
Engineering Education Project
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Improving students' mathematical sensemaking in engineering: Research and development
Abstract
For engineers, effective use of mathematics is more than manipulating equations and applying algorithms; it involves mathematical sensemaking, looking for coherence and meaning partly by translating between symbolic relations on the page and relations (causal and functional) in the world. Mathematical sensemaking is central to students’ success with modeling and design. Yet, many engineering students have trouble with it.
Typical engineering students first grapple extensively with mathematical descriptions of the world in the introductory physics courses they take as prerequisites for their majors. This project, a collaboration among the University of Maryland Departments of Physics, Mechanical Engineering, and Electrical & Computer Engineering, addresses two research questions:
 What factors contribute to students’ difficulties with mathematical sensemaking?
 Can redesigned introductory physics courses improve students’ mathematical sensemaking — and overall performance — in their later engineering courses?
Previous research suggests that, to address (1), we must probe not just for mathematical skill deficiencies but also for students’ lack of understanding of the relevant physics/engineering concepts, lack of ability or propensity to translate between formalism and realworld relations, and naïve beliefs about how to learn and apply math. We tease these factors apart and explore interactions among them using multiple methods, including analysis of video of students solving challenging problems.
To address (2), we draw on our previous work in the algebrabased introductory physics sequence for life science majors. There, we developed materials and teaching techniques focused on changing students’ beliefs about how to learn and apply conceptual and mathematical knowledge. The courses produced substantially improved conceptual learning and mathematical sensemaking.
We plan to follow students from the redesigned as well as unchanged (control group) introductory physics courses, into the Basic Circuit Theory course in Electrical & Computer Engineering and a Fluid Mechanics course in Mechanical Engineering. To see if the redesigned physics courses lead to better mathematical sensemaking and overall performance in those engineering classes, we will analyze students’ exam answers and scores, survey responses, and course grades.
Awards
 Mike Hull, Graduate Student Travel Award for The Conference on Transforming Research on Undergraduate STEM Education, Maine, June 2010.
 Eric Kuo, Graduate Student Travel Award for The Conference on Transforming Research on Undergraduate STEM Education, Maine, June 2010.
Publications: Peerreviewed Journal and Conference Proceedings
 Danielak, B. A., Gupta, A., & Elby, A. (Under Review). The Marginalized Identities of Sensemakers: Reframing Engineering Student Retention. Journal of Engineering Education.
 Hull, M., Kuo, E., Gupta, A., & Elby, A. (2013) Problematizing problemsolving rubrics: Enhancing assessments to include blended mathematical and physical reasoning throughout the solution. Physical Review Special Topics  Physics Education Research, 9(1), 010105.
 Kuo, E., Hull, M., Gupta, A., & Elby, A. (2013) How Students Blend Conceptual and Formal Mathematical Reasoning in Solving Physics Problems. Science Education, 97(1), 3257.
 Gupta, A., & Elby, A. (2011). Beyond Epistemological Deficits: Dynamic Explanations of Engineering Students' Difficulties with Mathematical Sensemaking. International Journal of Science Education, 33(18), pp. 24632488.. (Email ayush@umd.edu for PDF.)
 Gupta, A., Danielak, B. A., & Elby, A. (2010). Understanding Students' Difficulties in Terms of Coupled Epistemological and Affective Dynamics. Proceedings of the 2010 Frontiers in Education Conference (ASEE/IEEE). [PDF]
 Danielak, B. A., Gupta, A., & Elby, A. (2010). The Marginalized Identities of Sensemakers: Reframing Engineering Student Retention. Proceedings of the 2010 Frontiers in Education Conference (ASEE/IEEE). [PDF]
 Gupta, A., & Elby, A. (2010). Beyond Epistemological Deficits: Incorporating flexible epistemological views into finegrained cognitive dynamics. Proceedings of the International Conference of the Learning Sciences 2010. [PDF]
 Danielak, B., Gupta, A., & Elby, A. (2010). Incorporating Affect in Engineering Students’ Epistemological Dynamics. Proceedings of the International Conference of the Learning Sciences 2010. [PDF]
 Redish, E. F. & Gupta, A. (2010). Making Meaning with Math in Physics: A Semantic Analysis, GIREP Conf. Proc., Leicester, UK, August 20, 2009 [PDF]
Invited Talks
 Gupta, A., Stevens, R., Elby, A., and Hammer, D. Video Analysis Workshop: Reconciling Cognitivist and Interaction analysis methodologies, Invited Workshop presented at the AAPT 2012 Summer Meeting, Philadelphia, PA.
 Danielak, B. A. (2012, March 20). Using finegrained code and finegrained interviews to understand how students learn to program. Invited Talk, University of New Mexico  Department of Computer Science Colloquium.
 "Analyzing Interviews Suggests Patterns of Reasoning with Math in Physics," Kuo, E., Hull, M., Elby, A., & Gupta, A., Invited Poster, FFPER, Bar Harbor, ME.
 "Integrating Emotions with Fine Grained Dynamics of Students' Reasoning" Gupta, A., & Elby, A., Invited Poster, Foundations and Frontiers in Physics Education Research Conference, Bar Harbor, ME, June, 2011
 Cognitive Dynamics of Mathematical Sensemaking, Ayush Gupta, Invited Talk at the Center for Engineering Education and Outreach, Tufts University, Boston, MA, March, 2011.
 Cognitive Dynamics of Mathematical Sensemaking, Ayush Gupta, Invited Talk at the Winter Meeting 2011 of the American Association of Physics Teachers, Jacksonville, Fl, January 2011 (Session on Making Meaning with Mathematics; Organizer: Elizabeth Gire)
 Incorporating Emotions in FineGrained Cognitive Dynamics, Ayush Gupta, Invited talk at the The Conference on Transforming Research on Undergraduate STEM Education, Maine, June 2010.
 'The Role of Epistemology in Explaining Students' Troubles with Math in Science and Engineering Courses', Ayush Gupta and Andy Elby, Invited Talk at Mathematics Education Colloquim Series, EDCI, University of Maryland (College Park), May, 2009
 'The Dynamics of Constructing Meaning in Physics', Ayush Gupta, Invited Talk at University of Colorado, Boulder (Physics Education Group), March, 2009.

'Students' Intellectual Resources for Understanding Science', Ayush Gupta, Invited Talk at Northup Grumman (Classroom Readiness for STEM Volunteers Program), Baltimore, MD, March, 2009.
 'Improving Engineering Students' Mathematical Sensemaking', Ayush Gupta and Andy Elby, 2009 NSFEngineering Education Centers PI Meeting (Poster), Feb. 2009 (click to download PDF)
Contributed Talks/Posters
 Kuo, E. (2013). Not Just “PlugandChug”: How Physics Students Make Sense with Equations. Contributed poster at the Foundations and Frontiers in Physics Education Research (FFPER) Conference 2013, Bar Harbor, ME.
 Gupta, A., & Elby, A. Coupling Identity and Epistemology to Understand Differences in Students' Learning Experiences. Contributed Talk presented at the AAPT 2012 Summer Meeting, Philadelphia, PA.
 Kuo, E., Gupta, A., and Elby, A. (2012). When do I use symbolic forms? Contributed talk at AAPT summer 2012, Philadelphia, PA.
 "Linking the Dynamics of Student Reasoning to Epistemology," Kuo, E., Hull, M., & Gupta, A., Contributed Talk, Jean Piaget Society Annual Meeting, Berkeley, CA, June, 2011.
 "Integrating Emotions with Fine Grained Dynamics of Students' Reasoning" Gupta, A., & Elby, A., Contributed Poster, Jean Piaget Society Annual Meeting, Berkeley, CA, June, 2011.
 "Toward Expert Problem Solving: Blending Conceptual and Symbolic Reasoning," Kuo, E., Hull, M., Elby, A., & Gupta, A., Contributed Poster, 2010 Conference on Transforming Research in Undergraduate STEM Education (TRUSE), Orono, ME,
 "Beyond epistemological deficits: Incorporating flexible epistemological views into finegrained cognitive dynamics", Gupta, A., & Elby, A., Contributed Poster, International Conference of the Learning Sciences 2010, Chicago, IL. [PDF]
 "Explaining Student Expertise with Mathematical SenseMaking", E. Kuo, M.M. Hull, A. Gupta, and A. Elby, Contributed Talk, AAPT National Meeting, Washington, DC February 16, 2010. [PDF]
 "I'm doing what my teacher says, why aren't I expertlike?", M.M. Hull, E. Kuo, A. Gupta, and A. Elby, Contributed Talk, AAPT National Meeting, Washington, DC February 16, 2010.
 "The Role of Affect in Stabilizing Judy's Approach Toward Circuits", A. Gupta, B.A. Danielak, and A. Elby, Contributed Talk, AAPT National Meeting, Washington, DC February 16, 2010.[SLIDESHOW IN QUICKTIME]
 "Affect and Identity in Engineering Students' Approaches to Learning, Problem Solving", B.A. Danielak, A. Gupta, and A. Elby, Contributed Talk, AAPT National Meeting, Washington, DC February 16, 2010.
 'Undergraduate Engineers' Sensemaking of Math'', Mike Hull, Eric Kuo, Ayush Gupta, & Andy Elby, Contributed Poster, 2009 Physics Education Research Conference, Ann Arbor, MI, July 2009. [PDF] [PPT]
Current Activities
 Data Analysis
 Interviews with students
 Students who took two or more semesters of physics with us, and then took ENEE204; with the focus on investigating if and how the reformed physics effort contributes to their approaches to learning in other "regular" engineering courses
 Phys161 students on how they make sense of mathematics in their physics course
Past Activities
 Teaching Introductory Physics courses for Engineering majors with special emphasis on mathematical sensemaking (AE, AG, DH, EK, MH)
 Conducting and analyzing clinical interviews (AE, AG, BD, EK, MH)
 with engineering majors in introductory physics courses  focusing on diagnosing student difficulties with mathematical sensemaking in the context of physics problems and possible interventions to prompt sensemaking in those contexts
 with engineering majors in ENEE204 (Basic Circuit Theory)  focusing on how they make sense of mathematics in the context of circuits and exploring their epistemology around sensemaking in engineering contexts
 Survey validation interviews with engineering students in the circuits course.
 Designing and administering surveys (AE & AG)
 to determine students' attitudes and expectations towards sensemaking in physics and in engineering contexts (modified from MPEXII, and EBAPS).
 Surveys have been administered to about 350 students in Phys161 (Spring 2009) and 52 students in ENEE204 (Spring2009)
 Designing curriculum materials (AE, AG, DH, WL)
 for introductory physics courses for engineering majors  with emphasis on mathematical sensemaking
 for ENEE 204 (Basic Circuits Theory), providing opportunities for mathematical sensemaking and conceptual reasoning
 Analyzing mathsense making in exams (AE, DH, AG, BD, EK, MH)
 comparison of traditional versus reformed physics classes on students' mathematical sensemaking in specific questions (final exams for Phys161 and Phys260).
 Collecting Video data (EK & MH)
 (Spring 2010) Videotaping students engaged in discussions and problemsolving in small groups for Phys270
 (Spring 2010) Videotaping discussion sessions for ENEE204 (Basic Circuit Theory)
 (Fall 2009) Videotaping students engaged in discussions and problemsolving in small groups for Phys260
 (Spring 2009) Videotaping students engaged in discussions and problemsolving in small groups for Phys161
 (Spring 2009) Videotaping discussion sessions for ENEE204 (Basic Circuit Theory)
 (Spring 2009) Videotaped lecture sessions of Phys161 (General Physics: Mechanics and Particle Dynamics) to document how engaging students in mathematical sensemaking can be made an integral part of a large classroom lecture
Personnel Contact Information
NAME 
AFFILIATION 
PHONE (30140+) 
OFFICE 
EMAIL 
Andy Elby (PI) 
Physics, Education

55983 

elby@umd.edu 
David Bigio (CoPI) 
Mechanical Engineering 
55258 
Martin 2184 
dbigio@umd.edu 
David Hammer (CoPI) 
Education, Tufts University

** 
**

** 
Wes Lawson (CoPI) 
Electrical and Computer Engineering 
54972 
AVW 2325 
lawson@eng.umd.edu 
Edward F. Redish (CoPI) 
Physics, Education

56120 
Toll 1308 
redish@umd.edu 
Ayush Gupta (PD) 
Physics 
51669 
Toll 1320 
ayush@umd.edu 
Mike Hull (GRA) 
Physics 
56185 
Toll 1322 
mhull12@umd.edu 
Eric Kuo (GRA) 
Physics 
56185 
Toll 1322 
erickuo@umd.edu 
Brian Danielak (GRA) 
Education



briandk@umd.edu 
Mailing Address
Ayush Gupta
Room 1320, Physics Building
University of Maryland
College Park, MD 207424111
Fax: 3013149531
Ph: 3014051669
Work supported in part by a grant from the US National Science Foundation.
Engineering Education Project

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