,7] means. What is the smallest number in the interval?
The largest?
From a College Algebra class:
From a Precalculus class:
2 sin x.
These short writing assignments help students clarify concepts, and show the instructor where more work is needed.
Background and Purpose
I teach mostly at the precalculus level at Columbus State Community College and have used the following technique in my Intermediate Algebra, College Algebra, and Precalculus classes. Our students come to us very often with extremely weak mathematical backgrounds, with a history of failure at mathematics, with a significant amount of math anxiety, and frequently after being out of school for several years. I feel that my students need a variety of assessment techniques in order to learn the material and also to communicate to me what they have learned. Often my students have only a surface understanding of a particular concept. The assessment technique described in this paper helps students learn how to use mathematical language precisely. It helps me check whether they have absorbed the distinctions that language is trying to establish and the details of the principal concepts.
Method
One week before each test I hand out ten or twelve questions which require students to explain, describe, compare, contrast, or define certain mathematical concepts or terms. The goal is to have my students discuss these concepts using correct terminology. I expect the students to answer the questions using correct sentence structure, punctuation, grammar, and spelling. This strategy also serves as a review of the material for the students as it covers concepts that are going to be on my traditional test.
Examples:
From an Intermediate Algebra class:
,7] means. What is the smallest number in the interval?
The largest?
From a College Algebra class:
From a Precalculus class:
2 sin x.
Emphasizing why sin 2x2 sin
x by referring to geometric transformations and graphs helps students when they
must learn the double angle formulas. It is so tempting for
students to assume sin 2x = 2 sin x.
In order to receive full credit for this example, students must explain that the function y = sin 2x has a horizontal compression by a factor of 2 which makes the period of the function p and leaves the range at [-1, 1] whereas the function y = 2 sin x has a vertical stretch by a factor of 2 which makes the range [-2,2] but leaves the period at 2p. They must also sketch the graphs of each function. Requiring students to both verbalize and visualize a concept is an excellent way to strengthen understanding.
Findings
I have been using this strategy for more than six years. Many students have told me that these assignments help them make connections and gain a deeper understanding of the concepts that are being taught. One student recently told me that I forced him to think; his other mathematics instructors only made him work problems. Students who go on to take higher level mathematics courses frequently come back to thank me for the rigor that this type of strategy demands. They claim that the higher level courses are easier for them because of this rigor. At Columbus State half of the final exam in College Algebra and Precalculus is a department component which all instructors must use. I compared the results on the department component from my students for the last eight quarters with the results from the entire department and found that, on average, my students scored 4% higher than the students from the department as a whole. I believe that this assessment strategy is a contributing factor for this difference. A few examples will illustrate the type of misconceptions that I have found:
From Intermediate Algebra: When I ask the students
to explain the meaning of (-,7], I am reminded that
students frequently confuse the use of ( and [. I am also
reminded how fuzzy students ideas of the meaning of
and - are. One student told me that "the smallest number in the
interval was the smallest number that anyone could think of."
From College Algebra: When asked to respond to "If
x = 2 is a vertical asymptote of the graph of the
function y = f(x),
describe what happens to the x- and
y-coordinates of a point moving along the graph, as
x approaches 2," one student responded, "as
x gets closer, nothing specific happens to
the y-value but the graph approaches
-." This kind of an answer gives me insights that I would never get by asking
a traditional test question.
Use of Findings
When evaluating the results, I make extensive comments to my students trying to help clarify the concepts that have confused them and to deepen their understanding. I rarely just mark the result incorrect. These assignments have helped me improve my teaching. They have given me a better understanding of what it is about a specific concept that is confusing. I then carefully tailor my examples in class to address the ambiguities. So for example, when I discuss the vertical asymptotes of the graph of a rational function, I very carefully find the x- and y-values for a number of points on the graph for which the x-value is approaching the discontinuity. I have found certain concepts that I thought were obvious or easy to grasp needed more emphasis such as when to use "(" and when to use "[." I give more examples in class to illustrate the difference between using "(" and "[." When I return the papers, I only discuss the responses with the class if a number of students have made the same error or if a student requests more clarification.
Success Factors
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