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Table 6 Sample responses for sub-level “Mechanism” for all levels of the framework

From: Cognitive framework for blended mathematical sensemaking in science

Level

Acceleration

Beer’s law

Energy conversion

1 Qualitative

Acceleration would stop when you stop pushing, and you could see speed decrease

Interviewer: What is the main cause for what you observe?

Student: the amount of particles dissolved and the types of substance

I think everything has thermal in it, it is not all transferred into the kind of energy you actually want. There is going to be some thermal to it

2 Quantitative

Student: Let’s see, in table A (data table) we did 250 divided by 50, we get five. So, it’s applied force minus friction over mass gives the acceleration

Interviewer: Why did you state it in this form?

Student: I guess because acceleration is the furthest right on the table (the data table), and usually in the equation what you try to figure out is on the right of the equal sign

Interviewer: what is the cause for all of these outcomes?

Student: I think the applied force is the most important variable, it’s the only

thing that is really changing

Student: I was able to figure out that the absorbance is the molar absorption coefficient times the concentration times the container width

Interviewer: and why did you choose to express it that way?

Student: mostly because with a compound, the molar coefficient will be a constant, and because the math worked out for the molar coefficient times the concentration time container width to equal the absorption

Interviewer: what do you think is the main causal factor behind changing absorbance and transmittance?

Student: I’d say the concentration of the solution is the main changer

Student: I am assuming, the LED, since it has more energy output as light, the incandescent light bulb is going to have more thermal energy than the LED

Interviewer: what is the percent efficiency now (bottom right table)?

Student: It’s now a 125%

Interviewer: Does that supports your model?

Student: yes, it still supports the model, it’s just different

Interviewer: why does it support the model?

Student: because this percentage, in this case it’s like 125%, the model took into account that there is a constant. So, like, it’s a constant because it is 125% for this one, and 125% for this one (going down the column), and so on and so forth. It took into account that there is a constant, it just the constant can change depending on the situation

3 Conceptual

Student: I think it’s F = ma because the more mass that you get or the greater acceleration that you are going at, the greater the force will be. So, the force is directly related to the mass and the acceleration

Interviewer: Do you think force is the outcome?

Student: Actually no, force is not the outcome. The cause mechanism is a force which is causing the acceleration to go in a negative or positive direction, increase or decrease in magnitude

“Maybe the absorbance has to do with multiplying by whatever the concentration is because multiplying by zero (when the concentration is zero) will give you zero absorbance. Because no matter what all the other variables are, when concentration is zero absorbance is always zero. If it was addition or subtraction of the thickness of the container or the wavelength, it wouldn’t matter what those are, if they get multiplied by concentration which is zero, you would still get zero. If a concentration is zero, it’s just water; adding anything to water makes it less clear; if anything is less clear, it makes it harder for the light to go through

In every equation there is going to be a different ratio of thermal energy in the right side of the equation. Example- the light bulb (LED with solar panel). This one is very efficient. So, 1/8th of the total energy would be thermal, and the 7/8th that’s produced will be light”

  1. *Bold text indicates key evidence in student responses for the specific sub-level
  2. **Italics text indicates clarification comments from the authors