Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study
Abstract
:1. Introduction
2. Theoretical Framework
3. Objectives
- (O1)
- Study competence in solving verbal problems in different situations of divisibility involving the concept of g.c.d.
- (O2)
- Analyze whether some contexts present greater difficulties than others do.
- (O3)
- Find different erroneous resolution patterns and inquire about their origin.
4. Methodology
- (1)
- Problems with solutions obtained directly through the calculation of the g.c.d. or the g.c.d. and a division operation.
- (2)
- Problems in five different contexts: length, area, volume, discrete sets, and decontextualized numbers.
- GCDDQ problem: problem whose data refer to discrete sets and in which the g.c.d. must be obtained for its resolution, and division must be performed.
- GCDDL problem: problem whose data refer to continuous sets measuring length (meters) and in which the g.c.d. must be obtained for its resolution and division must be performed.
- GCDDC problem: problem whose data refer to continuous sets measuring volume (liters) and in which the g.c.d. must be obtained for its resolution and division must be performed.
- GCDDA problem: problem whose data refer to area and in which the g.c.d. must be obtained for its resolution and division must be performed.
- GCDQ problem: problem whose data refer to discrete sets and in which only the g.c.d. must be obtained for its resolution.
- GCDL problem: problem whose data refer to continuous sets measuring length (meters) and in which only the g.c.d. must be obtained for its resolution.
- GCDC problem: problem whose data refer to continuous sets measuring capacity (liters) and in which only the g.c.d. must be obtained for its resolution.
- GCDA problem: problem whose data refer to areas and in which only the g.c.d. must be obtained for its resolution.
- GCDD problem: problem whose data refer to discrete decontextualized quantities and in which only the g.c.d. should be obtained for its resolution.
5. Preliminary Analysis
6. Qualitative Analysis: Case Study
6.1. Case of Context: Areas
“You want to parcel out a field, 36 m long and 45 m wide, in square plots as large as possible, how long will these plots measure?”
- Student C:
- Buff, this problem is like the one the other day, buff, I don’t know… (referring to the similar problem they did in the written test)
- Student X:
- Okay, but let’s see what you ask for…
- Student C:
- I don’t know how to do it, ask what the plots will be like
- Student X:
- Let’s see…
- Interviewer:
- Why do you find this problem difficult?
- Student C:
- It is asking what the plots measure.
- Interviewer:
- Ok, can you describe the problem in your own words?
- Student C:
- We have a piece of land and…
- Student X:
- We want to divide it into large squares
- Interviewer:
- Ok, what is the data?
- Student C:
- The land is 36 and 45 m (write the data on the board)
- Interviewer:
- Are the data of the same type?
- Student X:
- Uh… yeah… it’s meters.
- Interviewer:
- What is required?
- Student C:
- Ehh… what the plots measure.
- Interviewer:
- Is the unknown the same type as the data?
- Student C:
- Well, ask for the area, I think…
- Student X:
- Good or the side of the plot.
- Student C:
- Yes, now.
- Interviewer:
- Is it the same type?
- Student C:
- It will be meters too.
- Interviewer:
- Draw the terrain (Student X draws a square on the board).
- Interviewer:
- Now put in a possible plot (Student X draws a small square in the upper right corner of the plot of land).
- Interviewer:
- What is required?
- Student C:
- This… (pointing to the small square).
- Interviewer:
- So what is asked is the side of the plot, right?
- Student C:
- Oh sure, okay.
- Interviewer:
- So, can you explain in your words what you have to do?
- Student C:
- Yes, we have to calculate the side of the plots, knowing that the total land is that (indicating the drawing).
- Interviewer:
- How are the data and the unknown related?
- Student C:
- Everything is meters.
- Student X:
- The plots are smaller.
- Interviewer:
- So is there a concept that relates the data and the unknown?
- Student C:
- What operation could we do?
- Interviewer:
- Do you think that the g.c.d. of the two quantities could be the solution?
- Student C:
- Ehh… no.
- Interviewer:
- Why?
- Student C:
- Well, the 18 m are here and the others are below (indicating the vertical and horizontal lines of the drawing of the terrain on the board).
- Interviewer:
- And that, what does it mean?
- Student C:
- Well, there cannot be common divisors.
- Interviewer:
- Why do you think they have no common divisors?
- Student C:
- They are not the same, they are in different places…
- Interviewer:
- So if it were 2 strings…
- Student C:
- So yeah
- Interviewer:
- And you, what do you think?—referring to Student X-
- Student X:
- The same, if it were two strings we could do the g.c.d.
6.2. Case of l.c.m.
“Marc has 18 stickers for soccer players and 30 for basketball players. If you want to put them in envelopes with the same number of stickers without mixing the soccer players with the basketball players and getting the greatest number of players per envelope, how many envelopes do you have to prepare?”
- Student G:
- Let’s see, fifteen… And of the others, twenty-four.
- Student L:
- Same envelopes.
- Student G:
- And they ask us how many envelopes.
- Student L:
- Yes.
- Student G:
- As we distribute the cards, we draw… it seems to me that the greatest common divisor, but…
- Student L:
- Uh… okay.
- Student L:
- Get 6 stickers.
- Student G:
- But you ask us the number of envelopes, right?
- Student L:
- Ah… and then what?
- Student G:
- Buff, then it will be the l.c.m.
- Student L:
- It could be
- Interviewer:
- Do you think the result is in line with what was requested?
- Student L:
- Yes, that’s 90 envelopes.
- Interviewer:
- If you read the statement again, knowing the resulting envelopes, is it possible?
- Student G:
- Let’s see if we have 18 for soccer and 30 for basketball and we put them in 90 envelopes… oh!… It can’t be.
- Student L:
- They have to be less envelopes, of course.
- Student G:
- Buff… well I don’t know
6.3. Case of g.c.d.
“Marc has 18 stickers for soccer players and 30 for basketball players. If you want to put them in envelopes with the same number of stickers without mixing the soccer players with the basketball players and that there is the greatest number of players per envelope, how many envelopes do you have to prepare? “
- Student M:
- I put the data, okay?
- Student S:
- Ok.
- Student S:
- That’s it, there is no more data.
- Student M:
- We have to find out the envelopes…
- Student S:
- So we have to divide, find the greatest common divisor, right?
- Student M:
- Well, but… yes, okay.
- Student S:
- Get 6 envelopes.
- Student M:
- Good.
- Interviewer:
- Just one question, is the result envelopes or stickers?
- Student S:
- They ask us for the envelopes.
- Student M:
- Of course… you have to distribute in envelopes, therefore divide. It is a divisor, … the g.c.d. are the envelopes.
7. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Key | Operation | Context | Statement |
---|---|---|---|
GCDDQ | GCD + div | DISCRETE SETS | Carla has 36 strawberry candies and 60 chocolate candies. She wants to separate them in identical boxes with the largest number of candies without mixing flavors. How many boxes does she need? |
GCDDL | GCD + div | LENGTH | For one of the camp tests, a scout group needs to prepare equal length ribbons as long as possible from two strings, one of 15 cm and another of 24 cm, how many ribbons can they get without leftover strings? |
GCDDC | GCD + div | VOLUME | Calculate how many containers of maximum volume are needed to be able to measure exactly a 12-L and a 40-L container. |
GCDDA | GCD + div | AREA | A carpenter wants to cut a wooden board that measures 18 cm long and 75 cm wide, into squares as large as possible. How many squares does he get? |
GCDQ | GCD | DISCRETE SETS | Carla has 18 strawberry candies and 30 chocolate candies. If Carla wants to put them in boxes with the same number of chocolates without mixing flavors, how many candies can you put maximum in each box? |
GCDL | GCD | LENGTH | A group of scouts needs to prepare ties for the camp trial. If you have two strings, one 18 cm and the other 24 cm, what is the largest possible size that you can cut the ties from both strings, so that they are all identical? |
GCDC | GCD | VOLUME | Calculate the maximum volume that a container must have in order to measure exactly the contents of two containers of 12 and 40 L. |
GCDA | GCD | AREA | A carpenter wants to cut a wooden board that measures 18 cm long and 75 cm wide, into squares as large as possible. What is the squares? |
GCDD | GCD | Decontextualized | Find the largest number that can exactly divide both 54 and 30. |
Type of Problem | Total Correct |
---|---|
GCDDQ | 43.7% |
GCDDL | 40.8% |
GCDDC | 35.2% |
GCDDA | 26.8% |
GCDQ | 67.6% |
GCDL | 69.0% |
GCDC | 56.3% |
GCDA | 31.0% |
GCDD | 70.4% |
All | 19.7% |
Type of Problem | Total Incorrect | l.c.m. | g.c.d. | Others | Blank |
---|---|---|---|---|---|
GCDDQ | 56.3% | 19.7% | 21.1% | 11.3% | 4.2% |
GCDDL | 59.2% | 15.5% | 31.0% | 9.9% | 2.8% |
GCDDC | 64.8% | 22.5% | 32.4% | 8.5% | 1.4% |
GCDDA | 73.2% | 21.1% | 21.1% | 11.3% | 19.7% |
GCDQ | 32.4% | 12.7% | - | 19.7% | 0.0% |
GCDL | 31.0% | 9.9% | - | 21.1% | 0.0% |
GCDC | 43.7% | 15.5% | - | 18.3% | 9.9% |
GCDA | 69.0% | 19.7% | - | 14.1% | 35.2% |
GCDD | 29.6% | 8.5% | - | 21.1% | 0.0% |
Average | 51.0% | 16.1% | 26.4% | 15.0% | 8.1% |
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Martinez, S.; Valverde, J.C. Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study. Mathematics 2022, 10, 1325. https://doi.org/10.3390/math10081325
Martinez S, Valverde JC. Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study. Mathematics. 2022; 10(8):1325. https://doi.org/10.3390/math10081325
Chicago/Turabian StyleMartinez, Silvia, and Jose C. Valverde. 2022. "Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study" Mathematics 10, no. 8: 1325. https://doi.org/10.3390/math10081325
APA StyleMartinez, S., & Valverde, J. C. (2022). Influence of Context on Greatest Common Divisor Problem Solving: A Qualitative Study. Mathematics, 10(8), 1325. https://doi.org/10.3390/math10081325