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Education 50G3
Module 10 – Research Report
Abstract:
This study examined the effectiveness of introducing the game of cribbage to a group of
Grade 10 Mathematics Essentials students as a non-traditional approach of decreasing
their dependence on calculators when performing basic mathematical computations such
as addition, subtraction and estimation. There were 21 students involved in this project,
12 of which required adaptations such as a reduced workload, preference seating, cues for
staying on task and marking spontaneous writing. Student pre-testing was preformed at
the beginning of the course along with both student and teacher survey/questionnaires. As
the study moved forward, progress was monitored through a number of cribbage
worksheets that were administered on a weekly basis. At the end of the study follow-up
testing along with second student questionnaire was conducted to determine the
effectiveness of the project. The results of the test suggested a positive increase in both
addition and subtraction skills and the questionnaire suggested that the cribbage method
was enjoyable.
Nature of the Problem:
Many lunchtime conversations with math colleagues focus on the concern over student
dependence on the calculator and/or their lack of simple mental math skills. For many
higher-level classes, the calculator is used so often in daily lessons that some of those
skills have become “rusty”. Although it is important for these students to undergo some
math exercises to clean out the mental math cobwebs, that the majority of them still have
the skills.
Unfortunately there are many groups of students who do not have those mental math
skills. The Math 10 Essentials course is designed to support those students who have
struggled with mathematics for most, if not all, of their school career. Many of these
students have external issues that make learning difficult and the peer pressures of high
school only compounds the problem. I’ve worked with approximately 8 groups of Math
10 Essential students and I am continually surprised at how weak their mental math skills
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are. When I mention mental math skills, many teachers think of multiplication facts. The
truth is that the majority of students I work with in this class have difficulty even adding
numbers that have answers in the double digits. Subtraction provides an even greater
problem.
Improving mental math skills is an important part of the Math 10E curriculum. Many
traditional approaches to increase mental math skills such as the “mad math minute”,
flash cards, and even power point lessons have been tried with varying degrees of
success. The problem with the traditional approach is that the daily practice soon looses
its bite as many in the class do not take it seriously or are not motivated enough to try and
improve their previous score. Therefore, it is important for teachers of the essentials level
program to find non-traditional methods of teaching mental math skills that might hook
essentials students and provide them with an interesting yet effective approach to
improving skills.
Literature Review
If I had $7 in my pocket and found a five-dollar bill on the ground, how much
money would I have in total?
For many students in my Math 10 Essentials class, this type of basic addition question
can cause an automatic trigger response. They either begin counting on their fingers or
reach for a calculator. Very few will mentally calculate the answer. If I ask them to solve
the problem without the aid of their fingers or the calculator, I may get some random
guessing occurring or, in some cases, refusal to answer. This is a typical response when
their fear of mathematics overpowers their ability in mathematics. For many, aids such as
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the calculator provide them with the edge necessary to navigate through math class
without having to deal with the hurdles that can arise from mental mathematics.
To deal with these issues, mental math is one of the outcomes in the Math 10 Essentials
curriculum. It is important to establish reasons why mental math and estimation are
important skills to develop and master. This has been my math hurdle for a number of
years. It is one thing to state that we need to improve mental math skills, but convincing
students, whom many of which have given up on math, that they need to practice doing
calculations in their head, is quite another. There are many traditional approaches to
improving mental math skills that can be effective with some groups of students.
Unfortunately, I have not had any success with these methods when implementing them
with the essential-level students. It is my hope that this literature review will help bring
some focus to my research question and perhaps provide some strategies to implement
and pitfalls to avoid. This review is based on the following research question:
Can using a non-traditional approach of teaching my Math 10 Essentials class
the game of cribbage, increase their mental math skills and allow them to work
through the addition, subtraction and estimation modules without the aid of a
calculator?
Gaming and Mathematics
Games provide opportunities to explore basic number concepts such as counting, quantity
and one-to-one correspondence along with higher levels of mathematical reasoning such
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as combinations, patterns and place value (Olson, 2007). For the past two decades, the
National Council of Teachers of Mathematics (NCTM) has been pushing the importance
of manipulatives as way of having students visualize number concepts and provide a
concrete reference to difficult mathematical operations. It can be argued that games are a
form manipulatives as they help students solidify mathematical operations. Unlike
traditional manipulatives such as algebra-tiles, games have the added bonus of disguising
their mathematical relevance. It is important to note that even though students may not
immediately recognize the connection between the game and a mathematical concept, the
importance or benefit of the manipulative is not lost. Baker (1999) reasoned that the
content-centered use of manipulatives could yield positive results regardless of how
immediate the connection is made. My essential students are more apt to continue using
the manipulative if they perceive it to be a game and not a teaching tool.
Gaming and Students
Games can provide students with an opportunity to explore mathematical concepts in an
engaging and non-threatening environment (Olson, 2007). Students in my Math 10
Essentials class have experienced limited success in mathematics and have come to
accept that their number sense and mathematical ability have not developed beyond basic
elementary operations. While each student can provide a number of unique reasons for
their weaknesses in math, the common theme that unites them all is that they have
developed an aversion for mathematics. Baker (1999) states that games can present an
alternative approach for students that may have become disenchanted with traditional
approaches to the mathematical curriculum. Games have the potential to be a very
powerful tool for teachers especially when dealing with students at the foundation (or
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essentials) level. Students are more likely to internalize and remember material when
they are actively engaged in the learning process (Bonwell & Sutherland, 1996).
Games have the distinction of giving the appearance that they are not connected to
educational goals and outcomes. For most of us, gaming was introduced before schooling
and thus we are able to justify their separation. Even games that are solely educationally
based can pass as a simple fun activity. An added bonus for teachers is that ideas and
gaming rules that are obvious to adults can often create mathematical discussions in the
classroom (Olson. 2007)
The Importance of Mental Math
O’Nan (2003) argues that “students who are equipped with a greater variety of mental
computation strategies will solve problems more quickly and accurately because they can
choose the strategy that will be most effective for the given problem.”(p. 10.) In my
experience, students who have strong mental computational skills also develop a comfort
level for mathematics and are less likely to form barriers when more difficult concepts
are introduced. Yang (2002) suggests that number sense is a valuable skill that leads to
improved efficiency in solving mathematical equations. If number sense is the key to
improving problem solving then Beishuizen, Van Putten, and Van Mulken (1997) insist
that the only way to increase number sense is to place an increased emphasis on mental
mathematics.
Another important reason for focusing on mental mathematics is the role it plays in the
curriculum. The Nova Scotia curriculum document for Math 10 Essentials lists the
following mental math outcomes:
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
know the double facts in addition; their extension to 2, 3 and 4 digits; their
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connections to subtraction, multiplication by 2 and by , division by 2 and
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multiplication by 50%

know the addition and subtraction facts and extend them to 2, 3, and 4 digit

numbers

estimate appropriate sums, differences, products and quotients

mentally calculate 1%, 10%, 15% and 50% of quantities that are compatible with
those percents

estimate percents of quantities
Why Cribbage
This proved to be the most difficult part of my literature review. Although I have a strong
sense for the potential role that cribbage can play in the essentials classroom, finding
articles to either defend or negate my feelings proved difficult. As Baker (1999) indicated
in his paper, “An exhaustive review of the literature, both electronic and text-based, led
to the sure conclusion that there are no papers or reports published in the past couple of
years that relate the standard deck of playing cards to the standard mathematics
classroom.” (p. 4)
I was able to locate one NCTM article that was written in April 2005 that directly ties in
the benefit of cribbage with mathematical thinking. Markel (2005) suggests that cribbage
hands are an excellent application of mathematical reasoning and goes on to talk about
how surprising their omission in the classroom is. It is rare to find any discussion of
cribbage in modern math textbooks especially when compared to how often poker is used
in probability and statistics.
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My goal is to teach my essentials class how to play an entire game of cribbage. This
would include teaching them the ability to count the total points in their hand using
mental math. Markel (2005) points out that for younger students, such as junior high
students, computing the point value of a hand is a good combinatorial problem. A more
difficult problem is determining which total points are possible and looking at the
different ways of obtaining these scores. Another aspect of cribbage problems is that
there are often several ways, or several orders in which the counting may be done.
Teachers can use cribbage hands to:

Evaluate the points of randomly chosen hands

List crib hands that can produce a certain points total.

Count the number of ways a given configuration can occur

Find all configurations that give a particular point value

Learn to play the game. Remember its fun and a good teacher of mathematical
thinking.
Games are fun and create an environment for developing mathematical reasoning. All the
articles that I have read provide overwhelming support for the use of games in the
classroom to enhance the mathematical experience and to provide students, especially
those that have become disillusioned, a fun way to develop computational skills. They
also outlined the importance of providing repeated opportunities to play games so that
mathematical ideas will emerge and pattern recognition can occur. As students begin to
understand the intricacies of the game, they will begin to form mathematical relationships
and strategies for winning the game. They will explore mathematical ideas and prioritize
the importance of these ideas based on how successfully they can implement them. This
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is an important part of developing math skills but one that takes time and patience.
Many articles also point out that it is important to choose appropriate games. Some
games encourage children to memorize facts, such as multiplication flash cards. These
games do not engage children for long as they either quickly memorize the facts or
realize that they have difficulty in memorizing and turn off completely. Another pitfall is
that teachers may use games that encourage recall speed. While this may be the desired
outcome of the game, it is important to stress that being “smart” in math does not
necessarily mean getting the answer the quickest. We need to place value on the process
of thinking.
Through playing and analyzing games, students gain computational reasoning skills.
Discussing relationships and strategies for winning the game help develop number sense
and strengthen ones mathematical confidence. Cribbage can provide students with an
interactive experience that allows them to develop addition, subtraction and estimation
skills and more importantly shows them that these mathematical methods can be
valuable. Although card games are not the only source of teaching mathematics, their use
in the classroom is justifiable.
Data Collection
A series of lesson plans were developed that break the game of cribbage down to
manageable pieces for Math 10 Essential students. (See Appendix) These worksheets
slowly introduce the many aspects of scoring and working with the cribbage board.
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To acquire some baseline data for my action research problem I decided to use both
questionnaires and pre-tests. My focus group for the action research is my second
semester Math 10 Essentials class. There are 21 students in this class and I was able to
use this group to obtain some of my baseline data. I also surveyed 5 teachers that work
with students in this level of mathematics. All data was collected during the first week of
February 2008.
Math 10 Essentials Student Questionnaire
On the first day of class I always hand out a questionnaire to my students that focuses on
their feelings about mathematics. This year, to coincide with my action research, I
adapted the questionnaire to include both questions about their feeling for mathematics
and their feelings about the use and/or their dependence on the calculator. I asked the
students to read the statements and make an honest choice based on their overall feeling.
To encourage truthful responses, I told the students that the survey was used to provide
me with a general overview of the whole class and it was not necessary for them to
include their name on the sheet.
Directions: Students were asked to read the statements and to check either AGREE or
DISAGREE depending on their feelings.
Participation: 21 Students
% Of Students
that Agree
% Of Students
that Disagree
I enjoy seeing how rapidly and accurately I can
work math problems.
48%
52%
I think about mathematics problems outside of
school and like to work them out.
19%
81%
I don’t feel sure of myself in mathematics
unless I use a calculator.
81%
19%
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I prefer to do math with a calculator even if the
questions involve simple addition.
76%
24%
I am faster at solving math problems if I use a
calculator
86%
14%
I feel more confident with my answer if I used a
calculator to find it.
86%
14%
When I go shopping, I am able to estimate what
my bill will be with tax included.
48%
52%
I find it easier to add two numbers than to
subtract two numbers.
81%
19%
I have always been afraid of math.
57%
43%
The results from the student questionnaire, while worrisome, are not surprising. Students
lack confidence in their mathematical ability when they are asked to perform tasks
without the aide of a calculator. Questions 3-6 all focused on the anxiety that students
experience in mathematics when dealing with topics that we assume they have mastered
in elementary school. There is a clear indication from this survey that students have
developed a calculator comfort zone that will continue to reduce their ability to perform
simple mathematical computations without some apprehension. It is also evident that I
need to focus more on subtraction skills. This also became apparent through the pre-tests.
Math 10 Essentials Teacher Questionnaire
I felt that it was important to survey teachers that work with students who are working the
essentials level of mathematics. Unfortunately, there are not many Math Essentials
teachers in our school. Thus, in order to get a larger sample, I surveyed Math 10
Essentials and Math 10 Foundations teachers. Both of these levels encounter the same
hurdles when it comes to mental mathematics.
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Directions: Teachers were asked to read the statements and to check either AGREE or
DISAGREE depending on their feelings.
Participation: 5 Teachers
I find that my students depend on the
calculator to help them solve basic math
questions (addition, subtraction, estimation)
I feel that my students loose confidence when
they are asked to solve a math problem without
a calculator
Students often use estimation to determine if
their final answer is appropriate
Students would do just as well on a basic
addition/subtraction test if they did not use a
calculator
Students enjoy traditional methods for
improving mental math skills (i.e. Mad
Minutes, Flash Cards, Memorizing etc.)
% Of Teachers
that Agree
% Of Teachers
that Disagree
100%
0%
100%
0%
0%
100%
40%
60%
40%
60%
Students need to improve their mental math
skills especially skills involving addition,
100%
0%
subtraction and estimation.
The results from the teacher questionnaire seem very one sided. It is very evident that
teachers feel students have a heavy dependence on calculators and that it is very
important to sharpen their mental math skills. There was some discrepancy when it came
to an approach to improve the math skills. Although I have not had much success with
traditional methods for mental math, 40% of the teachers interviewed feel that students
still enjoy those methods. Perhaps as part of my research, I need to inquire further into
the traditional methods that these teachers use.
Math 10 Essentials Pre-Tests (4 different tests were administered)
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I knew that pretests would provide an important piece of evidence for my action
research but I did not want to base all my conclusions on one test. I decided to administer
4 different tests to my students. One the first day of class (Wednesday February 6, 2008)
I gave my students a 25-question basic addition test. They had to complete the test
without the aide of a calculator but there was no time restrictions placed on them. They
could take as long as they wanted. When they completed this test, I handed them a 25question basic subtraction test. Again, they had to complete the test without the aide of a
calculator but they could take as much time as they needed.
The reason for administering the no time limit tests is that I wanted to assess whether or
not they had the basic skills. The goal of mental math strategies is to sharpen addition and
subtraction skills not to create these skills. Although students didn’t have a calculator
they had enough time to fall back on some other aides such as using their fingers for
counting.
On the second day of classes (Thursday February 7, 2008) I handed out an addition test
and a subtraction test but this time they had to complete the 25 questions in a set amount
of time. From the previous class, I knew that they had the necessary addition and
subtraction skills but now I wanted to create an environment where they had to complete
the problems mentally instead of relying on their fingers or other tools.
Directions: Students were given 4 tests .An addition test and a subtraction test with no time
limit and an addition test and subtraction test with a time limit.
Test: Each test consisted of 25 questions involving 1 - 2 digit addition and subtraction.
Participation: 21 Students
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Class Average
Addition Test – No Time Limit
98%
Subtraction Test – No Time Limit
88%
Addition Test – Time Limit
65%
Subtraction Test – Time Limit
20%
When students were given as much time as they needed to complete the tests, their
overall class averages indicate that their 1-2 digit addition and subtraction skills are quite
strong. I did observe many students using their fingers to count and/or making tick marks
on the paper to help them count. When the time limit was introduced, the averages
dropped significantly. One reason for this drop is that many of the students that relied on
finger counting could not complete the test in the given amount of time. Although they
were solving the problems without a calculator, they were not necessarily using mental
math to solve the problems. It is also very clear that while students are fairly consistent
when it comes to basic addition and subtraction skills, there is a large discrepancy
between their ability to apply mental subtraction skills and mental addition skills.
Math 10 Essentials Post-Tests (4 different tests were administered)
As a way to measure the effectiveness of the cribbage game, I administered 4 tests that
were similar in size and difficulty to the pre-tests that were given during the baseline data
collection at the beginning of the study. An addition test and subtraction test were first
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written without a time limit and then similar tests were written under a 1-minute time
restraint.
Directions: Students were given 4 tests .An addition test and a subtraction test with no time
limit and an addition test and subtraction test with a time limit.
Test: Each test consisted of 25 questions involving 1 - 2 digit addition and subtraction.
Participation: 21 Students
Class Average
Addition Test – No Time Limit
98%
Subtraction Test – No Time Limit
85%
Addition Test – Time Limit
99%
Subtraction Test – Time Limit
68%
Results
The results of the testing provided positive evidence that the Math 10 Essentials students
increased their addition and subtraction skills without the aide of a calculator. Although
there was little increase in their average scores on the “Addition Test – No Time Limit”,
and the “Subtraction Test – No Time Limit”, results from the other 2 tests showed
significant increases.
Pre-Test
Post-Test
98%
98%
Addition Test – No Time Limit
Change
0%
15
Subtraction Test – No Time
Limit
88%
92%
65%
99%
20%
68%
Addition Test – Time Limit
Subtraction Test – Time Limit
4% increase
34% increase
48% increase
The “No Time Limit” pre-tests show that students have the skills necessary to add and
subtract 1-2 digit numbers. Therefore it is not surprising that these scores remained
consistent during post testing. The “Time Limit” tests challenge students to use mental
math skills without counting aides such as fingers. The fact that there was such a drastic
increase in the scores on both the addition and subtraction post tests seems to suggest that
students have acquired some exercises that sharpened their mental math skills. I am
convinced that these positive results are directly related to the introduction of the
cribbage game over the past two months. Unfortunately, do to my current teaching load, I
was unable to bring a control group into this study which would allow me to compare
groups who received cribbage training with those who did not receive the training. If one
was to conduct further research into this topic, I would certainly suggest working with
two classes so that a better comparison of results could be obtained. That being said, I
believe there is enough evidence in this study to suggest that the non-traditional approach
to increasing mental math skills warrants further investigations.
It is my hope to create a small handbook on introducing students to the game of cribbage.
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I would like to make the worksheets available to other teachers who a searching for an
alternative method of increasing mental math skills in the classroom. The cribbage boards
can be purchased for a small fee and an entire classroom set, along with an appropriate
number of playing card decks can be purchased for under $50.
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Appendix
Learning Cribbage Worksheet #1
1. Scoring “15’s” is one of the main objectives in Cribbage. To obtain “15” you
simply have to be able to determine which cards in your hand add up to
fifteen.
If this was your hand, how many ways could you make different 15’s ?
9D + 3C + 3D = 15
9D + 3C + 3S = 15
9D + 3D + 3S = 15
Each time you make 15, it is worth 2 points. This hand has 6 points in “15’s”
How many “15’s” can you make from each of the given hands?
(a)
___________________________
___________________________
___________________________
(b)
____________________________
____________________________
(c)
_________________________
_______________________________
_________________________
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(d)
________________________
________________________
________________________
________________________
(e)
________________________
________________________
________________________
(f)
This hand has five “15’s” Can you find them all?
__________________________
__________________________
__________________________
__________________________
__________________________
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References
Baker, Robert N. (1995) Cards in algebra and pedagogy: Representation and analysis of a
common discrete set. Unpublished master’s paper. The University of Montana.
Baker, Robert N. (1999). Cards in the Classroom; Mathematics and Methods. Ketchikan,
Alaska: Report for the University of Alaska Southeast-Ketchikan Campus. (ERIC
Reproduction Service No. ED 428786)
Beishuizen, M. Van Putten, C.M., Van Mulken, F. (1997) Mental Arithmetic and
Strategy Use With Indirect Number Problems Up to One Hunder. Learning and
Instruction, 1, 87-106
Bonwell, C.C. & Sutherland, T.E. (1996) The active learning continuum; Choosing
activities to engage students in the classroom. New Directions for Teaching and
Learning, 67, 3-15.
Markel, William D. (2005). Cribbage: An Excellent Exercise in Combinatorial Thinking.
Mathematics Teacher, 98, 519-524
Nova Scotia Department of Education (2005) Math 10 Essentials Curriculum Document.
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Olson, Jo Clay. (2007) Developing Students’ Mathematical Reasoning through Games.
Teaching Children Mathematics, May 2007, 464-471
O’Nan, Mindy. (2003). Daily Number Talks and Development of Computational
Strategies in Fourth Graders. Knoxville, Tennessee: Dissertation/Theses. (ERIC
Reproduction Service No. ED 479330)
U.S. Department of Education (1987). Help Your Child Learn Math Manual.
Washington DC: Leaflet for parents. . (ERIC Reproduction Service No. ED
280676)
Weisskirch, Robert S. (2003). Dealing with Piaget; Analyzing Card Games for
Understanding Concepts. Monterey Bay, California: Annual Conference of the
American Phsychological Association. (ERIC Reproduction Service No. ED
481012)
Yang, D. (2002) Teaching and learning number sense. School Science and Mathematics,
4, 152-158