# Project Learning in Mathematics

## Program Level

### Mathematics senior high (Grade 10-12)

#### Program goals for senior high mathematics

• The learners have to independently apply mathematical concepts to familiar and unfamiliar situation
• The learners have to make precise and logical use of mathematical language
• The learners have to perform mathematical operations confidently with both speed and accuracy
• Concretize and symbolize problems from everyday situation
• The learners have to interpret results of an analysis, draw conclusions and then make predictions
 Grade 10 Sequence, series and set theory Algebra, Variables and functions Geometrical analysis and measurement Probability and data analysis Grade 11 Sequence, series and set theory Algebra, Variables and functions relationship Introduction to calculus, Probability and data analysis Grade 12 Sequence, series and set theory Linear Algebra, matrices and vectors Variables and functions Calculus, integration and differentiation of geometrical and trigonometric functions Advanced probability and statistical data analysis.

## Unit Level

Program: Senior high mathematics

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Course: Linear Algebra

Course objectives:

By the end of this unit, the learner should be able to:

1. Reduce matrices to echelon form
2. Find a rank of a matrix
3. Calculate both inverse and determinant of a square matrix (2 × 2)
4. Solve a given set of equation by:
• Reduction to echelon form
• Method of determinant
• Inverse matrix form
• Determine the length of a given vector
• Add given vectors in the Rn
• Determine whether or not a given equation is linearly independent or linearly dependent
• Define the terms vector space, vector subspace, spanning sets, and dimension of a vector space
• Define and give example of linear transformation
• Determine the kernel, image, rank and nullity of a linear operator

## Topic outline

Matrices: reduction to echelon form, rank of a matrix, determinants and inverses of a 2 × 2 and a 3 × 3 matrix. Systems of linear equation: solution by reduction to echelon form, method of determinants (Crammer’s rule), inverse matrix method (2 × 2).

Vectors: co-ordinate systems, vectors and their components, length of a vector, addition of a vector in Rn, linear dependence and independence of vectors, planes and line in R3, vector spaces and subspaces: definition, subspace, spanning sets, basis and dimension, Linear transformation, kernel and image, rank and nullity.

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## Description and list of instructional material

• Resource books: these include the list of all the course books selected for this course. The students could also use any additional book that is relevant to the course
• Technological aids: these include scientific calculators that are used for matrix and vector related calculations, and computer software programs for teaching vectors and matrices.

## Required instructional activities

One important instructional activity for this course is conducting the lessons using computer software program, mat lab. The students will have to be familiarized with using this software. This can be done during the process of instruction as they cover each of the topics. Since this is a grade k12 unit, it is also expected that most of the students should have interacted with the software from other previous units.

## Evaluation plan

Evaluation for this unit will incorporate both summative and formative evaluation. Two continuous assessment tests will be administered. The first one will be done after completion of half of the course content and it will cover the topics completed by then. The second continuous assessment test will be done after completion of every topic. Assignments for each topic will also be given after completion of the topic to test the level of understanding of that topic. For summative evaluation, an end of the course examination will be done at end of the course.

## Critical analysis

This section provides a critical analysis of the curriculum documents created in parts I and II above as important elements of instruction in mathematics for enhancing effective learning in students. Coming up with the program goals for senior high mathematics needs to observe a selection criteria. To enable curriculum developers and instructors to select objectives, one important thing that needs to be considered is the needs of the society and the needs of the learners. This formed the basis of the program objectives stated in section I above.

The students are part of the society and they too have needs. It is expected that whatever knowledge they gain out of a program, be utilized to solve societal problems. The students have to know the contribution that mathematics provides in the understanding of natural phenomena. There are so many mathematical concepts that are directly applicable in everyday life. These concepts have to be presented clearly to the students in order to develop an understanding of nature and its activities. The students have to be able to be developing a skill of problem solving using mathematical techniques. This means that students need to know how mathematics can help them in decision making (Bloom, 1986).

Another important factor to consider is how mathematics can make the students appreciate the culture of their present society. One way to achieve this is through the use of advanced technologies in the teaching and learning process. The students also need to be prepared for their vocational and further studies. Mathematics is one of the core subjects that have broad application in vocational studies. This therefore means that communicating mathematical ideas is an important achievement from a senior high school mathematics program. The above mentioned program goals, strictly meet this criteria for a comprehensive program goals.

A good program is one that has to stimulate the students to develop interest in the course and hence be motivated to learn. This is achieved in the set objectives for the program. The program has to distinguish between facts that are relevant and facts that are irrelevant. The primary focus should be presenting materials that meet the needs of the both the society and the students (Gerald, 1999). The above mathematic program not only helps achieve this objective but also stimulates curiosity in the student; an important aspect for motivating students.

The objectives for the program are behavioral. They state the intended outcome that is expected from the learners once they complete their learning experience in practical and measurable terms. Each of the objectives uses an action verb to describe exactly what the learners will be doing to demonstrate attainment of the objective. The objectives clearly describe the conditions under which the learners must be subjected to in order to demonstrate achievement of the each goal. Since the goals are for an entire program, they have to be stated in general terms. Even though they describe a specific behavior, they are fairly wide in scope.

Instructional objectives of the linear algebra unit are more specific. They measure immediate behaviors that do not need to be developed over a long period of time. It should take just a lesson or two for a student to demonstrate attainment of the objectives. The instructional objectives are based on the topics that need to be covered and not an entire learning program. But they are also made with reference to the program objectives.

The unit, linear algebra, is essentially important for the students to learn because it has a wide range of areas where it is applied. Computer linear programming and economics are the two major areas that concepts from this unit are applied. The curriculum therefore is one that would help students appreciate the unit especially those who intended to develop a career in these two areas. There are also other areas in which linear algebra prepares students for. However these areas are not as major as the two mentioned. Since mathematical techniques in algebra are generally used for solving equations, knowledge gathered from the unit is applied in problem solving. In algebra, there is a general emphasis in creating ways of representing mathematical relationships including quantities and functions.

The instructional activities need to develop certain important skills in the students. This depends with the nature of activity. Some activities are done individually while others are done in groups. The activities mentioned above for the selected course are done both by individual students and as a group. The ones done individually focus on building operational and decision making skills. It is expected that the students undertaking the unit use technological teaching aids that will familiarize them to operating computers and related devices (Kerr, 1996). Assignments that are given at the end of each class are expected to be discussed in groups and completed individually. Such an activity is important for building good relationships among the students and a spirit of co-operation.

The sequence of topics as shown in the scope and sequence chart creates a connection in the relationship between the topics. This is essentially important for the gradual development and attainment of the intended learning outcomes from the students. It also helps the students not to have confusion caused by the inability to figure out how one topic is related to another. The sequence of topic is designed to lead the students to achieve each of the program objectives. There is coherence in the sequence and these helps the student to know that the knowledge they gain deepens as they proceed with the program. It also helps the students to know that their ability to apply concepts of mathematics in familiar and familiar situations continues to expand.

Lastly, an evaluation plan provides the curriculum developers and instructors with an assertion that the program goals have either been achieved or not. Curriculum documents must give provision for evaluation. This is the most effective way to determine the effectiveness of a curriculum or a learning program. Evaluation has to be carried out consistently. This is because there is need to identify sections of the program that need to be changed in order to maintain the initial focus and achievement of the program goals (Joint Committee on Standards for Educational Evaluation, 1994). The curriculum documents in parts I and II above ensure that evaluation is conducted consistently through assignments, continuous assessment tests, and final examinations.

## References

1. Bloom, B. (1986). Taxonomy of Educational objectives: the classification of educational goals. New york: David Mackay Company.
2. Gerald, B. (1999). Curriculum development: a text book for students. London: McMillian Publishers.
3. Joint Committee on Standards for Educational Evaluation. (1994). Program evaluation standards: how to assess evaluations of educational programs. Thousand Oaks: Sage Publication.
4. Kerr, J. (1996). Changing the curriculum. London: University of London Press.
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