This document discusses problem-based learning (PBL) as an approach to teaching mathematics. It highlights how PBL shifts the traditional classroom dynamic to one where students have more ownership over their learning through problems that allow for creativity, argument, and multiple perspectives. The document also outlines some key aspects of PBL, including the use of discourse, scaffolding problems, and assessing problem-solving skills rather than just answers. Overall, the document advocates for PBL as a way to make mathematics more meaningful, thought-provoking, and invested for students.

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Problem-Based Learning
Empowering & Engaging Students in
Mathematics
Carmel Schettino, ISOMA, 2013

2. ©Carmel Schettino.org
2011 Grant Wiggins Survey –
“The Sstdens Voice”
Mathematics was both the subject that
Math as a
attained the highest percentage for both the
subject favorite subject and LEAST favorite subject.
Reasons
Not good at it, too much memorization
for not Boring, not relevant, no use for it
liking math
Rarely related Only 1 in every 100 responses note the
to the teacher teacher is the key to favorite subject
From a sample of 7300 student in the US , http://grantwiggins.wordpress.com/2011/11/17/the-student-voice-our-survey-part-1/

3. ©Carmel Schettino.org
What was the most interesting work?
Generalization from student
responses
Making the
abstract
Meaningful, concrete or an
Active,
thought- example of a
Hands-on situation
provoking
preparing for
another
http://grantwiggins.wordpress.com/2011/11/30/the-student-voice-part-3-the-most-interesting-work-of-the-past-year/

4. ©Carmel Schettino.org
Traditional Mathematics Classroom
Single
Owner of
Mathematical
Knowledge
Authority
Teacher
Director of Passive Uninvested
learning Learners Learning
Source of
Questions & Dependent
Answers Thinkers

5. ©Carmel Schettino.org
What would mean more invested learners?
Ownership of Material/Content
Your Text Here
Student Voice & Shared Authority
Co-Construction of Learning through discussion and inquiry
Multiple Perspectives on Solutions
Presentation Investment in
of Student Connections learning
Student between Relational
ideas and
generated ideas and Learning
partial
questions topics
solutions

6. ©Carmel Schettino.org
“Learning in my class was a matter of becoming convinced that
your strategy and your answer are mathematically legitimate. In
mathematics, knowing that something is true requires reasoning
from agreed-upon assumptions to their logical conclusions in the
context of some given conditions, practical or theoretical. Studying
mathematics this way involves my students in finding out what kind
of activity mathematics is; it provides them an opportunity to learn
and use the concepts, tools, and procedures that the field has
developed.”
-Magdalene Lampert, Teaching Problems and
the Problems of Teaching
What does it mean to teach with
problems?

7. ©Carmel Schettino.org
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 Use appropriate tools strategically
5 Look for and express regularity in repeated reasoning
6 Look for and make use of structure
Common Core Standards for
mathematical practice (U.S.)

8. ©Carmel Schettino.org
What is Problem-Based Learning?
An approach to curriculum and pedagogy where student
learning and content material are co-constructed by
students and teachers through problems that are
• both contextually-based and abstract
• discussed and presented by students in a non-
hierarchical environment
• deliberately scaffolded based on rstdensr’ prior
knowledge
Schettino, 2010

9. Whas do I mean bx ‘Scaffolding’
Summarizing Students Summarize larger
concept
Constructing Creation of
statements of new
knowledge
Analyzing/Validating
Problems allow for creativity, argument
and multiple interpretations in order for
discussion to happen.
Conjecturing Problems Posed with use of prior knowledge
needed
Triggering Students are reminded of prior knowledge
Determine what students should know and understand
Based on Tomlinson & Edison (2003)

10. What is the difference between
mathematical engagements?
Emergent
Preauthentication
Authenticity
Authentic practice Pre-analysis of
within mathematics activities attempting to
by engaging in activity simulate authentic
germane to field problem in learning
environment
Jonassen, D(2011) Supporting Problem Solving in PBL. IJPBL, 5(2), 95-119

11. Problem-Based vs. Project-Based
Problem-Based Project-Based
• Deliberately scaffolded
whole-problem curricula • Larger picture projects with
direct instruction units
• Preauthentication of
problems for mathematical • Emergent authenticity in
experience mathematical experience
• Student Agency stems from • Student Agency stems from
student presentations of directing project with
problem solutions from knowledge from instruction
prior knowledge and and collaboration
collaboration

12. Why is this different?
1 Discourse Driven by Students
2 Spiraled & Paralleled Topics
3 Built on Prior Knowledge
4 Multiple Representations
5 Ownership of Knowledge

13. ©Carmel Schettino.org
Aspects of Problem-Based Teaching
Discourse Moves Curriculum
Assessment of
Pedagogy
problem solving
Skills Metacognitive Student
Assessment Journaling Questioning
Facilitation of
Student Listening
Discussion
Safe
Technology Conjecturing Environment for
Risk-Taking

14. ©Carmel Schettino.org
Problem Purposes
Trigger prior
Inspire
Review material knowledge for an
construction of
from past courses upcoming
new knowledge
problem
Introduce new Challenge more
Practice new skill
terminology able students
See same new
idea from Introduce new Concretize an
different technology skill abstract concept
representation

15. ©Carmel Schettino.org
Triggering Prior Knowledge
A town’s building code does not permit building a house
that is more than 35 feet tall. An architect working on the
design shown would like the roof to be sloped so that it
rises 10 inches for every foot of horizontal length.
Given the other dimensions in the diagram, describe a way
that the builder could construct the house according to
town code.
22
30

16. ©Carmel Schettino.org
Inspire Construction of knowledge
In 1986, renovations were made to the stairs in the Statue of Liberty
in an overall celebration of its bicentennial. Frank Schettino was the
draftsman in charge of the calculations for the details of the
structural steel for those stairs. He found that the Lady’s pedestal
need to have a vertical rise of 7.8 inches and a horizontal width of
9.75 inches. The more complicated spiral stairs that led up the body
had a constant vertical rise of 9 inches and an average horizontal
width of 5.06 inches.
a. Why do we consider an “average” width for the spiral stairs?
b. Using graph paper or dynamic geometry software, create a
graphical representation of these two sets of stairs.
c. Which flight of stairs do you think is steeper? Why?
d. Calculate a height vs. width ratio for each set of stairs and verify
that the greater ratio belongs to the flight that you thought was
steeper. What does this ratio represent to you?

17. ©Carmel Schettino.org
Introduce new terminology
A two-dimensional side-view of a set of stairs goes from
point A(2,2) to B(6,4) to C(10,6) as shown in the diagram
below. One way to talk about rise/run ratio of the line AC
made by the points on the stairs is to call it the slope of the
line. What’s a way that you would calculate the slope of
AC? What do you think the slope of the line AC is in the
diagram?

18. Teaching for Higher Order Learning
©Carmel Schettino.org
Stop giving
Resist students the
Cognitive
temptation to final product
Apprenticeship
tell of our
thinking
Stop teaching Problems
Reevaluate
decontextualized first, teaching
Evaluation
content second
.
McCain, T. (2005) Teaching for Tomorrow, Corwin Press

19. Cognitive Apprenticeship
Redefining the role of the Teacher – Teacher
“Reporisioning”
Coaching Modeling Scaffolding Fading
Encouraging, Risk-taking, Revoicing,
Adding insight, Mapping ideas, Relinquishing/sharing
Questioning,
Asking probing Probing for authority, valuing
Making
questions, Holding explanation, multiple perspectives,
mistakes,
back, Supporting Requiring allowing students to
Finding
with information conjecturing, be expert
resources,
Conjecturing opportunities for
more student
interaction

20. ©Carmel Schettino.org
A continuum of PBL
Occarional “Problem of she Dax”
decompartmentalization
Math Competitions
Tradisional Tews book “btffes”
Traditional Textbook supplemented
vish “Mosiuasional Problemr”
PBL Units with Direct Instruction Classes
Whole Problem-Based Curriculum

21. ©Carmel Schettino.org
Where can you get good problems?
PEA course http://www.exeter.edu/academics/72_65
materials 39.aspx
The Opus
http://www.opusmath.com/
project
AERA SIG
http://tinyurl.com/aerasigpbl
for PBL
Problems
http://www2.edc.org/mathproblems/sea
with a rchTopic.asp
Point

22. ©Carmel Schettino.org
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23. Pros and Cons

24. ©Carmel Schettino.org
Open communication is important
The
teacher’s Just tell
What’s the not
answer?
teaching! me what
Did I get to do
it right?
Frustrated students
need a place to
express their feelings

25. ©Carmel Schettino.org
Are you inspired?
• Check out the PBL Resources List on my
website www.carmelschettino.org
• Visit schools where PBL is practiced
P B L

26. ©Carmel Schettino.org
Do you have
any
questions?
? ?

27. Reference List
©Carmel Schettino.org
• Grans Wigginr’ {log as
http://grantwiggins.wordpress.com/2011/11/17/the-
student-voice-our-survey-part-1
• Lampert, M (2001). Teaching Problems and the
Problems of Teaching. Yale University Press, New
Haven
• Jonassen, D(2011) Supporting Problem Solving in PBL.
Interdisciplinary Journal of Problem-Based Learning,
5(2), 95-119.
• McCain, T (2005) Teaching for Tomorrow: Teaching
.
Content and Problem-Solving Skills. Corwin Press,
Thousand Oaks, CA (44).
• Tomlinson. C & Cunningham, C. (2003). Differentiation
in Practice: A resource guide for differentiating
curriculum, Grades K-5. ASCD, Alexandria, VA.
• My website at www.carmelschettino.org
• My delicious account at
http://delicious.com/SchettinoPBL