Problem Solving and
Technology Implementation
in an Inclusion Classroom
Annie Fetter, The Math Forum @ Drexel
The INCLUDE Grant
The grant is designed to ensure that all students in the general
education classroom, including those with mild to moderate
disabilities, struggling students, and English language learners,
are provided the necessary accommodations in the general
mathematics classroom that will support their achievement of
the Core Curriculum Content Standards. The basis for the
INCLUDE grant is to improve academic achievement in
mathematics by using educational technology effectively.
Woodlynne Public School
Woodlynne, NJ (Camden County)
• Grades Pre – K through 8
• 497 students in district
• Ethnicity: Latino – 49% Black – 35%
Asian - 9% White - 7%
• Special Education – 21%
• Free Lunch – 57%
Reduced Lunch – 13%
• Median Income = $41,000
• Teacher : Student Ratio is 1 to 17
What is
The Math Forum?
... the leading online resource for improving math learning, teaching, and
communication since 1992.
We are teachers, mathematicians, researchers, students, and parents using
the power of the Web to learn math and improve math education.
We offer a wealth of problems and puzzles; online mentoring; research;
team problem solving; collaborations; and professional development.
Students have fun and learn a lot. Educators share ideas and acquire new
skills.
mathforum.org
Today we may…
• Explore activities focused on problem solving and
communication
• See samples of students’ work
• Learn how we used technology to support and
enable problem solving
• Get ideas for designing activities with multiple
levels of support
• Find out how problem solving helped all our
students make connections between different
areas of mathematics
Understanding
the
Problem
Greta’s Garden
(Scenario Only)
• Greta has a vegetable garden. She sells her extra
produce at the local Farmer’s Market. One Saturday she
sold $200 worth of vegetables — peppers, squash,
tomatoes and corn.
• Greta received the same amount of money for the
peppers as she did for the squash.
• The tomatoes brought in twice as much as the peppers
and squash together.
• The money she made from corn was $8 more than she
made from the other three kinds of vegetables
combined.
Greta’s Garden
What do you notice?
Greta’s Garden
What do you wonder?
The Procedure
What do you notice?
• Independent Thinking (1 min)
• Collaborative Sharing in Heterogeneous Groups (1 min)
• Made Class List of Noticings - input from all groups
What do you wonder?
• Independent Thinking (1 min)
• Collaborative Sharing in Heterogeneous Groups (1 min)
• Made Class List of Wonderings - input from all groups
What do you Notice? What do you Wonder? What do you Wonder?
Greta sold $200 worth of
vegetables
4 vegetables: tomatoes, peppers,
squash & corn
Peppers & squash cost the same
amount of money
Why are we doing this?
Why did she grow vegetables instead
of fruit?
- Not Proficient
When are we going to use this in
the world?
How much more will she make if
she increased the value?
- Resource Student – Proficient
If you know the peppers price you What would happen if she added
can figure everything out
more peppers and less corn?
I wonder why she made $200 if the
highest price is corn at $8?
- Self-Contained (except math, science &
specials)
How much more money would she
make if she increased the volume?
How much money you will spend if
you buy one of every produce?
The tomatoes cost twice as much
as the peppers and squash
together
Why does she sell them when she
What would happen if another
could eat them herself?
vegetable was added?
Multiplication - It doubles
Why is corn the most expensive?
- Resource Student - Proficient
- Resource Student – Not Proficient
- Resource Student – Not Proficient
Do less people like peppers and
squash?
The money she made on corn was
$8 more than what she made with How much money did she make on I wonder if the peppers are cheaper
the other 3 kinds of vegetables
each vegetable?
or she just didn’t sell many?
- Not Proficient
- Advanced Proficient
combined
- Not Proficient
What did Greta use the money for? Is Greta single?
What's the answer?
Students practiced solving the
problem using guess and check.
Guess and Check
Vocabulary
Constraints - Something that is true about
the quantities in the problem
Quantities – Something in the problem
that is measured
Kaytee’s Contest
The Constraints:
1.) Two cows, Bertha and Billy, together weigh a
total of 1, 696 pounds.
2.) Bertha weighs 848 pounds more than Billy
All work from this problem was done by 6th graders.
Each grade has one general education class and one inclusion class.
Name: __________________________
Identify the constraints.
1
2
Guess and Check
Use the constraints & show your work for at least 3 guesses below.
Guess 1
Guess 2
Guess 3
Keep going until you find a guess that works.
Kaytee’s Contest
• 1st problem where we had students identify the
constraints instead of noticings & wonderings
• Most students in 6th grade were successful in
identifying the constraints but they didn’t know
how to use them to guess and check
• We discovered we needed to model how to use
the constraints to make a guess then check for an
answer. (created four steps for guess and check)
• Students did not know how to work together in
groups and stay on task.
Work shows limited organization, challenging to understand
General Education Students – Both are Proficient
It was a common occurrence for students to erase their guesses and
work if it was not correct. This happened regardless of whether they
were a general or special education student.
No work or explanation, no guesses, student only wrote one
answer which they thought was correct
Resource Student – Proficient
No work or explanation, only guesses – Student did not know
how to use their guess to check for an answer
General Education Student – Advanced Proficient
Student understood the constraints and how to use their
guess to determine if it was accurate.
General Education Student – Not Proficient
This student explained why his guess did not satisfy a constraint in the
problem. NOTE: This student does very little in math class most days,
but on this day he had one of the best papers in 6th grade.
General Education Student – Not Proficient
Guess & Check Steps
1.) Identify the Constraints
2.) Make a Guess
3.) Check Your Guess
4.) Repeat as Necessary
A Cranberry Craving
The Constraints:
1.) Each day Carissa ate 7 more cranberries than
she ate the day before, beginning on Thursday,
Thanksgiving Day.
2.) Carissa ate a total of 161 cranberries in 1 week.
Work from this problem was done by 6th,7th and 8th graders.
Each grade has one general education class and one inclusion class.
A Cranberry Craving
• We found that students were able to understand the
constraints and some were even able to begin guessing and
checking without a teacher having to model the first guess.
• This problem allowed students to focus on using one
constraint to solve for an answer.
• Our goal in this problem was to have students use their
previous guesses to make more accurate subsequent
guesses.
• We started to see more organization from students when
guessing and checking their responses.
• Some 7th graders in the general education class attempted
to solve this problem using their knowledge of algebra.
With this problem, students began to show some type of
organization in their responses when guessing and checking.
In fact, many students chose to make a table.
General Education Student – Not Proficient
Student’s first guess was too high to satisfy one constraint, yet he still
guessed a larger number for his second guess. Our aim was to have
students use previous guesses to make more accurate future guesses.
General Education Student – Not Proficient
This student went beyond our
expectations of using previous
guesses to make more
accurate subsequent guesses.
The student’s first guess was very close to satisfying the second
constraint. Since the first guess had a total of 154 cranberries and she
needed 161, she was only short 7 cranberries. Since a week consists of
7 days, the student determined that Carissa ate one more cranberry
each day than she originally thought in her first guess.
General Education Student – Proficient
This class
consists of
general
education
students, most
are proficient
in math.
Some 7th graders were very eager to
try solving the problem using their
knowledge of algebra. Although they
needed prompting, many students
were able to write an expression to
show the number of cranberries that
were eaten each day. Some students
were able to write an equation from
the expressions and there were even
a few that were able to solve the
equation using algebra skills.
The student writes an expression for each day that relates to the
number of cranberries Carissa ate each day. The student then collects
like terms and writes an algebraic expression in simplified terms.
General Education Student – Proficient
This work was completed after he solved it using guess and check. In my
experience, it is a challenge for resource students to show each step
when solving an algebraic equation. For this reason I accept any work
that these students write as long as they also have the correct answer.
The work this student demonstrates shows a clear understanding of the
problem in algebraic terms. By analyzing the work closely, I can deduce
what steps the student took to solve the algebraic equation.
ESL and Resource Student – Not Proficient
This student
had an unique
approach to
writing an
equation and
collecting like
terms.
General Education
Student – Advanced
Proficient
Ostrich Llama Count
The Constraints:
1.) There are 47 heads.
2.) There are 122 legs.
The Question
How many ostriches and llamas are on the farm?
Work from this problem was done by 7th and 8th graders.
Each grade has one general education class and one inclusion class.
Ostrich Llama Count
• We chose this problem as one that would deter students
from using algebra to solve for the answer.
• Students had to satisfy the first constraint, then check to
see if the second constraint was also satisfied.
• Most students demonstrated some type of organization
when solving for the answer.
• Students started to feel successful with problem solving.
We started to hear comments such as, “I get it now!”
• We saw some unique problem solving strategies – in one
case, students had two very different approaches to
solving this problem even though both drew a picture.
The student demonstrates knowledge of the problem and constraints.
Although it is hard to follow, there is some type of organization of ideas.
Grade Resource Student – Proficient
Student demonstrates knowledge of the problem and constraints.
The paper shows organization of guesses and checks.
General Education Student – Not Proficient
FRONT
BACK
In certain situations, this student can be a challenge because of poor choices regarding his
behavior. On this day, the student isolated himself at a desk and worked diligently for 20
minutes. He produced guess after guess and yet did not give up. He did not stop, not even to
talk to his peers. When finally solved, he commented, “I was always a little more or a little less
and had trouble getting the right answer.” Since he clearly shows ability to use the constraints
to check his guesses and the patient to do the thorough checking, he’s ready to focus on making
better next guesses.
General Education Student – Not Proficient
Since this student solved the problem before most of his peers, we
asked him to write an explanation of how he solved the problem.
General Education Student – Advanced Proficient
This was written on a student’s paper.
General Education Student – Proficient
Resource Student
Collaborative Group
Consisted of one resource,
one self-contained and one
general education student.
The self-contained student is
the only group member
proficient in math.
Self-Contained Student (except for math, science & specials)
Student drew a picture and, in doing so, was able to solve the
problem. This was a different approach from other students.
General Education Student – Not Proficient
Here is a student who drew a picture as a strategy to solve the problem.
Interestingly, the student’s drawing became the guess and she then
needed to check her drawing to see if her guess was correct.
Resource Student – Not Proficient
Wheels R Us
The Constraints:
1.) ¼ are tricycles & the rest are bicycles
2.) total of 45 wheels
Work from this problem was done by 6th & 7th graders.
Each grade has one general education class and one inclusion class.
Wheels R Us
• The student must understand the constraints in
order to solve the problem.
• This problem also requires an understanding of
number sense and fractional parts.
• Students who previously hesitated when asked to
problem solve, quickly engaged themselves in
finding the answer using guess and check.
• We started to see reasonable guesses from students
that were close to the actual answer, resulting in
students attempting a fewer number of guesses to
solve the problem.
This student did not understand the constraints in the
problem or have a clear understanding of parts of a whole.
ESL Student – Not Proficient
The student concluded that…
This student either did not understand the constraints or
does not have a concrete understanding of fractions.
General Education Student – Proficient
This is the same student who drew a picture to solve a
previous problem yet still used a guess and check approach to
solve the problem. The student is clearly a visual learner.
Resource Student – Not Proficient
Attempt 1
Attempt 2
One of the 8th grade students who resisted the strategy guess and check
and thought he could rely on his algebra skills to problem solve, could
not write an algebraic equation to solve for the answer.
General Education Student – Proficient
This 7th grade student has a strong foundation of number sense.
Therefore, he was easily able to use his knowledge of number sense and
the problem solving strategy guess and check to solve for the answer.
Resource Student – Proficient
Eating Contest
The Constraints:
1.) Josh ate 8 more beans than Caleb.
2.) Elsie ate 15 fewer than twice as many as Caleb.
3.) Sol ate 30 more than ½ of Caleb.
4.) The children ate a total of 176 beans.
Work from this problem was done by 6th, 7th and 8th graders.
Each grade has one general education class and one inclusion class.
Eating Contest
• In this problem, students needed to decide which
person they were going to guess first. Many students
were able to read their constraints and identify this as
Caleb. They told us it was because most of the
constraints related to the number of beans he ate.
• Students began to feel success with the guess and
check strategy.
• We started to challenge students to write expressions
that would describe the constraints and then use them
to write an equation that would solve the problem.
• Students across all grade levels were becoming
problem solvers.
Even though we have been doing this for months, this student
still crosses out her guesses because they weren’t right.
General Education Student – Proficient
Many students are able to identify the constraints, make an
educated guess, check for correctness and then use it to make
more accurate subsequent guesses. Almost all students also
demonstrated some type of organization in their work.
Resource Student – Not Proficient
This is a clear example of a student using previous guesses to
make more accurate subsequent guesses. The end goal is to
make as fewest guesses as possible to solve a problem.
General Education Student – Proficient
The student didn’t know what to do when he got a decimal number. He
automatically assumed it was wrong and didn’t finish with his guess.
General Education Student – Not Proficient
This student divided the
total number of beans
eaten by the number of
children in the eating
contest and came up
with the number 44. The
student then attempted
to use the number to
find the number of beans
each child ate. He was
unsuccessful due to the
fact that some of the
constraints had more
than one operation.
General Education Student
– Proficient
This student also wrote
an algebraic expression
for each constraint.
Although every problem we solved had whole number answers, some
students were not discouraged by decimals. Twice this student chose to
guess an odd number of beans for Caleb, resulting in a decimal because
of the constraint Sol ate 30 less than ½ of Caleb.
General Education Student – Proficient
We started to
challenge students
to write expressions
that would describe
the constraints and
then use them to
write an equation
that would solve the
problem.
Resource Student
– Proficient
Since the student already solved the problem by guessing and checking,
she knew x = 34, the number of beans Caleb ate. She immediately knew
she made an algebraic mistake when she got a different answer for x.
General Education Student – Proficient
Two 8th graders continually resisted
using guess and check as a problem
solving strategy. This was because
they wanted to use algebra to solve the
problem. On this day, these two
students solved the problem very
quickly compared to their peers. On
their paper I saw that they only had
one guess and it was the correct one.
Although possible, I questioned their
strategy. It was then that they
produced a crumpled up piece of paper
with algebraic expressions all relating
to C, for Caleb, and the equation they
used to solve for the answer. We found
it interesting that the students thought
they had done something wrong.
General Education Students
– Proficient & Advanced Proficient
Using Technology to Foster
Problem Solving Skills
1.) Excel spreadsheet
2.) Interactive applets
3.) Jing Software– Allows for students to
describe the strategy they used to solve a problem
Student work is kept in their online portfolios.
Thanks!
Catch me afterwards if
you’d like to talk about
this or have more
questions