Assessing the physical science midterm
Galileo wrote,
La filosofia è scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l' Universo'), ma non si può intendere, se prima non il sapere a intender la lingua, e conoscer i caratteri ne quali è scritto. Egli è scritto in lingua matematica,...
which might be translated today as,
[Science] is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language,...
This would be echoed by physicist Freeman Dyson,
For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created... ...equations are quite miraculous in a certain way. I mean, the fact that nature talks mathematics, I find it miraculous. I mean, I spent my early days calculating very, very precisely how electrons ought to behave. Well, then somebody went into the laboratory and the electron knew the answer. The electron somehow knew it had to resonate at that frequency which I calculated. So that, to me, is something at the basic level we don't understand. Why is nature mathematical? But there's no doubt it's true. And, of course, that was the basis of Einstein's faith. I mean, Einstein talked that mathematical language and found out that nature obeyed his equations, too.
Max Tegmark argues that the universe does not just obey mathematics, the universe is mathematics.
...the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object—one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical—including you.
For 500 years mathematics has been the guiding light of physical science. The mathematics leads to models, theories, predictions, and replicability. Physical science should never be viewed as being primarily composed of accumulated facts. Physical science is, and has always been, a mathematical exploration of Galileo's universe.
Today I live in an age where the mathematics of physical science are either misunderstood, dismissed, ignored, or denied. Global warming, sea level rise, climate change, the inflationary universe and big bang, are argued against based primarily on belief that these are false. One set of factoids is seen as being as valid as any other choice of factoids. Scientifically established mathematical models are not a matter of opinion, but science as taught in schools is too often a collection of memorized facts, facts taken on faith. The speed of sound, the acceleration of gravity, the index of refraction, the conservation of momentum. Memorized for tests and quizzes. Calculations made based on memorized or provided equations, reducing physical science to a mathematics problems class.
SC 130 Physical Science, a non-science major general education science with lab course, has a curriculum built around gathering data and exploring mathematical models for that data using Desmos. The midterm, however, remained based on calculations made using provided equations. As noted in a previous blog, this term the midterm was a more open ended exploration of measurements made in a novel system.
The data is from a sinusoidally propelled object. The system is known to be roughly linear up to 3 Hertz. The system does not usually produce the highly correlated relationship of time versus distance for an object moving at a constant speed. The scatter away from a line is useful for leaving open other possible interpretations for a mathematical model.
The students were given the following framing for the midterm:
In the demonstration and data gathering phase you will gather data on the swizzle rate and velocity of the RipStik. You are to use this data to explore the relationship between the swizzle rate and velocity of the RipStik. Note that swizzle rate is in swizzles per second, velocity will be in meters per second. Show your data on this paper, make a hand sketch of the graph you make in Desmos. You should provide data based support for your discussion, analysis, and conclusions. Write up your work on this paper. Work alone or in pairs, put both names on the paper if working in a pair. Consider the following:
The students were to write up their results by hand in class. The students were encouraged to use Desmos to analyze their data. Eighteen students working in pairs produced nine analyses of the data.
All nine wrote their data in a table, and all nine included a sketch of the graph. Four explicitly stated whether they were treating the data as linear data or non-linear data. The other five pairs included mathematical models that implied whether they were treating their data as linear or not. All nine pairs sought to explain how the swizzle rate was related to the velocity. Some of these explanations were more coherent than others. All nine pairs recorded equations. Only four pairs arrived at the expected slope value for a direct linear relationship. Two pairs chose to model the data with a quadratic relationship.
Two pairs reversed the independent and dependent variables in their equations. Desmos will regress to essentially any equation, or attempt to do so. Both of these reversed equations were in the form of x₁~y₁+b, a form that does not produce a line that fits the data as the slope coefficient has been omitted. This reveals deeper misunderstandings as to the role of the slope in equations, as well as the independent and dependent variable relationship.
The midterm provided insight into the students abilities to analyze a system, to "do science", rather than memorized knowledge that will vanish within days of the end of the term. The students are learning a process, and learning science as based on observables, measurables. Just as one remembers how to do something learned long ago, so too I would expect these students will long remember how to engage in exploration of data.
This midterm provided insight into where the students are at in terms of their ability to carry out an exploration of data and reach a reasonable conclusion in the form of a mathematical model. I can also now see where I can perhaps make this process more comprehensible. The midterm was an assessment that provides me information on what I can adjust, what I should seek to explain better, what is clear and what is still confusing.
Thus the midterm provided both formative assessment: telling me where I can work on improving learning, and summative: the midterm was scored using a rubric using the criteria seen in the chart above.
SC 130 Physical Science, a non-science major general education science with lab course, has a curriculum built around gathering data and exploring mathematical models for that data using Desmos. The midterm, however, remained based on calculations made using provided equations. As noted in a previous blog, this term the midterm was a more open ended exploration of measurements made in a novel system.
The data is from a sinusoidally propelled object. The system is known to be roughly linear up to 3 Hertz. The system does not usually produce the highly correlated relationship of time versus distance for an object moving at a constant speed. The scatter away from a line is useful for leaving open other possible interpretations for a mathematical model.
The students were given the following framing for the midterm:
In the demonstration and data gathering phase you will gather data on the swizzle rate and velocity of the RipStik. You are to use this data to explore the relationship between the swizzle rate and velocity of the RipStik. Note that swizzle rate is in swizzles per second, velocity will be in meters per second. Show your data on this paper, make a hand sketch of the graph you make in Desmos. You should provide data based support for your discussion, analysis, and conclusions. Write up your work on this paper. Work alone or in pairs, put both names on the paper if working in a pair. Consider the following:
- What is the nature of the relationship between the swizzle rate and the velocity of the RipStik
- How is the swizzle rate related to the velocity of the RipStik?
- What is the mathematical equation for the relationship between the swizzle rate and the velocity of the RipStik?
The students were to write up their results by hand in class. The students were encouraged to use Desmos to analyze their data. Eighteen students working in pairs produced nine analyses of the data.
Two pairs reversed the independent and dependent variables in their equations. Desmos will regress to essentially any equation, or attempt to do so. Both of these reversed equations were in the form of x₁~y₁+b, a form that does not produce a line that fits the data as the slope coefficient has been omitted. This reveals deeper misunderstandings as to the role of the slope in equations, as well as the independent and dependent variable relationship.
The midterm provided insight into the students abilities to analyze a system, to "do science", rather than memorized knowledge that will vanish within days of the end of the term. The students are learning a process, and learning science as based on observables, measurables. Just as one remembers how to do something learned long ago, so too I would expect these students will long remember how to engage in exploration of data.
This midterm provided insight into where the students are at in terms of their ability to carry out an exploration of data and reach a reasonable conclusion in the form of a mathematical model. I can also now see where I can perhaps make this process more comprehensible. The midterm was an assessment that provides me information on what I can adjust, what I should seek to explain better, what is clear and what is still confusing.
Thus the midterm provided both formative assessment: telling me where I can work on improving learning, and summative: the midterm was scored using a rubric using the criteria seen in the chart above.
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