Motivation & Objectives

Methodology & Execution

The Classroom Activities

1.  Identify the real-world phenomenon

2.  Simplify or idealize the phenomenon 3.  Express the idealized phenomenon mathematically 4.  Perform the mathematical manipulations

5.  Interpret the mathematical solution in real-world terms 6.  Test the interpretation against reality1

Learning Activities & Materials

Mathematical Modeling Framework:

1.  Gainsburg, J., 2006, "The Mathematical Modeling of Structural Engineers." Mathematical Thinking and Learning, 8(1): p. 3-36.

2. Cole, Linsenmeier, Glucksberg, and McKenna, “Investigating Engineering Students’ Mathematical Modeling Abilities in Capstone Design.” Proceedings of the American Society for Engineering Education 2010 Annual Meeting, Louisville, KY, 2010.

3. Cole, Linsenmeier, Molina, Glucksberg, and McKenna, “Assessing Engineering Students’ Abilities at Generating and Using Mathematical Models in Capstone Design,” Proceedings of the American Society for Engineering Education 2011 Annual Meeting, Vancouver, BC, Canada, 2011.

4. Cole, Linsenmeier, Miller, and Glucksberg, “Using instruction to improve mathematical modeling in capstone design,” Proceedings of the American Society for Engineering Education 2012 Annual Meeting, San Antonio, TX, 2012.

Affiliations a.  Department of Chemical and Biological Engineering, Northwestern University b.  Departments of Biomedical Engineering, Neurobiology, and Ophthalmology, Northwestern

University c.  Department of Engineering, College of Technology and Innovation, Arizona State University d.  Department of Biomedical Engineering, Florida International University e.  Department of Bioengineering, State University of New York at Binghamton

ACTIVITY  2  –  BME  Design  Scenario  –  Phototherapy  Device  

To  help  design  their  device,  the  design  team  decided  to  create  a  mathema4cal  model  of  the  transmission  of  light  from  the  device  to  the  infant’s  skin.  The  intended  purpose  of  the  mathema4cal  model  is  to  generate  predic4ons  of  how  different  parameters  will  affect  the  performance  of  the  device.    Recall  that  the  team’s  research  has  shown  them  that  the  design  should  meet  the  AAP  guidelines  for  effec4ve  dosage  of  phototherapy  in  order  to  provide  successful  treatment  for  jaundice.    (See  aCached  problem  statement  for  details.)    

Assignment:  

As  advisors  to  the  design  team,  your  advice  is  again  needed.  Your  job  today  is  to  help  the  team  begin  to  set  up  the  model  by:  (a)      sketching  the  system  you  plan  to  model  (b)  iden4fy  and  list  the  relevant  parameters  and  variables  with  a  brief  explana4on  of  why  each  

one  is  important  and  if  they  are  related  in  any  way  to  the  other  listed  parameters  or  variables  

Please  also  provide:  (c)  Any  assump4ons  you  would  have  to  make  about  the  model  as  a  descrip4on  of  the  physical  

situa4on  (d)      A  descrip4on  of  the  geometries  you  are  considering  for  the  device  (e)      A  proposal  for  a  possible  mathema4cal  approach  to  the  problem  

Research partially funded by the National Science Foundation Engineering Education Program (EEP) Grant #0648316 and Grant #0851930.

Acknowledgements

References

• Capstone design instructors frustrated that students weren’t applying mathematical analysis from prior coursework to design solutions.

• Mathematical models can be used to predict physical behavior based on patterns in real situations (or hypotheses of physical behavior when too little information is available).

• Modeling allows students to be flexible in applying disciplinary knowledge in the process of design.

• Our research explored how instruction in modeling impacts students’ ability to create mathematical models.

• Our goal is to use instructional tools to improve students’ abilities to: • Translate real-world phenomena to mathematical representation • Manipulate mathematical models • Understand model output

• The design scenario concerned modeling the interaction of light with a baby’s skin. The students’ actual design projects required completely different models.

• Examples of BME design projects: • Urinary incontinence – for detection of absorbent failure • Placenta simulator – for training midwives • CPAP – for premature infants in conjunction with KMC • Ultraviolet Germicidal Irradiation – for use in developing countries

• Rubrics consisted of question sets and decision trees that matched up to the modeling steps

• Rubrics had to be flexible enough to apply to different projects. They contained many branch points allowing for different possible outcomes.

At left: an example of a section of the rubric for a portion of framework step 2 – identifying parameters

Improvement in Identifying Model Elements

Improvement in Relationships between Model Elements

Fewer students in BME09 were able to develop relationships between model parameters, leading to poor performance on subsequent modeling steps.

Implementation Course BME09 BME10 BME11

Biomedical (BME) capstone course (Fall quarter)

12 teams (38 students)

17 teams (77 students)

13 teams (52 students)

• Classroom activities using the design scenario (‘09, ’10, ‘11) • Scenario was a realistic design problem that all students could

solve • One 45-minute activity per week (4 weeks total)

• Lectures related to the design scenario (‘10 and ‘11) • 15-25 minute lectures/discussions in the class period following

the classroom activity2,3

• Analysis of final design projects (’09, ‘10, ‘11) • Final projects analyzed with rubrics for extent of completion of

mathematical modeling steps4

• Design Scenario: Phototherapy device for treating jaundice in premature infants, compatible with Kangaroo Mother Care (KMC)

Mock up of design from Phototherapy Team WQ08

• Goal of Activities: Develop a model for an array of LEDs in terms of height from the baby’s skin and distance between the LEDs

Make simplifications to begin with equations for a single, point-source LED

Develop model for a linear array of point-source LEDs Use model outputs to

inform design decisions

The Worksheets for the Classroom Activities

The Lectures/Discussions following Classroom Activities

Example of student work from BME FQ 2010. Activity 2 covers steps 1 and 2 of framework.

•  Lecture 1 (follows activity 1) •  Importance of mathematical modeling • Differences between mathematical

and physical modeling •  Lecture 2 (follows activity 2)

• Correct misconceptions about light distribution

• How to identifying relevant parameters and variables and make assumptions

•  Lecture 3 (follows activity 3) • How to generate model equations and

assumptions • Single LED model versus an array of

LEDs

Instructor: We want to get to the absolute simplest problem first. We have an LED, a point source. (sketch)

W is the power output of the LED. We are at a distance r (radial, in spherical coordinates). At this distance r, there is some amount of light. But the baby is not a constant distance from the LED. How do we sketch a baby into this system?

Student15: a plane

Instructor: Why did you say a plane?

Student15: it is one dimension

Instructor: So is a cylinder.

Student15: (no response)

Instructor: I’ll tell you why you thought a plane would be best. If we approximate our baby as a cylinder, what would the radius of our premature baby be?

Student16: 5cm

Instructor: How about our blanket thickness?

Student16: ½ cm

Instructor:

You can go back and check your assumption. How much difference does it make to have curvature versus a flat plane?

(adds flat plane to sketch)

We have an LED some distance from this plane. What is the relationship between light intensity and distance from an LED?

At right: An excerpt from the transcript of Lecture 2. This discussion covers a portion of step 2 of the framework – making assumptions.

The Rubrics

Instruction in mathematical modeling leads to an improvement in identifying the model parameters and the assumptions related to the model parameters. (2009 no instruction. 2010-11 included instruction.)

Using Model Outputs to Make Design Decisions • No discernable improvement in the later steps of model creation. This may be partially

due to the difficulty of completing all the elements of the design in one quarter. • Of the teams that were able to complete a mathematical model, there was a high

success rate in utilizing the model output to inform design.

Results: Assessing Impact on Capstone Projects

2012  Fron4ers  of  Engineering  Educa4on  Symposium    

Irvine,  California    October  14  -­‐  17    

0   10   20   30   40   50   60   70   80  

Parameters  Fully  Stated  

Parameters  Par4ally  Stated  

Parameters  Not  Stated  

Do  students  state  the  parameters?  

BME11  

BME10  

BME09  

0   10   20   30   40   50   60   70   80   90  

Assump4ons  Fully  Stated  

Assump4ons  Par4ally  Stated  

Assump4ons  Not  Stated  

Do  students  state  the  assumpBons?  

BME11  

BME10  

BME09  

Percent  of  teams  Percent  of  teams  

0   10   20   30   40   50   60   70   80  

Equa4on  Rela4onship-­‐    Developed  

Equa4on  Rela4onship-­‐    Undeveloped  

Non-­‐Equa4on  Rela4onship  

No  Rela4onship  Provided  

Do  students  provide  a  relaBonship  between  parameters  ?  

BME11  

BME10  

BME09  

Percent  of  teams