Topic: Cognition and Mathematics Instruction

Purpose: The National Center for Cognition and Mathematics Instruction has a core goal of redesigning components of a widely-used middle school mathematics curriculum—Connected Mathematics Project (CMP), and evaluating the efficacy of the redesigned curriculum materials.

Bringing together leading experts in cognition, instruction, assessment, research design and measurement, mathematics education, and teacher professional development, the Center team will apply research-based design principles to revise mathematics curricular materials for the grade span of 6 to 8, when fundamental concepts required for algebra and advanced mathematics are addressed. The redesign will be based upon principles derived from experimental studies in classrooms and controlled laboratory settings to enhance the conditions of instruction and improve learning outcomes for students. The Math Center will conduct an integrated series of design studies; develop and test practical guidelines that will enable mathematics teachers, curriculum developers, and publishers to apply the design guidelines; as well as conduct supplementary studies on important issues in mathematics teaching and learning. The Math Center will first complete a series of controlled experiments (RCTs) aimed at examining the effects of revised curricular units with 50 participating teachers, and then a large-scale, school-level random assignment efficacy study to examine the effects of the redesigned CMP in 78 schools. The Math Center will also widely disseminate findings and provide leadership to the education field.

Established through a five-year, $10.0 million grant from the Institute of Education Sciences (IES) of the U.S. Department of Education, the National Center on Cognition and Mathematics Instruction is housed at WestEd, and operated in collaboration with partners at the University of Illinois at Chicago, Carnegie Mellon University, Temple University, University of Wisconsin-Madison, and Worcester Polytechnic Institute.

Key Personnel: Steven Schneider, James Pellegrino, Ken Koedinger, Neil Heffernan, Julie Booth, Mitchell Nathan, Martha Alibali, Susan Goldman, Diane Briars, Shandy Hauk, Jodi Davenport, Kim Vivani

Center Website: http://www.iesmathcenter.org

IES Program Contact: Dr. Elizabeth Albro
Email: Elizabeth.Albro@ed.gov
Telephone: (202) 219-2148

Products and Publications

Book chapter

Booth, J.L., McGinn, K.M., Barbieri, C., Begolli, K.N., Chang, B., Miller-Cotto, D., Young, L.K., and Davenport, J.L. (2017). Evidence for Cognitive Science Principles that Impact Learning in Mathematics. In D. Geary, D.B. Berch, R. Ochsendorf and K. Koepke (Eds.), Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts (pp. 297–325). Academic Press.

Clinton, V., Cooper, J. L., Michaelis, J. E., Alibali, M. W., and Nathan, M. J. (2017). How Revisions to Mathematical Visuals Affect Cognition: Evidence from Eye Tracking. In C. Was, F.J. Sansoti, and B.J. Morris (Eds.), Eye-Tracking Technology Applications in Educational Research (pp. 195–218). Hershey, PA: IGI Global.

Hawkins, W.J., Heffernan, N.T., and Baker, R.S. (2014). Learning Bayesian Knowledge Tracing Parameters With a Knowledge Heuristic and Empirical Probabilities. In S. Trausan-Matu, K.E. Boyer, M. Crosby, and K. Panourgia (Eds.), Intelligent Tutoring Systems, Lecture Notes in Computer Science, Volume 8474 (pp. 150–155). Switzerland: Springer International Publishing.

Kelly, K., Heffernan, N., Heffernan, C., Goldman, S., Pellegrino, J., and Goldstein, D.S. (2013). Estimating the Effect of Web-Based Homework. In H.C. Lane, K. Yacef, J. Mostow, and P. Pavlik (Eds.), Artificial Intelligence in Education, Lecture Notes in Computer Science, Volume 7926 (pp. 824–827). Berlin Heidelberg: Springer.

Selent, D., and Heffernan, N. (2014). Reducing Student Hint use by Creating Buggy Messages From Machine Learned Incorrect Processes. In S. Trausan-Matu, K.E. Boyer, M. Crosby, and K. Panourgia (Eds.), Intelligent Tutoring Systems, Lecture Notes in Computer Science, Volume 8474 (pp. 674–675). Switzerland: Springer International Publishing.

Wang, Y., and Heffernan, N.T. (2014). The Effect of Automatic Reassessment and Relearning on Assessing Student Long-Term Knowledge in Mathematics. In S. Trausan-Matu, K.E. Boyer, M. Crosby, and K. Panourgia (Eds.), Intelligent Tutoring Systems Lecture Notes in Computer Science, Volume 8474 (pp. 490–495). Switzerland: Springer International Publishing.

Journal article, monograph, or newsletter

Alibali, M.W., Stephens, A.C., Brown, A.N., Kao, Y.S., and Nathan, M.J. (2014). Middle School Students' Conceptual Understanding of Equations: Evidence From Writing Story Problems. International Journal of Educational Psychology, 3(3): 235–264.

Booth, J.L., and Davenport, J.L. (2013). The Role of Problem Representation and Feature Knowledge in Algebraic Equation-Solving. The Journal of Mathematical Behavior, 32(3): 415–423.

Booth, J.L., McGinn, K.M., Young, L.K., and Barbieri, C. (2015). Simple Practice Doesn't Always Make Perfect Evidence From the Worked Example Effect. Policy Insights From the Behavioral and Brain Sciences, 2(1): 24–32.

Clinton, V., Alibali, M.W., and Nathan, M.J. (2016). Learning About Posterior Probability: Do Diagrams and Elaborative Interrogation Help?. The Journal of Experimental Education, 84(3): 579–599.

Clinton, V., Morsanyi, K., Alibali, M.W., and Nathan, M.J. (2016). Learning About Probability From Text and Tables: Do Color Coding and Labeling through an Interactive User Interface Help. Applied Cognitive Psychology, 30(3): 440–453.

Cooper, J. L., Sidney, P. G. and Alibali, M. W. (2018). Who Benefits from Diagrams and Illustrations in Math Problems? Ability and Attitudes Matter. Applied Cognitive Psychology, 32(1): 24–38.

Goldman, S.R., and Pellegrino, J.W. (2015). Research on Learning and Instruction Implications for Curriculum, Instruction, and Assessment. Policy Insights From the Behavioral and Brain Sciences, 2(1): 33–41.

Heffernan, N.T., and Heffernan, C.L. (2014). The ASSISTments Ecosystem: Building a Platform That Brings Scientists and Teachers Together for Minimally Invasive Research on Human Learning and Teaching. International Journal of Artificial Intelligence in Education, 24(4): 470–497.

Heffernan, N.T., Ostrow, K.S., Kelly, K., Selent, D., Van Inwegen, E.G., Xiong, X., and Williams, J.J. (2016). The Future of Adaptive Learning: Does the Crowd Hold the Key?. International Journal of Artificial Intelligence in Education, 26(2): 615–644.

McGinn, K.M., Lange, K.E., and Booth, J.L. (2015). A Worked Example for Creating Worked Examples. Mathematics Teaching in the Middle School, 21(1): 26–33.

Ostrow, K. S., Wang, Y., and Heffernan, N. T. (2017). How Flexible is Your Data? A Comparative Analysis of Scoring Methodologies Across Learning Platforms in the Context of Group Differentiation. Journal of Learning Analytics, 4(2): 91–112.

Ostrow, K.S., Heffernan, N.T., and Williams, J.J. (2017). Tomorrow's EdTech Today: Establishing a Learning Platform as a Collaborative Research Tool for Sound Science. Teachers College Record,, 119(3): 1–36.

Proceeding

Adjei, S.A., Botelho, A.F., and Heffernan, N.T. (2016). Predicting Student Performance on Post-Requisite Skills Using Prerequisite Skill Data: An Alternative Method for Refining Prerequisite Skill Structures. In Proceedings of the Sixth International Conference on Learning Analytics and Knowledge (pp. 469–473). New York: ACM.

Clinton, V., Alibali, M.W., and Nathan, M.J. (2013). Individual Differences in Calculating Posterior Probability: Do Statistics Education and Math Poficiency Matter?. In In Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 374). Chicago: University of Illinois-Chicago.

Clinton, V., Alibali, M.W., and Nathan, M.J. (2013). The Effects of Diagrams and Questioning While Reading on Learning From a Statistics Lesson. In In Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 341–348). Chicago: University of Illinois-Chicago.

Clinton, V., Alibali, M.W., and Nathan, M.J. (2013). Why do Diagrams Increase Learning From Lessons?. In In Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 318). Chicago: University of Illinois-Chicago.

Clinton, V., Nathan, M.J., and Alibali, M.W. (2013). The Influence of Visual Representations on Learning From Lessons on Functions. In Proceedings of the 34th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education(pp. 122). Kalamazoo, MI: Western Michigan University.

Cooper, J.L., Clinton, V., Riggs, E.A., Brey, E., Alibali, M.W., and Nathan, M.J. (2013). Contextual Visual Information in Middle School Problem Solving: A Puzzling Situation. In In Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education(pp. 555). Chicago: University of Illinois-Chicago.

Davenport, J., Kao, Y.S., and Schneider, S.A. (2013). Integrating Cognitive Science Principles to Redesign a Middle School Math Curriculum. In In Proceedings of the 35th Annual Conference of the Cognitive Science Society (pp. 364–370). Austin, TX: Cognitive Science Society.

Davenport, J.L., Kao, Y, Hubbard, A, and Schneider, S.A. (2014). Testing Cognitive Science Principles in a Middle School Mathematics Curriculum. In Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 3212). Austin, TX: Cognitive Science Society.

Gu, J., Cai, H., and Beck, J.E. (2014). Investigate Performance of Expected Maximization on the Knowledge Tracing Model. In Intelligent Tutoring Systems Lecture Notes in Computer Science, Volume 8474 (pp. 156–161). Switzerland: Springer International Publishing.

Gu, J., Wang, Y., and Heffernan, N.T. (2014). Personalizing Knowledge Tracing: Should we Individualize Slip, Guess, Prior or Learn Rate?. In Intelligent Tutoring Systems (pp. 647–648). Switzerland: Springer International Publishing.

Jiang, Y., Baker, R.S., Paquette, L., San Pedro, M., and Heffernan, N.T. (2015). Learning, Moment-by-Moment and Over the Long Term. In Proceedings of the 17th International Conference, AIED 2015 (pp. 654–657). Switzerland: Springer International Publishing.

Kehrer, P., Kelly, K.M., and Heffernan, N.T. (2013). Does Immediate Feedback While Doing Homework Improve Learning?. In Proceedings of the Twenty-Sixth International Florida Artificial Intelligence Research Society Conference (pp. 542–545).

Kelly, K., Arroyo, I., and Heffernan, N. (2013). Using ITS Generated Data to Predict Standardized Test Scores. In In Proceedings of the 6th International Conference on Educational Data Mining (pp. 322–324). Memphis, TN: Educational Data Mining Society.

Kelly, K., Heffernan, N., Heffernan, C., Goldman, S., Pellegrino, J.W., and Soffer-Goldstein, D. (2014). Improving Student Learning in Math Through Web-based Homework Review. In Proceedings of the Joint Meeting of the International Group for the Psychology of Mathematics Education (PME 38) and the North American Chapter of the Psychology of Mathematics Education (PME-NA 36). Vancouver, BC: University of British Columbia.

Kelly, K.M., and Heffernan, N.T. (2016). Optimizing the Amount of Practice in an On-Line Platform. In Proceedings of the Third ACM Conference on Learning at Scale (pp. 145–148). New York: ACM.

Koedinger, K.R. and Mclaughlin, E.A. (2017). Closing the Loop with Quantitative Cognitive Task Analysis. In Proceedings of the 9th International Conference On Educational Data Mining (pp. 412–417). Raleigh, NC.

Lang, C., Heffernan, N., Ostrow, K., and Wang, Y. (2015). The Impact of Incorporating Student Confidence Items Into an Intelligent Tutor: A Randomized Controlled Trial. In Proceedings of the 8th International Conference on Educational Data Mining (pp. 144–149). Madrid, Spain: Educational Data Mining.

Li, N., Stampfer, E., Cohen, W.W., and Koedinger, K.R. (2013). General and Efficient Cognitive Model Discovery Using a Simulated Student. In Proceedings of the 35th Annual Conference of the Cognitive Science Society (pp. 894–899). Austin, TX: Cognitive Science Society.

Lomas, J. D., Forlizzi, J., Poonwala, N., Patel, N., Shodhan, S., Patel, K., and Brunskill, E. (2016). Interface Design Optimization as a Multi-Armed Bandit Problem. In Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems (pp. 4142–4153).

Ostrow, K., and Heffernan, N.T. (2014). Testing the Multimedia Principle in the Real World: A Comparison of Video vs. Text Feedback in Authentic Middle School Math Assignments. In Proceedings of the Seventh International Conference on Educational Data Mining (pp. 296–299). London: Educational Data Mining.

Ostrow, K., Donnelly, C., Adjei, S., and Heffernan, N. (2015). Improving Student Modeling Through Partial Credit and Problem Difficulty. In Proceedings of the Second (2015) ACM Conference On Learning@ Scale (pp. 11–20). New York, NY: ACM.

Ostrow, K., Donnelly, C., and Heffernan, N. (2015). Optimizing Partial Credit Algorithms to Predict Student Performance. In Proceedings of the 8th International Conference on Educational Data Mining (pp. 404–407).

Ostrow, K., Heffernan, N., Heffernan, C., and Peterson, Z. (2015). Blocking vs. Interleaving: Examining Single-Session Effects Within Middle School Math Homework. In Proceedings of the 17th International Conference, AIED 2015 (pp. 338–347). Switzerland: Springer International Publishing.

Ostrow, K.S., and Heffernan, N.T. (2015). The Role of Student Choice Within Adaptive Tutoring. In Proceedings of the 17th International Conference, AIED 2015 (pp. 752–755). Switzerland: Springer International Publishing.

Ostrow, K.S., and Heffernan, N.T. (2016). Studying Learning at Scale with the ASSISTments TestBed. In Proceedings of the Third ACM Conference on Learning at Scale(pp. 333–334). New York: ACM.

Ostrow, K.S., Selent, D., Wang, Y., Van Inwegen, E.G., Heffernan, N. T., and Williams, J.J. (2016). The Assessment of Learning Infrastructure (ALI): The Theory, Practice, and Scalability of Automated Assessment. In Proceedings of the Sixth International Conference on Learning Analytics and Knowledge (pp. 279–288). New York, NY: ACM.

Selent, D., and Heffernan, N. (2015). When More Intelligent Tutoring in the Form of Buggy Messages Does not Help. In Proceedings of the 17th International Conference, AIED 2015 (pp. 768–771).

Stampfer, E., and Koedinger, K.R. (2013). When Seeing Isn't Believing: Influences of Prior Conceptions and Misconceptions. In In Proceedings of the 35th Annual Conference of the Cognitive Science Society(pp. 1384–1389). Austin, TX: Cognitive Science Society.

Van Inwegen, E., Adjei, S., Wang, Y., and Heffernan, N. (2015). An Analysis of the Impact of Action Order on Future Performance: The Fine-Grain Action Model. In Proceedings of the Fifth International Conference on Learning Analytics and Knowledge (pp. 320–324).

Van Inwegen, E., Wang, Y., Adjei, S., and Heffernan, N.T. (2015). The Effect of the Distribution of Predictions of User Models. In Proceedings of the 8th International Conference on Educational Data Mining (pp. 620–621). Madrid, Spain: Educational Data Mining.

Walkington, C., Clinton, V., and Mingle, L. (2016). Considering Cognitive Factors in Interest Research: Context Personalization and Illustrations in Math Curricula. In Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 89–98). Tuscon, AZ: The University of Arizona.

Wang, Y., Heffernan, N.T., and Heffernan, C. (2015). Towards Better Affect Detectors: Effect of Missing Skills, Class Features and Common Wrong Answers. In Proceedings of the Fifth International Conference on Learning Analytics And Knowledge (pp. 31–35). New York: ACM.

Wang, Y., Ostrow, K., Adjei, S., and Heffernan, N. (2016). The Opportunity Count Model: A Flexible Approach to Modeling Student Performance. In Proceedings of the Third (2016) ACM Conference on Learning at Scale(pp. 113–116). New York: ACM.

Wang, Y., Ostrow, K., Beck, J., and Heffernan, N. (2016). Enhancing the Efficiency and Reliability of Group Differentiation Through Partial Credit. In Proceedings of the Sixth International Conference on Learning Analytics and Knowledge (pp. 454–458). New York: ACM.

Wiese, E., Patel, R., Olsen, J.K., and Koedinger, K (2015). Transitivity is not Obvious: Probing Prerequisites for Learning. In Proceedings of the 37th Annual Meeting of the Cognitive Science Society (pp. 2655–2660). Austin, TX: Cognitive Science Society.

Wiese, E.S., Patel, R., and Koedinger, K.R. (2016). Why Sense Making Through Magnitude may be Harder for Fractions than for Whole Numbers. In Proceedings of the 38th Annual Meeting of the Cognitive Science Society (pp. 1229–1234). Philadelphia, PA: Cognitive Science Society.