Prospective Elementary School Teachers in a Mathematics
Department
Leo Jonker
The challenge of mathematical stewardship
University research mathematicians regard themselves as stewards of their discipline.
They are passionate about its development, enthralled with its beauty ([15], [9], [21]),
eager to share their excitement with students, and concerned about the place of
mathematics in the world. Their passion for the development of mathematics is amply
demonstrated in the volume of published high level research and in the frequency of
research conferences; to share the excitement with students, national research funding
agencies provide ample support for undergraduate students who show promise; most of
them also promote mathematics by funding math fairs, math camps, and mathematics
contests; and there are countless books highlighting mathematical ideas and problems for
the general reader.
Among all the options, ensuring high quality mathematics instruction in elementary
schools is probably the best way to ensure that mathematics is valued and understood by
the average citizen; and yet this seems to be the one area where mathematics departments
manage to do very little. In fact, it is surprisingly difficult to determine what anyone can
do to ensure high quality instruction. On the one hand, it is clear that teachers play a
crucial role in students’ academic development [20]; on the other, it appears that taking
more university mathematics courses has little or no effect on the quality of mathematics
teaching, especially at the elementary level ([1], [5], [6], [10] ). Even for high school
teachers, Monk showed that “gross measures of teacher preparation (such as degree
levels, undifferentiated credit counts, …) offer little useful information for those
interested in improving pupil performance" ([17] ).
Mathematical knowledge for teachers
The Study of Instructional Improvement research project, begun in 1999 at the University
of Michigan, seeks to understand the impact of various school improvement measures on
instruction and student performance in elementary schools. One of its principal
investigators, Deborah Ball, who has studied the mathematical preparation for teachers
for a long time, believes that the question of how much mathematics prospective teachers
need to know should be replaced by the question of what kind of mathematical
knowledge they need ([1], [2], [3], [7], [8], [13], [14]). This rephrasing of the question
dates back at least to Lee Shulman ([22], [23]), who introduced the concept “pedagogical
content knowledge” to supplement “content knowledge” and “knowledge of pedagogy”.
“Pedagogical content knowledge” is the specialized knowledge of a subject needed by a
teacher in her practice of teaching. It is a type of discipline content knowledge, but it is
not necessarily the content knowledge acquired in a regular university classroom.
Refining this idea in the context of mathematics, Hill, Rowan, and Ball (2005) coined the
term “Mathematical Knowledge for Teaching” (MKT) for the kind of mathematics
teachers need to know. They describe this as consisting of two components. The first of
these, Subject Matter Knowledge, is further divided into “Common Content Knowledge”
and “Specialized Content Knowledge”. Most people know how to multiply two two-digit
numbers – this is common content knowledge. Teachers have to know how to do this,
but so do many others. A teacher, however, when faced with a non-standard solutions to
a multiplication of two-digit numbers needs to be able to interpret what the student was
doing, and whether doing it that way generalizes. This is specialized content knowledge
required only of teachers – a type of applied mathematics for teaching. The second
component of MKT is Pedagogical Content Knowledge. The authors also divide this
into two components: “Knowledge of Content and Students” and “Knowledge of Content
and Teaching”. Knowledge of Content and Students includes familiarity with the kinds
of ways students typically misunderstand a question or a concept. Knowledge of Content
and Teaching refers to the ability to choose good examples to start a particular
discussion; to choose problems that will take students deeper into a discussion; when to
follow up a student suggestion, and when to postpone that to a later occasion. Each of
the four components of Mathematical Knowledge for Teaching involves deep
understanding of mathematics, even if much of it is quite different from “standard”
subject knowledge.
Hill, Rowan and Ball ([10]) designed an instrument to measure teachers’ MKT. They
applied this test to a group of teachers, asked them about their own mathematical and
pedagogical training, and measured the progress of their grade 1 and grade 3 students.
They discovered that there was almost no correlation between MKT scores and the
amount of training received by the teachers; and they found that MKT scores were a
significant predictor of student gains at both grade levels.
One might conclude from this that the role universities can play, especially in the training
of elementary school teachers, is minimal. The performance of Chinese teachers,
described in Liping Ma’s Ph.D. thesis ([16]) seems to support this as well. In Ma’s
terms, their “profound understanding of fundamental mathematics” is significantly better
than that of a group of U.S. teachers used for comparison. Yet, the Chinese teachers went
from high school directly into normal school to learn the teaching trade there, and later on
the job, much as did our elementary school teachers fifty years ago. Given that we are
not likely to go back to that system, however, we should pay closer attention to the
research literature when we design courses intended for prospective teachers; we would
do well to work toward mathematical knowledge for teaching.
It could be argued that this job should be left to our faculties of education. There is no
question that for a mathematics department to work closely with a faculty of education is
likely to benefit both units as well as their students, but leaving the task entirely to
education faculties is not a good idea. Typically students spend just one year in an
education faculty. This is not enough time to prepare prospective elementary school
teachers in all the subjects and to introduce them into theories of education, classroom
management skills, and so on. Furthermore, inasmuch as members of mathematics
departments care how mathematics is taught and how it is viewed, they should be willing
to play a role in ensuring a good outcome. In fact, I find that teaching a course focused
on basic mathematics taught appropriately for prospective elementary school teachers,
raises questions that help me reflect on all my other teaching.
A mathematics course that works well
So what kind of university mathematics course will provide students with something like
what Ball and her co-authors refer to as “Mathematical Knowledge for Teaching”? One
of the conclusions we can draw from the research is that a good course manages to blur
the traditional distinction between content and method even as it keeps the careful
development of mathematical content in focus. Over the past eight years I have
developed a course in the Department of Mathematics and Statistics at Queen's
University that, I believe, includes many of the right elements. 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. This enrichment program is
provided for the benefit of grade 7 and 8 students in the city’s schools. The university
students form pairs at the start of the course; each pair visits a local school for one 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 textbooks ([11], [12]) used for the course are two enrichment manuals 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 below; a detailed syllabus is available on the StepAhead website
http://www.queensu.ca/stepahead/ .
An essential feature of the course is that it presents mathematics that the students are able
to understand at some depth. Mathematics that is not well understood cannot function in
on-the-spot classroom decisions so critical to good teaching. Most of the course centers
on problems, since that is often the best way to engage students ([18], [19]). The
problems are carefully selected to be accessible and interesting to the students and so that
together they provide a gradual development of the course’s key mathematical concepts.
The pace of the course is relaxed. It is essential that students be given lots of time to
discuss problems in small groups, to present solutions ideas to the rest of the class, and to
examine each others’ misconceptions. These misconceptions are as important as correct
solutions, for two reasons: In discussing opposing conceptions students are forced to
seek more fundamental forms of understanding (reasoning, explanation, proof) to settle
their differences; and, once my students become teachers, their teaching skills will be
needed most when they are confronted with their students’ misunderstanding.
Opportunities to explain solutions are also crucial in other ways. The ability to articulate
a solution is valued in its own right as a communication skill, and as a way to develop
appropriate mathematical language ([19] , [4]). The public presentation of solutions also
helps the class to see that mathematics problems can often be approached in several
different ways – that there is often no unique solution. The ‘level’ of the mathematics
will sometimes seem low compared to a typical university course, but the engagement is
intense, and for my students the mathematics is relevant, not only to their future teaching,
but more immediately to their enrichment teaching assignment. For most of the students
in the course this is precisely the kind of mathematics that will optimize their engagement
and their development as practitioners of mathematics.
I try, to a very great extent, to avoid algebraic methods in the course. This is easy to do,
for when a student suggests a solution that uses a formula remembered from high school
days, I counter by asking how that solution can be understood by students in grade 7 or 8,
who are only just learning some very basic algebra. A second, and deeper, reason for
discouraging algebra is that many of my students did not learn their mathematics well
when they were in high school. For such students, algebra can too easily become a set of
completely formal procedures that obscure what is really happening. Explaining,
discussing, and proving are essential elements in the course.
It is very important for the success of the course that students know they have a very
good chance of success. Most of my students are majoring in the humanities, and many
have had a bad experience with mathematics in high school, or even in elementary
school, and have come to dislike mathematics. Sometimes they are deeply anxious about
the subject. In some cases, the reasons for a student’s dissatisfaction are actually
commendable. Thus one of them commented in a questionnaire completed at the start of
the course:
“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."
This student wants to understand, but feels she has not been given the chance. She would
not take a university mathematics course if she did not feel that there was a chance that
this time she would understand; and taking a ‘standard’ university mathematics course
would almost certainly reinforce her feeling that mathematics is not something she can
grasp.
The most important, and most attractive, element of the course is the opportunity for
students to teach enrichment classes to young teens. Most of the students in the course
would not take it if it were not for that feature. Students who want to become elementary
school teachers are not often motivated by abstraction and generalization – they want to
work with children, and nothing is more particular and concrete than a child. The
enrichment program is also the course feature that plays the most important role in
ensuring that at least a good part of what the students learn fits the definition of
mathematical knowledge for teaching. Class discussions are frequently re-focused by the
question, “but how will my enrichment students understand this?” and students’ efforts to
learn material are always efforts to understand it well enough so that it can be explained.
Participating schools are asked to select students from grades 7 and 8 who are deemed
most ready for a mathematics enrichment program, and invite these to take the
enrichment courses. We suggest 25% of a class as a guideline. Schools are allowed to
combine students from the two grades into a single enrichment class. Our program
alternates between a number pattern focus one year and a geometry program the next so
that grade 7 students taking the enrichment course one year can come back the next as
grade 8 students. This means, incidentally, that the university course could be thought of
as a pair of distinct courses which prospective teachers could take in successive years.
The advantage of working with a select group of 7 and 8 students is that my students get
to teach a relatively motivated group of teens without having to deal with difficult class
management issues which they are not yet competent to handle. This allows a strong
focus on the mathematics itself and the on the pedagogical questions raised in its
presentation.
The fact that the students are paired off for their enrichment classes is also important. On
the one hand, since I do not have the resources to monitor their teaching, the two
instructors combine their insights to prepare lesson plans. On the other, they monitor
each other’s work and provide ready feedback. At the same time, their working together
to articulate their lesson plans and to assess the outcomes provides an early experience of
consultative professional development.
Curriculum
The curriculum of the university course not only alternates between number patterns and
geometry; it also varies somewhat from year to year, depending on the class and the
circumstances. The focus is not on covering a particular list of topics (though there is a
definite sequence to the material), but rather on creating the optimal classroom learning
experience. The way the material is presented and understood is more important than the
material itself. The variation between what enrichment topics get covered in one
elementary school and what is taught in another is even greater, due to a large variation in
the readiness of students in different schools, and because occasionally a school event
conflicts with a mathematics enrichment session.
To provide a picture of the course, we will describe the geometry version of the course as
it was given recently. One of the main focuses of the geometry course is to develop
students’ ability to visualize three-dimensional situations, and to analyze them using twodimensional subsystems, calculations and measurements. We do this by exploring the
Theorem of Pythagoras, similarities, polyhedra, the Euler number of a polyhedron,
regular polyhedra, areas and volumes, and areas and circumferences of disks.
The course begins with an exploratory discussion of the possible shadows cast by a disk.
From there we proceed to a problem requiring the Theorem of Pythagoras. Though it is
presented as a story (usually involving Harry Potter) it comes down to this: A sphere
circumscribes a cube, which in turn circumscribes a smaller sphere. If the diameter of the
smaller sphere is 6 cm, what is the diameter of the larger sphere? Without solving the
problem immediately (the class usually does not see a solution right away) we discuss
problem solving strategies, and, applying one of these, replace the problem with one
involving a circle around a square around a circle. We then use this problem to introduce
the Theorem of Pythagoras. We remind ourselves what the theorem claims. We collect
measured evidence using right-angled triangle drawn on graph paper, discuss the matter
of experimental error, and question whether these experiments suffice to establish the
theorem. I then get the class to work in groups to “discover” a proof of the theorem. This
leads to a discussion of inductive versus deductive evidence and the role of proof in
mathematics.
Once the Theorem of Pythagoras is well understood, and we have done some practice
problems, we return to the spheres-and-cube problem for its solution. This is followed by
a series of other problems requiring Pythagoras. These include the spider and fly in a
rectangular room, grandmother’s crochet needle stuck in a 90º turn of the central vacuum
system’s pipes, and packing candy rolls in the smallest possible rectangular box. These
problems are engaging; they require a variety of visualization skills, including the net of
the rectangular room in the spider-and-fly problem; they allow for a variety of solutions;
and they consistently take us back to the Theorem of Pythagoras. In all cases students are
encouraged to work in small groups and to present their solutions to the rest of the class.
Some of the practice problems using Pythagoras are multi-step problems in which the
Theorem has to be used repeatedly to calculate more and more lengths in a complex
figure. In these exercises the focus is not on the answer but on the sequence of steps, the
strategy, used to find the lengths. This skill will be needed later when we discuss the
number π.
Once the Theorem of Pythagoras is well understood and has been used in a variety of
situations, we turn our attention to similarity transformations in 2-d and in 3-d. These
constitute a family of transformations that tend to be neglected in the middle school
geometry program, even though we zoom in and out all the time when we look at
computer files and pictures. Finding the largest equilateral triangle that can be cut out of
a standard 8.5 × 11 inch sheet of paper is one problem that can be solved using a
combination of similarity properties and the Theorem of Pythagoras. Another problem,
assigned for homework at this point of the course, asks students to find the shape and the
measurement of the shadow cast by a 160 × 200 cm table top supporting a table lamp of
height 40 cm if the legs of the table are 80 cm. The solution to this problem becomes
beautifully simple if we use the similarity of 3-d figures.
A discussion of polyhedra, regular polyhedra, and the Euler number is next in the course.
We discuss the cube, the tetrahedron and the octahedron as examples of regular
polyhedra. We examine the profiles they produce when held at different angles, and their
cross-sections when sliced. We then begin a search for other regular polyhedra, using the
Euler number. The search ends with the identification of the dodecahedron and the
icosahedron, and sometimes with their construction out of cardboard.
The discussion of area begins with a reflection of what we mean when we speak of the
area of a figure, and how that understanding relates to formulas for areas of rectangles,
parallelograms, triangles and trapezoids. One of the problems in this unit asks students to
determine the shape and the area of the shadow cast by a rectangular wooden fence from
a light mounted high on the side of a nearby building. Other problems involve volumes
and surface areas.
The final unit of the course centers on the number π, with emphasis on its origin as the
area of a circle of radius 1. We consider Archimedes’ method of exhaustion, in which the
circle is inscribed and circumscribed by regular polygons whose areas can (in principle)
be calculated using Pythagoras and will then give us upper and lower estimates for the
number π. There are several excellent web sites that support this calculation. The
formula for the area of a circle of radius r is then obtained using our grasp of similarities;
and the formula for the circumference is found by imagining the circle cut up into many
very thin congruent wedges which are then re-assembled to form an (almost) rectangle.
The problems used to apply the formulas for the area and the circumference of a circle
include one in which I bring rolls of toilet paper to class in a package that informs the
buyer of the number of paper segments on each roll. The challenge to the class is to
calculate the thickness of the toilet paper. This wonderful problem, with a number of
different solutions, is described more fully below.
It is important in this course that the focus is not on covering a certain number of topics,
but rather on carefully choosing and implementing learning situations that will introduce
and reinforce particular kinds of understanding. It is essential as well that the instructor
imagines that she is addressing middle school students when working with her class of
would-be teachers. This is not done primarily because their understanding of
mathematics is at that level, though their misunderstanding of mathematics does often
reach back to that stage in their schooling, but because we want to discuss the
mathematics as mathematics for teaching. The question of its appropriateness for
presentation in the enrichment program is kept front and center.
An example of a class discussion
In the fall term of 2008, the final of the year was the problem that asks students to
calculate the thickness of toilet paper. Each group of 3 or 4 students was handed a roll of
toilet paper at the beginning of the activity. The class was informed that the plastic
wrapping around the toilet paper rolls indicated that there were 200 sheets of paper per
roll. Students were asked to work in groups to find the thickness of the paper. Since this
was the last problem discussed in the course, it served as a nice summary of the extent to
which the students had learned to work together and to connect mathematical ideas.
The variety of solutions presented at the end of about 40 minutes was delightful. The
simplest solution involved measuring the inner radius (the radius of the cardboard roll),
say at 2cm, and the outer radius of the roll of paper at 5cm, to calculate the area of the
circle representing the hole, and the area of the larger circle representing the roll of paper.
The difference (π× (52 – 22) ≈ 66 cm2) represented the area of the (cross section of the)
paper in the roll. The length of a single sheet of toilet paper was found to be (roughly)
10cm. For this solution, students then considered the cross section of a single sheet to
have area 10 × t cm2, where t is the unknown thickness. Since there are 200 sheets in the
roll, 200 × 10 × t = 66, so that t = 66 ÷ 2000 ≈ 0.03 cm is the thickness of the paper.
Another group of students calculated the circumference of the circle mid-way between
the inner and outer circles to get 2 × π × 3.5cm ≈ 22cm; they then divided that by the
length of one sheet of paper to conclude that ‘on average’ each layer of toilet paper in the
roll had 2.2 sheets in it. Thus there were 200 ÷ 2.2 ≈ 91 layers of paper in a roll. Since
the thickness of the roll (from inside circle to outside) was 3 cm, this meant that the
thickness of the paper was 3 ÷ 91 ≈ 0.03 cm.
Several groups related the problem to an earlier discussion about areas of parallelograms
and trapezoids. They imagined cutting the roll of toilet paper radially and then unrolling
it to get a pile of paper whose cross section had the shape of a trapezoid of height 3 cm,
and whose base and top measured 4 × π and 10 × π respectively. They then used the
formula for the area of a trapezoid, and divided that by the cross-sectional area of a single
sheet as in the first solution.
Other groups used combinations of these ideas. One student, who as a fourth year
engineering student had considerably more mathematical training than the rest of the
class, tried to think of each layer of paper as the hypotenuse of a right angled triangle
whose right angle sides were the thickness t and the length of the previous layer of paper.
This allowed her to sum a series of successive lengths and solve for t. It did not matter
that this did not work as a model for the problem. What mattered was that she was not
content to make the connections that seemed immediate to her, but instead explored more
remote possibilities.
Does the course belong in a mathematics department?
I have already suggested that as mathematicians we should be concerned about the way
mathematics is taught in the schools and the way it is perceived by the public. I have also
suggested that learning the mathematics needed for good teaching, even at the elementary
school level, cannot be accomplished during a single year in a faculty of education.
There are too many other things that need to be done during that year.
Even so, some of my colleagues may feel that the mathematics content of the course is
too elementary for a university program, and that teaching such a course would be very
uninteresting for a mathematician trained to do mathematics research. In fact, it surprises
me again and again how much fun it is to find really good problems and to create fruitful
learning situations even for mathematical ideas that have been around for more than two
millennia. The beauty of the material did not stop shining through when I first
understood it years ago. It seems instead to glow more brightly and to present new
challenge each time it is taught to a new group of students.
For students, too, seeing it for the second or third time can be very rewarding. At the end
of a recent instance of the course, one of the students, who had started out in first year
engineering with marks in the mid 70’s, and who was now in the fourth year of a science
program, commented that he had learned more in the StepAhead course than in any other
he had taken in his four years at university. It says something about the importance and
the enormous satisfaction of learning basic concepts very well.
A science course on the StepAhead model
With the support of an NSERC grant we have now also developed a science course on the
StepAhead model. This course is offered through the Physics Department at Queen’s
University and is based on experiments that extend and enrich the regular school
curriculum for grades 7 and 8. The reasons for the creation of the course are very similar
to those that led to the mathematics course. Prospective teacher tend to have a weak
understanding of physical science and they do not often choose to take a physics course
at university. The intent of the course is to give prospective teachers a chance to learn
basic physics concepts well by conducting and discussing simple demonstrations and
experiments, by forming lesson plans based on these activities, and by using these to
teach enrichment lessons to grade 7 and 8 students in 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, pages 1-48),
Greenwich, CT: JAI Press.
[2] Ball, D.L. (2000). Bridging Practices - Intertwining Content and Pedagogy in
Teaching and Learning to Teach. Journal of Teacher Education, Vol. 51, no. 3, MayJune 2000, pp. 241-247.
[3] Ball, D.L. & Cohen, D.K. (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] Barker, W., D. Bressoud, S. Epp, S. Ganter, B. Haver, and H. Pollatsek. 2004.
Undergraduate Programs and Courses in the Mathematical Sciences: CUPM
Curriculum Guide 2004. Washington, DC: MAA. Available online at
www.maa.org/cupm . Accessed June 1, 2009.
[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. & Geeslin, W. (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. & Ball, D.L. (1999). Beyond Being Told not to Tell. For the
Learning of Mathematics 19, 2.
[8] Cohen, D. K. & Ball, D. L. (2001). Making Change: Instruction and its
Improvement. Phi Delta Kappan, September 2001, pp. 73-77.
[9] Dreyfus, T. and Eisenberg, T. (1986). On the aesthetics of mathematical thought. For
the learning of mathematics, 6(1), 2-10.
[10] Hill, H.C., Rowan, B. & Ball, D.L. (2005). Effects of Teachers' Mathematical
Knowledge for Teaching on Student Achievement, American Educational Research
Journal, Vol. 42, No. 2, pp. 371-406
[11] Jonker, L. & Aguilar, D. (2007), Enrichment Mathematics for
Grades Seven and Eight (Part I - Numbers), available for download at
http://www.queensu.ca/stepahead/english/index.html
[12] Jonker, L. & Aguilar, D. (2007), Enrichment Mathematics for
Grades Seven and Eight (Part II - Geometry), available for download at
http://www.queensu.ca/stepahead/english/index.html
[13] Lampert, M. & Ball, D. L. (1998) Teaching, Multimedia and Mathematics:
Investigations of real practice. New York: Teachers College Press.
[14] Lampert, M. & Ball, D. L. (1999). Aligning Teaching Practice with Contemporary
K-12 Reform Visions. In G. Sykes & L. Darling-Hammond (Eds.), Teaching as the
Learning Profession: Handbook of Policy and Practices (pp. 33-53). San Francisco:
Jossey Bass.
[15] Le Lionnais, F. (1948/1986). La beauté en mathématiques. In F. Le Lionnais (Ed.),
Les grands courants de la pensée mathématique (pp. 437-465). Paris: Editions Rivages.
[16] Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United
States. Mahwah, NJ: Erlbaum.
[17] Monk, David H. (1994) Subject Area Preparation of Secondary Mathematics and
Science Teachers and Student Achievement. Economics of Education Review, 30(2), pp.
125-145.
[18] National Council of Teachers of Mathematics (1989), Curriculum and
EvaluationSstandards for School Mathematics, Reston, VA
[19] National Council of Teachers of Mathematics (2000), Principles and Standards for
School Mathematics, Reston, VA
[20] Sanders, William L., and Horn, Sandra P. (1998) Research Finding from the
Tennessee Value-Added Assessment System (TVAAS) Database: Implications for
Educational Evaluation and Research, Journal of Personnel evaluation in Education 12:3
247-256.
[21] Sinclair, N. (2006). The aesthetic sensibilities of mathematicians. In N. Sinclair, D.
Pimm and W. Higginson (Eds.). Mathematics and the aesthetic: New approaches to an
ancient affinity (pp. 87-104). New York: Springer.
[22] Shulman, L. S. (1986). Those who understand: Knowledge growth in
teaching. Educational Researcher, 15(2) pp. 4-14.
[23]Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.
Harvard Educational Review, 57 pp. 1-22.