although not as synchronous as a traditional classroom environment, the cloud graphic organizer gives opportunities for students to support each other in their learning process. This is all while students make personal sense of the mathematic concepts through individual reflection as well as through exposure to differing perspectives. In a truly collaborative environment, it has been shown that knowledge is socially constructed and that students learn best in ‘learning communities’ (Scardamalia and Berieter, 1994).

In the application phase, students use their new ideas to match graphs with their corresponding equation in an online game. They then construct a polynomial to represent their roller coaster design. This dual activity helps students to use their new ideas in a familiar context of using textbook-like questions in a gaming platform as well as in a new setting of applying their knowledge across the curriculum into the realm of science. This phase is different from a traditional setting in two regards. First of all, traditional textbook questions are non-interactive and non-competitive. Also many textbook questions are focused solely on the math concepts rather than the application of these concepts in other fields and in authentic situations. This activity ends with the students self-evaluating their polynomial function which helps students realize the concepts and processes they should have encountered in this activity.

Finally students are asked to reflect on four review questions at the end of the activity which is important in helping abstract concepts commit to long term memory according to cognitive theorists like Piaget (1977). In order to make it into long term memory, the new information must be actively processed in a meaningful way to match an existing memory structure (known as assimilation by Piaget). It can also change a conflicting misconception (known as accommodation by Piaget) or replace an old memory structure entirely. As a result, the act of having students write down the patterns observed or sketch the graphs in the restructuring or review phase provides a meaningful strategy to help match, change, or replace any of their pre-existing memory structures to successfully assimilate or accommodate the abstract new memory structures.

In conclusion, I believe that by incorporating all the aspects listed above my lesson has inherited many constructivist elements found in the Constructivist Instructional Model (CIM), the Predict-Observe-Explain Model (POE) (White & Gunstone, 1982) and the Conceptual Change Model (CCM) (Posner et al., 1982). The personalized student-centered approach that is supported by dynamic, guided, and reflective activities would make it easy for students to construct knowledge both independently and collaboratively. Students are asked to assess their prior knowledge in the orientation and pre-structuring phases of the lesson; thereafter, students are asked to use the tools to explore through observation and to create a dissatisfaction with their existing knowledge so that they can modify and reconstruct their ideas. The students are able to see the varying applications of the mathematical concepts and apply the learned concepts to a situated and meaningful real world problem. By doing so, the new concept has become intelligible, plausible and fruitful (Posner, 1982). By then committing these to their long term memory, students are able to benefit greatly from the constructivist elements of my revised lesson plan.