Math Course for Pre-service El Ed Students

[Jerry Becker]

**************************
From redOrbit News, Friday, September 5, 2008.
See
http://www.redorbit.com/news/display/?id=1544535# 
. Our thanks to Kenneth Teitelbaum for bringing
this piece to our attention.
**************************
A Mathematics Course for Prospective Elementary School Teachers

By Jonker, Leo

Abstract: It is possible to get prospective
elementary school teachers to learn interesting
mathematics well by not only requiring that they
learn the material in a mathematics course but
also requiring that they teach it as part of an
enrichment program at the middle school level.
This article discusses the reasons for such a
course, and describes a university mathematics
course that combines these elements.
Keywords: Preparation of elementary teachers,
mathematics enrichment, service learning.

1. TEACHING AND LEARNING MATHEMATICS

One of the paradoxes in the mathematics
preparation of elementary school teachers is
that, though it seems obvious that knowing more
mathematics should lead to better teaching, it
has been difficult to find empirical evidence to
support mis belief. In [1] Ball summarizes
attempts made through the second half of the 20th
century to establish a connection between the
learning of elementary school students and
characteristics of their teachers. Studies
conducted before the mid-1970s, reviewed in Begle
and Geeslin [6] and in Begle [5], found that no
single teacher characteristic proved to be
"consistently and significantly correlated with
student achievement." In particular, it seemed
that having taken more university mathematics
courses was of no benefit. Even for high school
teachers the picture is far from simple. In a
study published in 1994, Monk [16] concludes
that, for the teaching of secondary school
mathematics and science, it appears that "gross
measures of teacher preparation (such as degree
levels, undifferentiated credited counts, or
years of teacher experience) offer little useful
information for those interested in improving
pupil performance."

Clearly then, simply taking more mathematics
courses is not going to help prepare elementary
teachers teach better mathematics classes. Yet,
it cannot be true that understanding mathematics
better is of no benefit to elementary school
teachers. One feels that the difficulty finding a
strong correlation between mathematics courses
taken by teachers and subsequent performance of
their students has to do with the kinds of
courses taken and the depth at which the course
material is understood rather than the number of
courses.

In the 1980s education researchers began to study
teachers' mathematical understanding as it
exhibited itself in the context of their
classrooms. They hoped that by focusing on
teacher practice in the rough and tumble of
teaching situations they might discover
significant effects of teacher preparation and
characteristics on meaningful student outcomes.
Lee Shulman [18, 19], wishing to describe the
kind of teacher knowledge that leads to good
teaching, introduced the notion of "pedagogical
content knowledge." This is knowledge of the
subject as it presents itself in the complexity
of the classroom. Ball, Lampert and others [1, 3,
2, 13, 12, 10, 7, 8] developed this further and
conducted extensive classroom studies to detect
and describe situations where subject matter
knowledge plays a critical role in the classroom.
Their intention was to build a repository of
videotaped and transcribed material that could be
used to focus teacher education courses as
deliberately as possible on the kind of knowledge
of mathematics and pedagogy that really matters
in the elementary classroom.

Several international studies [21, 15, 4, 14],
inspired by the relatively poor performance of
U.S. students on TIMSS, confirm the importance of
mathematical subject knowledge focused on
classroom situations as well as the benefits of a
culture of collaborative professional
development. By and large, these studies are less
focused on the psychology of learning, and more
on the depth of teachers' understanding of
mathematics and their familiarity with strategies
for presenting ideas in the classroom. In her
widely read study [15, Chapter 5], Ma speaks of
"profound understanding of fundamental
mathematics." Her comparison of American
elementary school teachers with Chinese
counterparts, who have had less formal education
than their American colleagues, makes it clear
once again that adding more mathematics courses
to the requirements of teacher education does not
help unless the courses themselves are better
tuned to the practicalities faced in the
elementary classroom.

Kennedy, Ball, and McDiarmid, in [11], developed
instruments for measuring the kind of subject
knowledge that matters in the classroom. A recent
study reported by Rowan, Chiang, and Miller [17]
found some correlation between teachers'
responses to items designed to measure teachers'
mathematical knowledge as displayed in the
context of their teaching, and the performance of
their students. It seems that we are beginning to
identify the kind of knowledge we should be
aiming for when we design our mathematics courses
for pre- service teachers. A key goal is deep
understanding of basic mathematics rather than
surface understanding of a large amount of
advanced material; and, as much as possible, this
understanding of basic mathematics should be
qualified and honed by use in the classroom.

2. CREATING COURSES FOR PROSPECTIVE ELEMENTARY
TEACHERS IN A DEPARTMENT OF MATHEMATICS

When we try to create a university program that
takes these insights seriously we find ourselves
facing a number of challenges. In the first place
there is the challenge of creating courses that
are at a level appropriate for the students who
should be taking them, and yet recognized by the
institution as university level courses. In the
second place, these courses should be designed to
ensure that the material is understood by the
students at a deeper level than would be the case
if they took a more traditional mathematics
course. Thirdly, the material should be presented
as much as possible in a form that connects to
the ways in which the subject comes up in the
elementary classroom. Fourthly, the courses
should be such that they motivate and engage
students who have come to fear mathematics and
mistrust their own abilities to understand it at
all. Finally, the courses should involve
opportunities and requirements for communicating
understanding of mathematics.

In many North American post-secondary
institutions there is a separation (unfortunate
in view of the findings reported above) between
the mathematics courses and the pedagogy courses
taken by pre-service teachers. The former are
taught in a mathematics department by research
mathematicians, while the latter are taught in a
school of education. In some cases, the
mathematics courses are taken at a university
that does not have an education program; students
at these schools have to apply for study
elsewhere upon their graduation to obtain their
teaching qualifications.

Faculty members in a mathematics department have
ambitions for their students; becoming an
elementary school teacher is not usually high on
that list. It is not a given that a course
appropriate for pre-service elementary school
teachers will be perceived by a mathematics
department as a genuine mathematics course rather
than a course in pedagogy. To accomplish this it
will be necessary to build the course around good
mathematics that is challenging and yet
accessible to the average pre-service elementary
school teacher. To convince colleagues that such
a course has a place in the undergraduate program
will require a careful explanation of its goals.
Fortunately, most colleagues can be persuaded
that mathematics does not have to be advanced in
order for it to be interesting and worthy of
study; and, when pressed, they will acknowledge
that we deceive ourselves if we imagine that most
of our undergraduates understand most of what we
teach them in our regular mathematics courses.

Often we content ourselves with the thought that
the students who did not understand the material
as well as we hoped will learn it better when
they see it again in the next course. For
intending elementary school teachers there
probably will not be a next mathematics course.
In a very short time they will be teaching in an
environment where (in North America, at least)
opportunities for sustained further learning are
more limited than they should be, and where a
culture of collegial support does not seem to be
well developed. In other words, it is essential
that the course contain good mathematics that can
be learned at a deep level by all the students
taking it.

Many of the students in a training program for
elementary school teachers, or planning to enter
one, have had little or no mathematics since high
school, and have found their high school
mathematics difficult. These students will
approach any university mathematics course with a
high degree of apprehension. In questionnaires
administered prior to one of my courses, about a
third of the students attest to their anxiety.
Usually they point to one of the early years of
high school as the point at which mathematics
became a burden to them. In some cases these
students will report that at that stage their
mathematics marks began to slide.

"It got to the point in high school that if I
could not grasp a concept on my own there was no
one who could teach it to me and that was very
frustrating."

Others will say that they continued to get good
marks even though they did not understand what
they were doing. "I really dislike the fact that
I feel as though I have just squeaked through
math all my life rather than really understanding
it. Although I have always gotten good marks in
the math courses I have taken, I do not feel
comfortable with the subject in any way."

In nearly all cases these students indicate in
their responses that one of the obstacles to
their enjoyment of mathematics was their refusal
to continue without satisfactory understanding of
the ideas behind the formulas.

A second challenge, therefore, is to ensure that
the course is accessible to students with a weak
mathematics background and a negative attitude
toward the subject, and to present the course in
such a way that these students will feel
confident that the material can be understood.
There will have to be adequate support for the
struggling student, with emphasis on student
engagement with the material, and avoidance of
evaluation schemes that create non- productive
anxiety. At the same time, the course should be
such that students are not able to get away with
the superficial learning habits that have helped
some of them survive mathematics through high
school, and put them in this predicament.

The third issue is the matter of motivation. As
illustrated by the quotes, many of the students
who intend to become elementary school teachers
have been "turned off mathematics. Perhaps they
were not naturally inclined toward mathematics in
the first place, or came across a teacher who did
not have a good sense of the power and beauty of
the discipline. Repeated frustration at not
getting the necessary time and help to attain the
level of understanding the student himself wishes
for only adds to the aversion. We should not
imagine that for these students a good college
instructor and a well- planned course are enough
to guarantee interest in the mathematics. If it
is rekindled along the way we may be grateful,
but there should be other elements in the course
to motivate the students, elements more closely
related to their professional aspirations.

A related issue is the matter of courage. To
instill in the students the confidence, perhaps
for the first time, that it is safe to explore
mathematics, that they can do it, it is essential
that the course give them the opportunity and
sufficient time. The best way to achieve this is
to choose mathematics that does not rely on
material that was learned poorly, but instead
begins with topics that the students feel secure
about.

Finally, the course should have a component in
which students are asked to communicate their
understanding. Speaking of the ubiquity of
superficial, illusory understanding, Shulman
writes [20]: "if we can create conditions where
they can discuss what they know with others, we
significantly raise the likelihood that the
problems [of superficial understanding]
diminish." This deeper level of understanding
that enables communication is especially
important for would-be teachers. Not only will
they be required to discuss mathematics with
their students, but they will be looked to as
resources, and will be required to react quickly
and imaginatively to learning opportunities as
they present themselves.

Furthermore, if the teacher is to receive maximal
benefit from participation in continued
professional development activities, knowing how
to talk comfortably about mathematics will be
very important. Both Ma [15] and Stigler and
Hiebert [21] stress the importance of a
professional teaching culture that emphasizes
discussion of teaching strategies that focus on
subject content and presentation. To the extent
that opportunities for professional development
are available in North America (apparently less
than in China or Japan), it is important that
teachers feel as comfortable talking about the
content of their mathematics courses as they do
discussing other pedagogical concerns.

3. A COURSE THAT SEEMS TO WORK

Over the past five years I have developed a
course in the Department of Mathematics and
Statistics at Queen's University that attempts to
achieve these goals. The course was developed
together with, and is now organized around and
essential to, StepAhead, a mathematics enrichment
program taught by the students in the course, for
the benefit of grades 7 and 8 students in local
schools. The university students each visit a
local school, in pairs, for an hour per week over
ten weeks. Most of the mathematics discussed in
the university classes serves as preparation for
these school visits; and as instructor, I tell
the students that as much as possible I will
attempt to model ways in which the mathematics
can be presented to young teenagers, and conduct
the classes as if I were teaching that group. The
textbook [9] used for the course is a two-volume
enrichment manual written for the purpose, and
informed by many years' personal experience
teaching enrichment mathematics to students in
grades 7 and 8. The syllabus for the enrichment
program is outlined at the end of this article.

By using mathematics that is accessible to good
middle school students I try to ensure that the
course does not, at the outset, involve material
that is threatening to the university students
taking it. Since the enrichment program is fairly
ambitious for students at the middle school level
(I ask the classroom teachers to identify the
students who would benefit from the program,
using 25 percent of the class as a guideline),
the mathematics is interesting for me to teach
and challenging for the pre-service teachers to
learn. Furthermore, the enrichment material is
largely independent of the high school
curriculum, so university students with a
stronger mathematics background do not feel they
are under- challenged by the course, and weaker
students are not tempted to fall back on
formulaic understanding remembered from their own
high school years.

It is significant that, since the university
students have to present enrichment classes based
on the material they are learning in class, and
since many of the students they will face are
very quick, it is not possible to get away with
superficial understanding. From the first of
their ten weekly enrichment visits my students
realize that they have to be ready for unexpected
questions and developments during their classroom
visits. The university classroom is significantly
transformed by the fact that the students have to
deliver the material they are learning. They
constantly ask themselves (and the instructor)
how they will explain the mathematics to their
students. Every week we set a block of time aside
to review the school visits together.

To maximize the communication component in the
class, the university students are required to
prepare their lessons in pairs and to visit the
schools together. This also helps ensure that the
program delivered to the schools is of high
quality, and minimizes the potential for problems
caused by sending out inexperienced teachers on
their own.

The format of the course takes care of the
problem of motivation. When university students
are offered the opportunity to teach a group of
young teens, they have much less trouble focusing
on the course than they would without that
incentive. The format of the course also works
well in providing feedback to the students. The
pressure of that weekly school visit is much
greater than looming assignment deadlines or
exams could ever be. The students gain confidence
as they find they can understand the material.
They gain respect for the real challenges of the
classroom, and for the extent to which good
students are able to learn difficult material.

As an added bonus, the local schools are
enormously grateful for what they consider to be
an excellent enrichment program. At the end of
the course I routinely write to the principals of
the schools, asking them to use an enclosed
questionnaire to provide feedback on the school
visits. One of the questions I put is "Did the
grade 7/8 students appear to enjoy the classes? I
have yet to see a negative answer. The following
are typical:

"The students were always eager to get to class
and told me they loved the challenging/difficult
math they were working on."

"From the participants: 'We should have this for a longer time!' "

To the question "Did the classes seem useful for
their program?" one teacher who sat in on all the
classes wrote:

"The classes have become an important part of our
fall mathematics enrichment programming for the
Challenge Program students."

Comments about the university students are also
invariably positive. One vice principal wrote:
"The two students who took part in your
program/class this year were nothing short of
amazing." It appears that while creating an
important opportunity for the university
students, we have also stumbled upon a deeply
felt need.

4. COURSE CONTENT AND DESIGN

Course content is constrained by two simple
criteria: The material should be accessible and
yet challenging to students at the grade 7/8
level; and the material should have the potential
for engaging the university students. We try to
avoid giving middle school students accelerated
access to high school mathematics. In particular,
we avoid using algebra other than at a very basic
level, not only because students in grades 7 or 8
are only just beginning to learn algebra, but
also because with the use of algebraic methods
there is a temptation for students to suspend
reflection in favor of calculation. This applies
to both the university students and the young
teenagers they teach, but it is particularly
important for the university students, who will
remember some of the algebra they took in high
school but may have a very weak understanding of
the mathematics underlying the mechanics. It is
fortuitous that you can tell these university
students that they must find solutions that do
not use algebra because the middle school
students have not had the necessary techniques.

Because some of the enrichment classes are at
small schools where one grade alone would not
provide a sufficiently large group of students
for an enrichment program, we allow schools to
combine grade 7 and 8 students in one class. To
ensure that these students do not run into the
same material on two successive years, we have
two distinct programs, used in alternate years.
One program focuses on numbers and number
patterns, while the other focuses on geometry. To
provide a picture of the course's content we will
describe the topics covered in a recent year when
the focus was on numbers and number patterns.
While the material changes somewhat from year to
year, the core remains fairly constant. You will
notice that there is a high degree of coherence
between the concepts included in the course. As
much as possible without giving up on that
coherence, we center the lessons on problems, and
encourage a habit of student exploration and
discovery.

We began the course with a short exploration of
patterned sequences, such as 1, 3, 5, 7, . . .,
1, 4, 9, 16, . . ., and 1, 3, 6, 10 ..... This
allowed us to explore the idea of infinity and
students' understanding of pattern rules and
functions. We used cases where different pattern
rules produce the same infinite sequence, to
motivate and explore the role of a proof.

We then switched to a discussion of prime
factors. Assuming the uniqueness of prime
factorization without questioning it, we explored
the role of these prime factors in giving each
number its unique characteristics. As part of
this discussion we reviewed the "tricks" for
determining divisibility by 2, 3, 4, 5, 6, 8 and
9, focusing on the reasons these techniques work.
In most cases both the method and the explanation
had been seen before, but not for 3 and 9. We
promised to explain these later. We continued the
divisibility unit with a discussion of large
primes. After drawing students' attention to the
website for the "Great Mersenne Prime Search"
(www.mersenne.org/prime.htm) we discussed the
largest prime known to date and challenged
students to estimate the number of digits in 2" -
1. We also discussed the relation between prime
factors and greatest common divisors and least
common multiples. In the schools these topics are
typically taught by writing down the lists of
factors (respectively multiples) of the two
numbers and then identifying the greatest (resp.
least) common entry by inspection. The connection
to prime factors was new to the students.

Following this discussion we focused on rational
and irrational numbers. We began by reviewing the
conversion from fractions to decimal expressions.
We considered why a fraction of integers always
results in a terminating or repeating decimal. We
studied which fractions produce terminating
expressions. We discussed the relationship
between the length of the repeating decimal
pattern and the denominator of the fraction. We
then discussed examples of numbers that are not
rational. This included a proof that [the square
root of]2 is irrational. We included a
probabilistic argument to show that (essentially
in terms of the Lebesgue measures of the sets)
there are more irrational numbers than rational
numbers.

The unit on rational and irrational numbers was
followed by a short and simple discussion of
modular arithmetic. We did this partly because it
was fun, and partly because it allowed us to give
the promised explanation of the tricks for
divisibility by 3 and 9.

At the end of the course we had some time to
discuss counting techniques (essentially a very
basic introduction to permutations and
combinations) followed by a simple discussion of
probability.

5. MATHEMATICS AND IMAGINATION

There is a wonderful richness in this material,
even if approached in very simple form. The
discussion of infinity is a case in point. It is
easy to find good geometric examples to stimulate
students' imaginations. Because they are not
visual, we do not easily associate numbers with
imagination. Nevertheless, there were two places
in the course where the idea of infinity played a
particularly important role in igniting students'
imaginations.

The first appearance of the idea of infinity came
in the context of the exploration of number
sequences. It is one thing to think of a number
sequence in terms of its pattern rule (the rule
that generates the sequence); it is quite another
to think of the outcome, the infinite sequence,
as an object in its own right.

We have found that Hilbert's (well-known)
Infinite Hotel is an excellent way to get
students' imaginations going around the idea of
infinity. We started the discussion with the
infinite hotel and one infinite bus. The seats in
the bus are numbered, as are the hotel rooms. To
effect orderly transfer of the bus passengers to
the hotel, the hotel manager had to decide on a
single instruction to the passengers: "find the
room whose number corresponds to the number on
your seat". Once the importance of a single rule
that can be applied to all the passengers had
become clear, we changed the problem by bringing
an additional single-occupant car on the scene,
followed later by additional infinite buses. In
each case the single, or at least finite,
instruction came in the nature of a formula, and
so constituted a natural instance of simple
algebra. The other virtue of the infinite hotel
story is that it brought out very naturally some
of the nonintuitive features of infinite
cardinals. How many infinite buses will fill the
hotel? Two? Five? Infinitely many?

Infinity appeared a second time when we discussed
non- terminating decimals and irrational numbers.
We spent at least a week exploring the
relationship between (real) numbers and points on
the number line. We located points corresponding
to infinite decimal expressions. We identified
particular irrational numbers, such as pi.

The discussion that followed this was
fascinating. Without commenting on the solution,
I asked the rest of the class to say whether the
student was right. Almost immediately there were
dissenting voices, and within minutes, the
student who had written out that very convincing
argument announced that he took it back: "The
numbers are not the same." I participated freely
in the discussion, but made sure to participate
by asking questions rather than speaking
definitively to the issue.

At the end of about 20 minutes, the students were
each given a sheet of paper and asked to
summarize and react to what they had learned over
the past week about rational and irrational
numbers. The results were revealing in that they
showed how difficult it was for students to think
of an infinite sequence as a completed entity as
opposed to a process. One student wrote: "I'm
still a little confused how 0.999. . . = 1
because 0.999. . . shrinks smaller and smaller,
getting infinitely close to one but should
theoretically never actually get there or become
one." Another student added: "I don't understand
pi as a number either. It's an expression."

In mathematics teaching we make lots of demands
on student imagination. Though all of these
demands have a potential for giving delight, they
are not all of the same order. The richest
challenges to the imagination are the ones that
bring together quite different ideas and invite
us to see them connected. Completed infinities
such as inherent in discussions of countable sets
or representations of the continuum are wonderful
for students who know little mathematics
precisely because of the way they challenge
students to extend their thinking. One student
commented at the end of the class discussion "I
had no idea mathematics could be so much like
philosophy." In a class of students who have been
accustomed to thinking of mathematics as a large
but finite set of difficult abstract facts and
formulas, that is not a bad point to get to.

6. SUMMARY

By inviting intending elementary school teachers
to teach enrichment mathematics classes for
students in grades 7 and 8 it is possible to
create a university mathematics course that is
effective for learning good mathematics, for
acquiring the beginnings of pedagogical content
knowledge, and for providing a mathematics
enrichment program that is appreciated by local
schools.

REFERENCES

1. Ball, D. L. 1991. Research on Teaching
Mathematics: Making Subject-Matter Knowledge Part
of the Equation. In: J. Body (Ed.) Advances in
Research on Teaching: Teacher's Subject Matter
Knowledge and Classroom Instruction, Volume 2,
pp. 1-48. Greenwich, CT: JAI Press.

2. Ball, D. L. 2000. Bridging
practices-Intertwining content and pedagogy in
teaching and learning to teach. J. Teacher Edu.
51(3): 241-247.

3. Ball, D. L., and D. K. Cohen. 1999. Developing
Practice, Developing Practitioners. In: L.
Darling-Hammond and G. Sykes (Eds.) Teaching as
the Learning Profession-Handbook of Policy and
Practice. San Francisco: Jossey-Bass.

4. Bass, H., Z. P. Usiskin, and G. Burrill 2002.
Studying Classroom Teaching as a Medium for
Professional Development- Proceedings of a
U.S.-Japan Workshop. Washington, DC: National
Academy Press.

5. Begle, E. G. 1979. Critical Variables in
Mathematics Education: Findings from a Survey of
Empirical Literature. Washington, DC: Mathematics
Association of America and the National Council
of Teachers of Mathematics.

6. Begle, E. G, and W. Geeslin. 1972. Teacher
Effectiveness in Mathematics Instruction
(National Longitudinal Study of Mathematical
Abilities Reports: No. 28). Washington, DC:
Mathematical Association of America and National
Council of Teachers of Mathematics.

7. Chazan, D., and D. L. Ball. 1999. Beyond Being
Told not to Tell. For the Learning of Math.,
19(2): 2-10.

8. Cohen, D. K., and D. L. Ball. 2001. Making
Change: Instruction and its Improvement. Phi
Delta Kappan, September 2001: 73-77.

9. Jonker, L. 1999. Enrichment Mathematics for
Grades 7 and 8, Parts I and II. Department of
Mathematics and Statistics, Queen's University,
Kingston, Canada. 10. Lampert, M. 2001. Teaching
Problems and the Problems ofTeaching. New Haven,
CT: Yale University Press.

11. Kennedy, M. M., D. L. Ball, and G. W.
McDiarmid. 1993. A Study Package for Examining
and Tracking Changes in Teachers' Knowledge
(Technichal Series 93-1). East Lansing, MI:
National Center for Research on Teacher Education.

12. Lampert, M., and D. L. Ball. 1999. Aligning
Teaching Practice with Contemporary K-12 Reform
Visions. In G. Sykes and L. Darling- Hammond
(Eds.), Teaching as the Learning Profession:
Handbook of Policy and Practices, pp. 33-53. San
Francisco: Jossey Bass.

13. Lampert, M., and D. L. Ball. 1998. Teaching,
Multimedia and Mathematics: Investigations of
Real Practice. New York: Teachers College Press.

14. Leung, F., and K. Park. 2002. Competent
students, Competent teachers? International J.
Educ. Res. 37: 113-129.

15. Ma, L. 1999. Knowing and Teaching Elementary
Mathematics: Teachers' Understanding of
Fundamental Mathematics in China and the United
States. Mahwah, NJ: Erlbaum.

16. Monk, David H. 1994. Subject Area Preparation
of Secondary Mathematics and Science Teachers and
Student Achievement. Economics of Educ. Rev.,
30(2): 125-145.

17. Rowan, B., F.-S. Chiang, and R. J. Miller.
1997. Using research on employees' performance to
study the effects of teachers on students'
achievement. Sociology of Educ. 70(4): 256-284.

18. Shulman, L. S. 1986. Those who understand:
Knowledge growth in teaching. Educational
Researcher. 15(2): 4-14.

19. Shulman, L. S. 1987. Knowledge and teaching:
Foundations of the new reform. Harvard Educ. Rev.
57: 1-22.

20. Shulman, L. S. 2000. Teacher development:
Roles of domain expertise and pedagogical
knowledge. J. Applied Developmental Psychology.
21(1): 129-135.

21. Stigler, J., and J. Hiebert. 1999. The
Teaching Gap: Best Ideas from the World's
Teachers for Improving Education in the
Classroom. New York: Free Press.

Address correspondence to Leo Jonker, Department
of Mathematics and Statistics, Queen's
University, Kingston ON K7L 3N6, Canada. E- mail:
Leo@...

BIOGRAPHICAL SKETCH

Leo Jonker did his doctoral work at the
University of Toronto. He is currently a
professor in the Department of Mathematics and
Statistics at Queen's University, and holds a
Queen's Chair in Teaching and Learning. His
mathematical interest is in the area of dynamical
systems. In recent years he has been interested
in the teaching of mathematics in elementary and
high schools, in teacher preparation, especially
for the elementary school level, and in teaching
and learning in university courses.

Copyright PRIMUS Jul/Aug 2008

(c) 2008 Primus : Problems, Resources, and Issues
in Mathematics Undergraduate Studies. Provided by
ProQuest LLC. All rights Reserved.
Story from REDORBIT NEWS:
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Published: 2008/09/05 03:00:24 CDT

© RedOrbit 2005
*****************************************************

--
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL  62901-4610
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[John Clement-5]

Today's NY Times science section has an important article which shows that
student ability to estimate without calculation is a strong predictor of
success in math.

<http://www.nytimes.com/2008/09/16/science/16angi.html?_r=1&ref=science&oref
=slogin>

If the link does not work go to
www.nytimes.com then click on science times.

Gut Instinct's Surprising Role in Math
By NATALIE ANGIER
Published: September 15, 2008
You are shopping in a busy supermarket and you're ready to pay up and go
home. You perform a quick visual sweep of the checkout options and
immediately start ramming your cart through traffic toward an appealingly
unpeopled line halfway across the store. As you wait in line and start
reading nutrition labels, you can't help but calculate that the 529 calories
contained in a single slice of your Key lime cheesecake amounts to
one-fourth of your recommended daily caloric allowance and will take you 90
minutes on the elliptical to burn off and you'd better just stick the thing
behind this stack of Soap Opera Digests and hope a clerk finds it before it
melts.

One shopping spree, two distinct number systems in play. Whenever we choose
a shorter grocery line over a longer one, or a bustling restaurant over an
unpopular one, we rally our approximate number system, an ancient and
intuitive sense that we are born with and that we share with many other
animals. Rats, pigeons, monkeys, babies - all can tell more from fewer,
abundant from stingy. An approximate number sense is essential to brute
survival: how else can a bird find the best patch of berries, or two baboons
know better than to pick a fight with a gang of six?

When it comes to genuine computation, however, to seeing a self-important
number like 529 and panicking when you divide it into 2,200, or realizing
that, hey, it's the square of 23! well, that calls for a very different
number system, one that is specific, symbolic and highly abstract. By all
evidence, scientists say, the capacity to do mathematics, to manipulate
representations of numbers and explore the quantitative texture of our world
is a uniquely human and very recent skill. People have been at it only for
the last few millennia, it's not universal to all cultures, and it takes
years of education to master. Math-making seems the opposite of automatic,
which is why scientists long thought it had nothing to do with our ancient,
pre-verbal size-em-up ways.

Yet a host of new studies suggests that the two number systems, the bestial
and celestial, may be profoundly related, an insight with potentially broad
implications for math education.

One research team has found that how readily people rally their approximate
number sense is linked over time to success in even the most advanced and
abstruse mathematics courses. Other scientists have shown that preschool
children are remarkably good at approximating the impact of adding to or
subtracting from large groups of items but are poor at translating the
approximate into the specific. Taken together, the new research suggests
that math teachers might do well to emphasize the power of the ballpark
figure, to focus less on arithmetic precision and more on general reckoning.


"When mathematicians and physicists are left alone in a room, one of the
games they'll play is called a Fermi problem, in which they try to figure
out the approximate answer to an arbitrary problem," said Rebecca Saxe, a
cognitive neuroscientist at the Massachusetts Institute of Technology who is
married to a physicist. "They'll ask, how many piano tuners are there in
Chicago, or what contribution to the ocean's temperature do fish make, and
they'll try to come up with a plausible answer."

"What this suggests to me," she added, "is that the people whom we think of
as being the most involved in the symbolic part of math intuitively know
that they have to practice those other, nonsymbolic, approximating skills."

This month in the journal Nature, Justin Halberda and Lisa Feigenson of
Johns Hopkins University and Michele Mazzocco of the Kennedy Krieger
Institute in Baltimore described their study of 64 14-year-olds who were
tested at length on the discriminating power of their approximate number
sense. The teenagers sat at a computer as a series of slides with varying
numbers of yellow and blue dots flashed on a screen for 200 milliseconds
each - barely as long as an eye blink. After each slide, the students
pressed a button indicating whether they thought there had been more yellow
dots or blue. (Take a version of the test.)

Given the antiquity and ubiquity of the nonverbal number sense, the
researchers were impressed by how widely it varied in acuity. There were
kids with fine powers of discrimination, able to distinguish ratios on the
order of 9 blue dots for every 10 yellows, Dr. Feigenson said. "Others
performed at a level comparable to a 9-month-old," barely able to tell if
five yellows outgunned three blues. Comparing the acuity scores with other
test results that Dr. Mazzocco had collected from the students over the past
10 years, the researchers found a robust correlation between dot-spotting
prowess at age 14 and strong performance on a raft of standardized math
tests from kindergarten onward. "We can't draw causal arrows one way or
another," Dr. Feigenson said, "but your evolutionarily endowed sense of
approximation is related to how good you are at formal math."

The researchers caution that they have no idea yet how the two number
systems interact. Brain imaging studies have traced the approximate number
sense to a specific neural structure called the intraparietal sulcus, which
also helps assess features like an object's magnitude and distance. Symbolic
math, by contrast, operates along a more widely distributed circuitry,
activating many of the prefrontal regions of the brain that we associate
with being human. Somewhere, local and global must be hooked up to a party
line.

Other open questions include how malleable our inborn number sense may be,
whether it can be improved with training, and whether those improvements
would pay off in a greater appetite and aptitude for math. If children start
training with the flashing dot game at age 4, will they be supernumerate by
middle school?

Dr. Halberda, who happens to be Dr. Feigenson's spouse, relishes the work's
philosophical implications. "What's interesting and surprising in our
results is that the same system we spend years trying to acquire in school,
and that we use to send a man to the moon, and that has inspired the likes
of Plato, Einstein and Stephen Hawking, has something in common with what a
rat is doing when it's out hunting for food," he said. "I find that deeply
moving."

Behind every great leap of our computational mind lies the pitter-patter of
rats' feet, the little squeak of rodent kind.