Inquiry-Based Learning: Pedagogy, Research and Practice · Inquiry-Based Learning: Pedagogy, Research and Practice Philip K. Hotchkiss Julian F. Fleron Westfield State University

www.artofmathematics.org

Inquiry-Based Learning:

Pedagogy, Research and

Practice

Philip K. Hotchkiss

Julian F. Fleron

Westfield State University

PBL Math Teaching Summit

July 17, 2015


There are several groups at college/university level promoting IBL:

• The Academy of Inquiry-Based Learning (AIBL)

• http://inquirybasedlearning.site-ym.com/

• The Educational Advancement Foundation (EAF)

• http://www.eduadvance.org/

• The Legacy of R. L. Moore

• http://legacyrlmoore.org/

• 4 centers

• University of Michigan

• University of Chicago

• University of Texas at Austin

• University of California, Santa Barbara

• Primary focus on mathematics majors


• IBL Calculus

• http://www.iblcalculus.com/

• POGIL

• http://pogil.org/

• IBL for Pre-service Teachers

• UC Santa Barbara

• Mathematics for Liberal Arts

• Discovering the Art of Mathematics (DAoM)


Research supporting IBL:

• In a study on the effectiveness of IBL at the college level that was conducted by S. Laursen,

M. L. Hassi, M. Kogan, A. B. Hunter and T. Weston, it was found that

1. Students in the study reported deeper learning due to having to figure the mathematics

out for themselves; in fact, these learning gains were statistically significantly correlated

to the fraction of time spent on student centered activities and anti-correlated to the

faction of time listening to instructors talk.

2. Students in IBL courses self-report higher gains in persistence than students in non-IBL

courses.

3. Students also reported gains in independence, enjoyment, and confidence as well as

gains in thinking and problem solving skills that they believed would benefit them in

other areas.

4. IBL courses provided better support for the lower achieving students while not harming

the higher achieving students.

http://www.colorado.edu/eer/research/documents/IBLmathReportALL_050211.pdf


Research supporting IBL continued:

• Among the conclusions of a meta-analysis of 225 studies involving active learning in science,

technology, engineering and mathematics (STEM) disciplines by Scott Freeman, Sarah L.

Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary

Pat Wenderoth were the following:

1. Active learning increases student performance across the STEM disciplines.

2. Average examination scores improved by about 6% in active learning sections.

3. Students in active learning sections were 1.5 times less likely to fail then students in

traditional lecturing sections.

http://www.pnas.org/content/early/2014/05/08/1319030111


There are four of us on the Discovering the Art of Mathematics project. Every semester

each one of us teaches 1 or 2 sections of 30-40 students of a general education course

called Mathematical Explorations.

Generally students in this course are non-mathematics and science majors; students in

this class are primarily drawn form the following majors:

• English

• History

• Music

• Psychology

• Criminal Justice

• Social Work

• Art

• Mass Communications

This type of course is what is called a Mathematics for Liberal Arts (MLA) course.


According to a 2010 survey by the American Mathematical Society

(AMS), the MLA student cohort was the 5th largest cohort of

mathematics students with 232,000 each semester.

In 2005, 79% of MLA courses at community colleges were taught using

the standard lecture mode. By 2010, this percentage had increased to

85%.

We are going in the wrong direction.

We want to empower these disenfranchised students by using IBL.


Discovering the Art of Mathematics (DAoM), is an innovative

approach to teaching mathematics to liberal arts and humanities students,

that offers the following vision:

MLA students will be actively involved in authentic mathematical

experiences that

• are both challenging and intellectually stimulating,

• provide meaningful cognitive and metacognitive gains, and,

• nurture healthy and informed perceptions of mathematics,

mathematical ways of thinking, and the ongoing impact of

mathematics not only on STEM fields but also on the liberal arts and

humanities.


Imagine that you are a student and this is the first day of your required

mathematics class.

1. What might this feel like to you?

2. What type of energy do you have?

3. Do you have discussions with your students about their mathematical

experiences and their feelings about mathematics? How?


Let me just start with the fact that I’ve attempted college at least three

times. I swear every failed attempt is due to Mathematics.

As it stands, I'm terrified of this class.

For me personally doing math is immediately associated with struggle

and hardship. It feels like no matter how I try I am unable to firmly grasp

math as a whole or as its individual formulas. Math has always felt like a

weight I am unable to lift.

Quotes from Student Biographies





Mathematics is the art of explanation. If you deny students the

opportunity to engage in this activity - to pose their own problems, to

make their own conjectures and discoveries, to be wrong, to be creatively

frustrated, to have an inspiration, and to cobble together their own

explanations and proofs - you deny them of mathematics itself. So no,

I’m not complaining about the presence of facts and formulas in our

mathematics classes, I’m complaining about the lack of mathematics in

our mathematics classes.

-Paul Lockhart A Mathematician’s Lament


Mathematical Question: For integers a, b > 0, what are all possible

outputs of the equation 3a+5b?


1. What did you observe?

2. Were there things that you think were empowering to the students?

3. Where was the teacher?


1. What did you observe?

2. How does the mathematical argument in the student’s presentation

compare to the group’s original discussion?

3. What role do the student comments play?



Where/How has IBL played a central role in students’ investigation of

this problem?


Guiding Principle 1: Learning

Mathematical ideas should be explored in ways that stimulate curiosity,

create enjoyment of mathematics, and develop depth of understanding.

Massachusetts Curriculum Frameworks

Guiding Principles for Mathematics Programs


1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Massachusetts Curriculum Frameworks

Standards for Mathematics Practice


The Big Question:

How do you create environments where this type of inquiry happens with

students whose mathematical skills, interest levels and

attitudes/perceptions of mathematics are so challenging?


http://www.artofmathematics.org/classroom


http://www.artofmathematics.org/blog


















College students study the best paintings, the most glorious music, the

most influential philosophy, and the greatest literature of all time.

Mathematics departments can compete on that elevated playing field by

offering and making accessible to all students intriguing and powerful

mathematical ideas... Indeed, these courses [general education and

introductory mathematics courses] should be developed and offered with

the philosophy that the mathematical component of every student's

education will contain some of the most profound and useful ideas that

the student learns in college.

-Committee on the Undergraduate Program in Mathematics, 2004


Does your mathematics class contain “the most profound and useful

ideas that students learn”?

• If so, share one.

• If not, name one that could/should be included.


Discovering the Art of Mathematics

http://artofmathematics.org

Julian Fleron

[email protected]

Philip Hotchkiss

[email protected]

Volker Ecke

[email protected]

Christine von Renesse

[email protected]

mailto:[email protected]




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Inquiry-Based Learning: Pedagogy, Research and Practice · Inquiry-Based Learning: Pedagogy, Research and Practice Philip K. Hotchkiss Julian F. Fleron Westfield State University