The Effects of Developing Mental Math Skills in Pre-service Elementary Teachers
Sally Kleinknecht
Education 671: Integrated Seminar
The Effects of Developing Mental Math Skills in Pre-service Elementary Teachers
Having a love and understanding of math begins and is
nurtured in elementary school. Therefore, elementary teachers are key players
in this scenario. It is important then that elementary teachers have confidence
in their own mathematical skills so that they can pass it on to their students.
One way to increase elementary teachers’ confidence in their mathematical
skills is to develop mental math skills of pre-service elementary teachers.
What is the best way to develop mental math skills for these preparatory teachers? Many students today are
dependent on their calculator for even simple arithmetical calculations.
Instead of using the calculator as a tool, they treat their calculator as the
ultimate authority. Will developing mental math skills affect their dependency on
their calculator? What other effects will developing mental math have on
pre-service elementary teachers? This research examines the above questions.
Review of Literature
The
goals of the National Council of Teachers of Mathematics (NCTM, 2000) include
both a student’s proficiency in estimation skills and mental math, and the
student’s expertise of the use of the available technology (e.g. the
calculator). Teachers must not only be practiced in these areas, but must be
able to train their students in the use of these skills. The training of
students in estimation skills, mental math, and the use of the calculator as a
tool should begin in elementary school, but Ralston (1999) believes that most
elementary teachers are afraid of math. Therefore, pre-service teachers (and,
in particular, pre-service elementary teachers) need a mathematical education
that will not only increase their mathematical knowledge, but also their
confidence in their ability and their attitudes toward math (Quinn, 1998;
Glasgow, 1998). This review of literature will address the areas pertinent to
this study: mathematics and the pre-service teacher, the standards of the NCTM,
mental math and estimation skills, and the use of the calculator.
Pre-service
teachers
The
mathematical education methods courses of pre-service teachers will not only
increase their knowledge of mathematical content, but will also improve their
attitude toward math (Quinn, 1998). Improving these two areas will greatly
enhance their students’ understanding and learning of mathematics.
Mathematical
education of pre-service teachers. According to
Quinn
(1998) found that even though an elementary mathematical methods course greatly
improved the pre-service teacher’s knowledge of mathematical content, their
scores were still only 70.4% on a sixth-grade-level test. The problems areas
for pre-service elementary teachers were fractions, long division, geometry,
statistics, and probability. He felt that spending more time in mathematical
methods courses, in these different content areas, would be beneficial for the
pre-service teachers, and thus improve mathematical education (Quinn, 1998).
The
mathematical attitudes of pre-service teachers. The mathematical attitudes of pre-service teachers are very important
because if they do not have a good understanding of mathematics, they will be
unable to create an atmosphere for their students to appreciate mathematics (
Pre-service elementary
teachers’ attitudes toward math, according to Kelly & Tomhave
(1985, as cited in Quinn, 1998) and Rech, Hartzell, & Stephens (1993, as cited in Quinn, 1998),
are less favorable than that of the general university population. Ralston
(1999) believes that most elementary teachers are afraid of math and that also
they are less interested in math than the other subjects they are required to
teach.
Quinn (1998) found that an
elementary mathematics methods course improved the pre-service teachers’
attitudes toward math. To improve math education, not only do pre-service
teachers need to understand mathematical content, but they also need to have a
positive attitude toward mathematics.
Reflection of their own thought processes along with observing the
thought processes of others is a key in changing the pre-service teachers’
attitude toward mathematics (Hart, 2002).
National
Council of Teachers of Mathematics
The
Principles and Standards for School Mathematics put a high priority on both
using technology to its full advantage and on learning estimation and mental
math skills (NCTM, 2000).
Mental
math and estimation skills. From early elementary school through high
school, computing fluently and making reasonable estimates are key standards in
mathematics. Given any problem, students need to be able to select appropriate
methods of solving: mental calculation, estimation, calculator (computer), or
pencil and paper. They also need to develop and use strategies for estimation,
so that they will be able to judge the reasonableness of their numerical
computations and results (NCTM, 2000).
Use
of calculators. The NCTM (2000) states that technology is an
indispensable tool for teaching, and ultimately learning and doing mathematics.
Students’ understanding of abstract mathematical concepts can be greatly
enhanced by using the calculator. The effective use of calculators depends
greatly on the teacher. Technology should be used as a tool, not as a
replacement for teaching. Graphing, visualizing concepts, and computing are
enhanced when using the calculator. They can also be used to help teachers
bring together for the students the skills of mathematics with the overall
understanding of mathematics (NTCM, 2000).
Mental
math and estimation skills
Pomerantz (1997) says that mental math, along with pencil
and paper and estimation, is essential in the development of mathematical
learning skills. It is not only necessary if one does not have a calculator,
but, more importantly, it is necessary to check the reasonableness of the
calculator answer. In fact, mental math and estimation is even more crucial
when using a calculator (Ralston, 1999).
Sowder and Kelin (1993, as cited
by Reys & Reys, 1998)
show that understanding of mathematical concepts increase as students use
estimation and mental math – especially when teachers encourage student
discovery. An added benefit for the student using estimation and mental math is
increased attention span.
The use of mental math and estimation
skills. Mental math should be the
first choice in problem solving if possible. If not, then estimation of the
answer is the second choice. If an exact answer is needed, than a calculator or
a standard written computational algorithm (pencil and paper method) would be
used, but the estimation of the answer is still crucial for accuracy (Reys & Reys, 1998). If mental
math is not encouraged and a student only resorts to a standard written
computational algorithm, then the student will see mathematics as only
algorithms.
The timetable in teaching mental math and
estimation skills. Elementary
students should be encouraged to invent their own computational strategies,
which involves a great deal of mental math. Estimation should only be used in
gaining a sense of numbers for the elementary students such as guessing how
many marbles may be in a jar. It is recommended that computational estimation
be introduced later in the intermediate grades after the students have a good
grasp on large numbers (Reys & Reys, 1998).
As students move into the
intermediate grades, they should continue sorting out different strategies as new
problems arise, increasing their mental math skills and thus their
understanding of mathematical concepts. By the middle grades, the students
should have a good grasp of whole numbers and should be extending their
conceptual knowledge to include fractions by using mental math of the student’s
invention. Estimation should be a major focus at this time. (Reys & Reys, 1998).
Rubenstein (2001) says that
mental math and estimation should be taught not only in elementary and middle
school, but also in the high school and college math courses. Adults use mental
math and estimation in their daily lives at home and at work such as interest
on loans or investments, shopping, taxes, tips for waiters, traveling, etc.
Sharing mental math strategies gives many opportunities to study mathematical
properties and to understand them, for example, the inverse operations and the
distributive property. Mental math also combats calculator-dependency as
students learn calculator-free strategies. Mental math skills give them flexibility
as they see the many options before them in problem solving. It gives the
students the feeling of empowerment when they are confident in their estimation
and mental math skills (Rubenstein, 2001). He believes that every mathematical
course has built-in mental math strands that can be used and taught.
Mental math. Ralston (1999) argues that mental math in many
instances is less time-consuming than using a calculator and can be very
efficient, for both computing an answer and checking an answer (Reys & Reys, 1998). Mental
math encourages a student to design his own personal algorithms and thus it
promotes a deeper understanding of the concepts (Ralston, 1999). The student
who uses mental math well not only improves his number sense, but also he knows
how to organize his thought processes which is a useful life skill.
Estimation skills. Estimation skills play a key role in mathematical
reflection. Mathematics teachers value reflection of the computed answer,
whether the computation was done via calculator or pencil and paper. By
estimating an answer, the student can compare his exact results with his
estimated results to see if his answer is in a reasonable range (
Most
students only use a few strategies that were taught to them in the classroom to
estimate answers, such as rounding numbers off. The best estimators are those
that form their own strategies from their understanding of the concept. A
teaching method that presents a wide variety of estimation strategies, most of
them student-driven based on the understanding of number sense and problem
solving strategies, will be much more effective (McIntosh, Reys
& Reys, 1992; Sowder,
1992 as cited in Glasgow, 1998).
Many
students are not confident in their own estimation skills, especially when it
conflicts with a calculator-produced result. The students place more trust in
the calculator than in their own estimation skills (
Calculators
There
is still much disagreement about the usage of calculators in the classroom.
Many purport that calculators have great advantages for students, allowing them
to visualize the mathematical concepts without sacrificing the time and energy
for tedious computations (Glasgow, 1998; Pomerantz,
1997). Others argue that using calculators will weaken students’ ability to
perform math or understand its processes (Hunsaker,
1997). To eliminate rote memorization and learning algorithms will only
increase mathematical illiteracy. Bracey (1998) argues that to say students
need to know how to calculate by rote instead of using a calculator is the same
reasoning that Socrates used for oral recitation vs. writing. Socrates argued
that learning to write would destroy people’s memory. It appears, Bracey (1998)
says, that in this country we seem to be of two opinions about the use of the
calculator.
The
advantages of the use of the calculator.
“Calculators are valuable educational tools that allow the students to reach a
higher level of mathematical power and understanding” (Pomerantz,
1997, p. 1). When students use calculators, they are able to focus on
understanding the concept, setting up the problem and then interpreting the results
instead of worrying about tedious calculations (Dick, 1992; Hopkins, 1992 as
cited in Beckmann, Senk & Thompson, 1999). Meel (1997, as cited in Bracey, 1998) and Glasgow (1998)
agree that by freeing up students from having to expend a lot of time and energy
in doing calculations, the use of calculators actually gives the students more
time in solving and conceptualizing problems. In addition, Pomerantz
(1997) states that the use of calculators allows students more time in
developing number sense and mathematical reasoning. Calculators are more
efficient, accurate, and faster for laborious computations (Glasgow, 1998; Pomerantz, 1997). Therefore, teachers are able to give
students more “real life” problems, even at a younger age that would be
otherwise too difficult to grasp without the calculator.
Calculators should not
replace mental skills or pencil and paper methods – they should complement them
by giving students the ability to solve problems in multiple ways (Pomerantz, 1997). They are also a mathematical equalizer.
For students who have always been frustrated with long computations or have
given up on math, calculators allow them the ability to experience mathematics
and cultivate an understanding of mathematics without being bogged down or
hating it.
Bracey
(1998) believes that one actually has to know more to use a calculator, since
the student has to determine whether the answer is reasonable or not. Students
have better attitudes and are more confident when they are able to use
calculators on assessments (Meel, 1997 as cited by
Bracey, 1998). Students who use a calculator on the SAT score slightly higher
than those that do not because of less computational error (Lee, 1999).
The
disadvantages of the use of the calculator. Glasgow (1998) states that his research found
that students showed a disinclination to question the results of the
calculator, that is, the student gives the calculator too much authority
instead of using it as a tool. The attitude of the calculator-driven student is
that if the calculator says it, then it must be right. Hunsaker
(1997) affirms these findings. He says that a calculator encourages a student
to try every combination of mathematical operations to find an answer, instead
of thinking through and deciding which operation would be the best. Calculators
prohibit the student from seeing the principle behind the mathematical process,
such as long division. Therefore, the student misses the “inherent structure
and beauty in math” (Hunsaker, 1997, p. 20). A
student who has grown up using a calculator will not only struggle with the why
of math, but also the how of math. They have no number sense and
very little estimation skills, and in many cases, cannot generalize
mathematical principles. Pomerantz (1997) calls such
arguments myths, and argues that calculators used as a tool are of great value.
Hunsaker (1997) disagrees and believes strongly that
the prolonged use of the calculator will stagnate the mind of the child. She
thinks there are many uses of the calculator, but not as an educational tool,
unlike
Teachers
must change how they teach and assess their students who are using a
calculator. Students can now program their calculators to do much of the
mathematical processes for them. Thus, many teachers clear all programs in the
memory of the students’ calculator prior to exams. In addition, when the
students are bored, they can play games on their calculators during lectures
(Lee, 1999). This can be a disruption in the classroom.
Discussion
of the Review of Literature
As
a teacher, I have the seen both the use and misuse of the calculator in
mathematics classes. As the literature has borne out, the calculator is a great
tool in the hand of a wise student. This places the burden on the teacher to
train their students to use the calculator as a tool and not as a crutch. I
think the reason why our students rely on their calculator as the ultimate
authority is that their estimation skills and mental math strategies are poor
to none. These important skills are taught beginning in elementary school.
However, many elementary teachers are not confident in their own math abilities
and therefore teaching math can become just a job and not a passion. As this
cycle progresses, we see students entering high school and college who rely
totally on their calculator for their answers because they do not trust their
own judgment. Research has shown that mathematical methods courses have
improved pre-service teachers knowledge of math and attitudes about math. If
pre-service elementary teachers were confident in their own mental math and
estimation skills and knew how to use the calculator as an educational tool to
visualize math, elementary students may turn into stronger high school math
students. Mental math and estimation skills should be taught (or discovered)
not only in elementary school, but also in middle school, high school, and
college. Therefore, we should not put the burden on the elementary teacher
alone, but we (i.e. high school and college teachers) should train our own
students to be proficient in these skills. Researchers differ on the solution:
some say to throw out the calculators, others to abandon pencil and paper. I
think we should train our students in the use of all tools available: pencil
and paper, estimation skills, mental math strategies, and the calculator. The
student has thus reached proficiency and maturity when, first of all, he
reflects on the problem itself to determine which strategies will be the most
effective and secondly, he judges the reasonableness of the computed answer.
Procedures
School Setting
I chose to conduct my
research within the Math Department at the
Class Subject and Section
Two classes of Math 104 (Basic
Elementary Mathematics I) were chosen for my research. Math 104 is the first content area math class for
pre-service elementary teachers. According to USI’s online course catalog, Math
104 teaches “fundamental concepts in mathematics selected for the elementary
education major. Topics included are problem solving processes and strategies,
sets, numeration systems relating to real number operations and computational
algorithms, functions and their graphs, logic, and selected topics in
statistics and probability. This course is taught with a contemporary approach
to problem solving and requires participation in small and large group
manipulation-based activities. Enrollment open only to students in elementary
and/or middle school programs” (
Therefore, the students learn not only the
basics of elementary math, but also many different approaches in learning math
that will be instrumental for them as teachers in the elementary classroom.
Developing mental math skills fit very well into the established curriculum.
Sections 002 and 003 were selected for three reasons: both sections were taught by Mrs. Wells, they were back-to-back in the schedule and they
were both in the same classroom. We randomly chose the first class to be the
control group (22 students) and the second class the experimental group (24
students). Math 104 is a four-hour class meeting on Monday through Thursday for
50 minutes each. It is a first semester freshman course for students who test out of preparatory Math 100.
Students
Only13 of the 22 students finished the course in
the control group, which consisted of three males and ten females.
They were all considered traditional students (between the ages of 19 through
22). There were five first-semester freshmen, 2 second-semester freshmen, two
first-semester sophomores, one second-semester sophomore, one second-semester
junior, and two first-semester seniors (see Appendix B).
Only 15 of the 24
students in the experimental group completed the course. This group consisted
of one male and fourteen females. Two
out the fifteen were considered non-traditional students (one was 28 years old
and the other one 38 years old). The other 13 students were between the
ages of 18 and 21. There were eight first-semester freshmen, two first-semester
sophomores, two second-semester sophomores, two first-semester juniors, and one
second-semester junior.
Classroom procedures
On the first day of class, a pre-test was given to both groups, one for student attitudes toward math and another for math aptitude. These tests were without a calculator and not timed. The students finished both tests easily during the 50-minute period. After the experimental group finished their tests, they were told that they would be given mental math exercises to practice for ten minutes every day. The students would be expected to keep a daily notebook of the mental math exercises that would be turned in at the end of the semester. Every two weeks, a mental math quiz would be given over the material learned in the previous two weeks. In addition to the mental math exercises, questions pertaining to mental math, the calculator, and education would be posted on Blackboard for them to respond to every two weeks.
During the semester, students in the experimental group practiced forty-seven different mental math exercises, kept a mental math notebook, and took seven mental math quizzes. Each student in the experimental group answered seven discussion board questions.
On the second to
last day of the semester, both classes met in a conference room for their
posttests and for a calculator experiment conducted individually by current
math instructors at the
Assessment
Instruments
Aiken Revised Math Attitude
Scale
I used the Aiken’s
Revised Attitude Scale (Aiken, 1963) to measure the students’ attitudes toward
mathematics. This instrument consists of 20 statements concerning math that the
students must rate on a 5-point Likert-type scale
from Strongly Disagree to Strongly Agree. Sample items include: “I
do not like mathematics, and it scares me to have to take it,” and “Mathematics
is very interesting to me, and I enjoy math courses” (see Appendix C).
I scored each question with a 0, 1, 2, 3, or 4 points depending on the level of the response. I reversed the negatively stated questions so that the higher score would indicate a more positive response toward mathematics. The possible range of scores was 0 points (indicating a very strong dislike of mathematics) to 80 points (indicating a very strong like of mathematics).
Revised Essential Elements of
Elementary Mathematics Test
Meaningful knowledge of
mathematical concepts was tested using 19 of the 50 multiple-choice questions
on the Essential Elements of the Elementary School Mathematics Test developed
by M.A. White (1986). The 19 questions were chosen to emulate the topics
covered in the Math 104 curriculum. White’s test was developed to measure
conceptual and intuitive understanding of mathematics. The questions require
more than a calculator, memorization of certain formulas, or applying
algorithms. The test covers a wide variety of concepts such as place value,
addition, subtraction, multiplication, division, fractions, rounding, percent,
statistics, and probability. Test items may be seen in Appendix D. I
scored each correct answer with one point, for a total of nineteen points.
I added a twentieth
question concerning the students’ use of the calculator. The question was “I
would have felt more comfortable using a calculator on this test.” The students
responded with same 5-point Likert-type scale from Strongly Disagree to Strongly Agree as the Aiken test, only
in multiple-choice form. I scored this question using 0, 1, 2, 3, or 4 points
from Strongly Agree to Strongly Disagree respectively. Zero points indicated a strong dependency on
the calculator, and four points a strong independency of the calculator.
Mental Math Exercises and Quizzes
I developed forty-seven mental math exercises on addition, subtraction, multiplication, division, fractions, and percents (see Appendix E). These exercises were to coincide loosely with the lessons being presented in Math 104. The experimental group was split into six small groups based on proximity within the classroom. Each day (except for quiz and test days) the exercises were placed on the overhead five minutes prior to class. As the students came into class, they would begin doing the exercises and recording the problems and strategies in their mental math notebooks. After working on the exercises individually, they began to discuss the problems within their small group. After ten minutes, the mental math exercises would end with the whole class discussing their strategies in an exchange of ideas. At the end of every week, the students could access the daily mental math exercises learned that week on Blackboard to use for studying, especially in case of absences.
Every two weeks, the students were given a timed mental math quiz over the material covered the previous two weeks. The quizzes were timed (usually 2 ½ minutes per 15 problems) so that the students would be encouraged to use the algorithms or “tricks” they had learned or discovered instead of relying on pencil and paper. The purpose of the quizzes was to ascertain whether the students were learning the daily mental math skills and to keep them on task. These quizzes were graded in two ways: the percent attempted and the percent correct out of the number attempted. The goal for each student was to attempt 70% of the questions with 70% correct out of the number attempted. The scores were not used to determine their grade in class, but to give them an indication of their improvement or their understanding of the material.
Mental Math Notebook
Each student in the experimental group was required to keep a mental math notebook of their daily work. The purpose of these notebooks was to write down ideas, strategies, and tricks learned each day to help them increase their mental math skills. These were collected and graded for thoroughness and completion. The range of scores was from 0 to 41 points.
Discussion Board Questions
I posted seven different
discussion board questions on Blackboard every other week (see Appendix F).
Each student was to respond in essay form to the question the first week the
question was posted. During the second week, the students were to read their
classmates’ answers and respond to two of their postings. The purpose of the
discussion board questions was to encourage the students to think through some
educational issues such as the importance of mental math and how it relates to
using a calculator, how and when a student should use a calculator in school,
and how to teach a subject (such as math) if one does not like it.
I did not score the responses, though I did keep track of whether students completed the assignment. At the end of the semester, each of their responses were printed out and attached to their tests.
Calculator Experiment
I replicated an
experiment used by
The instructors were
given a sheet that told them exactly what to say during the experiment. They
were to instruct the students that the math department was conducting an
experiment on how students estimate. They informed the students that they would
be taking notes and that there was no time limit. For the first three problems, they did not
make any comments. After the fourth and fifth estimation (assuming the student
had not questioned the calculator), the instructor asked the student how she
make her estimate, prior to the student using the calculator. On the last two
problems, the instructor asked the student to rate her confidence level of this
particular estimate on a 6-point Likert-type scale (absolutely not confident to very confident) and what the upper-bound
for the calculator should be prior to her using the calculator.
I met with each instructor to go through the protocol and to give him a programmed calculator. Each instructor was placed in a separate room, so that there would be strict privacy. As the students were taking their posttests, the instructor would call one student out at a time to interview. Each interview took about ten minutes.
Student Interviews
I designed a student questionnaire that I used to personally interview randomly selected students from the experimental group as they finished their posttests (see Appendix H). The purpose of this questionnaire was to ascertain what the students thought of the mental math exercises, the mental math quizzes, the discussion board questions, and the calculator experiment. I also wanted to know what suggestions they had to improve the teaching of mental math skills and if they thought their own mental math skills had improved.
Faculty Interview
I designed a faculty
questionnaire that I used to personally interview Mrs. Wells (see Appendix I). The purpose of this questionnaire was to
ascertain what Mrs. Wells
thought of the mental math exercises and the quizzes that she had used in her
classroom. I wanted to know her reactions as she watched the students work through these exercises. I also thought
it was imperative to find out what changes she would make and if she would use
these exercises/quizzes again in the future.
Other Measurements
Three other measurements
of each student will be used in the statistical analysis: SAT scores, PLEA
scores, and the Math 104 semester grades. If two SAT scores were found on the
application, the highest one was used. Some students transferred into the
university or had taken the ACT instead and therefore their SAT scores were not
available. The PLEA is taken by every incoming freshman at USI to place him/her
in the correct mathematics course. Those students transferring from other
universities do not have to take this test. The PLEA and the SAT scores were
used to equalize the control group and the experimental group in my statistical
analysis. Unfortunately, only 8 students out of 15 in the experimental group
and 9 out of 13 in the control group had both scores available which
significantly lowered my already small sample size.
Findings
Aiken Revised Math Attitude
Scale
Using SAT and PLEA
scores to equalize the groups, there was no significant difference between the
groups regarding their posttest
scores using ANCOVA. The mean of
the control group decreased from 40.62 on the pre-test to 38.85 on the posttest
– a 4.4% decline. Seven out of 13 students either stayed the same or increased
in their positive attitude toward mathematics. The mean of the experimental
group also decreased from 45.73 to 38.27 – a 16.3% decline. Only three out of
15 either stayed the same or increased in their positive attitude toward
mathematics. This decline can be attributed to several things: one might be burnout at
the end of the semester. One thing it does say – this Math 104 class (with or
without mental math exercises) did nothing to help the students have a more
positive attitude toward math.
Revised Essential Elements of
Elementary Mathematics Test
Using ANCOVA with
the SAT and PLEA scores to equalize the
groups, there was no significant difference between the groups regarding their
posttest scores. The mean of the control group increased from 10.69 on the
pre-test to 12.15 on the posttest – a 13.7% rise. Eleven out of 13 students
either stayed the same or increased in their understanding of essential
elementary mathematics. The mean of the experimental group also increased from
10.2 to 11.67 – a 14.4% rise. Twelve out of 15 either stayed the same or
increased in their understanding of essential elementary mathematics. These
results do show that the Math 104 class did improve their understanding, but
they do not show if the mental math exercises were part of that increase.
The last question of the
Revised Essential Elements of Elementary
Mathematics Test measured the student’s independence of (or mature use of)
a calculator. There was again, using ANCOVA, no significant difference between the groups regarding their posttest
scores. The mean of the control group increased from 1.69 on the pre-test to
2.38 on the posttest – a 40.8% rise. Eleven out of 13 students either stayed
the same or increased in their independent use of the calculator. The mean of
the experimental group stayed the same with 1.73 on the pre- and posttest.
Eleven out of 15 students either stayed the same or increased in their
independent use of the calculator. These results are only based on one
question, so it says very little statistically, but it does hint that
developing mental math skills did little for increasing a mature use of the
calculator in the experimental group in this study. A list of all test scores
for each student may be found in Appendix J.
Calculator Experiment
Because of the amount of
error while conducting the calculator experiment (
Putting these errors
aside, two students from the experimental group questioned the calculator prior
to the last (or seventh) problem as opposed to none from the control group. I
would like to think this was from doing the mental math exercises daily and the
quizzes and discussion board questions every two weeks, but I cannot prove that
statistically.
Student Interviews
I was a little surprised with the results from the student interviews. None of the students complained about doing the mental math exercises daily or keeping a mental math notebook. I expected some of them to say it was too much for them, but they did not. None of them would change the way the exercises were presented. A few did not like the timed quizzes. They would have liked more time or no time limit. They all felt they had improved their mental math capabilities. They said it was most evident in real life situations – shopping, balancing a checkbook, etc. Some of them said it helped in their homework and they also felt they were using their calculators less often. Only one interviewed questioned the calculator in the calculator experiment. The rest said that they thought the calculator’s results were strange, but they still did not feel confident enough in their own answers to question the calculator. I do not think, though, they will look at the calculator again in the same way. Most of them did not study the mental math exercises outside of class except possibly before the quizzes. They liked having access to the mental math exercises on Blackboard. Most of them felt the discussion board questions were beneficial in thinking through some educational issues concerning mental math, calculators, and teaching elementary math. In summary, according to the students interviewed (8 out of 15), they benefited from the mental math exercises, especially outside of the classroom setting. Their only recommendation would be not to time the quizzes. The timing of the quizzes was to encourage students to use the new and quicker algorithms (or “tricks”) that they had learned instead of relying on their former pencil and paper methods - it was not to cause stress. In the future, I would lengthen the timing of the quizzes so as not to cause undue stress on the students. I want them to have success in using what they have learned.
Faculty Interview
Mrs. Wells believed that the mental math exercises fit very nicely
with the curriculum. Sometimes she was able to refer back to the mental math
exercise for that day during her lecture. The students responded well, though
there were complaints about the timed quizzes. Having them timed stressed out
some of the students. The positive aspects of the exercises were twofold:
giving the students many mental math strategies and allowing them to try them.
Negatively, sometimes the exercises extended into more class time which cut down
on lecture time. In addition, the students did not seem to “tackle them as
vigorously at the end as at the beginning of the semester.” She will definitely
use the mental math exercises in her next semester Math 104 class along with
the pre-tests. The only change she will make is cutting down the frequency of
the exercises from four times a week to two times a week. She feels it may make
the students work harder, stay fresh and on task.
Other Measurements
Regarding their grades
in the Math 104 class, there was no significant difference between the groups using
ANCOVA. I calculated the mean by using
1 point for D, 1.5 for D+, 2 for C, 2.5 for C+, 3 for B, 3.5 for B+, and 4 for
A. The mean grade for the control group was 2.0 and for the experimental group,
1.9.
Discussion
Even though the
statistical analysis did not show there was any difference in the students’
posttests or grades in developing mental math skills, I think developing mental
math skills made a difference in the students’ lives and complimented the Math
104 curriculum. This study should be repeated with a larger sample size to
statistically determine what kind of effects developing mental math skills on preservice teachers will have. Mental math is the most practical form of
math and could be used in most math courses from grade school through college
to the benefit of the students. I will definitely incorporate some form of
developing mental math skills in the classes I teach. Conducting
this study was helpful to me personally in many ways, but the sample size was too small to make any concrete statements or
predictions concerning my research question.
I would do several
things differently if I repeated
this study. I like the Revised Essential
Elements of Elementary Mathematics Test, but I would like to develop my own
test that corresponds exactly to the course and includes more mental math and
estimation questions. The test is very good in evaluating the students’
understanding of mathematical concepts, but it did not have enough mental math
questions to determine if the experimental group had developed better mental
math skills. I should have given a separate test to each group specifically
testing their mental math skills – prior to the class and at the end of the
semester. I love the calculator experiment and will use that in the future.
Next time I will have a training session for the instructors to reduce errors
in procedure. The instructors were surprised to see such a dramatic display of
dependency on the calculator. I think this experiment may change how
instructors teach and hopefully how students use the calculator.
In conclusion, there are
many benefits of developing mental math skills in preservice
elementary teachers. One such benefit is confidence in their own math skills.
As they gain more confidence in their own mathematical ability, it will only
enhance their students’ learning and love for math.
References
Aiken,
L. (1963). Personality correlates of attitude toward mathematics. The Journal
of Educational Research, 56(9), 476-480.
Beckmann, C., Senk, S., & Thompson, D. (1999). Assessing students’ understanding of functions in a graphing calculator environment. School Science & Mathematics, 99(8), 451-456.
Bracey, G. (1998). Calculations about calculators. Phi Delta Kappan, 79(6), 473.
Dick,
T. (1992). Super calculators: Implications for calculus curriculum,
instruction, and assessment. In J.T. Fey (ed.), Calculators in mathematics
education, 1992 yearbook (pp. 145-157).
Glasgow, B. (1998). The authority of the calculator in the minds of college students. School Science and Mathematics, 98(7), 383-387.
Hart, L. (2002). Preservice teachers’ beliefs and practice after participating in an integrated content/methods course. School Science & Mathematics, 102(1), 4-24.
Hunsaker, D. (1997). Ditch the calculators. Newsweek, 130(18), 20.
Kelly,
W., & Tomhave, W. (1985). A study of math
anxiety/math avoidance in preservice elementary
teachers. Arithmetic Teacher, 32(5), 51-53.
Lee,
J. (1999). Calculators throw teachers a curve.
McIntosh, A., Reys, B., & Reys, R. (1992). A proposed framework for examining
basic number sense. For the learning of mathematics, 12(3), 2-8, 44.
Meel, D. (1997). Calculator-available assessments: The why,
what and how. Educational assessment 4(3), 149-174.
National
Council of Teachers of Mathematics (2000). Principles for school mathematics:
The technology principle. Retrieved
Pomerantz, H. (1997). The role of calculators in math
education. Retrieved
Putney, L., & Cass, M. (1998). Preservice
teacher attitudes toward mathematics: Improvement through a manipulative
approach. College Student Journal, 32(4), 626-632.
Quinn, R. (1998). Effects of mathematics methods courses on the mathematical attitudes and content knowledge of preservice teachers. Journal of Educational Research, 91(2), 108-113.
Ralston, A. (1999). Let’s abolish pencil-and-paper arithmetic. The Journal of Computers in Mathematics and Science Teaching, 18(2), 173-193.
Rech, J., Hartzell, J., & Stephens, L.
(1993). Comparisons of mathematical competencies and attitudes of
elementary education majors with established norms of a general college population.
School Science and Mathematics, 93(3), 141-145.
Reys, B., & Reys, R. (1998). Computation in the elementary curriculum: shifting the emphasis. Teaching Children Mathematics, 5(4), 236.
Rubenstein, R. (2001). Mental mathematics beyond the middle school. Mathematics Teacher, 94(6), 442.
Sowder, J. (1992). Estimation and number sense. In D. Grouws (ed.), Handbook for research on mathematics
teaching and learning (pp. 371-389).
Sowder, J.,
& Kelin, J. (1993). Number sense and
related topics. In D. Owens (Ed.), Research ideas for the classroom: Middle
grades mathematics (pp. 41-57).
University of Southern
White, M. (1986). Preservice teachers’ achievement in the essential elements of elementary school mathematics: The development of an evaluation instrument (Doctoral dissertation, University of Houston, 1986). Dissertation Abstracts International, 47, 2068A.