ES-342: TEACHING OF MATHEMATICS

Answer the following questions:

i) Explain factors affecting mathematics curriculum at secondary level.

(250 words)

INTRODUCTION :

In education, there is frequently a mismatch between the intended, the implemented and the attained curriculum (Cuban, 1993). The intended curriculum is the one prescribed by policy makers, the implemented curriculum is the one that is actually carried out by teachers in their classrooms, and the attained curriculum is the one learnt by students (Howson & Wilson, 1986). Part of the mismatch is due to the fact that teachers and students work on more limited goals than those proposed by curriculum developers, teacher educators, writers of syllabuses, and textbook authors (Handal, 2001). Mathematics teachers, for example, are concerned only with students acquiring facts and performing skills prescribed by the syllabus rather than being concerned about broader educational goals.

FACTOR AFFECTING MATHEMATICS CURRICULUM :

Factors affecting curriculum alignment and change in mathematics education have been extensively discussed by Anderson and Piazza (1996), Clarke (1997), Memon (1997), and Mumme and Weissglass (1991). In the context of a school based curriculum development project, Clarke (1997) identified 12 factors that appeared to influence the change process:

(a) the reform movement in general;

(b) the principal and school community;

(d) the spirit of collegiality, collaboration, and experimentation;

(e) the gradelevel team of teachers;

(f) innovative curriculum materials;

(g) the in-service program;

(h) external support personnel;

(i) the researcher acting as a participant observant and critical friend;

(j) outcomes valued by the teacher;

(k) day-to-day conditions under which teachers work; and

(l) teacher knowledge.

Memon (1997) suggested a more comprehensive list of factors affecting curriculum change that are grouped as curricular, instructional, and organisational factors and reproduced in Table 1. It is clear that curriculum change is a complex process and while there are many resource and support factors that appear to influence change, it is apparent that any successful reform will need to take into account mathematics teachers’ beliefs about the intended, the implemented, and the attained curriculum.

Factors Affecting Educational Reform in Mathematics Education :

Mathematics teachers’ beliefs can play either a facilitating or an inhibiting role in translating curriculum guidelines into the complex and daily reality of classroom teaching (Haynes, 1996; Jackson, 1968, 1986; Koehler & Grouws, 1992; Sosniak, Ethington, & Varelas, 1991). If teachers hold beliefs compatible with the innovation then acceptance will be more likely to occur. However, if teachers hold opposing beliefs or perceive barriers in enacting the curriculum, then lowtake up, dilution and corruption of the reform will likely follow (Burkhardt,Fraser, & Ridgway, 1990). Prawat (1990) has affirmed that teachers can be either conveyances of, or obstacles to, change. No matter how much is expected of them to support reform, it is always possible that their views do not coincide

with those underpinning the reform and therefore become a major impediment in that effort. Hart (1992) adds that when teachers consider new tasks to be trivial and superficial they tend to mistrust other innovations.

Unfortunately, innovations can create disunity because groups of ‘resisters’ are formed (Fullan, 1993). Hall (1997) explained that any innovation represents an encounter of two cultures in which conflict of values and goals needs to be minimised and hopefully blended. Aborted reforms affect teachers’ morale causing stress, cynicism, burnout syndromes, anxiety and scepticism (Fullan, 1993; Sinclair & McKinnon, 1987).

The high rate of failure of educational innovations (Fullan, 1993) has drawn researchers to look more closely at teachers’ beliefs as a significant mediator in curriculum implementation. Fullan and Stegelbauer (1991) have stated that it is very unlikely that teachers can modify their teaching practices without changing their values and beliefs. Change can also be cosmetic, that is, a teacher can be using new resources, or modify teaching practices, without accepting internally the beliefs and principles underlying the reform (Fullan, 1983). Burkhardt,Fraser, and Ridgway (1990) warn that even innovative programs that boast of having attained changes on a large scale, have accomplished these changes with a ‘travesty’ of the explicit and original principles underlying the innovation.

This mismatch between curriculum goals and teachers’ belief systems is a factor that affects current curriculum change in mathematics education. Anderson and Piazza (1996, p. 54) argued that “teachers, who must be the agents of change, are products of the system they are trying to change” and proposed that teachers’ feelings, beliefs, and values that are opposite to constructivism are a barrier to reform in mathematics education. Sosniak, Ethington, and Varelas (1991) have described the complexity of this mismatch in the context of changing beliefs, teaching approaches and resources in the United States in the 1950s and 1960s. These authors argued that the success of innovative mathematics programs was constrained by inconsistencies between the content of new materials and the working requirements of that content by teachers. The degree of change was limited, due to the fact that the beliefs about mathematics underlying the innovation did not match teachers’ beliefs. In addition, these programs required new roles and teachers’ responsibilities that were too demanding. Not only did teachers feel unfamiliar with the content change, they had to align to a new way of teaching.

According to Martin (1993a, 1993b) curriculum implementation approaches that do not consider teachers’ beliefs have a temporary life. Unfortunately, many educational reforms in mathematics have had a top-down approach (Kyeleve & Williams, 1996; Martin, 1993a, 1993b; Moon, 1986) that did not take into account mathematics teachers’ beliefs and belittle the fact that “the ultimate fate of an innovation would seem to depend upon user decisions” (Doyle & Ponder, 1977, p. 3). These reforms were often disseminated using a traditional approach in which teachers were presented with a prepared product and a rigid set of procedures to follow. The major cause of failure of these programs was their negligence in failing to take into account teachers’ pedagogical knowledge and

beliefs as well as the contexts in which these teaching behaviours occurred (Knapp & Peterson, 1995). In other words, curriculum change in the last several decades relied on the simplistic assumption that teachers will, machine-like, alter their behaviours because they were simply told what was good for them and for their students (Grant, Hiebert, & Wearne, 1994). Current approaches to curriculum implementation need to rely on more

realistic assumptions about teachers’ beliefs, recognising that it is difficult to change teaching styles because changing practices demands a process of unlearning and learning again (Mousley, 1990). It also needs to be recognised that change will cause feelings of discomfort that can be unpleasant and intimidating (Martin, 1993b).

__________________________________________________________________________________

ii) Differentiate Inductive and Deductive methods of teaching Mathematics.

INTRODUCTION :

Students have different intellectual capacities and learning styles that favour or hinder knowledge accumulation. As a result, teachers are interested in ways to effectively cause students to understand better and learn. Teachers want to bring about better understanding of the material he/she wants to communicate. It is the responsibility of the educational institutions and teachers to seek more effective ways of teaching in order to meet individual’s and society’s expectations from education. Improving teaching methods may help an institution meet its goal of achieving improved learning outcomes.

Teaching methods can either be inductive or deductive or some combination of the two.

The inductive teaching method or process goes from the specific to the general and may be based on specific experiments or experimental learning exercises. Deductive teaching method progresses from general concept to the specific use or application. These methods are used particularly in reasoning i.e. logic and problem solving. To reason is to draw inferences appropriate to the situation.

Inferences are classified as either deductive or inductive.

For example, “Ram must be in either the museum or in the cafeteria.” He is not in the cafeteria; therefore he is must be in the museum. This is deductive reasoning. As an example of inductive reasoning, we have, “Previous accidents of this sort were caused by instrument failure, and therefore, this accident was caused by instrument failure.

The most significant difference between these forms of reasoning is that in the deductive case the truth of the premises (conditions) guarantees the truth of the conclusion, whereas in the inductive case, the truth of the premises lends support to the conclusion without giving absolute assurance. Inductive arguments intend to support their conclusions only to some degree; the premises do not necessitate the conclusion.

Inductive reasoning is common in science, where data is collected and tentative models are developed to describe and predict future behaviour, until the appearance of the anomalous data forces the model to be revised.

Deductive reasoning is common in mathematics and logic, where elaborate structures of irrefutable theorems are built up from a small set of basic axioms and rules. However examples exist where teaching by inductive method bears fruit.

EXAMPLES: (INDUCTIVE METHOD):

A) Ask students to draw a few sets of parallel lines with two lines in each set. Let them construct and measure the corresponding and alternate angles in each case. They will find them equal in all cases. This conclusion in a good number of cases will enable them to generalise that “corresponding angles are equal; alternate angles are equal.” This is a case where equality of corresponding and alternate angles in a certain sets of parallel lines (specific) helps us to generalise the conclusion. Thus this is an example of inductive method.

B) Ask students to construct a few triangles. Let them measure and sum up the interior angles in each case. The sum will be same (= 180°) in each case. Thus they can conclude that “the sum of the interior angles of a triangle = 180°). This is a case where equality of sum of interior angles of a triangle (=180°) in certain number of triangles leads us to generalise the conclusion. Thus this is an example of inductive method.

C) Let the mathematical statement be, S (n): 1 + 2 + ……+ n =. It can be proved that if the result holds for n = 1, and it is assumed to be true for n = k, then it is true for n = k +1 and thus for all natural numbers n. Here, the given result is true for a specific value of n = 1 and we prove it to be true for a general value of n which leads to the generalization of the conclusion. Thus it is an example of inductive method.

EXAMPLES: (DEDUCTIVE METHOD):

A) We have an axiom that “two distinct lines in a plane are either parallel or intersecting” (general). Based on this axiom, the corresponding theorem is: “Two distinct lines in a plane cannot have more than one point in common.” (Specific). Thus this is an example of deductive method.

B) We have a formula for the solution of the linear simultaneous equations as and(general). The students find the solutions of some problems like based on this formula (specific). Thus this is an example of deductive method.

Aims of Teaching Mathematics
Education is imparted for achieving certain ends and goals. Various subjects of the school curriculum are different means to achieve these goals. So with each subject some goals are attached which are to be achieved through teaching of that subject. According to Sidhu (1995) the goals of teaching mathematics are as below:

To develop the mathematical skills like speed, accuracy, neatness, brevity, estimation, etc.
To develop logical thinking, reasoning power, analytical thinking, critical-thinking.
To develop power of decision-making.
To develop the technique of problem solving.
To recognize the adequacy or inadequacy of given data in relation to any problem.
To develop scientific attitude i.e. to estimate, find and verify results.
To develop ability to analyze, to draw inferences and to generalize from the collected data and evidences.
To develop heuristic attitude and to discover solutions and proofs with the own independent efforts.
To develop mathematical perspective and outlook for observing the realm of nature and society.

DIFFERENTIATE BETWEEN INDUCTIVE METHOD AND DEDUCTIVE METHOD :

INDUCTIVE METHOD	DEDUCTIVE METHOD
1. It gives new knowledge	1. It does not give any new knowledge.
2.It is a method of discovery.	2. It is a method of verification.
3. It is a method of teaching.	3. It is the method of instruction.
4. Child acquires first hand knowledge and makes use of it.	4. Child gets ready made information and information by actual observation
5.It is a slow process.	5. It is quick process
6. It trains the mind and gives self confidence and initiative	6. It encourages dependence on other sources
7. It is full of activity	7. There is less scope of activity in it.
8. It is an upward process of thought and leads to principles.	8. It is a downward process of thought and leads to useful results.

CONCLUSION :

We can say that inductive method is a predecessor of deductive method. Any loss of time due to slowness of this method is made up through the quick and time saving process of deduction. Deduction is a process particularly suitable for a final statement and induction is most suitable for exploration of new fields. Probability in induction is raised to certainty in deduction. The happy combination of the two is most appropriate and desirable.

There are two major parts of the process of learning of a topic: establishment of formula or principles and application of that formula or those principles. The former is the work of induction and the latter is the work of deduction. Therefore, friends, “Always understand inductively and apply deductively” and a good and effective teacher is he who understands this delicate balance between the two. Thus: “his teaching should begin with induction and end in deduction.”

__________________________________________________________________________________

iii) Select a topic inMathematics and develop lesson plan mentioning the essential steps involved in it. (1000 words)

Step #1 - Objectives and Goals :

In the Objectives section of my lesson plan, write precise and delineated goals for what i want my students to be able to accomplish after the lesson is completed.

Be Specific. Use numbers where appropriate.

To define my lesson's objectives, consider the following questions:

· What will students accomplish during this lesson?

· To what specific level (i.e. 75% accuracy) will the students perform a given task

in order for the lesson to be considered satisfactorily accomplished?

· Exactly how will the students show that they understood and learned the goals of your lesson? Will this occur through a worksheet, group work, presentation, illustration, etc?

Additionally, I will want to make sure that the lesson's objective fits in with their district and/or state educational standards for their grade level.

By thinking clearly and thoroughly about the goals of my lesson, i will ensure that I making the most of my teaching time.

Step #2 - Anticipatory Sets :

In the Anticipatory Set section, I will outline what I will say and/or present to mystudents before the direct instruction of the lesson begins.

The purpose of the Anticipatory Set is to:

· Provide continuity from previous lessons, if applicable

· Allude to familiar concepts and vocabulary as a reminder and refresher

· Tell the students briefly what the lesson will be about

· Gauge the students' level of collective background knowledge of the subject to help inform my instruction

· Activate the students' existing knowledge base

· Whet the class's appetite for the subject at hand

· Briefly expose the students to the lesson's objectives and how I will get them to the end result

To write my Anticipatory Set, consider the following questions:

· How can I involve as many as students as possible, piquing their interests for the subject matter to come?

· How should I inform my students of the lesson's context and objective, in kid-friendly language?

· What do the students need to know before they can delve into the lesson plan itself and direct instruction?

Anticipatory Sets are more than just words and discussion with my students. I can also engage in a brief activity or question-and-answer session to start the lesson plan off in a participatory and active manner.

Step #3 - Direct Instruction :

Direct Instruction could include reading a book, displaying diagrams, showing real-life examples of the subject matter, using props, discussing relevant characteristics, watching a movie, or other hands-on and/or presentational steps directly related to my lesson plan's stated objective.

When determining your methods of Direct Instruction, consider the following questions:

· How can I best tap into the various learning modalities (audio, visual, tactile, kinesthetic, etc.) to meet the learning style preferences of as many students as possible?

· What materials (books, videos, pneumonic devices, visual aids, props, etc.) are available to me for this lesson?

· What relevant vocabulary do I need to present to my students during the lesson?

· What will my students need to learn in order to complete the lesson plan's objectives and independent practice activities?

· How can I engage my students in the lesson and encourage discussion and participation?

Think outside the box and try to discover fresh, new ways to engage your students' collective attention to the lesson concepts at hand.

Avoid just standing in front of my students and talking at them. Get creative, hands-on, and excited about my lesson plan, and your students' interest will follow.

Before I move on to the Guided Practice section of the lesson, check for understanding to ensure that my students are ready to practice the skills and concepts I have presented to them.

Step #4 - Guided Practice :

In the Guided Practice section of my written lesson plan, outline how my students will demonstrate that they have grasped the skills, concepts, and modeling that I presented to them in the Direct Instruction portion of the lesson.

While I circulate the classroom and provide some assistance on a given activity (worksheet, illustration, experiment, discussion, or other assignment), the students should be able to perform the task and be held accountable for the lesson's information.

The Guided Practice activities can be defined as either individual or cooperative learning.

As a teacher, I should observe the students' level of mastery of the material in order to inform my future teaching. Additionally, provide focused support for individuals needing extra help to reach the learning goals. Correct any mistakes that I observe.

Step #5 – Closure

Closure is the time when I wrap up a lesson plan and help students organize the information into a meaningful context in their minds. A brief summary or overview is often appropriate. Another helpful activity is to engage students in a quick discussion about what exactly they learned and what it means to them now.

Look for areas of confusion that I can quickly clear up. Reinforce the most important points so that the learning is solidified for future lessons.

It is not enough to simply say, "Are there any questions?" in the Closure section. Similar to the conclusion in a 5-paragraph essay, look for a way to add some insight and/or context to the lesson.

Step #6 - Independent Practice :

Through Independent Practice, students have a chance to reinforce skills and synthesize their new knowledge by completing a task on their own and away from the teacher's guidance.

In writing the Independence Practice section of the Lesson Plan, consider the following questions:

· Based on observations during Guided Practice, what activities will my students be able to complete on their own?

· How can I provide a new and different context in which the students can practice their new skills?

· How can I offer Independent Practice on a repeating schedule so that the learning is not forgotten?

· How can I integrate the learning objectives from this particular lesson into future projects?

Independent Practice can take the form of a homework assignment or worksheet, but it is also important to think of other ways for students to reinforce and practice the given skills.

Get creative. Try to capture the students interest and capitalize on specific enthusiasms for the topic at hand.

Once I receive the work from Independent Practice, I should assess the results, see where learning may have failed, and use the information I gather to inform future teaching. Without this step, the whole lesson may be for naught.

Step #7 - Required Materials and Equipment :

In the Required Materials section, consider:

· What items and supplies will be needed by both the instructor and the students in order to accomplish the stated learning objectives?

· What equipment will I need in order to utilize as many learning modalities as possible? (visual, audio, tactile, kinesthetic, etc.)

· How can I use materials creatively? What can I borrow from other teachers?

Keep in mind that modeling and the use of hands-on materials are especially effective in demonstrating concepts and skills to students. Look for ways to make the learning goals concrete, tangible, and relevant to students.

The Required Materials section will not be presented to students directly, but rather is written for the teacher's own reference and as a checklist before starting the lesson.

Step #8 - Assessment and Follow-Up :

Learning goals can be assessed through quizzes, tests, independently performed worksheets, cooperative learning activities, hands-on experiments, oral discussion, question-and-answer sessions, or other concrete means.

Most importantly, ensure that the Assessment activity is directly and explicitly tied to the stated learning objectives.

Once the students have completed the given assessment activity, I must take some time to reflect upon the results. If the learning objectives were not adequately achieved, I will need to revisit the lesson in a different manner.

Student performance informs future lessons and where I will take my students next.

IGNOU b.ed STUDENT

ASSIGNMENT FOR 2012 B.ed

B.ed 1st YEAR [ ES - 342 ]