Assessment in algebra and trigonometry

MS 101 Algebra and Trigonometry now includes a course wide student learning outcome common math assessment (CMA) along with a more traditional final examination (FX). This provides the opportunity to compare performance on the common assessment versus the final examination.

Summer 2015 fourteen students enrolled in and completed the course. As with any course, the focus and parameters vary by instructor. My own approach includes an open book evaluation, the elimination of all rote memorization, a de-emphasis on trigonometric manipulations, the inclusion of exponential and logarithmic regressions using spreadsheets, the use of trigonometry in contexts such as scalable vector graphics programming, and the use of technologies such as the WolframAlpha engine to aid in solving problems.

Student using WolframAlpha app on Android 

The common math assessment is intended to evaluate the following student learning outcomes:

1.1 Solve and evaluate exponential equations and functions.
1.2 Solve and evaluate logarithmic equations and functions.
1.3 Interpret the inverse relationship between exponentials and logarithms.
1.4 Use exponentials and logarithms to model and solve real-life problems.
2.1  Sketch and recognize angles using degree mode.
2.2  Sketch and recognize angles using radian mode.
2.3  Convert angles between degree and radian mode
3.1  Use right triangles to evaluate trigonometric ratios of acute angles.
3.2  Use reference angles to evaluate trigonometric ratios of nonacute angles.
3.3  Use the unit circle to evaluate trigonometric ratios of quadrantal angles
3.4 Use a calculator to evaluate trigonometric ratios of angles.
3.5  Use right to triangles to model and solve real life problems.
4.1  Recognize and sketch the basic sine and cosine curves.
4.2   Find the period, amplitude and midline of sinusoidal curves
4.3  Use sinusoidal functions to model real-life phenomena.

While students in the course could and did use computer based technologies to assist in solving problems, on the common math assessment the students were restricted to the use of scientific calculators. While in class tests were open book, the common math assessme
nt was closed book.

The common math assessment is multiple choice, the final examination was paper based with all answers in a "fill-in-the-blank" format. The common math assessment was programmed into Schoology for delivery to the students. This provided automated tabulation of the results.

Multiple choice provides a random chance of being right even when the student does not know how to answer a question. The answers also provide information to the student on the form of answer which in turn may assist in solving the problem.

Rendering the common math assessment in Schoology meant generating diagrams. To gain full control over a diagram's design, the diagrams were done in Scalable Vector Graphics, imported into GIMP, and then saved as PNG files for uploading in Schoology. Some answers, as seen above, were done using the preformatted element in HTML. Schoology provides support for HTML in multiple choice answers.

Schoology provided detailed analysis of the student's responses including how many chose which particular right or wrong answer.

Due to an absence, thirteen students sat the common math assessment. All fourteen students completed the final examination. Note that the shift in emphasis of the course and the coverage choice on the final meant that the final examination did not cover all of the same outcomes as the common math assessment.

The average correct on the common math assessment was 51%, on the final examination the average correct based on an item analysis was 73%, a difference of 21%. Noting that the two instruments are not fully aligned, performance on outcomes on the common math assessment which were also on the final examination saw a 59% performance, a difference of 14% against the final examination. Bear in mind that the final examination permitted the students the use of the full array of tools encountered during the course and was open book, the common math assessment restricted the students to scientific calculators and was closed book.

For those outcomes which were evaluated by both the common math assessment and the final examination, the correlation was 0.76. The correlation is significant (p-value: 0.047), or to return closer to the language used by Fisher, the correlation would be surprising for someone who presumed that the two data sets are unrelated.

Source SLO CMA% Source SLO FX% Diff zDiff
CMA 1.1 0.31 FX 1.1 0.75 0.44 1.54
CMA 1.2 0.54 FX 1.2 0.68 0.14 -0.49
CMA 1.3 0.77 FX 1.3 0.77 0.00 -1.44
CMA 1.4 0.23 FX

CMA 2.1 0.77 FX

CMA 2.2 0.69 FX

CMA 2.3 0.77 FX 2.3 0.95 0.18 -0.20
CMA 3.1 0.54 FX

CMA 3.2 0.46 FX

CMA 3.3 0.31 FX

CMA 3.4 0.85 FX 3.4 0.98 0.14 -0.51
CMA 3.5 0.31 FX 3.5 0.38 0.08 -0.91
CMA 4.1 0.08 FX

CMA 4.2 0.56 FX 4.2 0.57 0.01 -1.33
CMA 4.3 0.54 FX


0.73 0.2125
Stdev (diff) 0.23

0.21 0.1490

0.73 0.1415

Diff = FX - CMA
zDiff = z-score for the difference = (Diff-0.73)/0.1490
StDev (diff) is the standard deviation of the difference values, not the difference in the standard deviations of the CMA and FX
MeanSelct is the mean for selected outcomes, those found on both the CMA and FX

As a side note, the final examination asked the students to perform a series of routine sine and cosine calculations. The students answered these correctly at 98% rate on the final examination.

When asked, as in 19A and 19B seen above, to make essentially the same calculations as presented in a context involving SVG graphics, the success rate fell to 39%. Question 19h* then asked the student to plot the heptagon vertices on their computer using SVG. 36% of the students were able to generate the heptagon on their computer with the vertices correctly placed. For students in the Computer Information Systems major, SVG is a language they could conceivably have to work with in the future. There is an application to the knowledge in the course, I view this as mathematics in a context. These problems are hard and all of the tools of WolframAlpha or Mathematica do not help a student solve such a "real world" problem. For a CIS major, I would argue that this constitutes a form of authentic assessment.

Measuring pi out on the lawn

*Note that 19h is a complex task.The students only had to plot the vertices using circle elements, some worked on adding the outline of the heptagon using the path element.

The code to generate the heptagon 

The course is open book, open note, allowing the students to use pre-existing class work as a starting point for the diagram. Thus some portions of the SVG code are boiler plate that the students already know to leave alone. Note too that a transform moves the center to (0,0) which means that the calculations such as 100 sin (39) generate the correct SVG grid values except for the flipped y-axis. The students were also working in SVG from within HTML using the HTML5 doctype to avoid the problems that full SVG (XML) bring (yellow screens of death). Notes for the students covers how to set up these files.

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