Barrier 1: Students who are anxious or who lack a sense of self efficacy have trouble focusing and staying on task.
Barrier 2: Students who feel inferior are less likely to be engaged in their lessons. In early primary school, children start to believe some children are superior or "smarter" in math.
Barrier 3: Students who believe that success depends on innate ability do poorly compared to those who believe that success depends on effort.
Barrier 4: Research has shown that students need extensive practice to master new concepts and skills, but they aren't always motivated to practice.
Barrier 5: The brain is easily overwhelmed by too much new information; math problems that are too complex or overly contextualized or texts that have too many new ideas on a page can discourage and confuse students.
Barrier 6: Weak readers and ESL students can be overwhelmed by too much text, making their language challenges a barrier to achievement in math.
Barrier 7: It is important to teach mathematics using models, but sometimes concrete materials can be distracting or confusing: students don't necessarily learn efficiently from using manipulatives in unstructured lessons.
Barrier 8: Students who haven't mastered basic number facts and operations and committed them to long term memory must use short term memory to do so, leaving inadequate short term memory capacity for problem solving. Students who haven't mastered basic number facts also have trouble seeing patterns and making estimates and predictions.
Barrier 9: Students often memorize rules or procedures without understanding. This may enable them to answer narrowly put questions, but without promoting true understanding: math doesn't always make sense to them.
- Carraher, T. N. Carraher, D. W. & Schliemann, A. D. (1985). Mathematics in the streets and in schools, British Journal of Developmental Psychology, 3, 21-29.
Solution: Use "guided discovery" to teach for deep understanding: deliver lessons in well-scaffolded steps, so one concept naturally leads to the next and students have enough practice to master the concepts, but let students discover the connections between ideas themselves as much as possible.
- Sweller, J. (1988). Cognitive load during problem-solving: Effects on learning. Cognitive Science, 12, 257-285.
- Sweller, J., van Merriënboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional change. Educational Psychology Review, 10, 251-296.
- Van Merriënboer, J. J. G., & Sweller, J. (2005). Cognitive load theory and complex learning recent developments and future directions. Educational Psychology Review, 17, 147-177.
- Ross, P. (2006). The Expert Mind, Scientific American, July.
Barrier 10: To succeed in later grades, students must master the concepts and skills taught in the elementary curriculum. But many students never master these skills and concepts, even though the vast majority are capable of doing so.