Supporting Research for JUMP Approach

External Research that Supports JUMP Math's Approach

The JUMP Math program delivers the mathematical curriculum through the method of "guided discovery." In JUMP lessons, students explore and discover mathematical concepts independently in manageable steps, while the teacher provides sufficient guidance, examples, feedback and scaffolding for all students to meet their full potential.

JUMP Math is recommended by the Canadian Language and Literacy Research Network as a program that "offers educators... complete and balanced materials as well as training to help teachers reach all students".

  • J. Bisanz et al. (2010) Foundations for Numeracy: An Evidence-based Toolkit for the Effective Mathematics Teacher. Canadian Child Care Federation and Canadian Language and Literacy Research Network, p. 44.

In 2011, L. Alfieri et al. conducted a meta analysis of 164 studies of discovery-based learning and concluded that "Unassisted discovery does not benefit learners, whereas feedback, worked examples, scaffolding and explicit instruction do." The authors recommend "enhanced discovery" (discovery with the instructional supports mentioned above) as the most effective approach to instruction in mathematics.

  • Alfieri, L., et al. (2011) Does Discovery Based Instruction Enhance Learning? Journal of Educational Psychology, Vol. 103, Issue 1, p 1-18.
  • See also the references below for evidence that discovery needs to be balanced with rigorous guidance: Anderson (2000), Gobet (2005), Van Merrienboer (2005), Ross (2006), Kirshner (2006).

JUMP's extensive Teacher Materials are the core of the JUMP program. The lesson plans in the guides cover the full curriculum and include ideas for contextualizing mathematical concepts, mini-assessments and questions for formative assessment, problems and challenges that allow students to investigate and develop concepts, mental math exercises to help students develop computational fluency and automatic recall of facts, extension questions for students who finish their work early, and a variety of games and activities with concrete materials. The JUMP workbooks, which are used at the end of a lesson, help teachers assess whether students understood the lesson and give students sufficient practice to consolidate skill and concepts.

JUMP lesson plans and materials allow teachers to differentiate instruction by providing extra practice, scaffolding and continuous assessment for students who need it, and more advanced work for students who finish their work early. But while instruction is differentiated, the significant majority of students are expected to meet the same standards.

A growing body of evidence in education and cognitive science suggests that, with proper instruction, children can develop abilities in subjects for which they previously showed no real aptitude or gift. (see for instance, Ross, P. E. (2006) "The Expert Mind." Scientific American, July.) But research also shows that new abilities are more likely to emerge when a child's brain is attentive and engaged.

Supporting Research:

  • Posner, M., Rothbart, M. (2005) Influencing brain networks: implications for education, TRENDS in Cognitive Science, v.9, no. 3
  • Gathercole, S. E., Alloway, T. P., Kirkwood, H. J., Elliott, J. G., Holmes, J., & Hilton, K. A. (2008). Attentional and executive function behaviours in children with poor working memory. Learning and Individual Differences, 18(2), 214-223.
  • Schwartz, J., Bregley, S. (2002) The Mind and the Brain. NY, Regan Books.

In designing the JUMP program we asked: What barriers prevent students from paying attention and being engaged in math lessons?

 

Barrier 1. Students who are anxious or who lack a sense of self efficacy have trouble focusing and staying on task.

Supporting Research:

  • Ashcraft, M. H. & Kirk. E. P. (2001) The Relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130
  • Ashcraft, M., Krause, J. A. (2007) Working memory, math performance, and math anxiety, Psychonomic Bulletin and Review, 14 (2), 243-248.
  • Beilock, S. (2008). Math performance in stressful situations. Current Directions in Psychological Science, 17, 339-343.
  • Pacheco-Unguetti, A. P., Acosta, A., Callejas, A., & Lupianez, J. (2010). Different Attentional Functioning Under State and Trait Anxiety. Psychological Science, 21, 298-304.
  • Lussier, G. (1996). Sex and mathematical background as predictors of anxiety and self-efficacy in mathematics. Psychology Reports, 79, 827-833.
  • Bandura, A. (1986) Fearful expectations and avoidant actions as coeffects of perceived self-inefficacy, American Psychologist, December, 1389-1391
  • Pajares, F., and Miller, D. (1994). Role of self-efficacy and self-concept beliefs in mathematical problem solving: A path analysis. Journal of Educational Psychology, 86, 193-203.

Solution: Provide students with regular confidence building exercises that look challenging but enable all students to do well.

Supporting Research:

  • Margulis, H., Mccabe, P. (2006) Improving Self Efficacy and Motivation, Intervention in School and Clinic, 2006, Vol.41 No. 4, 218-227.
  • Humbre, R., (1990). The nature, relief and effects of mathematics anxiety. Journal for Research in Mathematics Education, 12(1), 33-46.

 

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.

Supporting Research:

  • Henderlong Corpus, J., et. al. (2006). The effects of social-comparison versus mastery praise in children's intrinsic motivation. Motivation and Emotion, 30, 335-345
  • Hong, E., Peng, Y., & Rowell, L. L. (2009). Homework self-regulation: Grade, gender, and achievement-level differences. Learning & Individual Differences, 19(2), 269-276.
  • Dweck, C.S. (1975). The role of expectations and attributions in the alleviation of learned helplessness, Journal of Personality and Social Psychology, 31, 674-684.
  • Dweck, C. S. (1999). Self-Theories: Their role in motivation, personality, and development. Philadelphia, PA: Psychology Press.
  • Dweck, C. S. (2006) Mindset: The New Psychology of Success, Random House, NY, 2006
  • Ma, X. & Xu, J. (2004). Determining the causal ordering between attitude toward mathematics and achievement in mathematics. American Journal of Education, 110, 256-280.

Solution: Review basic skills and concepts thoroughly before introducing new topics, so all students can start on an equal footing, and target less confident students with special "bonus" questions that allow them to experience success. When students feel equally capable, their brains work efficiently, and they tend to become equally capable.

Supporting Research:

  • Pashler, H., Bain, P., Bottge, B., Graesser, A., Koedinger, K., McDaniel, M., and Metcalfe, J. (NCER 2004-2007). Organizing Instruction and Study to Improve Student Learning. Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education
  • Geary, D. C. (2006). Development of mathematical understanding. In W. Damon (Ed.), Handbook of child psychology (6th ed., Vol. 2, D. Kuhl & R. S. Siegler (Eds.). Cognition, perception, and language, pp. 777-810). New York: John Wiley & Sons.
  • Hutton, L. A., and Levitt, E. (1987). An academic approach to the remediation of mathematics anxiety. In R. Schwarer, H. M. van der Ploeg and C. D. Spielberger (Eds.), Advances in test anxiety research, Vol. 5, 207-211.
  • Humbre, R., (1990). The nature, relief and effects of mathematics anxiety. Journal for Research in Mathematics Education, 12(1), 33-46.

 

Barrier 3: Students who believe that success depends on innate ability do poorly compared to those who believe that success depends on effort.

Supporting Research:

  • Usher, E. L. (2009). Sources of middle school students' self-efficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275-314.
  • Kamins, M. L. & Dweck, C. S. (1999). Person versus praise and criticism: implications for contingent self-worth and coping. Developmental Psychology, 35, 835-847.
  • Mueller, C. M. & Dweck, C. S. (1998). Praise for intelligence can undermine children's motivation and performance. Journal of Personality and Social Psychology, 75, 33-52.
  • Licht, B. G. & Dweck, S. C. (1984). Determinants of academic achievement: the interaction of children's achievement orientations with skill area. Developmental Psychology, 20, 628-636. Canadian Child Care Federation (CCCF).

Solution: Give students problems that get incrementally harder to show them they can surmount any challenge through their work. Students learn best when they are allowed to take moderate risks with positive feedback.

Supporting Research:

  • Pashler, H., Bain, P., Bottge, B., Graesser, A., Koedinger, K., McDaniel, M., and Metcalfe, J. (NCER 2004-2007) Organizing Instruction and Study to Improve Student Learning. Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education.
  • Clifford, M. (2009). Students need challenge not easy success, Educational Leadership, September.

 

Barrier 4: Research has shown that students need extensive practice to master new concepts and skills, but they aren't always motivated to practice.

Supporting Research:

  • Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer.
  • Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2010). The effects of strategic counting instruction, with and without deliberate practices on number combination skill among students with mathematics difficulties. Learning and Individual Differences, 20, 89-100.
  • Trautwein, U., & L�dtke, O. (2009). Predicting homework motivation and homework effort in six school subjects: The role of person and family characteristics, classroom factors, and school track. Learning & Instruction, 19(3), 243-258

Solution: Practice doesn't need to be "drill and kill." Turn practice into games that engage students and make math enjoyable, by embedding it in exercises in which students are challenged, constantly reach higher levels of success, and receive immediate feedback.

Supporting Research:

  • Pashler, H., Bain, P., Bottge, B., Graesser, A., Koedinger, K., McDaniel, M., and Metcalfe, J. (NCER 2004-2007). Organizing Instruction and Study to Improve Student Learning, Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education
  • Anderson, E. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, Vol. 100, No. 3, 363-406.

 

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.

Supporting Research:

  • 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.

Supporting Research:

  • 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.

Supporting Research:

  • Nesher, P., Hershkovits, S., & Novotna, J. (2003). Situation model, text base and what else? Factors affecting problem solving. Educational Studies in Mathematics, 52, 151 - 176.
  • Adler, J. (1998). A language of teaching dilemmas: Unlocking the complex multilingual secondary mathematics classroom. For the Learning of Mathematics, 18(1), 24-33.
  • Kotsopoulos, D. (2007). Mathematics discourse: "It sounds like hearing a foreign language." Mathematics Teacher, 101(4), 310-305.
  • Hayfa, N. (2006). Impact of language on conceptualization of the vector. For the Learning of Mathematics, 26(2), 36-40.
  • Jordan, N., Hanich, L., & Kaplan, D. (2000). A longitudinal study of mathematical competence in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 7, 834-850.
  • Rasanen, P. & Ahonen, T. (1995). Arithmetic disabilities with and without reading difficulties: a comparison of arithmetic errors. Developmental Neuropsychology, 11, 275-295.
  • Jordan, N. C., Huttenlocher, J. E., & Levine, S. C. (1992). Differential calculation abilities in young children from middle and low income families. Developmental Psychology, 28, 644-653.
  • Jordan, N. C., Huttenlocher, J. E., & Levine, S. C. (1994). Assessing early arithmetic abilities: effects of verbal and non-verbal response types on the calculation performance of middle- and low- income children.
  • Jordan, N. C., Levine, S. C., & Huttenlocher, J. E. (1994). Development of calculation abilities of middle- and low- income children after formal instruction in school. Journal of Applied Developmental Psychology, 15, 223-240.
  • Fuchs, L., Fuchs, D. & Prentice, K. (2004). Responsiveness to mathematical problem-solving instruction: comparing students at risk of mathematics disability with and without risk of reading disability. Journal of Learning Disabilities, 37, 293-306.

Solution: Minimize the use of text in student materials, and introduce language gradually and rigorously. Place activities or exercises that require lengthy descriptions in the Teachers Guides. Ask students to communicate their understanding, but allow pictures, numbers or oral answers when writing is a challenge.

Supporting Research:

  • Jordan (1992, 1994, 1994). The three papers referenced immediately above, show particularly well that allowing alternative (non-verbal) response modalities can reveal knowledge that would not be revealed if responses relied heavily on language.

See also:

  • Uttal, D. H., Liu, L. L., & DeLoache, J. S. (2006). Concreteness and symbolic development. In L. Balter and C. S. Tamis-LeMonda (Eds.), Child Psychology: A Handbook of Contemporary Issues (2nd Ed.) (pp.167-184). Philadelphia, PA: Psychology Press.
  • Chamot, A., & O'Malley, J. M. (1994). The CALLA handbook: Implementing the cognitive academic language learning approach. Mass: Addison-Wesley Publishing Co.

 

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.

Supporting Research:

  • Fuchs, L.S., Fuchs D., et al (2006) The cognitive correlates of third grade skill in arithmetic, algorithmic computation and arithmetic word problems, Journal of Educational Psychology, 98, 29-43
  • Geary, D.C., Mathematical difficulties: cognitive, neuropsychological and genetic components, Psychological Bulletin, 1993, Vol.114, No.2, 345-362
  • Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers. The Journal of Mathematical Behavior, 23, 443-463.
  • K. S. McGrew, R. W. Woodcock, Woodcock-Johnson III Technical Manual (Riverside Publishing, Itasca, IL, 2001). The technical manual for the Woodcock-Johnson III Test of Achievement Battery - a widely used and recognized battery of achievement tests - shows that performance on math fluency and calculation subscales is significantly, positively correlated with performance on the applied problems subscale, suggesting that fluency may be critical for freeing up resources to attend to conceptual processing and to making mathematical discoveries (e.g. the correlations are .46 and .57 for math fluency and calculation, respectively for children aged 9-13, see page 170). This reference is provided as an example - these findings are not unique to the WJ-III.

Solution: Make mental math exercises, like the ones in the JUMP Teachers Guide, part of the lesson. Let students develop their own strategies for computation, but don't neglect to teach the basic operations rigorously so students master them and understand how they work.

Supporting Research:

  • Foundations for numeracy: An Evidence-based Toolkit for the effective mathematics teacher. Ottawa: CCCF. Child 92. Psychology and Psychiatry, 35, 283-2Canadian Child Care Federation (CCCF), 2010.
  • Murata, A., & Fuson, K. (2006). Teaching as assisting individual constructive paths within an interdependent class learning zone: Japanese first graders learning to add using 10. Journal for Research in Mathematics Education, 37(5), 421-456.

 

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.

Supporting Research:

  • 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.

Supporting Research:

  • Sweller, J. (1988). Cognitive load during problem-solving: Effects on learning. Cognitive Science, 12, 257-285.
  • Sweller, J., van Merri�nboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional change. Educational Psychology Review, 10, 251-296.
  • Van Merri�nboer, 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.

Supporting Research:

  • Rittle-Johnson, B., & Kmicikewycz, A. O. (2008). When generating answers benefits arithmetic skill: The importance of prior knowledge. Journal of Experimental Child Psychology, 101, 75-81.

Solution: Provide teachers with materials and training that allow them to continuously assess what their students know and to teach essential skills and concepts effectively (and explicitly if necessary).

Supporting Research:

  • Feifer, S, et al. (2005). The Neuropsychology of Mathematics: Diagnosis and Intervention. Riverside, CA: RET Centre.
  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Educational Psychologist, 41(2), 75 - 86.
  • J. Bisanz et al. (2010). Foundations for numeracy: An Evidence-based Toolkit for the effective mathematics teacher. Canadian Child Care Network and Canadian Language and Literacy Research Network.

Related Information

Research

Research Goals and Purposes

Research Reports about JUMP