The language of physics is mathematics, and physics ideas, laws and models describing phenomena are usually represented in mathematical form. Therefore, an understanding of how to navigate between phenomena and the models representing them in mathematical form is important for a physics teacher so that the teacher can make physics understandable to students. Here, the focus is on the “experimental mathematization,” how laws are established through quantifying experiments. A sequence from qualitative experiments to mathematical formulations through quantifying experiments on electric current, voltage and resistance in pre-service physics teachers’ laboratory reports is examined. The way students reason and justify the mathematical formulation of the measurement results and how they combine the treatment and presentation of empirical data to their justifications is analyzed. The results show that pre-service physics teachers understand the basic idea of how quantifying experiments establish the quantities and laws but are not able to argue it in a justified manner.
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Appendix: Excerpt from Electric Current Part from Report of Group 5
Appendix: Excerpt from Electric Current Part from Report of Group 5
1.1 Experimental concept formation
1.1.1 Qualitative experiments of electric current
(a) The following experiments can be used for reifying the different interactions of electric current to pupils. You cannot see or observe the electric current directly; instead you can observe the electric current’s effects with its surroundings.
Electric current causes light when, for example, a bulb is connected to an electric circuit. The filament produces light, because it is heated to the temperature of a few thousand degrees due to resistance and it radiates almost a spectrum of black body.
The heat effect can be noted in the same experiment by touching the glass surface of the bulb when the bulb is on. It is clearly warmer than its surroundings.
The magnetic effect of electric current can be noted by taking a compass near the current-carrying wire. Then, it is observed that the compass needle turns subsequent to the magnetic field produced by the electric current in the wire.
The chemical effect of the electric current can be noted by soaking electrodes in salt water. There the water starts to separate into hydrogen and oxygen. Hydrogen can be collected in the test tube at the anode, and it can be observed to be burning gas by setting it on fire with a match.
[Two Photos: 1. Photo of closed electric circuit with a bulb, which is turned on, and a compass, 2. Photo of closed electric circuit with an illuminated bulb and electrodes in salt water.]
Figure 1. Electric current produces light and causes a chemical reaction in salt water.
(b) The next experiments can be used for showing to the students that the electric current can only flow in a closed electric circuit, and its direction can affect the nature of the phenomenon.
A closed circuit is built with a battery and a bulb. When the other terminal of the battery is disconnected, the bulb turns off, because now the circuit is open and the current does not flow. The properties of the electric current are changed when the battery’s terminal are swapped. The forming of hydrogen relocates to the former cathode, which is naturally the anode after the swapping. Thereby, the electric current is a directed phenomenon. Changing the direction of current changes also the direction of the magnetic field caused by the circuit. This can be observed with the help of a compass and by changing the direction of current by swapping the terminals of the battery.
(c) These experiments can be used for showing to the students that the magnitude of electric current can be reified by the brightness differences of bulbs connected to the circuit.
Let us assume that we have an ideal constant current source and two identical lamps and eyes that have the ability to observe “precisely” brightness differences. As an additional assumption, one must know that the brightness of the lamp is nonlinearly proportional to the electric current in such way that the bigger current produces more light.
The equal currents can be observed, when first one bulb is connected to the circuit and its brightness is determined. Then, the bulb is removed from the circuit and another bulb is connected to the circuit. If the brightness is the same, then in both cases, the equal currents have flown in the circuits. The equality or inequality can be noticed at the same time by connecting the bulbs in parallel. Then, the differences in the brightness depict the differences in the currents. In addition, there is an experiment where current A flows through two bulbs and current B divides between two bulbs. Now when all bulbs burn with equal brightness, we know that current B is twice as big as A.
[Photo of the circuit, which is presented as a diagram in Fig. 5].
Figure 2. Estimating the differences of currents through comparing the brightness of the bulbs.
(d) The electric circuits in Figure 3 (see Fig. 5) can be used for explaining to students that the amounts of bulbs and batteries affect the magnitude of current but their order does not matter.
Connecting different amounts of bulbs in series in a way presented in Figure 3 (see Fig. 5), it can be observed that they turn off and on simultaneously, and they burn equally bright. Thereby, a current of the same magnitude flows through them. The more bulbs are connected in series, the dimmer they burn. It can also be noticed that adding voltage sources in series makes the bulbs burn brighter. Therefore, the magnitude of current depends on the amount of batteries. In the series circuits, the order of the components does not affect the electrical properties of the direct current circuit.
(e) The circuit in Figure 4 (see Fig. 6) can be used for reifying to students how the electric current divides when the circuit branches and how the magnitude of current changes when it goes through two routes at the same time. When the switch in Figure 4 (see Fig. 6) is closed, the current divides in two branches. There is now a parallel connection between two bulbs, which have smaller brightness than the third bulb of the circuit because the total current goes through it.
The quantification of the electric current
(f) In this experiment, a quantity that describes the magnitude of electric current is introduced to students. The meaning of the new quantity is defined to be proportional to the measurable property of the phenomenon’s effect.
The experiment is done applying magnetic interaction. The setup of experiment is presented in Figure 5. Similar bulbs are connected in parallel. The brightness of the bulbs is the same; therefore, same amount of current goes through the bulbs. A coil is connected as a part of the electric circuit so that the current in coil is always the same as the current through the chosen bulbs, which are connected in parallel. The current can be connected through one, two, three, four or five bulbs. Thus, the current in coil correlates with the current through the certain bulbs. The magnetic force in the coil caused by the electric current is measured by putting a bar magnet on scale, and the coil attracts the magnet upward. The force can be determined in a static situation by balancing the scale every time when the attraction of the coil changes due to different bulb configurations. Thus, it can be observed that the interaction of magnetic force is linearly proportional to the magnitude of electric current.
[See Fig. 1]
Figure 5. Diagram of the quantifying experiment of current.
[Photo of the measurement system]
Figure 6. The measurement, where the current in the coil goes through all five bulbs.
The resistance in the coil has to be a constant, i.e., not depending on the current. In other words, it cannot warm up so that it would not change the current flowing in the circuit in different circumstances. Hence, the cross-sectional area of the conductor has to be big enough, and it has to have limited amount of turns.
The bar magnet has to be in same position in relation to the coil in every measurement that the shape of the magnetic field does not affect in the measurement of the force. In Table 1 (see Table 5), the measured values are presented. In Figure 6, the measurement system is presented. In Figure 7 (see Fig. 7), the magnitude of force is presented as a function of currents of the bulbs.
Because the electric current is now quantified, we can start to use ammeters, whose measurements are based on the measurement of the magnetic force caused by the electric current.
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Mäntylä, T., Hämäläinen, A. Obtaining Laws Through Quantifying Experiments: Justifications of Pre-service Physics Teachers in the Case of Electric Current, Voltage and Resistance. Sci & Educ 24, 699–723 (2015). https://doi.org/10.1007/s11191-015-9752-z
- Electric Current
- Empirical Formula
- Prospective Teacher
- Concept Formation
- Laboratory Report