EVALUATING THE PROCESS OF LEARNING MATH
One of the most difficult tasks a teacher has is evaluating what the student has learned. The evaluating process centers in three general aspects: concepts, procedures and attitudes.
From experience, the math department has come up with the following elements that should be considered when evaluating the learning process of a student: skills development to solve algorithms, concept understanding, and applications. The most important tools needed to evaluate these three main areas are: tests and quizzes, homework, projects, and the students´ attitude towards learning. Each one of these tools is analyzed as to the way it helps evaluate the students learning process.
a) Tests and quizzes:
A test may be divided into three parts. One that demonstrates the skills a student needs to develop to solve an algorithm, another where the student demonstrates that he/she understands the concepts, and the third part in which the student applies the new concepts and relates them with previous knowledge (problem solving situations and real world applications).
· When evaluating skills to solve algorithms the teacher must consider accuracy and process. Some examples of the type of problems used to evaluate accuracy are:
1) b-2b4 2)12x8 3) 8(k-2)2
4x3 4k-2
4) -27w3t4 5) 3xy + 5
-3w3t12 3xy
Working with these types of problems, the student should apply certain math rules to simplify what is given and the answer must be correct to get full credit.
Other skills require a multistep process to get the correct answer. Such problems include, for example:
a) Systems of equations with two variables which can be solved using different methods, such as: elimination, substitution, graphing, or determinants.
b) Solving absolute value equations
c) Solving quadratic equations
d) Finding the values of the six trigonometric functions given a point
e) Finding algebraic or trigonometric derivatives
For these types of problems the teacher can give credit for the process, even though the student does not get the correct answer.
· For concept understanding the student should be able to analyze the new concept and relate it to different situations in order to give a meaning based on logic. Examples of these types of problems are open questions such as:
Answer the following questions, be as clear as possible and use examples if necessary.
1. What does it mean when you are solving by substitution and at the end the variables cancel out and you get that 3 = 5 or that 3 = 3?
2. When solving by Cramer’s rule, how do you know there is no solution or an infinite number of solutions?
3. Describe solutions, the system of equations and the lines when you solve by graphing.
4. What does it mean if x and y cancel out at the same time when you are using elimination method to solve a system of equations?
The answers to these questions can be graded based on how well the student understands and relates the concept to the problem. Full or partial credit can be given to the answers.
· When solving a problem the student should be able to prove that he/she can apply the skills learned and therefore understand the concepts. In a word problem situation, the student must:
- Identify the variables of the problem (the student analyzes the problem and understands what is being asked).
- Construct equations or use formulas (the student has found a strategy to solve the problem).
- Solve the equation or formula to find an answer (the student is using mathematical skills to solve the algorithm).
- Analyze the answer, give a meaning to the solution (the student checks the answer and analyzes if it’s a logical solution to the problem).
An example is the following real world application:
The steepest railway in the world is the Katoomba Scenic Railway in Australia. The passenger car is pulled up the mountain by twin steel cables. It travels along the track 1020 feet to obtain a change in altitude of 647 feet,
i. find the angle of elevation of the railway
- How far does the car travel in a horizontal direction
In a word problem a teacher can give credit for:
a) Identifying and defining variables.
b) Drawing a picture to help solve the problem.
c) Construction of equations or well chosen formula.
d) Process to solve the equation or formula.
e) Correct answer with its analysis.
In general, quizzes have only been used to demonstrate that a student has developed the necessary skills to solve algorithms.
The math department evaluates students with two tests per quarter, each one worth 20% of the grade. A teacher can do as many quizzes as necessary with a total worth of 15% of the quarter grade. Occasionally a teacher assigns take home quizzes or group quizzes.
b) Homework
Homework is mostly graded on completion. It is important that a student practices the mathematical concept that was taught in class since one of the ways to learn is by doing. Through homework, the student develops understanding of the math concepts.
Homework should always be corrected and if there are any doubts or questions they should be answered before starting a new objective.
Each teacher has his/her own method for checking that the homework is being done and understood.
Homework is worth 15% of the quarter grade.
c) Projects
A student is expected to do a project per quarter. The rubrics to evaluate a project depend on the objective. There are mainly three areas that are evaluated on any project:
a) The making of the project: creativity, presentation, directions followed, etc.
b) The math: if it includes process and correct answers.
c) Writing: conclusions on how this project was made, what was learned, what could make it better, etc.
An example of a project with its rubrics is:
Pre-calculus Project
First Quarter
Ma. Leticia G. de Espinosa
In this project you will relate different types of strategies you can use to solve triangles with real world applications. Invent or look for real world applications in internet or in other books for each of the cases below, if you know:
1. the lengths of 3 sides of a triangle
2. the lengths of two sides and the measure of the included angle
3. the measure of two angles and the lengths of the included sides
4. the measure of two angles and the lengths of a non-included side.
5. a right triangle if you know one side and one angle different from 90°
6. a right triangle if you know two sides
7. an angle of depression and other needed information
8. an angle of elevation other needed information
For each problem you need to write down the formula and the process followed to solve the problem.
Grading system:
1. Presentation 10 points
2. Conclusions 10 points
3. Bibliography or internet sites 5 points
4. Problem difficulty 10 points
5. Correct answers with procedure 10 points for each type of problem (total 80 pts.)
6. Project will be graded over 115 points.
The student should know before hand what will be evaluated in the project. The project is 20% of the quarter grade.
d) Attitude towards learning:
An attitude is very difficult to evaluate, therefore there are several aspects that should be considered:
· Responsibility
· Respect
· Active participation
· Willingness to help others
· Academic integrity
· Organization skills
Attitude towards learning is 10% of the quarter grade.
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