**Karim Ezzatkhah**

**Qualify Exam**

**C&I499**

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**Teaching**

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**1-The statement of issue****:**

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Teaching and learning are very closed to each other, that; sometimes is not possible to divide their rules and regulations. Because; teaching initially based on the learning situation; that generally set-up by teachers to interact by students.

**Definition:** I would like
to focus on teaching mathematics and define teaching with regarding different
views in mathematics understanding. Teaching as a transmitting information
(Skill approach)-teaching to understand the meaning of mathematics concepts
(conceptual approach)-teaching to think and solve mathematical problems
(problem solving approach) and finally fostering mathematical power of students
to understand mathematics meaningfully and better reasoning, connecting
mathematical concepts; communicating to understand and solve problems (NCTM
1989). Some of teachers agreed that; teaching is a collection of purposeful
activities, which occur with awareness. Mestre (1994) Argued; Students should
be assisted in linking new knowledge to previously learned knowledge and in
building a hierarchical structure where ancillary concepts are placed below,
and linked to major concepts. In any teaching situation we have three
components:

1-Teacher.

2- Students.

3- Subject Matters.

Interaction between these three components results teaching. Teaching methods vary during interaction among these factors and particularly the role of teacher and students in the teaching situation. Therefore; there are two major roles for teacher and students in any teaching situation.

1- Students centered teaching refers to students being active and construct their own knowledge by teacher guidance.

2- Teacher centered teaching refers to teacher being active in the classroom to teach students. In this case students are passive instead teacher is active and try to transmit knowledge to students.

3-

**Philosophical background**:
Plato and his student Aristotle were the fist teachers have established
Scholastic method in teaching. Most of teacher still is applying this method in
the teaching situation. Herbart the German philosopher mentioned that student’s
previous knowledge could be as a base to construct new knowledge. He has
suggested five steps in teaching situation for teachers.

1- Preparation.

2- Presentation.

3- Association.

4- Generalization.

5- Application.

Most of lesson plan formats are still following Herbart’s modal. Realist philosophers believe that teaching should base on the practical situation and give opportunity to children to discover realities.

In eighteen century Jean Jacque Russo French philosopher in his famous book (Emil) has mentioned. Humans will educate by nature, surroundings and other persons. He says children should be free and choose own interests to learn and understand.

The last group of philosophers that influenced in teaching is pragmatists. William James says; teaching is art and teacher should be as artist. John Dewy; the famous American educator established some rules and beliefs in teaching. Some of them are as below:

1- Teaching should be child centered.

2- Teacher is as a guidance to let students interact in their natural and social environment.

3- Class management should be democratic and have to consider students rights.

4- Students have to be active in all teaching situation.

5- Teacher should consider student’s everyday experiences. This is a French expression (Education pour la vie par la vie)

6- The school is a small society and students should learn their social behaviors in this society.

7- In elementary school, there is not gap between different subject matters. Teacher should connect all school subjects in teaching situation and give chance to students to understand the relationship between science, mathematics, reading, arts and other subjects.

**Psychological Background**: We
have several views about teaching and learning in most of schools of
psychology. Recently in the last century educational psychologists focused on
two major schools of psychology 1- Behaviorism, and 2- constructivism. These
two major views has influenced also in mathematics Teaching.

** 1-Behaviorist view**: In this view students to learn a complex process is to
break down the process into component parts, learn each part then, put together
the components to obtain desired behavior.

** 2-Constructivist**: that represents the latest model of child-centered
learning (Learman & Dengate1995). From the recent theory (constructivist
view) an investigative approach has emerged that involves purposeful inquiry
based on meaningful instruction and, thus, can foster all aspects of mathematical
power (Baroody 1998). According to Piaget, logical knowledge consists of the
cognitive structures that assimilate and accommodate to all incoming
information (Piaget and Inhelder 1960).

** What we are teaching in
Elementary School**? We are teaching
three kinds of subject matters in elementary Schools. In other word mathematics
curriculum has three different meanings:

1-
Students learn mathematics
and science, because they need to solve problems, communicate with mathematics
language and also they want to connect ideas and students need reasoning in
problem solving situation and so forth.

2-
Students memorize arithmetic
facts meaningfully and rules. For example three times nine equal 27 or in the
number sequence each number has an indicated place and value.

3-
Students get skill and
mastery in writing mathematical symbols like 1, 2, 3, 4 and so forth, also
drawing geometrical shapes like square.

Therefore in teaching situation we have to apply a lot of
techniques to let students understand mathematics and use their understanding
in new situation. For example in first grade students after understanding the
meaning of numerals they want to write and read by symbols. So they need to
have mental image and a motor plan to writ any of the symbols. Therefore when
writing them, they need practice and reinforcement to settle down the
behaviors. It means the teacher should apply behaviorist approach. Regardless a
lot of usefulness of new approaches in cognitive domain, in teaching situation
commitment to one particular approach is hard and sometimes impossible.

**2-Brief Review of Relevant Researches**:

Ann L. Brown (1994)
has different definition of teaching and also using manipulitives in teaching
situation. Based on Francis Bacon (1623), she argues, “Neither the hand nor the
mind alone would amount to much without aids and tools to perfect them”. Jerome
Bruner (1965) suggested three stages in teaching situation and classroom
activities (enactive, pictorial and symbolic experience). Using manipulitives
and today high levels technology in teaching situation is a fundamental idea
and help student to understand better and deeply.

Ann L. Brown (1994) also has suggested a new teaching
approach. Her experience in constructivist domain includes creating community
of learners in mathematics education. In order to work in community of learners
(COL); She emphasizes in term of method of teaching The Reciprocal Teaching.
The Reciprocal Teaching began as a method of conducting “ reading group” once
an established ritual of the grade-school class. I remember this method as
cooperative teaching method; we had in our traditional schools, but with
different organization. It also reminds me Scholastic method of teaching in
ancient Greek. It seems the idea of constructivism is not a new idea in cognitive
domain and many teachers experienced those methods in the past centuries.

* What is the
problem*? In the real classroom
situation looking to one of the three major components of mathematics education
(teaching, learning and curriculum) without considering the effects of the
others is impossible. Because teaching, learning and curriculum are
interrelated in practical situation. After Wertheimer’s (1959) visited some
mathematics and science classrooms and mentioned that, our mathematics teaching
is not preparing students as productive thinking mathematically or
scientifically. Teachers, educators, researchers tried to cathagorize the main
problems in teaching mathematics. For example Schoenfeld (1988) has indicated
that students did not pay attention to mathematical problem as multi solution
problems or they look at (key word) in word problems instead of reading,
analyzing, reasoning, connecting known parts of the problem and producing new
concepts.

1-
Conceptual knowledge.

2-
Procedural knowledge.

Comparing NCTM
standards with other industrialized country’s curriculum, Mathematics
conceptual knowledge in K-12 are similar, but with different perspective and
length of teaching. Researchers found that mathematics problems are multi-solution
problems, and students come to school with a lot of mathematical knowledge
Peterson (1994). Teachers should consider their knowledge and give them
opportunity to construct their knowledge based on what they already know.

* Are teachers professional*?
The main problem is that Teachers do teach mathematics in a way they have been
taught; and they have strong believes that make restriction to change ideas
about teaching procedures (Cooney 1996). He also argued, that reform at
classroom level is in its infancy. Teachers believes are the main issue in term
of changes in there teaching; but as Guskey (1986) suggests; changes in
teacher’s believes and attitudes follow changes in students outcomes which, in
term, follow changes in teacher practice.

* Mathematics Teachers Training program:* Thomas Cooney (1994) mentioned three major departments are
involved in teachers training program (mathematics, Education and Psychology)
in the Universities and teachers training colleges. While it may be a case that
many teachers need to know more mathematics, pedagogy or psychology, it is also
the case that the acquisition of more knowledge by itself is sufficient to
ensure change in the classroom Teaching situation). In the USA are a lot of
different programs for in-service and pre-service teachers programs. Indeed,
Weiss, et al. (1990) these programs are quite effective in enhancing teacher’s
knowledge of mathematics. There are a lot of plans and projects and also
suggestions by researchers about teachers In-service programs. Peterson,
Fennema and carpenter (1989) found, Teachers informed about recent research on
children’s learning could modify their instruction to help students construct
knowledge. Mestre (1994) argues that “ perhaps the behaviorist approach might
be better described as training not educating”. We should consider training is
part of educational program. Teachers apply a lot of strategies in teaching
situation and training is strategy.

Peterson (1988) in a research study found
that, in teachers training classes who learned about research findings, could
teach better than others. This study Clearfield research studies effect
teachers and they apply research findings in classroom activities. Peterson
(1994) has emphasized elite education and she gives us example from his first
grade daughter and Ball (1993, 1991) classes about Sean one of her students who
claimed and invented a new idea about odd and even numbers and also Feynman
Nobel prize winner and his discoveries in physiques. It seems Peterson is still
has commitment to last educational reform. Because NCTM standards insisted to
equity as a reform principle.

* Textbook*: Researchers argue, commercially available textbooks do
little to promote in term of conceptual change or skillful procedures in solving
mathematical problems Mestre (1994).

**
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** 3-Major Approaches:**

Teaching mathematics has roots in human-being history and
has varied in different times according to the educational goals and teaching
situation. Teachers, Philosophers and scientists have mentioned so many methods
and techniques for teaching. In the recent years child-centered teaching
presented by John Dewey was popular in the United States and contemporary in
other countries. Dewey and his followers in progressive education programs have
defined teaching as guiding students to construct their knowledge. He said:
teachers should prepare a real life situation at schools, to learn children
living in a democratic society. Pestalozy one of the famous European educator
mentioned “ * Education pour la vie par la vie*”.

William James (1892) in his book “talks to teachers” has
mentioned, psychology is science so it has rules but teaching is art. Therefore
teachers can apply educational psychology’s rules in their classroom. By
considering definitions of teaching and learning, teaching is a complex
phenomenon, multi-dimensions and has several components.

1-
Interaction among learners
and learning environments.

2-
Interaction between students
and teacher.

3-
Interaction among peer
students and also small groups of students.

4-
Preparing reliable resources.

5-
Choosing appropriate strategy
and commitment to this strategy during teaching.

6-
Applying appropriate
technology.

7-
Applying techniques and
procedures in different phases of teaching.

8-
Lesson planning, indicating
content and content analyzing.

9-
Indicating the level of
instruction according to Bloom taxonomy or other documents.

10- Evaluation.

In addition of these factors, there are several other
factors affect teaching and instruction and teachers should be familiar. For
example, student’s beliefs, teacher’s beliefs and classroom environment are
effective at teaching.

According my own experience and teachers views from
different countries also educator’s opinions; any individual teacher has
freedom in the classroom to deal-with those factors. Teachers do choose their
strategies and teaching techniques by own decision. It seems their previous
knowledge, experiences and teacher’s pre-service and in-service programs have
great role in this decision. Therefore there are a lot of methods, approaches,
strategies and teaching techniques based on educational psychology learning
theories and other educational issues as philosophy of education or teachers
experiences and beliefs from old civilization until this century; that teachers
apply in teaching situation.

* Models of
Teaching*: In the recent years of the
last century; researchers, educators and teachers build-up many teaching models
and have recommended to novice teachers and practitioners. Most of teaching
models based on learning theories of two major institutes of educational
psychology, behaviorist and cognitive domains. These models have advantages and
disadvantages. So teachers have options to apply appropriate model in their
teaching. For example most of mathematics teachers will apply problem-solving
model in mathematics classes. Bruce Joyce and Marsha Weil (1992) categorized
the models of teaching in four major families.

1- Social family.

2-
Information-Processing family.

3-
Behaviorist Family.

4-
Individual teaching (tutoring) Family.

In each family are some models, that teachers can apply the
appropriate model in her/his teaching situation. Apparently teachers are
willing to apply their own model rather than models that have been recommended.

**Four Major
approaches in Teaching Mathematics**:

*1-
** Skills Approach*:
Teachers are teaching students for skills mastery. These teachers believe
prescribed curriculum in teaching mathematics. They are trying to transit
rules, formulas, basic arithmetic facts to students. They are expecting quick
answer from students and students try to memorize instead of meaningful
understanding.

*2-
*__Conceptual Approach__**: **Teachers are teaching for understanding. In this approach
teacher is as a guide to ensure that students memorized meaningfully. Students
are not enough active in learning situation for example David Ausubel (1968)
meaningful verbal learning theory. He emphasized linking students previous
knowledge to new knowledge by advance organizers.

*3-
*__Problem Solving approach:__** **In this approach students are active and teacher is as a
consultant beside students problem solving activities. Teacher helps students
to device solution strategies and students got opportunities to construct their
mathematical knowledge.

*4-
*__Investigative approach:__** **Based on NCTM standards and social constructivist’s views,
investigative approach developed and grown-up by mathematics teachers in the
recent years of the last century. The main assumption is that basic skills,
concepts, and inquiry process are all necessary for mathematical power.
Therefore teachers teach for skills mastery, understanding and mathematical
thinking. It means teachers teach mathematics in a meaningful fashion.

Some teachers also some researcher translated mathematical power as capacity to apply mathematical knowledge to new situation. Investigative approach

Emphasizes deep understanding of mathematics, student’s
engagement in the process of mathematical inquiry and positive attitude, active
position to learn mathematics to apply for solving problems in different
situation.

__More about teaching
mathematics:__

American students in K-12 classes have enrolled with different backgrounds. Most of students are from five major Ethnic groups (Whit, African American, Asian American, Hispanic, and American Indian). Students in each ethnic group have different characteristics in term of socio-economic statues, language and cultural issues. Regarding to increasing the number of minority by year 2000 approximately 30% to 40% of total enrolment in American schools would be minorities (Secada, 1992). Therefore the issue is most important and considerable, because of its reflection to the society. For example in New York area 45.3 of over 16 years old were not eligible to work in the society (Philips, 1990). It means less than half of Americans in that area were unable to fit into productive part of this society. Moreover unfortunately, all assessments results particularly Achievement tests have emerged a big gap between students mathematics learning and understanding in K-12 Classes (NCTM 1989).

Robert B. Davis (1992) recommended many solutions to existing problems in mathematics education. He is arguing about students that have been observed in mathematics class that, students with a large amount of disparities have been participated in the class. Teacher was trying to impose formal algorithm to solve mathematics problem, in stead some of students had their own strategy to solve problem that teacher did not give any value or attention to student’s creative performances.

This kind of diversity is not only problem in the United States; in other countries we also can meet similar problems in mathematics education. For instance In Iran we have diverse society and students attend in the classes with cultural, social, and economical differences. Of course students differences in many socio-economic, cultural and language area are important and problematic. Therefore these problems reflect on teaching mathematics in K-12 classrooms.

To solve student’s disparities problem in mathematics classes, there are a large body of research in teaching mathematics in the U.S.A (Secada, 1992). Instead, some country follow well developed countries procedures or their teachers create several approaches according to their practical teaching experiences. These creativities are a topic of discussion in mathematics teacher’s education classes. Overall there is a lot of teaching methods that, teachers examined in the recent years. These methods of teaching are based on research findings, but they are not last solutions. Teaching to divers students has many difficulties. For example, there are a lot of variables that is not possible to control all of them in the same time. Children in a randomly assigned mathematics class have different socio-economic positions, different ethnic groups and also different languages. Therefore to eliminate all these problems is almost impossible, but by employing some models or method of teaching can reduce the influences of such factors from teaching environment.

** Direct instruction**:

Direct instruction is a highly structured form or teacher’s behaviors that are thought to support student’s engagement in and learning mathematics. For example Everston et al. (1980) found a consistent pattern of relationships between teacher behaviors and enhanced students achievement in mathematics. Timss (1996) study found that most of Japanese mathematics teacher’s use 90% of time to direct instruction instead American teachers only use 50% of the class time as active teaching. The best example for direct instruction is Active Mathematics Teaching (AMT) originally developed by Good and Grouws (1977 &1979). Direct instruction is effective for conveying a large amount of highly structured mathematics concepts for different student backgrounds (Grant 1989).

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** Cognitive Guided
Instruction**:

Cognitively Guided Instruction (CGI) has been found effective for enhancing first-grade students achievement on basic skills and problem solving (Carpenter, Fennema, Peterson, Chiang, & Loef, 1990). This approach based on four assumptions:

- Children construct their own mathematics knowledge.
- Mathematics instruction should be organized to facilitate children construct of mathematics knowledge.
- Children’s development of mathematical ideas should provide the basis of the sequencing topics of instruction.
- Mathematical skills should be taught in relation to problem solving and understanding of children.

Cognitively guided instruction is based on four interlocking principles (Carpenter, Fennema, Peterson, Chiang, & Loef, 1990).

- Teacher knowledge of how mathematical content is learned by their students.
- Problem solving as the focus of instruction.
- Teacher access to how students are thinking about specific problems.
- Teacher decision- making based on teachers knowing how their students are thinking.

CGI program has been examined in diverse learners classes, and some researchers as Peterson et al (1990) have proposed that CGI shows promise for the teaching of diverse learners.

__Grouping:__

Grouping students in mathematics classes is more popular way among teachers to meet diverse students needs. Teachers are grouping students either along lines of ability or in cooperative groups, (Slavin, 1990). In elementary and often middle school, for instance whole classes may be created along lines of ability (between class ability groups) or teachers may form ability groups within their classes. These kinds of groupings have advantages and disadvantages. The alternative to ability groups that is commonly proposed by some researchers is the use of small cooperative groups of heterogeneous ability. Many studies of small groups and their processes have used direct instruction or active mathematics teaching for the development of a mathematics lesson, but they have grouped children heterogeneously for some part of that lesson (Swing & Peterson, 1982).

Most of these approaches are thought to affect, first enhanced mathematics achievement for diverse population and second, closing the achievement gaps between those population. Over all researchers found that these methods work in diverse students classes and can solve student’s low achievement problem with high proportion (Secada, 1992).

**4-Conclusion**:

After second curriculum movement in the United
States (1950) teachers focused on teaching mathematics meaningfully and
purposefully. The main aim of the movement was to prepare students for high
standard and sophisticated industry and technology. The prosperous of the
attempts are today’s progress and US high technology in the world. Besides
these successful mathematics and science education in the higher education,
this country is still suffering from lack of nation wide mathematical
knowledge. Therefore the first principal of the NCTM standards is Equity
Principal. To overcome the problem it is necessary to choose appropriate
teaching strategy in term of teaching mathematics in the K-12. In the other
hand from 1970 a lot of researches have been done in this country to help
teachers to teach mathematics professionally (Kilpatrick 1992). Moreover Timss
study (1996) showed that USA students in all three levels of K-12 mathematics
study are under average. Indeed Timss study was warning to many countries
including USA, to realize that, student’s mathematical knowledge is not
adequate to live better in 21 century.

The question is:
Why school age children are not willing to understand mathematics
concepts, procedures and apply mathematics procedures in other situations? In
order to find answer, we have to look at some of educational, psychological,
social, political, scientific, and economical aspects of education system and
their influences in mathematics teaching in k-12 classes.

Virtually there are many positive points to consider and
hopefully challenge to solve existing problem. For example technology
availabilities, scientific standards, political supports, school funds and
so-forth are positive points to rely-on, and teach mathematics, with purpose of
understanding all students mathematics meaningfully. There are also some
constraints as segregation, teacher’s negative beliefs about students
understanding of mathematics, parents lack of cooperation in the teaching phase
and etc. But problem is solvable. The next question is: **what are the
solutions?**

I am not going to
prove that there is only one solution to the educational problems. Because
there are a lot of solutions to the problems. One of the major factor as a
solution is mathematics-teaching approach. Actually some of the mathematics
teachers are not teaching effectively?** **We
all as a teacher believe that, teachers have a lot of freedom to apply any kind
of teaching approach to teach mathematics in their classroom. Why are not they
using the best approach such as investigative approach?

1- teachers have to believe that the investigative approach
is the best and is appropriate to the mathematics instructional goals or
guarantees the students achievement. In order to conceive our mathematics
teacher, we should have some fixed agreements or some rules. Unfortunately
researchers are following different views from cognitive domain and all
research findings are reliable sources but how is it possible to apply those
findings in teaching fields are not enough clear. For example in cognitive
domain we are looking at gestalt educational psychologists findings,
social-constructivist, radical constructivist and Piaget followers and many
other views and recommendations without enough teacher’s training in all these
approaches

2- Teaching situations are not a fixed and unvaried
environment. Therefore it is impossible to teach mathematics within a specific
method or technique. For example when teacher teaches the numerical symbols in
the first grad, he/she has to give practice (skills approach) beside student’s
mental image and correct motor plan from the symbol.

We have to remember William James definition from teaching,
he says, teaching is art. It means teaching has only one rule. ** Teach meaningful
and applicable**.

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**References:**

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