A5-1
APPENDIX 5
MEANING OF UNCERTAINTY
CATCHING A QUANTUM MECHANICAL BUS
This appendix contains the detailed analysis of Question 2 of the primary instrument which was administered to 231 first year physics students at the University of Sydney in 1995.
Question 2 utilised a multiple choice tick-box-and-explain response format. The question was designed to elicit the students’ understanding of the term uncertainty in the context of quantum mechanics.
A5.1RESULTS
Question 2 surveyed 231 first year physics, resulting in 191 tick-box responses, 160 written responses and of these 154 with both a written and tick-box response.
A5.2RESULTS - QUESTION 2 - TICK BOX SECTION
Of the student sample 83% (191/231) ticked one of the choices. The responses were tabulated (see Figure A5-1) and at the same time a ‘level of certainty’ was assigned to the responses. Details of the correctness classification are presented below.
Figure A5-1 : The histogram shows the distribution of student tick-box responses. Additionally the correctness of the associated written responses is indicated.
The analysis revealed that the two most popular choices were average 9:00am (the correct response) and might be delayed (a distracter). To check the actual understanding of those students that ticked the correct response (on average 9:00am) the written responses were analysed for correctness. This revealed that only 28% (17/60) actually articulated a supporting ‘correct’ written response resulting in only 9% (17/191) of the total responses could actually be considered correct. Details of the correctness classifications and representative responses are presented below.
Correct — The response was categorised correct if the response included supporting ‘correct’ reasoning.
The analogy of a bus is not good, unless you assume it is an experiment, and hence catching the bus is an event that is repeated a number of times. Hence from a large number of events, the average of the arrivals is 9:00am. (sid 173)
Insufficient — The response was categorised insufficient if the response did not provided supporting reasoning
The probability of the bus arriving at anytime is greatest at 9:00am (sid 122)
Incorrect — The response was categorised incorrect if the response contained one or more of the parts from the other tick box options.
This is largely due to the fact that on any day there is no way of knowing exactly when or where the bus will be due to many reasons. It may break down, driver has a hangover, a bomb may prevent it going under 50 mph etc. However you do know that the bus is more likely to appear at the bus stop at 9:00am than anywhere else. (sid 206)
The majority of the incorrect written responses which had ticked the Average 9:00am option mentioned that there were causes that could delay or assist the bus during its journey. This implied that the bus could arrive earlier or later due to these events but on average it would arrive at 9:00am. In quantum mechanical terms this kind of uncertainty is considered an intrinsic property of nature. There are no ‘hidden variables’ necessary to explain it.
The insufficient response and no written response contained insufficient information to judge whether the student understood the meaning of uncertainty in the context of the quantum mechanics.
A5.3RESULTS - MEANING OF UNCERTAINTY
A5.3.1Phenomenographic Analysis
The analysis revealed six categories of description which were not exclusive (ie. the response could fall into a number of categories). Details of categories are listed below (see Table A5-1) and representative responses are presented in Table A5-2.
Question 2 - Uncertainty
Agreed Meanings
CATEGORY / AGREED MEANINGRange of values / The response associated uncertainty with a range in measured value.
On average 9:00am / The response stated that the bus would arrive on average at 9:00am.
Due to unknown causes / The response associated uncertainty with other events.
Instrument not exact / The response associated uncertainty with some form of instrument error.
Uncertainty Principle / The response focused on the inability of measuring two values simultaneously or the response stated the uncertainty principle.
Probability / The response equated uncertainty with probability.
Table A5-1 : Summary of the set of categories of description for the phenomenographic analysis
There was no clearly dominant category of description. Uncertainty Principle rated highest at 31% (49/160) with on average at 9:00am, due to unknown causes,instrument not exact and probability each scoring around 20% (~30/160).
It was also noted that the categories of due to unknown causes and instrument not exact were exclusive (no overlap) as were the categories of probability and instrument not exact.
There were various small overlaps between the categories. Some interesting examples are listed below
Due to unknown Causes (35) and on average 9:00am (5 overlap).
Uncertainty principle (49) and the categories of instrument not exact (2 overlap), due to unknown causes (4 overlap) and on average 9:00am (3 overlap).
Probability (29) and the categories of on average 9:00am (6 overlap), uncertainty (6 overlap), due to unknown causes (1 overlap) and range of values (1 overlap).
In summary the five categories of description on average 9:00am, due to unknown causes, instrument not exact, uncertainty principle and probability cover 90% (144/160) of the student responses and are therefore considered to encapsulate the major ideas students hold regarding uncertainty
.
Comparing results between individual classes
The only marked differences in the distributions in categories between streams was that the Physics 1A stream had 64% (18/22) in the uncertainty principle category whereas the average was 31% (49/160), and the Physics 1A stream had 0% (0/22) in the due to unknown causes category whereas the average was 23% (36/160).
This result would suggest that the Physics 1A stream was more aware of the meaning on uncertainty in relation to quantum mechanics.
Question 2 - Uncertainty
Phenomenographic Analysis
Category / Representative Examples of Responses / (n)%
1 / Range of values / No electrons, photons etc. act exactly the same, however when we are dealing with many of them, an accurate average can be taken. (sid 127) / (12)
7.5
2 / On average 9:00am / In quantum mechanical terms, the bus would have the least uncertainty arriving at 9:00am. ie It is most likely to arrive at 9:00am. But to say that it would arrive at 9:00am is wrong, as there is a certain “uncertainty” involved. You can never say that the bus will arrive at 9:00am, just that is more possible that it will. (sid 408) / (29)
18
3 / Due to unknown causes / One can never be entirely ABSOLUTELY certain of something yet to happen as there are many unknown and known stringencies that could effect the outcome slightly or greatly. (sid 406) / (36)
23
4 / Instrument not exact / Uncertainty is that nothing can be measured accurately - No two properties can be measured simultaneously. Whenever one property is sort, the “constant property “ is disturbed in the process of measuring. For example, when measuring the temperature of a glass of water, the thermometer absorbs some of the heat from the water, cooling the water. Therefore the temperature of the water is changed, hence an uncertainty. (sid 022) / (33)
21
5 / Uncertainty Principle / The uncertainty principle proposes the microworld is “fuzzy” in the sense that the “bus” doesn’t have a well defined position, velocity, momentum or energy until you observe it. Thus this uncertainty leaves the arrival time of the bus unknown, except that when the bus left, there quantities were known by an observer. (sid 301) / (49)
31
6 / Probability / There is a high probability of the bus being on time and mostly on time 9 am but it will vary. Hence quanta vary but a high probability may occur at a certain point. (sid 228) / (29)
18
Table A5-2 : Representative examples and percentages of student responses for the phenomenographic analysis.
A5.3.2Context Analysis
Six categories of context were utilised to encode the responses — Words, pictures, equation, evidence or example, analogy and did not address bus in answer. The criteria for inclusion into a category and an excerpt from a representative student response is provided in Table A5-3.
Question 2 - Uncertainty
Context Analysis
Category / Criteria for inclusionExample excerpt from student response / (n)
%
1 / Words / Response consisted of a sentence or more of text.
The bus would arrive mostly around 9:00am but could arrive earlier or later. (sid 209) / (160)
100
2 / Pictures / The response had a picture
Picture of Prob’ cloud (1)0.6 Picture Uncertainty (6)4 / (9)
5.6
3 / Equation / xph (sid 016) (7)4 ; EtH (sid 016) (3)2
Prob = 2 (sid 110 ) (2)1 ; E =hf- (sid 201) (1)0.6 / (13)
8.1
4 / Evidence or Example / Experimental evidence or example is provided to support the response
... For example - take a thought experiment where all the instruments are perfect ... (sid 002) / (29)
18
5 / Analogy / A metaphor was used to describe an aspect of the response
... It is like predicting the position of an electron... (sid 157) / (3)
1.9
6 / Did not address Bus in answer / The response did not mention or refer to the bus in any way. Three Categories were evident.
Find Electron (chance of finding an electron) (15)9.4
For an electron or quantum of energy you could never pin point the exact position ... (sid 179)
Uncertainty in Measure (cannot measure two things at once) (29)18
If time is known exactly then its energy may not be determined according to the uncertainty principle... (sid 325)
Pure probability (uncertainty is the probability) (6)3.8
Uncertainty describes the range in which a value could occur ... (sid 004) / (50)
31
Categories of context are not exclusive
Sample Size : 160 student responses analysed
Table A5-3 : Representative examples of student responses in the context analysis.
Overall Analysis
The following table (Table A5-4) and the bar graph (Figure A5-2, see next page) seeks to show the evidence that the students gave in support of the choice they made. In the graph students choosing each of the tick-box responses are identified by a unique shading. Those same shadings show how many of each fall into the phenomenological and context categories. As an example students who chose tick-box D (no accurate watch) gave explanations that that fell into the due to unknown causes, Instrument not exact and uncertainty principle x-t categories. None of them drew a picture in their answer.
A5.3.3Content Analysis
It was agreed that content analysis basically mirrored the Phenomenographic Analysis. The only notable feature was that the Physics 1A used more terminology.
Question 2 - Uncertainty
Distribution Summary
Table A5-4 : Table summarising the tick-box response versus the distribution of phenomenographic and contextual categories.
Figure A5-2 : Graphical representation of the tick-a-box, phenomenographic and context analyses. Refer to text for detailed explanation.
Question 2 - Uncertainty
Results Summary - Part 1
Table A5-5 : Tabular summary of analysis showing class and overall results - Part 1.
Question 2 - Uncertainty
Results Summary - Part 2
Table A5-6 : Tabular summary of analysis showing class and overall results - Part 2.
APPENDIX 5...... 132
MEANING OF UNCERTAINTY...... 132
CATCHING A QUANTUM MECHANICAL BUS...... 132
A5.1 RESULTS...... 132
A5.2 RESULTS - QUESTION 2 - TICK BOX SECTION...... 132
A5.3 RESULTS - MEANING OF UNCERTAINTY...... 134
A5.3.1 Phenomenographic Analysis...... 134
A5.3.2 Context Analysis...... 137
A5.3.3 Content Analysis...... 139
Figure A5-1 : The histogram shows the distribution of student tick-box responses. Additionally the correctness of the associated written responses is indicated. 133
Figure A5-2 : Graphical representation of the tick-a-box, phenomenographic and context analyses. Refer to text for detailed explanation. 140
Table A5-1 : Summary of the set of categories of description for the phenomenographic analysis.135
Table A5-2 : Representative examples and percentages of student responses for the phenomenographic analysis. 137
Table A5-3 : Representative examples of student responses in the context analysis...... 138
Table A5-4 : Table summarising the tick-box response versus the distribution of phenomenographic and contextual categories. 139
Table A5-5 : Tabular summary of analysis showing class and overall results - Part 1...... 141
Table A5-6 : Tabular summary of analysis showing class and overall results - Part 2...... 142