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In the Jamaican education system, mathematics has always been included in the core curriculum, that is, the subject mathematics is taught to all students at both the primary and secondary levels. What is it that is about mathematics why you think it is made a compulsory subject? (150-200 words)
Recent theories of learning have focussed on the learner and the need for teachers to deliver lessons that are student-centered. Why should the learner be the main focus of any classroom setting, in particular the mathematics classroom? (150-200 words)
(c) Much emphasis in the learning of school mathematics has been placed on the understanding of the subject.
(i) What, for you, is mathematical understanding?
(ii) Discuss, using your own experience and relevant literature, why it is important for learners to engage in mathematical thinking and develop mathematical understanding. (300 - 350 words)
(d) Bearing in mind your discussion in part (c), design a three part lesson plan (introduction, development & closing) to illustrate how you would facilitate your students in constructing their own mathematical understanding of a named topic/concept to a particular grade level. (You should include in your plan, the activities the students will be involved in.)
(e) Outline examples of tasks and questions you could use to determine whether or not your students had achieved an understanding of the topic/concept named in part (d). Also justify why these will help you to assess their understanding.
Students need to be able to navigate their lives in this complex modern world. This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. The study of mathematics is necessary for this. It has also led to many a fulfilling career. Actuaries, rocket scientists, physicists, and engineers all have credentials in math. Computer scientists were generally led to their field by an interest in higher mathematics. Scientists use probability and statistics to construct and evaluate studies, and usually express their findings in mathematical form (Grischy, n.d.).
On the other hand, students who are not interested in these careers often wonder what use is the quadratic equation or the Pythagorean Theorem to them. The fact is that mathematics trains and disciplines your mind. Just as the point of reading books is not to memorize vocabulary words, the point of math is not to memorize formulas. Students who are strong in mathematics develop an ability to analyze and identify relationships and patterns. They also develop critical and logical thinking skills and a great aptitude for problem solving. These skills prepare any student for their future. According to the College Board of America, students who take basic algebra and geometry in high school have about an 80 percent chance of ending up in college, regardless of race, religion, or family income ("Math: What is it good for?", 2010).
In the traditional approach to teaching, the teacher lectures and the students listen. The students work individually on assignments, and cooperation is discouraged. Student-centered teaching methods shift the focus of activity from the teacher to the learners. These methods include active learning, cooperative learning, and inductive teaching and learning.
According to Doyle (n. d.), learners need to feel some control of their learning. It is human nature to seek control over what is happening to us. A student who feels they have no control over the learning process may exert their control by choosing not to take part.
The traditional approach to teaching mathematics may lead to students passing exams, but it often sacrifices conceptual understanding for procedural knowledge. The ultimate goals of learning mathematics in school is that students become confident in their ability to do mathematics, become problem solvers, and learn to communicate and reason mathematically (Leonelli, 2007). Students should also see some beauty of mathematics and learn to value it. A student centered approach is best to accomplish this end.
"A problem is not necessarily solved because the correct answer has been made. A problem is not truly solved unless the learner understands what he has done and knows why his actions were appropriate."
-William A. Brownell, The Measurement of Understanding (1946)
(i) Mathematical understanding is the comprehension of mathematical concepts, operations, and relations. It occurs when new mathematical information is suitably connected to existing knowledge, when relationships are found between different mathematical ideas, and when links between schoolwork and aspects of everyday life are formed.
(ii) I have taught Mathematics in a private school in which students were repeating the CXC examinations due to poor performance. There were basically eight months available with which to cover the syllabus adequately and practise with past exam questions. The students were not interested in much abstract theory, but wanted to be shown algorithms so they could apply it to the problems which are tested.
I taught mathematics in a linear way, first touching on remedial skills, explaining a topic, doing a worked example and practicing with a series of questions in increasing order of difficulty. Once the students mastered one topic, we would move to another. I knew I could get the majority of my students to pass the exam with a respectable grade, but I felt something was lacking when they couldn't remember how to factorize an expression once we moved on to another math topic. Math class was also an abysmal affair when the "class clown" wasn't there to crack the occasional joke and liven up the period. I have learnt that many students struggle with mathematics. It is therefore essential that we understand what teachers can do to break this pattern. Procedural skills that are not eventually accompanied by some form of understanding are brittle and easily lost.
The beauty and usefulness of mathematics both stem from a mathematical thinking, which distinguishes it from simply the accumulation of information, or the application of practical skills. Mathematical thinking, or reasoning, connects the parts of mathematics to each other. Developing ideas, exploring phenomena, justifying results, and using mathematical conjectures help students see and expect that mathematics makes sense. A program of mathematics teaching that omits these connections causes the subject to become a matter of following a sequence of procedures and replicating examples without thought. People who have not been taught mathematical reasoning are not being saved from something difficult, but rather, being deprived of something easy (Raimi, 2002).
Class: Form 3 Mathematics
Topic: Trigonometric ratios
Overview: In this activity students use an induction method to understand the concepts of the trigonometric ratios. Students should have prior knowledge of working with ratios as well as decimal fractions.
Chalk, blackboard, pencils, rulers, paper, calculators
Each group will receive a set of triangles cut out of coloured cardboard. There are three pairs of similar triangles, all of which have right angles. An angle, Î± is labeled on each triangle.
Present the students with a large right-angled triangle with one angle marked.
Remind them about the hypotenuse from Pythagoras' theorem and show them the opposite and adjacent sides in relation to the marked angle.
Practise labeling right-angled triangles from the board
Discuss the concept of ratio
Explain that the session involves investigating of the ratios of pairs of sides of right-angled triangles
Divide class into groups of 3 students.
Hand out one a set of triangles to each group.
Each student takes two triangles. They measure the sides and calculate the ratios opp/hyp, adj/hyp and opp/adj for each triangle.
Members of the group compare and and discuss their findings. Teacher circulates the room, monitoring and assisting where necessary.
When every group is completed, one member of each group briefly reports their findings. They should have concluded that triangles which have the same ratios also have the same angles (similar triangles). Some groups may also have noted that the ratio increases as the angle increases for opp/hyp and opp/adj but it decreases as the angle increases for adj/hyp. Others may may have noticed that the sine of an angle is the cosine of its complement.
Explain to students that these ratios have special names:
opp/hyp is sine of the angle (sin)
adj/hyp is cosine of the angle (cos)
opp/adj is tangent of the angle (tan)
These ratios are used in a branch of mathematics called trigonometry, which deals with triangles.
Briefly discuss the uses of trigonometry such as map reading, surveying, navigation etc.
The final presentation is a manner of assessing understanding. In addition to this, the use of open-ended questions will encourage students to reflect on what they have understood and discovered.
Why did you do this?
What made you think of this?
What would have happened if you did this instead?
This group did something different. Is their way better?
Would you do anything differently?
Can we improve our findings?