Karim Ezzatkhah

Qualify Exam





1-The statement of issue:


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:


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.


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 studentís 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 Herbartís 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 studentís 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, ďNeither the hand nor the mind alone would amount to much without aids and tools to perfect themĒ. 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 ď reading groupĒ 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 Wertheimerís (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. Therefore we are not teaching mathematics to our students to think mathematically and produce new concepts and procedures. NCTM has accounted ten standards for mathematics content and procedure (1988). Moreover James Hiebert (1986) has divided mathematics curriculum to two main parts:

1-     Conceptual knowledge.

2-     Procedural knowledge.

Comparing NCTM standards with other industrialized countryís 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 teacherís 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 teacherís 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 childrenís learning could modify their instruction to help students construct knowledge. Mestre (1994) argues that ď perhaps the behaviorist approach might be better described as training not educatingĒ. 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).


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 ď Education pour la vie par la vieĒ.

William James (1892) in his book ďtalks to teachersĒ has mentioned, psychology is science so it has rules but teaching is art. Therefore teachers can apply educational psychologyís 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, studentís beliefs, teacherís beliefs and classroom environment are effective at teaching.

According my own experience and teachers views from different countries also educatorís 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 teacherís 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 constructivistís 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, studentís 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 studentís 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 studentís 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 teacherís 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 teacherís behaviors that are thought to support studentís 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 teacherís 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).


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:

  1. Children construct their own mathematics knowledge.
  2. Mathematics instruction should be organized to facilitate children construct of mathematics knowledge.
  3. Childrenís development of mathematical ideas should provide the basis of the sequencing topics of instruction.
  4. 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).

  1. Teacher knowledge of how mathematical content is learned by their students.
  2. Problem solving as the focus of instruction.
  3. Teacher access to how students are thinking about specific problems.
  4. 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 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 studentís low achievement problem with high proportion (Secada, 1992).




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 todayís 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, studentís 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, teacherís 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 teacherís 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 studentís 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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