A Trajectory for Advancing the Meta-Cognitive Solving of Mathematics-Based Programming Problems with Scratch
Abstract
:1. Introduction
2. Literature Review
2.1. Scratch in Problem Solving
2.2. Meta-Cognition in Problem Solving
2.3. Problem Solving and Negotiations as Tools for Educating Learners and Teachers
3. Research Rationale, Goals, and Questions
3.1. Research Rationale and Goals
3.2. Research Question
4. Materials and Methods
4.1. Research Context and Participants
4.2. Data Collection and Analysis
4.2.1. Data Collection Tools
Video Recording
The Prospective Teachers’ Solution Texts
4.2.2. Data Analysis Tools
4.2.3. Validity and Reliability of the Analysis
5. Results
5.1. Problem Solving for Setting the Educating Stage: Assessing the Meta-Cognitive Skills of the Prospective Teachers in Solving Mathematical-Based Programming Problems Using Scratch
5.2. A Trajectory of Problem Solving and Negotiation Processes as Means for Prospective Teachers’ Education
5.2.1. Negotiating the Skills Needed for Solving the Programming Problems: Programming Skills and Meta-Cognitive Skills
- PT:
- I write N but nothing happens in the program.
- PS1:
- You need to go to DATA and drag the variable N, not writing it.
- PS2:
- Salam, could you describe what you did in order to draw the shape?
- Salam:
- At the beginning, I draw the whole shape on a paper and wrote its mathematical properties on the drawing, so that I can begin to think about drawing it in Scratch.
- PS2:
- How did you program the drawing of the second square?
- Salam:
- Depending on the first.
- PS2:
- O.K. Can you tell us what you thought when you first saw the problem?
- Salam:
- At the beginning I thought how to arrive at the lengths of the sides and the measure of the angles of the second square in order to draw it.
- PS2:
- Have not you done anything before?
- Salam:
- No.
- PS2:
- Have not you written first the lengths of the first square.
- Salam:
- Yes, I did.
- PS2:
- We need to be aware of each step in our solution process.
- Salam:
- You are right. This awareness will help us keep track of our solution process.
5.2.2. Negotiating a Model of Meta-Cognitive Skills:
- PS1:
- Huda, could you describe what you did in order to solve the problem?
- Huda:
- Computations.
- PS1:
- Computations! O.K. What were the givens in the problem?
- Huda:
- A square.
- PS1:
- The drawing is part of the givens. We need to consider the unknowns.
- Huda:
- The length of the square.
- PS1:
- [The PS wrote on the board “Unknowns: length of the square = 2a”]. What else?
- Huda:
- The angle of rotation.
- PS1:
- [PS1 wrote on the board in the Unknowns item: “angle of rotation = β”]. What else?
- Huda:
- The number of squares.
- PS1:
- Huda:
- Right.
- PS1:
- You drew a square, and then drew the next square. [The PS drew the shapes on the board]. Huda, come and tell us the rest.
- Huda:
- [She went to the board] I named this “a” and this “a” [The PT pointed at the appropriate line segments]. What is the problem here? To know the angle of rotation = β, and to know the side length of the next square
- PS:
- [The PS put “?” beside the side of the next square]. We name this the representation of the problem.
5.2.3. Problem Solving for Developing the Prospective Teachers’ Knowledge of Scratch Programming
- PS1:
- O.K. Let us look at the solution of Salam to assess its accuracy. [The PS started to read every block of the Scratch program and discus its accuracy]. “ask what’s the length? and wait”. This block is to input the length of the square “2a”. O.K. seems good. “Set side length to answer”. The answer means the length of the side of the square. O.K. seems good. Afterwards “Repeat 10”. What is this?
- PTs:
- The number of squares.
- PS1:
- The number of squares. O.K. Afterwards “Repeat 4”. Each loop of the first repeat involves a repeat of 4 times doing some drawing.O.K. “move side length steps” and “turn 90 degrees”. What did we do here?
- PTs:
- This is to draw a square.
- PS1:
- O.K. what did we do after drawing a square?
- PTs:
- These blocks prepare the drawing of the next square starting at the midpoint on the side of the current square, and do rotation of 45 degrees. Then set the side length of the next square as the side length of the current square multiplied by . When performing the next loop in the first repeat, the next square will be drawn.
5.2.4. Negotiating the Meta-Cognitive Skills with Scratch Programming when the Prospective Teachers Solved Programming Problems Individually
- PS1:
- As you say, at the beginning we need to program the shape of a circle. How do you suggest doing this programming?
- Saed:
- My method starts from the center. It depends on using the radius. The goal is to draw the point on the circumference of the circle as the end point of the radius. The radius will not be drawn, for it is used to draw the end point only. We move from the center to the end point and return to the circle center. We turn each time one degree and repeat the procedure until we draw something like 360 points.
- PS1:
- This strategy seems promising. Can you please try to carry out the strategy described by Saed?
- PTs:
- [The prospective teachers got engaged individually with implementing the strategy described by Saed using Scratch].
- PS1:
- Can you write a Scratch description, not necessarily code, of the method described by Saed.
- PTs:
- [The prospective teachers got engaged in writing a more exact Scratch-version of the strategy described by Saed].
- Rola:
- I wrote the following: Go to (x,y) representing the center of the circle. Repeat 360 times the following steps: Move from the center a distance that equals the radius, when the pen is up. Put the pen down to draw a point, then turn 180 degrees, put the pen up and move a distance that equals the radius until you arrive at the center. Turn 180 degrees and then one more degree. Add a counter so that we see the frequency of the repetition during the drawing. This counter starts at one in the block “set a to 1”, and at the end of each loop, the value of “a” changes through “set a to a + 1”.
- PS1:
- Great Rola. Do you think there is another strategy to draw a circle using Scratch?
- Huda:
- I think I got one. It is trigonometry-based. We start from an angle Alpha that measures zero degrees. We repeat 360 times the following steps: define the value of x coordinate for a point on the circumference of the circle: x = the x coordinate of the center + Radius * cos (Alfa). We then define the value of y coordinate for the same point on the circumference of the circle: y = the y coordinate of the center + Radius * sin (Alfa). We put the pen down to draw the point. Then turn one degree and increase Alfa by one degree through “set Alfa to Alfa + 1”. We continue to draw the points one beside the other.
- PS1:
- Fine Huda. [PS1 talks to the whole class] What do we conclude from Saed’s and Huda’s strategies for drawing a circle using Scratch?
- PTs:
- We need to look for more than one strategy to work with Scratch?
- PS1:
- Why do we need to do that?
- PTs:
- One strategy could be easier or more efficient to perform with Scratch than the other.
- PS1:
- What is the difference between the two strategies of Saed and Huda? You can discuss the difference in pairs or groups.
- PTs:
- [The prospective teachers started to discuss the difference between the two strategies]
5.2.5. Negotiating, in Groups, the Meta-Cognitive Skills with Scratch
5.2.6. Negotiating, Independently, the Use of Meta-Cognitive Skills in Programming with Scratch
- Maha:
- It seems there is a problem in the rotation angle, probably in the length of the side too.
- Namarik:
- It could be that the problem is due to the direction of the rotation right/left, or the angle size 120 or −120, or due to a mistake in the algebraic expression of the rotation angle.
- Maisoon:
- Let us go back and check our calculations to be sure of the rotation angle size and the length of the new edge.
- Group:
- [The group members worked again on the calculations]
- Rand:
- We changed the angle size but the mistake is still there. The triangles got turned and moved out. We still have a problem.
- Namarik:
- Maybe we need to change the edge’s length.
- Group:
- [The group members manipulated the geometric shape drawn in Scratch, changing the edge’s length, but without overcoming the mistake.
- Namarik:
- Let us try again to change the angle.
- Group:
- [The group manipulated the angle size and its direction several times].
- Rand:
- [Rand monitored the algebraic expression of the new rotation angle]. Here is the mistake. We missed multiplying by “a”.
- Namarik:
- When the value of “a” becomes negative, the shape gets messed up. We need to add a condition to stop the drawing when “a” turns to be less than zero.
6. Discussion
7. Conclusions and Considerations for Practitioners
7.1. Conclusions
7.2. Considerations for Practitioners
Author Contributions
Funding
Conflicts of Interest
References
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Session | Topic | Goals |
---|---|---|
1, 3 | Pedagogic supervisor-based activity—Setting the stage for educating the prospective teachers to use Scratch programming. The participating prospective teachers drew a specific canonic-square shape using Scratch programming, in addition to writing the steps of this Scratch programming. | To introduce the students to Scratch programming. |
4, 5 | Pedagogic supervisor-based activity—Setting the stage for educating the prospective teachers to use meta-cognitive processes in Scratch programming. The participating prospective teachers presented their solution of the canonic-square shape with Scratch programming. During this presentation, the prospective teachers negotiated their solutions with the pedagogic supervisors, wherein this negotiation involved metacognitive skills. | To introduce the students to meta-cognitive processes. |
6–9 | Pedagogic supervisor-based activity—The prospective teachers, individually and through interaction with the pedagogic supervisors, solved a specific problem of drawing a tangent for two circles. The class had a rich discussion related to the solution. | The prospective teachers used advanced Scratch programming and were in control of this programming, in addition to discussing meta-cognitive processes for problem solution. |
10, 12 | Prospective teachers-based activity—The prospective teachers solved, in groups, mathematics-based programming problems. The interaction with the supervisor came through the solution process. | Advancing the knowledge of prospective teachers in using meta-cognitive skills to solve mathematics-based programming problems with Scratch. |
13, 16 | Prospective teachers-based activity—The prospective teachers solved, independently, mathematics-based programming problems. They worked in groups, and the interaction with the supervisor came after the solution process. | Advancing the knowledge of prospective teachers in using meta-cognitive skills to solve, independently, mathematics-based programming problems with Scratch. |
Variable | Word/Term | Examples |
---|---|---|
Meta-cognitive process | Encoding, representation, decomposition, planning, selecting strategy, monitoring, evaluating, and suggesting other strategies | “Let us plan our solution”, “let us choose an appropriate strategy”, “is our use of this sequence of Scratch command draws the requested shape?” |
Negotiation process | Initial situation, contribution from the participants in the dialogue regarding this initial state, and refinement of the initial state of affairs | “I think there is a problem with your suggestion”; “I think your suggestion can be improved”; “this result is acceptable, but we can make it better” |
Problem solving process | Reading, paraphrasing, visualizing, hypothesizing, estimating, computing, and checking | “We need first to read the problem”, “let us paraphrase the problem in terms of Scratch commands”, “we need first to compute this operation and decide its fitness” |
Phase | Frequency |
---|---|
Negotiating the skills needed for solving the programming problems: programming skills and meta-cognitive skills | 63 |
Problem solving for developing the prospective teachers’ knowledge of Scratch programming | 55 |
Negotiating the meta-cognitive skills with Scratch programming when the prospective teachers solve programming problems individually | 57 |
Negotiating, in groups, the meta-cognitive skills with Scratch | 69 |
Negotiating independently the use of meta-cognitive skills in programming with Scratch | 53 |
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Daher, W.; Baya’a, N.; Jaber, O.; Awawdeh Shahbari, J. A Trajectory for Advancing the Meta-Cognitive Solving of Mathematics-Based Programming Problems with Scratch. Symmetry 2020, 12, 1627. https://doi.org/10.3390/sym12101627
Daher W, Baya’a N, Jaber O, Awawdeh Shahbari J. A Trajectory for Advancing the Meta-Cognitive Solving of Mathematics-Based Programming Problems with Scratch. Symmetry. 2020; 12(10):1627. https://doi.org/10.3390/sym12101627
Chicago/Turabian StyleDaher, Wajeeh, Nimer Baya’a, Otman Jaber, and Juhaina Awawdeh Shahbari. 2020. "A Trajectory for Advancing the Meta-Cognitive Solving of Mathematics-Based Programming Problems with Scratch" Symmetry 12, no. 10: 1627. https://doi.org/10.3390/sym12101627
APA StyleDaher, W., Baya’a, N., Jaber, O., & Awawdeh Shahbari, J. (2020). A Trajectory for Advancing the Meta-Cognitive Solving of Mathematics-Based Programming Problems with Scratch. Symmetry, 12(10), 1627. https://doi.org/10.3390/sym12101627