### Five Alive! A Critical Thinking Pilot Program for Mathematics

```FIVE ALIVE! A CRITICAL
THINKING PILOT PROGRAM FOR
MATHEMATICS
Jim Rutledge and Carol Weideman
Faculty Co-Champions, Mathematics
INTRODUCTION

QEP Critical Thinking Initiative at SPC

2011—Mathematics


Math QEP Committee met during Spring and
Summer, 2011
Decided to focus on problem solving as an overarching theme and more specifically on critical
thinking skills pertaining to problem solving.
EXAMPLES OF CRITICAL THINKING SKILLS
PERTINENT TO PROBLEM SOLVING
Analyze data and search for patterns
 Analyze possible outcomes
 Construct a diagram
 Eliminate the impossible
 Guess, check and revise
 Identify relevant (and irrelevant) information
 Interpret graphical data
 Use lists or tables to order and display data

VISION, GOALS & SUPPORT



Vision: Each math class (college-wide) would
identify, illustrate and promote critical thinking
skills as an integrated part of the curriculum.
Goals: To impress upon students the importance
of critical thinking and to give them enjoyable
opportunities to exercise and develop these skills.
thinking in each class, faculty will underscore its
importance and validate the college-wide effort.
PROGRAM CONCEPT & DEVELOPMENT


Concept: Use the first five to ten minutes of
certain classes to introduce a teaching/learning
episode that illustrates a specific critical thinking
skill and then provide students with the
opportunity to exercise that skill in a problemsolving mode.
Five Alive! pilot program implemented in
Summer session and continued in Fall, 2011.
MGF 1106—Carol Weideman (online, blended)
 MGF 1107—Jim Rutledge (on campus)

FIVE ALIVE! OVERVIEW—MGF 1107




MGF 1107 program consisted of three two-week
segments (involving the first ten minutes of four
class meetings), each of which focused on a
specific critical thinking skill.
Students worked in groups of two to solve the
Five Alive! critical thinking challenges.
Students had an opportunity to present their
solutions to the class.
Certificates of achievement were awarded at the
end of the semester (based on achievement level).
RESULTS—SUMMER, 2011
Pretest average score: 1.6 (out of 10)
 Post-test average score: 4.0 (out of 10)

Optional (but highly encouraged) participation


Three Five Alive! assignments—total of 15
achievement points possible:
4 students earned 10-15 points
 19 students earned 5-9 points
 1 student earned 1-4 points
 5 students did not participate

CHANGES FOR FALL, 2011




Survey at end of Summer indicated that some
students objected to the fact that the Five Alive!
assignments did not contribute to their grade.
The MGF 1107 Fall pilot will award extra credit
for Five Alive! achievement (15 points maximum;
800 normal points in semester).
More emphasis will be put on having students
record their critical thinking processes. This was
done minimally by students in the Summer.
More encouragement will be given to students to
share their successful problem-solving efforts
with the class as a whole.
St. Petersburg/Gibbs Mathematics Department
presents the Summer, 2011
Five Alive! Award
John Doe
to
for
high achievement during the
Five Alive! critical thinking program
Presented on July 20, 2011
James Rutledge, Prof. of Mathematics
Pascal’s Triangle
Five Alive! Critical Thinking Skills
Critical thinking skill: Analyze data and search for patterns
Application illustration
The first several rows of Pascal’s Triangle are presented to Burt and Izzy and they are asked to
determine the entries in the next row.
1
1
1
1
1
1
2
3
4
5
1
1
3
6
10
1
4
10
1
5
1
Izzy says to Burt, “Well, Burt, it seems pretty obvious that the first and last entries in the next
row will be 1’s since that is the case for each row.”
“Good observation, Izzy,” replies Burt, “and it is also fairly obvious that the second and next-tolast entries will be 6’s since those entries always increase by one in each successive row. But
what about the middle part of the row?”
“Hmm.m.m…..” says Izzy. That’s a bit more difficult. How do you think they got the 10’s in
the middle of the current bottom row?”
“Aha!” exclaims Burt. “I see it! If you add the 4 and the 6 in the row above, you get 10 as the
result in the row below!” And that pattern works for the other middle entries as well. So 1+4=5,
and 4+6=10; in the same way, 6+4=10 and 4+1=5. And that’s how you get the middle entry
values! That’s pretty exciting!”
“Dude!” declares Izzy. “So the next row will have middle entry values of 1+5=6, 5+10=15,
10+10=20, 10+5=15, and 5+1=6.”
“Awesome,” says Burt. “So the next row consists of the entries: 1 6 15 20 15 6 1”
Five Alive! Critical Thinking Skills
Pascal’s Triangle
Critical thinking skill: Analyze data and search for patterns
Application illustration
The first several rows of Pascal’s Triangle are presented to Burt
and Izzy and they are asked to determine the entries in the next
row.
1
1
1
1
1
1
2
3
4
5
1
1
3
6
10
1
4
10
1
5
1
Izzy says to Burt, “Well, Burt, it seems pretty obvious that the first
and last entries in the next row will be 1’s since that is the case for
each row.”
“Good observation, Izzy,” replies Burt, “and it is also fairly obvious
that the second and next-to-last entries will be 6’s since those entries
always increase by one in each successive row. But what about the
middle part of the row?”
“Hmm.m.m…..” says Izzy. That’s a bit more difficult. How do you
think they got the 10’s in the middle of the current bottom row?”
“Aha!” exclaims Burt. “I see it! If you add the 4 and the 6 in the row
above, you get 10 as the result in the row below!” And that pattern
works for the other middle entries as well. So 1+4=5, and 4+6=10; in
the same way, 6+4=10 and 4+1=5. And that’s how you get the middle
entry values! That’s pretty exciting!”
“Dude!” declares Izzy. “So the next row will have middle entry values
of 1+5=6, 5+10=15, 10+10=20, 10+5=15, and 5+1=6.”
“Awesome,” says Burt. “So the next row consists of the entries:
1 6 15 20 15 6 1”
Five Alive! assignment
Using an analytical approach similar to Burt and Izzy’s,
determine the pattern involved in this variation of Pascal’s
Triangle and determine the entries in the next row:
1
1
1
1
1
1
3
5
7
9
1
1
5
13
25
1
7
25
1
9
1
Entries in next row:
_____________________________________________________________
In a paragraph, please describe the critical thinking process that
conjectures that you made (including those that turned out to be
incorrect) as you searched for the correct solution.
FIVE ALIVE! OVERVIEW—MGF 1106




MGF 1106 program was introduced in an online
summer section.
The program consisted of six discussion topics focused
on course material and critical thinking; students
worked in teams of 5-6 students.
Each week a new topic was introduced. Each topic
included several questions for student response.
Certificates of achievement were awarded at the end
of the semester (based on achievement level).
MGF1106: TOPICS



Course covers topics ranging from problem
solving and critical thinking, logic, statistics,
geometry
Course begins with a focus on problem solving
Five Alive Activities introduced with the concept
of problem solving
RESULTS—SUMMER, 2011
o Six Five Alive! Activities incorporated as
discussion topics in online course
 Each activity was worth 12 points, with lowest
score dropped: 60 points out of 380 points





14 students earned 51-60 points
1 student earned 41-50 points
4 students earned 31-40 points
5 students earned less than 30 points (3 of these
students failed the course)
Anonymous student survey: 93% felt these
activities reinforced the course concepts
MGF1106: CHANGES FOR FALL, 2011
Course offered in blended format
 Pretest included as part of course orientation
(mean = 53%)
 Posttest will be given at end of course




MGF 1106 Fall pilot will award extra credit for
Five Alive! achievement
More emphasis will be put on having students
record their critical thinking processes.
More encouragement will be given to students to
share their successful problem-solving efforts
with the class as a whole.
POLYA'S METHOD*
1.
Understand the problem
2.
Devise a plan to solve the problem
3.
Carry out the plan
4.
Check the results
*
From Polya’s book “How to Solve It,” published in
1945
FIVE ALIVE! PROBLEM SOLVING
APPROACH
1. State the question clearly. Work with one problem at a time
2. Understand and translate.
3. Work out a plan (or plans) for solving. Identify assumptions
and determine if they are reasonable.
4. Use the information provided to carry out the plan; make sure
you have sufficient information.
5. Check the results. If there are alternative plans for solving
check by using an alternative approach. If the results are not
reasonable, you can go back to step 2 and try again.
6. State the results; infer only what the evidence implies. Discuss
implications and consequences.
FIVE ALIVE! ACTIVITY: STACK OF CUBES
Identical blocks are stacked in the corner of the
room as shown:
How many of the blocks are not visible?
SOLVING THE PROBLEM
Step 1: State the question clearly.
Work with
one problem at a time
We want to determine the number of blocks we
cannot see in the stack of cubes
Step 2: Understand and translate.
Since it appears to be easy to calculate the
blocks we can see, we could calculate the total
number of blocks and subtract the visible blocks
to find the hidden blocks.
Alternatively we could calculate the total in each
row and subtract the front visible row; repeat for
each of the five row.
SOLVING THE PROBLEM
Step 3: Work out a plan (or plans) for
solving. Identify assumptions and
determine if they are reasonable.
Assumption; We assume that each row
extends fully in the corner.
Let x = total blocks in stack
Let y = visible blocks
Then: # of Hidden cubes = x – y
SOLVING THE PROBLEM
Step 4: Use the information provided to
carry out the plan.
We’ll use the first approach.
We’ll find x (total blocks)
Top row has one block.
Second row has 1 + 2 = 3 blocks
Third row has 1 + 2 + 3 = 6 blocks
Fourth row has 1 + 2 + 3 + 4 = 10 blocks
Fifth row has 1 + 2 + 3 + 4 + 5 = 15 blocks
Total blocks in the stack = 1 + 3 + 6 + 10 + 15 = 35 blocks
So x = 35 blocks
SOLVING THE PROBLEM
Step 4 (continued):
Top row has one block.
Second row has 1 + 2 = 3 blocks
Third row has 1 + 2 + 3 = 6 blocks
Fourth row has 1 + 2 + 3 + 4 = 10 blocks
Fifth row has 1 + 2 + 3 + 4 + 5 = 15 blocks
Notice another relationship among the rows:
Blocks in row (n + 1) = (# in row n) + (n +1)
Row 4 = # in Row 3 + 4 = 6 + 4 = 10
Row 5 = # in Row 4 + 5 = 10 + 5
SOLVING THE PROBLEM
Step 4 (continued):
Now we’ll find y (visible blocks)
Top row has one visible block.
Second row has 2 visible blocks
Third row has 3 visible blocks
Fourth row has 4 visible blocks
Fifth row has 5 visible blocks
Total visible blocks = 1 + 2 + 3 + 4 + 5 = 15 blocks
So y = 15 blocks
Total hidden blocks = x – y
= 35 – 15 = 20 blocks
SOLVING THE PROBLEM
Step 5:
Check the results
We can use the alternative plan to check our
answer. How many hidden blocks in each row?
Row 1 – none
Row 2: 3 – 2 = 1 hidden
Row 3: 6 – 3 = 3 hidden
Row 4: 10 – 4 = 6 hidden
Row 5: 15 – 5 = 10
Total Hidden : 1 + 3 + 6 + 10 = 20 blocks
SOLVING THE PROBLEM
Step 6:
State the results.
Cubes stacked in a corner with 5
rows have 20 hidden blocks.
TRY THIS YOURSELF!
If the stack had 6 rows, how many blocks are
hidden?
We know that stack with the five rows had 20
hidden so we only need to add the hidden
blocks in the 6th row.
Solution: 20 + 15= 35 hidden blocks
Notice that the hidden blocks a row are
equal to the total blocks in the row above
MGF1106 PROBLEM SOLVING CHECKLIST
REUSABLE LEARNING OBJECT
Stack of Cubes:
SoftChalk Activity
http://softchalkconnect.com/lesson/serve/LQFNbkp
gxodjtY/html
FIVE ALIVE ! GEOMETRY ACTIVITY
Samantha is thinking of buying a circular hot tub 12ft
in diameter, 4 ft deep and weighing 475 lbs. She
wants to place the hot tub in a deck built to support
30,000 lb. Use π = 3.14. NOTE: Round each answer
to the nearest whole unit.
Can the deck support the hot tub?
GEOMETRY FIVE ALIVE ACTIVITY
Samantha is thinking of buying a circular hot tub 12ft
in diameter, 4 ft deep and weighing 475 lbs. She
wants to place the hot tub in a deck built to support
30,000 lb.
1. Determine the volume of water in the hot tub in
cubic feet.
Hint: Volume
V == (3.14)
π r2 h (6)2 (4) = 144 (3.14) = 452.16 ft3
2. Determine the number of gallons of water the hot
tub will hold. NOTE: 1 ft3 = 7.5 gal.
Gallons = 452.16 ft3 * 7.5 gal = 3391.2 gallons
GEOMETRY FIVE ALIVE ACTIVITY
Samantha is thinking of buying a circular hot tub 12ft
in diameter, 4 ft deep and weighing 475 lbs. She
wants to place the hot tub in a deck built to support
30,000 lb. Use π = 3.14. NOTE: Round each answer
to the nearest whole unit.
3. Determine the weight of the water in the hot tub.
NOTE: Fresh water weighs about 8.35 lbs/gal.
Weight = 3392 gal (8.35 lbs/gal) = 28,316.52
4. Will the deck support the weight of the hot tub and
Yes: Hot tub + Water = 475 + 28317 = 28,792 < 30,000
5. Will the deck support the weight of the hot tub,
water and four people, whose average weight is
Yes: Hot tub + Water + 4(115) =
28,792 + 600 = 29,392 < 30,000
FIVE ALIVE! FUTURE PLANS




Recruit other math faculty to incorporate
activities
Expand to other courses: STA2023, MAC1105
Use Five Alive! Activities as Discussion Topics in
online, blended courses
We welcome your feedback and input!
Any Questions?
LOGIC FIVE ALIVE ACTIVITY
Conservative commentator Rush Limbaugh directed this
passage at liberals and they way they think about crime.
“Of course, liberals will argue that these actions
(contemporary youth crime) can be laid at the foot of
socioeconomic inequalities, or poverty. However, the Great
Depression caused a level of poverty unknown to exist in
America today, and yet I have been unable to find any
accounts of crime waves sweeping our large cities. Let the
liberals chew on that” (from See, I told You So, p. 83)
We can write this passage as an argument:
LOGIC FIVE ALIVE ACTIVITY QUESTIONS

es,
Is the argument valid? Identify the standard form
of the argument.
The argument is valid: Law of Contraposition
Where p = Poverty causes crime
q = Crime waves swept American cities during the Great
Depression
```