Dealing with misconceptions about uncertainty & measurement

Standard

This post is a follow up to my last post about  misconceptions regarding uncertainties and measurement. In this post, I will present some of the lab activities that my supervisor, Dr. Doug Bonn, and I have implemented to directly target some of these misconceptions in a first year lab course. Many of these ideas come from existing literature and curriculum and I highly recommend checking out the references I’ve included at the bottom, though I will highlight a few throughout the discussion.

The first key feature of how we teach uncertainties is to call them uncertainties rather than errors. As I discussed in the last post, many students will interpret the word error literally and so we stick with that meaning of the word: An error is a mistake and it should be fixed. On the same notion, there is no need for lists of unquantified sources of error (such as “human error”) at the end of a lab report. Any uncertainties should be quantified at the beginning of the lab and accounted for in all calculations and analysis. This, of course, requires significant understanding of how to quantify and understand those sources of uncertainty.

The University of Cape Town’s Introduction to Measurement Manual recommends that different forms of uncertainty (what they call single and repeated measurement uncertainties) should be taught separately. Single measurement uncertainties stem from the precision of digital or analog devices, similar to the “half the smallest division” rule that students often see in high school. Very deliberately, we do not start with this type of uncertainty. This is reinforced by the Guide to the Expression of Uncertainty in Measurement by the International Organization for Standardization [4], which introduces treatments of digital and analog uncertainties that require understanding of probability distributions, thus calling for a clear understanding of repeated measurement (or random) uncertainties first.  I will focus on teaching that uncertainty for the remainder of this post.

The first activities we conduct in the term focus on teaching students about histograms, standard deviation, and the standard uncertainty in the mean (often called the standard error, but we’ve replaced the word error, of course). We do this using a two-part Invention activity, where students invent a solution to a problem where the expert solution would be to design a histogram and calculate the standard deviation [6]. Removing jargon (the students are asked to invent an “accuracy index”) means that students are more apt to creatively explore the solution space to develop their invention. When followed up with a traditional lesson about the expert solution, students have a better understanding of the concept than if they’d just received a lesson (controlling for time on task) [7]. After the lesson about standard deviation and histograms, we get students to do an experiment to reinforce the idea and demonstrate it physically, which brings us to:

The period of the pendulum – the gift that keeps on giving

The first experiment involves measuring a single period of the same pendulum. We combine all the measurements and students build a histogram and calculate the standard deviation and standard uncertainty of the mean of the distribution. Since the data fails in a fairly normal distribution (if your class is big enough), comparisons between individual measurements help support the use of the standard deviation as the uncertainty in a single measurement, since the measurements fall, on average, within one or two standard deviations from each other. Similarly, comparisons between the different lab sections show that the means of the distributions fall, on average, within one or two values of the uncertainty of the mean from each other.

The next step is to discuss improving the measurement. Numerous activities can be used to motivate that measuring the time for multiple swings (N) and then dividing that time by the number of swings gives a more certain measurement than measuring N trials of single swing measurements. Mathematically this works as follows:

N trials of single swings:
uncertainty in a single measurement = σ, uncertainty in the mean = \frac{\sigma}{\sqrt{N}}

1 measurement of N swings:
uncertainty in a single measurement = σ, uncertainty in one period = \frac{\sigma}{N}

The important factor here is to motivate why the uncertainty in a single measurement, regardless of the number of swings is still the standard deviation from the previous measurements. The simplest explanation involves identifying that the source of the uncertainty in the measurement comes from starting and stopping the stopwatch. Thus, each time you start and stop the stopwatch you introduce an uncertainty, σ. So if you only start and stop the stopwatch once, but let it swing a whole bunch of times in the middle, you still only count the uncertainty, σ, once in the measurement, and thus \frac{\sigma}{N} times for a single period. This can then be done experimentally. That is, students can measure the time for 10 swings and then calculate that the standard deviation of that set of measurements will still be σ.

From there, a number of trials of 50 pendulum swing measurements can provide sufficient precision to pick out the angle dependence of the period of the pendulum (that is, students can measure the period for angles between 5° and 20° and pick out the second order quadratic behaviours!). Since students have seen in class that the small angle approximation means the angle/amplitude does not effect the period, this provides an important lesson that good quality measurements can actually introduce new physics or reveal limitations to models. This could then lead into a great discussion about the nature of scientific theories and models. It also demonstrates that when things don’t work out in the lab, it’s not necessarily because of bad equipment, which could present a teachable moment for students that retroactively inflated their uncertainties to get agreement. Other experiments could involve measurements at different lengths or masses of the pendulum to derive the relationship T = 2\pi\sqrt{\frac{L}{g}}, measuring the length of the pendulum as precisely as possible and then comparing the calculated period to the measured period, or even, once they get tired of counting, building an automated system to do the counting for them!

We have a number of research projects on the go evaluating these teaching methods and so I would love to have some feedback from others about experiences with these issues and any things they’ve tried teaching or measuring.

References:

[1] Allie, S., Buffler, A., Campbell, B., and Lubben, F. (1998). First year physics students’ perceptions of the quality of experimental measurements. American Journal of Physics, 20(4):447–459.

[2] Allie, S., Buffler, A., Campbell, B., Lubben, F., Evangelinos, D., Psillos, D., and Valassiades, O. (2003). Teaching measurement in the introductory physics laboratory. The Physics Teacher, 41(7):394.

[3] Buffler, A., Allie, S., and Lubben, F. (2001). The development of first year physics students’ ideas about measurement in terms of point and set paradigms. International Journal of Science Education, 23(11):1137– 1156.

[4] Buffler, A., Allie, S., & Lubben, F. (2008). Teaching Measurement and Uncertainty the GUM Way. The Physics Teacher, 46(9), 539. doi:10.1119/1.3023655

[5] Buffler, A., Lubben, F., and Ibrahim, B. (2009b). The relationship between students’ views of the nature of science and their views of the nature of scientific measurement. International Journal of Science Education, 31(9):1137–1156.

[6] Day, J., Nakahara, H., & Bonn, D. (2010). Teaching Standard Deviation by Building from Student Invention. The Physics Teacher, 48(8), 546.
[7] Schwartz, D. L., & Martin, T. (2004). Inventing to Prepare for Future Learning: The Hidden Efficiency of Encouraging Original Student Production in Statistics Instruction. Cognition and Instruction, 22(2), 129–184.

[8] Séré, M.-G., Fernandez-Gonzalez, M., Gallegos, J. A., Gonzalez-Garcia, F., Manuel, E. D., Perales, F. J., and Leach, J. (2001). Images of science linked to labwork: A survey of secondary school and university students. Research in Science Education, 31(4):499–523.

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