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.
- Lee, K., Ng, E. L., & Ng, S. F. (2009). The contributions of working memory and executive functioning to problem representation and solution generation in algebraic word problems. Journal of Educational Psychology, 101(2), 373-387.
- Sweller, J. (1988). Cognitive load during problem-solving: Effects on learning. Cognitive Science, 12, 257-285.
- Marzocchi, G. M., Lucangeli, D., De Meo, T., Fini, F., & Comoldi, C. (2002). The disturbing effect of irrelevant information on arithmetic problem-solving in inattentive children. Developmental Neuropsychology, 21, 73-92.
- McNeil, N. M., & Uttal, D. H. (2009). Rethinking the use of concrete materials in learning: Perspectives from development and education. Child Development Perspectives, 3, 137-139.
Solution: Research has shown that the "big ideas" must be built up, systematically, from smaller component ideas. Teach with the big picture in view, but start by ensuring that students master the component skills and concepts they need in manageable chunks. As students become more able, let them explore more complex or open-ended problems. Research also shows that conceptual and procedural knowledge develop iteratively, with increases in one type of knowledge leading to increases in the other. Effective lessons allow children to develop both kinds of knowledge concurrently.
- Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer.
- Gobet F. (2005) Chunking Models of Expertise: Implications for Education. Applied Cognitive Psychology, 19, 183-204.
- Rittle-Johnson, B. Siegler, R. S. and Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93, 346-362.
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.
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.