Categorizing student learning difficulties in mathematics
1. Cognitive conflicts arise when they choose, often unconsciously, between previously held ideas and newly acquired ones. Students entering college are expected to engage in mathematical thinking that extends beyond the application of fundamental concepts taught in secondary school. Conflict results when the student attempts to learn new ideas while trying to make sense of previous knowledge.
2. Epistemological obstacles arise from the complexity of mathematics knowledge. Many students struggle with concepts that exhibit higher levels of abstraction. I found this to be true when teaching trigonometry. When I moved from using the right triangle as a static form to using it as a reference angle that moves around the Cartesian plane, numerous student difficulties occurred.
3. Cognitive obstacles stem from a student’s inability to conceptualize or apply the mathematics. This may involve spatial difficulties or difficulties visualizing a concept, thereby preventing full understanding of the mathematics. This obstacle interconnects with the epistemological obstacles. Student difficulties increase proportionally to the level of mathematics complexity.
4. Didactic obstacles. These are issues that relate to the actual teaching process.
5. Semiotic conflicts. Students experience a “discordance, disparity or mismatch between the meanings attributed to the same expression.” Learning the language of mathematics is akin to learning the spoken language of a community with all its words, intonations, idioms, and gestures. Students lacking fluency in mathematical language to
describe and call upon representations at the college level face another barrier to effective learning of new mathematical concepts, concept application, and problem solving activity.
6. Mathematical discontinuities. Prior notions of how they are to think and do mathematics differ from current expectations. This may occur when students are moved from an interactive, exploratory, cooperative learning classroom into a classroom based on the traditional lecture format.
7. Transition obstacles. These are discrepancies in mathematical concepts, ideas, or representations taught between educational panels. Typically, a transition obstacle results in a gap in student’s knowledge base due to a curriculum mismatch. This is where I am focusing student difficulties. We will review the curriculum documents with respect to the college course outline to determine either a curriculum mismatch or a discontinuity in the concept development process..