MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem
Abstract
:1. Introduction and Theoretical Framework
1.1. Estimation, Mathematical Modelling and Fermi Problems
1.2. Difficulties, Mistakes and Obstacles When Solving a Modelling Task
1.3. Temporal Analysis of the Resolution Process
- -
- Can the classification made by Moreno, Marín and Ramírez-Uclés [39] be adapted to the problem proposed in our research? Additionally, if this is the case, can this classification be used as the basis to detect students’ mistakes?
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- Does the MAD+ diagrams allow us to see which phases are the most conflicting? What interpretation can we give to the presence of mistakes of a category in the different phases of the process described in the MAD+?
2. Materials and Methods
2.1. Sample
2.2. The Fermi Problem
Vincent takes his little brother to a ball pit to celebrate his ninth birthday. When they enter, they realise that the pool has been filled with new balls. Vicent’s brother thinks that they’ve bought a lot of balls, probably more than 1000! Do you think Vicent’s brother has exaggerated the number? Calculate approximately how many balls fit in the pool if you have the diagram shown in Figure 2.
2.3. Methodology
3. Results and Discussion
3.1. Mistakes and Misconceptions Detection Using the Transcriptions
00:07:46 GBS2 No, because it says: how many balls did they have to buy? I’m sure it’s more than 1000. It’s like they’ve added more balls.00:07:50 GBS1 Yeah, yeah.00:07:54 GBS3 So maybe there’s a catch there.00:07:57 GBS2 Of course that’s what it is. It’s not that there’s 1000, it’s that there’s 1000 more than there were before.
00:02:45 GBS1 And we could do it scaled because you have a drawing. I mean, from the drawing you can measure, and you do it scaled but I don’t know if that would be okay.00:03:05 GBS1 Because what scale do you use? I don’t know, I don’t remember how that worked.00:03:31 GBS1 We’ve said that you must calculate the volume by taking out the slide, the staircase, and doors, right?00:04:24 GBS3 Maybe it’s crazy to take a ruler and then scale it.00:04:46 GBS1 Of course, I think the right thing to do is to do the scale, but I don’t know, is there a fixed scale for the drawing or something?00:04:57 GBS2 No. You can do whatever scale you want. I don’t think you don’t gain anything by doing the scale because the scale is whatever you want.
00:19:35 GCS2 What I would say is that the total number of balls that we’ve got is 7458, but as the balls are not going to take up all the available space, we could subtract 2000 of those balls, at a guess…00:19:47 GCS3 Yeah…2000 will certainly be left out.00:19:51 GCS2 Because, anyway, we don’t know how many balls fit in 1 square metre, we’d have to assume that…Well, yeah, we could figure it out, you know, by having the diameter of the sphere…00:20:08 GCS3 Yeah, but you can’t work out how much the little gaps take up either…00:20:13 GCS2 No, no, no…The volume that it takes up in those holes, we can’t calculate that.00:20:24 GCS3 Subtract about…2000…00:20:27 GCS2 Yeah, it’s by estimation…as these are modelling problems…00:20:44 GCS1 …minus 2000, so 5458 balls. So…, so we could say that there’s room for more than 1000 balls.
00:48:03 GAS3 37 million 44,00000:48:11 GAS1 Shall we convert it to metres?00:48:14 GAS2 Are you doing it in centimetres?00:48:29 GAS3 Okay, now better 37,044 meters.00:48:36 GAS1 cubic meters.
00:35:14 GBS1 Okay, it is 13 point 5, the whole area [of the pool picture]00:35:20 GBS2 Thirteen point 5 square meters. Ok.
3.2. Analysis of the Mistakes’ Timeline. Construction of the MAD+ Diagrams
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
MAD | Modelling Activity Diagram |
MAD+ | Enhanced Modelling Activity Diagram |
TTT | Task Time Tracker |
Appendix A. Spanish Transcriptions of the Dialogues
00:07:46 GBS2 No, porque dice ¿cuántas bolas han tenido que comprar? Seguro que más de 1000. Es como que han añadido más bolas.00:07:50 GBS1 Sí, sí.00:07:54 GBS3 Entonces es que igual está la trampa ahí.00:07:57 GBS2 Claro es que es eso. Es que no es que hayan 1000, es que hay 1000 más las que ya más la que ya hubiera antes. Supuestamente
00:02:45 GBS1 Y lo podríamos hacer a escala porque tú tienes un dibujo. O sea a partir del dibujo puedes medir y lo haces a escala pero no sé si eso estaría bien.00:03:05 GBS1 Porque ¿qué escala utilizas? No sé, eso no me acuerdo cómo funcionaba00:03:31 GBS1 Hemos dicho que hay que calcular el volumen quitando tobogán, la escalera y puertas ¿no?00:04:24 GBS3 Igual es una locura pero coger una regla y luego pasarlo a escala.00:04:46 GBS1 Claro yo creo que lo más correcto es que vas a hacer la escala pero claro yo tampoco sé. ¿Hay una escala fija para los dibujos o algo?00:04:57 GBS2 No. Tú puedes poner la escala que tú quieras. Es que yo creo que haciendo la escala tampoco ganas nada porque la escala es la que tú quieras.
00:19:35 GCS2 Yo lo que pondría es que el total de bolas que hemos obtenido son 7458, pero como las bolas no van a ocupar todo el espacio disponible, podríamos restar 2000 de esas bolas, a ojo…00:19:47 GCS3 Sí…2000 seguro que se quedan fuera.00:19:51 GCS2 Porque, de todas formas, no sabemos cuántas bolas caben en 1 metro cuadrado, eso tendríamos que suponerlo…Bueno, sí, lo podríamos calcular, vaya, teniendo el diámetro de la esfera…00:20:08 GCS3 Sí, pero tampoco lo puedes calcular lo que ocupan los huequecillos esos…00:20:13 GCS2 No, no…El volumen que ocupa en esos huecos, no lo podemos calcular.00:20:24 GCS3 Réstales unas…2000…00:20:27 GCS2 Sí, es a ojo…como son problemas de modelización…00:20:44 GCS1 …menos 2000, o sea, 5458 bolas. Entonces…, por tanto podríamos decir que sí que caben más de 1000 bolas
00:48:03 GAS3 37 millones 44,00000:48:11 GAS1 lo pasamos a metros?00:48:14 GAS2 ¿Lo estáis haciendo en centímetros?00:48:29 GAS3 Vale, ahora mejor 37,044 metros.00:48:36 GAS1 metros cúbicos
00:35:14 GBS1 Vale, me da 13 con 5, toda el área [del dibujo de la piscina]00:35:20 GBS2 Trece con 5 metros cuadrados. Vale.
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Mistake Description | Students’ Considerations | |
---|---|---|
T1. Simplification | 1.1. Incomplete real model associated with the lack of consideration of elements of reality | Any of the following elements are not considered:
|
1.2. Incomplete real model due to inconsistencies in the relationships between the elements of reality considered |
| |
1.3. Does not develop an objective function for the real model that can then be expressed in mathematical terms. |
| |
1.4. Does not build a real model. |
| |
T2. Mathem. | 2.1. Mathem. model inconsistent with reality |
|
2.2 Incomplete mathematical model |
| |
2.3. Does not build any mathematical models |
| |
T3. Solve | 3.1. Conceptual mistakes: failure in relation to the associated mathematical object |
|
3.2. Procedural mistakes in calculations |
| |
3.3. Incomplete resolution of the proposed mathematical model. |
| |
T4. Interpret. | 4.1 The results obtained are not interpreted on the basis of the proposed model |
|
4.2. The limitations of the mathematical model in the original situation are not recognised. |
|
T0 | T1 | T2 | T3 | T4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reading | Simplification | Mathematisation | Solve | Validate | |||||||||
0 | 1.1 | 1.2 | 1.3 | 1.4 | 2.1 | 2.2 | 2.3 | 3.1 | 3.2 | 3.3 | 4.1 | 4.2 | |
Group A | x | x | x | x | x | x | |||||||
Group B | x | x | x | x | x | x | x | x | |||||
Group C | x | x | x | x | x | ||||||||
Group D | x | x | x | x | x | x |
MAD Categories | Description of Modelling Activities of Students from the Problem in Context |
---|---|
Reading | Understanding the task: The students must understand the context of the problem and to focus on the main question of the task. How many balls can fit into the pool of the ball pit? Is this number lower or greater than 1000? |
Modelling | Transitioning from a real-world context to a mathematical interpretation of the task: Students find out how to calculate the number of balls fitting in the pool, asking themselves how to obtain the result. They have to think if any of the unit iteration, density or grid models can lead to a solution, even if they are not explicitly aware of that decision. They list the quantitative variables to be estimated. |
Estimating | Making sense of the quantitative estimates in the problem in context: Students estimate the quantities involved in the model. |
Calculating | Using simple mathematical concepts to calculate the missing information on the sketched diagrams or figures: Students calculate the volume of the pool and the volume of the ball, convert cm into m (or vice versa), use the proportion to calculate the scaled measures, etc. |
Validating | Interpreting, verifying and validating the mathematical calculations: Students make sense of the mathematical results, including calculations, within the problem in context. They compare their calculation with the number given in the task formulation. |
Type of Mistake | Code |
---|---|
0. Reading | Square (∎) |
1. Simplification | Triangle (▲) |
2. Mathematisation | Diamond (♦) |
3. Solve | Cap (⯊) |
4. Validate | Star (★) |
T0 | T1 | T2 | T3 | T4 | |
---|---|---|---|---|---|
V | o | ||||
C | x | o | x | ||
E | o | x | |||
M | x | x | o | x | x |
R |
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Pla-Castells, M.; Melchor, C.; Chaparro, G. MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem. Educ. Sci. 2021, 11, 747. https://doi.org/10.3390/educsci11110747
Pla-Castells M, Melchor C, Chaparro G. MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem. Education Sciences. 2021; 11(11):747. https://doi.org/10.3390/educsci11110747
Chicago/Turabian StylePla-Castells, Marta, Carmen Melchor, and Gisela Chaparro. 2021. "MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem" Education Sciences 11, no. 11: 747. https://doi.org/10.3390/educsci11110747
APA StylePla-Castells, M., Melchor, C., & Chaparro, G. (2021). MAD+. Introducing Misconceptions in the Temporal Analysis of the Mathematical Modelling Process of a Fermi Problem. Education Sciences, 11(11), 747. https://doi.org/10.3390/educsci11110747