The Effects of Developing Mental Math Skills of Pre-service Teachers
Sally Kleinknecht
Education 671: Integrated Seminar
The Effects of Developing Mental Math Skills of Pre-service Teachers
Review of Literature
The
goals of the National Council of Teachers of Mathematics (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 National Council
of Teachers of Mathematics, 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 (National Council of Teachers of Mathematics, 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.
Questions to be
explored.
In
addition to current research, there needs to be other questions addressed.
Concerning the calculator, why, if the majority of the studies finds that calculators do not hinder the learning of
students, do those findings not play out in the classroom? If one would
interview math teachers, most would agree that students are far too calculator-dependent.
Why does the research not confirm that? Calculator-dependency is related to the
lack of estimation skills and mental math capabilities. Where does this
breakdown occur for students? Is it in the elementary grades? Is it because
many elementary teachers are not confident in their own math skills? Would
increasing the mathematical confidence of elementary school teachers affect the
student’s use of the calculator and their estimation and mental math skills? Is
it too late to change pre-service teachers’ attitudes and aptitudes toward math
in a mathematical methods class? If change is attainable, what is the best way
to affect a permanent change?
References
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).