Using Mathematical Modeling
to Engage All Learners
Doug Burge, Holmen School District
burdou@holmen.k12.wi.us
Dave Ebert, Oregon School District
dde@oregonsd.org
Why Modeling?
• Common Core State Standards for
Mathematics – Standard for Mathematical
Practice and Standard for Mathematical
Content
• Next Generation Science Standards
“These standards are not
intended to be new names for old
ways of doing business. They are
a call to take the next step.”
-CCSSM p.5
Why Modeling?
• Modeling helps build deeper
understanding and application of content.
• Modeling gives mathematics relevance –
answers the question “why do we need to
learn this?”
• Modeling engages all learners – involving
all students can be a Tier 1 Intervention.
• Modeling sustains and enhances student
interest.
Mathematics is, after all, a human
activity…. The pupil himself should
reinvent mathematics. During this
process, the learner is engaged in
an activity where experience is
described, organized, and
interpreted by mathematical means.
- Hans Freudenthal
Steve Rasmussen at NCSM IGNITE
The Wave
What do students gain from this?
• Students have a meaning of slope as a
rate of change.
• Students understand the meaning of
y = mx + b
It isn’t just an arbitrary formula.
• Students are engaged in their learning.
Pisa Example
At a rock concert, a rectangular field 100m
x 50m was reserved for fans to stand. The
concert was sold out. Approximately how
many fans were in attendance?
a. 2000
b. 5000
c. 20,000
d. 50,000
e. 100,000
There is a gap between research
and practice
• Research: Modeling is valuable
• Practice: Teaching modeling is difficult
- assessment
- planning
- classroom control
Which is less expensive, a
t-shirt at the local store for $15,
or a t-shirt at the mall for $12?
Things necessary for effective
modeling
1. Effective classroom management
- task-oriented work, not just
social group work
2. Learners are actively engaged
- balance between independence and
teacher guidance
3. Learners are engaged meta-cognitively
- students reflect on their work
Things necessary for effective
modeling
4. A broad variety of examples
- variation of real-world and
mathematical contexts
5. Multiple solutions are encouraged
- teacher’s solution is not the only, or
even the best, answer
6. Repeat and practice
- learning modeling is a long-term goal
Things necessary for effective
modeling
7. Assessment needs to reflect modeling
8. Positive beliefs and attitudes must be
developed
- students’ belief that math problems
take more than 3 minutes to solve
9. Technology can be used as a tool
10. Modeling can be learned if there is
quality teaching
If a lighthouse is 30 m tall, how
far away can a ship see the
light?
Steps to teach modeling
1. Understand the task – what does it
mean?
2. Search the mathematics – what do you
know and what do you need to know?
3. Use the appropriate mathematics.
4. Explain the result – does the answer
make sense?
If Beethoven’s 6th Symphony
takes an orchestra 40 minutes to
play, how long will it take to play
th
Beethoven’s 9 Symphony?
High Performing Countries
In high-performing countries, teachers
place greater cognitive demands on
students by encouraging them to focus on
concepts and connections among those
concepts in their problem solving.
In Our Classrooms
Teachers make mathematical tasks more
explicit by breaking them down into
smaller steps, specifying exact procedures
to be followed, or even doing parts of the
tasks.
This robs students of the opportunity to
develop meaningful mathematical
understandings.
Productive Struggle is
Necessary
Struggle means that students expend
effort to make sense of mathematics.
Struggle does not mean being presented
information to be memorized or being
asked only to practice what has been
demonstrated.
What Not to Do
Don’t use word problems in which the
premise is false.
Don’t ask questions that only a math
teacher would ask.
(Glencoe McGraw-Hill)
A better question…
How close must the cheetah get to
the gazelle in order to catch it?
What information do we need to
know to solve this problem?
How are we going to find this
information?
(Glencoe McGraw-Hill)
(Glencoe McGraw-Hill)
U is for Undertow
by Sue Grafton
“In both junior high and high
school, I had trouble staying
focused in classes where I was
doing poorly, math being my
weakest subject.”
“A train leaves Chicago for Boston traveling
sixty miles an hour, while a second train leaves
Boston, speeding toward Chicago at eighty
miles an hour. A bird flies back and forth
between the two… and that’s as far as I’d get.
I’d start wondering why the bird was behaving
so erratically, positing a virus affecting the bird’s
internal gyroscope. I’d daydream about who
was on the train and why they were going from
Chicago to Boston. Then I’d fret about what
was happening in Boston that residents had
crowded into the fastest train out. I’d never
been to Boston, and now I was forced to
scratch it off my list.”
Teachers matter most!