Solving for x’s and why’s:
Bringing Cognitive Science
into the Math Classroom
South Dakota Laptop Institute 2007
Kevin Judd
Western Regional Vice President
Carnegie Learning, Inc.®
Carnegie Learning
|
www.carnegielearning.com
Learning by Doing
“We learn by doing – that is the thing. For
though you think you know it, you have no
certainty until you try.”
- Sophocles, 270 B.C.
Carnegie Learning
|
www.carnegielearning.com
Sources




How People Learn and How
Students Learn from the National
Research Council – John Bransford
Larry Lowery from Lawrence Lab at
Berkeley and FOSS
Al Corbett, John Anderson and Steve
Ritter at Carnegie Mellon University
The Teaching Gap by James Stigler
and James Weisert.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners use what they already
know to construct new
understandings.
How People Learn: Brain, Mind, Experience and School
Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National
Academy Press.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research
An example of prior knowledge:
HOUSE
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learning with understanding is
enhanced when new knowledge is
built on the major concepts of a
discipline.
How People Learn: Brain, Mind, Experience and School
Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National
Academy Press.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners take control of their learning
with metacognitive strategies.
•
Write a sentence describing how
you found these answers.
•
You will be expected to share your
answers with the class.
•
Explain to your partner how you got
your answer
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners must be actively engaged.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners’ motivation and sense of self affect
what and how much they will learn.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners utilize different approaches,
strategies and styles based upon
individual learning opportunities
and prior experiences.
How People Learn: Brain, Mind, Experience and School
Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National
Academy Press.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learners cement learning through the
use of distributive practice (vs.
mass practice).
How People Learn: Brain, Mind, Experience and School
Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National
Academy Press.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research
Social interactions enhance learning.
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research

Learning is situated in activity and is
shaped by the context and culture in
which it occurs.
How People Learn: Brain, Mind, Experience and School
Bransford, J.D., Brown, A.L.& Cooking, R.R. (Eds) 1999, National
Academy Press.
Carnegie Learning
|
www.carnegielearning.com
Context is important
Carnegie Learning
|
www.carnegielearning.com
Cognitive Analysis
Start-unknown Problems
Result-unknown Problems
1. Story
4.Story
When Ted got home from his waiter
job, he took the $81.90 he earned that
day and subtracted the $66 he
received in tips. Then he divided the
remaining money by the 6 hours he
worked and found his hourly wage.
How much per hour does Ted make?
When Ted got home from his waiter
job, he multiplied his hourly wage by
the 6 hours he worked that day. Then
he added the $66 he made in tips and
found he earned $81.90. How much
per hour does Ted make?
2. Word-equation
Starting with some number, if I
multiply it by 6 and then add 66, I get
81.9. What number did I start with?
5. Word-equation
Starting with 81.9, if I subtract 66 and
then divide by 6, I get a number.
What is it?
6. Equation
3. Equation
6x + 66 = 81.90
(81.90 - 66) / 6 = x
Carnegie Learning
|
www.carnegielearning.com
Results
90
80
Story problem
Number Correct
70
Word-equation
60
Equation
50
40
30
20
Result-unknown
Carnegie Learning
Start-unknown
|
Koedinger, K.R., &
Tabachneck, H.J.M. (1995).
Verbal reasoning as a
critical component in early
algebra. Paper presented at
the annual meeting of the
American Educational
Research Association, San
Francisco, CA.
www.carnegielearning.com
“How People Learn” research
1.
Multisensory – 1st hand experience
2.
Pictorial – Video, photo, diagram, etc…
3.
Symbolic – Reading, solving equations, etc…
Carnegie Learning
|
www.carnegielearning.com
“How People Learn” research
Skills or Subjects???
 Skills:

Examples - piano, multiplication tables,
reading
 Generally learned by practice.

Subjects:
Examples – social studies, art, mathematics
 Generally learned by rehearsals.

Carnegie Learning
|
www.carnegielearning.com
Expert – Novice studies

Experts recognize relationships better.
 Experts generalize better.
 Experts notice details better.
 Experts have a tremendous amount of prior
knowledge in the area of their expertise.
 Experts are not more intelligent.
 All people can become experts.
 Experts often can not explain their expertise
to novices (Expert Blind Spot).
Carnegie Learning
|
www.carnegielearning.com
What’s important to learn?
Carnegie Learning
|
www.carnegielearning.com
Teaching Math– the current state

Typical class

Warm-up, check homework
 Demonstrate a procedure with definitions
 Students practice the procedure
 Homework


80% of the mathematics concepts are
only stated not developed
95% of seatwork engaged in practicing
routine procedures
Carnegie Learning
|
www.carnegielearning.com
Ramifications
– skill level changes
Skilled
Unskilled
20%
15%
Unskilled
60%
Professional
20%
1950
Professional
20%
Skilled
65%
2000
National Summit on 21st Century Skills for 21st Century Jobs
Carnegie Learning
|
www.carnegielearning.com
Ramifications in student performance

Typical urban school system





15,346
9th graders,
9,788
10th graders,
8,031
11th graders, and
6,461
12th graders
67% of African-American students and 77% of
Hispanic students failed the mathematics
state assessment
Carnegie Learning
|
www.carnegielearning.com
Another Urban System
Grade
9
10
11
12
Number of Students
68,599
52,317
38,908
28,132
Carnegie Learning
|
% of 9th
Graders
100%
76%
57%
41%
www.carnegielearning.com
Potential Earnings by Educational
Attainment
Education
Attained
Unemployment Rate
Median Earnings in
2001
Less than HS
8.5%
$22,350
HS grad
5.3%
$29,187
Some college
4.8%
$34,340
Associate
4.0%
$36,399
Bachelors
3.1%
$46,969
Masters
2.8%
$56,589
Doctorate
1.6%
$75,182
Professional
1.6%
$82,421
Bureau of Labor Statistics
Carnegie Learning
|
www.carnegielearning.com
The Solution?
Carnegie Learning
|
www.carnegielearning.com
The Real Solution
Aligning teaching to learning
Vs.
Making learning align with teaching
Carnegie Learning
|
www.carnegielearning.com
The Real Solution
Solving problems to learn mathematics
Vs.
Learning procedures to solve problems
Carnegie Learning
|
www.carnegielearning.com
Carnegie’s Curriculum
Real World Problem Solving
Procedures
Problem Solving
Carnegie Learning
|

Students LEARN math by
DOING math

Students KNOW more than
they SHOW

Students should never ask
THE question

Reverse the traditional
approach
www.carnegielearning.com
Classroom Curriculum
Real World Problem Solving
Classroom – 3 days per week (60%)

Collaborative learning
 Conversations about mathematics
 Writing about mathematics
 Student presentations
Carnegie Learning
|
www.carnegielearning.com
Classroom Curriculum
Your job at U.S. Shirts is to calculate costs for various tshirt orders. For each order, U. S. Shirts charges $8 per
shirt plus a one time set up charge of $15.
Task 1: Answer the following questions using
complete sentences.
1.
How much would you charge for an order
of 3 shirts? 10 shirts? 100 shirts? Write a
sentence describing how you found these
answers.
2.
How many shirts could a customer
purchase with $50? $100? $1000? $52.50?
Write a sentence describing how you
found these answers.
Carnegie Learning
|
Task 2: For this
problem we will need
to clearly identify the
problem situation,
table of
of
produce a table
values,
values construct a
graph, and find an
algebraic rule. Your
teacher will be giving
your group overlays
for you to put your
answers on to share
share
with the class.
www.carnegielearning.com
Bridge to Algebra
Carnegie Learning
|
www.carnegielearning.com
Computer Curriculum
Real World Problem Solving
Computer Lab – 2 days per week (40%)

Students discovering mathematics
 Coaching by intelligent Tutor
 One-on-one student-Tutor relationship
 Teacher spends time with students who need
it most
Carnegie Learning
|
www.carnegielearning.com
Computer Curriculum
Carnegie Learning
|
www.carnegielearning.com
Five “Dos” to help students learn





Do make students “do the math”
Do let students struggle
Do use word walls
Do create student work walls
Do provide students with ample classroom
time to do presentations of their solutions
Carnegie Learning
|
www.carnegielearning.com
Five “Don’ts” to help students learn





Don’t give students answers – always answer a
question with a question
Don’t make assumptions about levels of difficulty or
levels of ease
Don’t help students “set up” problem solving tasks
Don’t keep students from working together - allow
students to assist each other
Don’t let students give short answers - require
students to write all answers in complete sentences
Carnegie Learning
|
www.carnegielearning.com
Thank you!
Kevin Judd
kjudd@carnegielearning.com
1-888-851-7094 x477
Carnegie Learning
|
www.carnegielearning.com