**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.

**Supporting Research:**

*Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175-197.**McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171-184.**Ball, D. L. (1992). Magical hopes: manipulatives and the reform of math education. American Educator, Summer edition.**Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (in press). Transfer of mathematical knowledge: The portability of generic instantiations. Child Development Perspectives.**Kaminski, J. A., Sloutsky, V. M, & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320, 454-455.**Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2009). Concrete instantiations of mathematics: A double-edged sword. Journal for Research in Mathematics Education, 40, 90-93.**Kaminski, J. A., Sloutsky, V. M. & Heckler, A. F. (2006). Do children need concrete instantiations to learn an abstract concept. In R. Sun and N. Miyake (Eds.). Proceedings of the XXVIII Annual Conference of the Cognitive Science Society (pp. 411-416).**Peterson, L. A. & McNeil, N. M. (2008). Using perceptually rich objects to help children represent number: established knowledge counts. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society (pp. 1567-1572). Austin, TX: Cognitive Science Society.*

**Solution:** Use simple models or symbolic representations that allow students to see the math clearly, rather than being distracted by the details. Use a variety of concrete representations for concepts, but make sure each representation is rigorously taught.

**Supporting Research:**

*Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, (33), 202-233.**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.**Kaminski, J. A., Sloutsky, V. M, & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320, 454-455.**Neil, N. M. Mc& Jarvin, L. (2007). When theories don't add up: Disentangling the manipulatives debate. Theory Into Practice, 46, 309-316.**Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols; a new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37-54.**Moreno, R., Ozogul, G., Riessleir, M. (2011) Teaching with concrete and abstract representations: Effects on students' problem solving, problem representations and learning perceptions, Journal of Educational Psychology, Vol. 103, Issue 1, February. P 32-47.*

**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.

**Related Information**