# Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving

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## Abstract

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## 1. Introduction

[...] some draw attention to gaps or conflicts in the mainstream teacher knowledge discourse. Both Chapman (2012) and Foster, Wake and Swan (2014) take up a critique that Shulman’s [1986] framework and its derivatives focus on knowledge of mathematical concepts at the expense of problem solving proficiency.(p. 489) [2]

## 2. Theoretical Framework of Pedagogical PS Knowledge

#### 2.1. Knowledge of Student as Problem Solver

#### 2.2. Knowledge of PS as Worthwhile Task

#### 2.3. Knowledge of Non-Cognitive Factors Affecting PS

#### 2.4. Knowledge of PS Instructional Practice

## 3. Related Literature

## 4. Research Methods

#### 4.1. Participants

#### 4.2. Data Sources

#### 4.2.1. PS Learning Questionnaire

#### 4.2.2. PS Teaching Questionnaire

#### 4.3. Data Analysis

## 5. Results: PTs’ Knowledge of Teaching and Learning PS

#### 5.1. PTs’ Knowledge of PS Learning

#### 5.1.1. PTs’ Knowledge of Problem-Solver

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

Successful-problem-solvers’ characteristics | |||||

1 | His mathematical knowledge is connected and well organized. | 95.6 * | 5.4 | 97.8 * | 2.2 |

2 | He is persistent in maintaining the selected strategy already planned | 71.4 | 28.6 * | 60.2 | 39.8 * |

3 | He tends to focus on the problem’s structural characteristics and not on the superficial or obvious ones | 75.0 * | 25.0 | 62.4 * | 37.6 |

4 | He gets frustrated more easily by not getting the results quickly | 21.4 | 78.6 * | 22.6 | 77.4 * |

5 | He is aware of their strengths and weaknesses | 85.7 * | 14.3 | 93.5 * | 6.5 |

6 | He is able to control and monitor their own work | 96.4 * | 3.6 | 92.5 * | 7.5 |

7 | He has a concern about his resolution process is well done, using sophisticated strategies, being clear and reasonable in their process | 96.4 * | 3.6 | 92.5 * | 7.5 |

8 | He is less concerned about the details and more about finishing quickly | 8.9 | 91.1 * | 10.8 | 89.2 * |

Novice-problem-solvers’ characteristics | |||||

9 | He can distinguish relevant information from irrelevant | 30.4 | 69.6 * | 36.6 | 63.4 * |

10 | He keeps with his strategy even though it is not well-suited for the specific problem | 69.6 * | 30.4 | 64.5 * | 35.5 |

11 | He is impulsive in the choice of a solution strategy | 89.3 * | 10.7 | 82.8 * | 17.2 |

12 | He maintains his solution strategy even if he does not find partial results | 75.0 * | 25.0 | 72.0 * | 28.0 |

13 | He has poor clarity of the way forward to get the solution | 76.8 * | 23.2 | 80.6 * | 19.4 |

14 | He uses strategies not appropriate to the type of problem proposed | 66.1 * | 33.9 | 73.1 * | 26.9 |

15 | He finds a result without checking its accuracy or suitable | 76.8 * | 23.2 | 78.5 * | 21.5 |

- While both groups demonstrated more knowledge of successful than novice problem solvers, more GWo than GW demonstrated knowledge of more characteristics of both.
- More GWo than GW appropriately agreed with five of the eight characteristics of successful problem solvers and four of the seven characteristics of novice problem solvers.
- Thus, the findings suggested that with or without engaging the PTs in a course focused on PS, most of them held or developed appropriate knowledge of several characteristics of successful problem solvers during their teacher education program and many were also able to identify most characteristics for novice problem solvers.
- The main challenge for most of them in both groups was determining whether successful problem solvers would persist in maintaining the strategy they initially selected.
- Many of them thought that both successful and novice problem solvers will persist with their initial strategy. In general, most were better at identifying characteristics of successful than novice problem solvers.

#### 5.1.2. PTs’ Knowledge of PS as Worthwhile task

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

Problem selection | |||||

16 | The task allows for exploring and developing mathematical ideas | 96.4 * | 3.6 | 97.8 * | 2.2 |

17 | The task is contextualized in situations close to the student’s world | 98.2 * | 1.8 | 98.9 * | 1.1 |

18 | The task prompt the interest to seek for the solution and motivate the students to its resolution | 98.2 * | 1.8 | 98.9 * | 1.1 |

19 | The task offers different levels of solution and difficulty in solving the problem | 92.9 * | 7.1 | 91.4 * | 8.6 |

20 | The task answer is not direct or short | 53.6 * | 46.4 | 49.5 * | 50.5 |

21 | The task’s mathematical structure can be used in different situations so that students can make generalizations about problem families | 91.1 * | 8.9 | 92.5 * | 7.5 |

22 | The task is understandable and available to students, so students believe they could solve them and know by themselves when they have reached a solution | 100 * | 0 | 94.6 * | 5.4 |

Strategies | |||||

23A | Know “act it out” | 60.7 * | 39.3 | 57.0 * | 43.0 |

23A2 | Would use “act it out” | 78.6 * | 21.4 | 80.6 * | 19.4 |

23B | Know “check reasonableness of answer” | 92.9 * | 7.1 | 97.8 * | 2.2 |

23B2 | Would use “check reasonableness of answer” | 92.9 * | 7.1 | 98.9 * | 1.1 |

23C | Know “choose an operation” | 85.7 * | 14.3 | 91.4 * | 8.6 |

23C2 | Would use “choose an operation” | 62.5 * | 37.5 | 76.3 * | 23.7 |

23D | Know “draw a diagram” | 64.3 * | 35.7 | 80.6 * | 19.4 |

23D2 | Would use “draw a diagram” | 69.6 * | 30.4 | 83.9 * | 16.1 |

23E | Know “draw a picture” | 98.2 * | 1.8 | 100 * | 0 |

23E2 | Would use “draw a picture” | 94.6 * | 5.4 | 98.9 * | 1.1 |

23F | Know “estimate” | 91.1 * | 8.9 | 97.8 * | 2.2 |

23F2 | Would use “estimate” | 64.3 * | 35.7 | 82.8 * | 17.2 |

23G | Know “look for a pattern” | 76.8 * | 23.2 | 92.5 * | 7.5 |

23G2 | Would use “look for a pattern” | 62.5 * | 37.5 | 82.8 * | 17.2 |

23H | Know “make a graph” | 96.4 * | 3.6 | 98.9 * | 1.1 |

23H2 | Would use “make a graph” | 87.5 * | 12.5 | 93.5 * | 6.5 |

23I | Know “use manipulative” | 98.2 * | 1.8 | 100 * | 0 |

23I2 | Would use “use manipulative” | 100 * | 0 | 100 * | 0 |

23J | Know “building a table” | 98.2 * | 1.8 | 98.9 * | 1.1 |

23J2 | Would use “building a table” | 96.4 * | 3.6 | 98.9 * | 1.1 |

23K | Know “solve a simpler problem” | 87.5 * | 12.5 | 92.5 * | 7.5 |

23K2 | Would use “solve a simpler problem” | 66.1 * | 33.9 | 84.9 * | 15.1 |

23L | Know “solve an equivalent problem” | 91.1 * | 8.9 | 97.8 * | 2.2 |

23L2 | Would use “solve an equivalent problem” | 89.3 * | 10.7 | 87.1 * | 12.9 |

23M | Know “guess and check” | 85.7 * | 14.3 | 92.5 * | 7.5 |

23M2 | Would use “guess and check” | 66.1 * | 33.9 | 83.9 * | 16.1 |

23N | Know “sort and classify “ | 89.3 * | 10.7 | 96.8 * | 3.2 |

23N2 | Would use “sort and classify” | 82.1 * | 17.9 | 87.1 * | 12.9 |

23O | Know “use a model” | 75.0 * | 25.0 | 88.2 * | 11.8 |

23O2 | Would use “use a model” | 58.9 * | 41.1 | 83.9 * | 16.1 |

23P | Know “split the problem” | 94.6 * | 5.4 | 100 * | 0 |

23P2 | Would use “split the problem” | 91.1 * | 8.9 | 96.8 * | 3.2 |

23Q | Know “work backwards” | 26.8 * | 73.2 | 58.1 * | 41.9 |

23Q2 | Would use “work backwards” | 30.4 * | 69.6 | 44.1 * | 55.9 |

PS models | |||||

24 | Diagram 1 represents PS process in a real way because it shows that you can go back over what has been done or skip phases | 78.6 * | 21.4 | 63.4 * | 36.6 |

25 | Diagram 2 represents PS process in a genuine way because it shows the beginning and the end of the process, with the steps a student must follow | 78.6 | 21.4 * | 88.2 | 11.8 * |

26 | Diagram 1 is incorrect because its pointers indicate that you can go back on the work done | 14.3 | 85.7 * | 19.4 | 80.6 * |

27 | Diagram 2 is the best way to represent PS process because it shows as moves toward the solution | 51.8 | 48.2 * | 48.4 | 51.6 * |

Problem Posing | |||||

28 | Students must solve problems posed by themselves | 80.4 * | 19.6 | 84.9 * | 15.1 |

29 | Only the teacher should pose problems for the students | 5.4 | 94.6 * | 6.5 | 93.5 * |

30 | Posing problems can hamper the development of students’ mathematical knowledge | 10.7 | 89.3 * | 4.3 | 95.7 * |

31 | Problem posing can help students develop mathematical knowledge | 98.2 * | 1.8 | 100 * | 0 |

32 | Problem posing fosters mathematical creativity | 100 * | 0 | 98.9 * | 1.1 |

33 | Problem posing can be made before, during or after solving a problem | 82.1 * | 17.9 | 80.6 * | 19.4 |

34 | Problem posing can encourage the use of wrong strategies | 33.9 | 66.1 * | 24.7 | 75.3 * |

35 | Problem posing is reformulating a given problem | 32.1 * | 67.9 | 35.5 * | 64.5 |

36 | Problem posing is posing a new problem without a precondition | 39.3 * | 60.7 | 57.0 * | 43.0 |

- Overall, regarding the PTs’ knowledge of PS as worthwhile task, the two groups demonstrated similar level of knowledge for problem selection and different levels for strategies, PS models, and problem posing.
- About the same number of PTs for both groups correctly identified almost all of the characteristics for problem selection.
- More of the GW than the GWo appropriately identified almost all of the strategies and indicated they would use them.
- More GWo than GW identified three of the four factors for PS model. More of the GW than GWo correctly identified six of the nine characteristics of problem posing.
- Thus, while the additional focused work on problem solving seemed to give the GW an advantage over the GWo for strategies and problem posing, it did not for problem selection and PS model.
- However, there were some key issues for both. For example, while a majority of them identified knowledge of many of the PS strategies, some of them did not know act it out, draw a diagram, and work backwards.
- There was also not a one-to-one match between knowing a strategy and using it, for example, some GW knew of but will not use choose an operation, estimate, work backwards and some GWo knew but will not use look for a pattern, solve a simpler problem, and guess and check. For both groups, more of them will use act it out although they did not know it.
- Both groups also demonstrated contradictions in their knowledge of PS models regarding whether the process is cyclical or linear.
- Only about half of both groups indicated that a problem should not have a short and direct response and less than half demonstrated knowledge of problem posing as reformulation of a given problem and as posing a new problem without a precondition.

#### 5.1.3. PTs’ Knowledge of Non-Cognitive Factors That Influence PS

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

37 | Students should discover how to solve a problem by themselves without the teacher’s explanations | 60.7 * | 39.3 | 66.7 * | 33.3 |

38 | Students should know that the most important thing is to get the correct answer to a problem | 5.4 | 94.6 * | 4.3 | 95.7 * |

39 | Students have to assume that problems have only one correct answer | 10.7 | 89.3 * | 10.8 | 89.2 * |

40 | Once the students have solved the problem, they should know all the correct answers to the problems | 89.3 * | 10.7 | 90.3 * | 9.7 |

41 | Students who solve problems in different ways, end up getting confused | 14.3 | 85.7 * | 96.8 | 3.2 * |

42 | Students should use keywords (add, give, etc.) to solve word problems | 66.1 | 33.9 * | 73.1 | 26.9 * |

43 | It is better for the students to practice arithmetic calculations without context than use arithmetic calculations to solve word problems | 8.9 | 91.1 * | 14.0 | 86.0 * |

44 | Students should only solve problems once the mathematical concept has been taught | 33.9 | 66.1 * | 14.0 | 86.0 * |

45 | Students must solve problems as quickly as possible | 3.6 | 96.4 * | 1.1 | 98.9 * |

46 | It is better for the students that the teacher only teach PS after teaching mathematical concepts | 35.7 | 64.3 * | 11.8 | 88.2 * |

47 | To learn how to solve problems you must practice on an everyday basis | 58.9 * | 41.1 | 67.7 * | 32.3 |

- For eight of the eleven factors, about the same or fairly close amounts of both the GW and GWo indicted the appropriate factors, suggesting not a significant difference in the knowledge between them.
- However, while both groups held many beliefs that were appropriate to support students’ learning of PS, there were also many of the PTs who held some beliefs (five for GWo and four for GW) that were inconsistent with what were appropriate.
- Thus, there were mixed knowledge and contradictions demonstrated in their thinking about the beliefs that promoted an adequate development of PS in students.

#### 5.2. PTs Pedagogical Knowledge of PS Teaching

#### 5.2.1. PTs’ Knowledge of PS Teaching Approaches

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

1 | Class discussions should focus only on the answer to the problem instead of the process to get the answer | 3.6 | 96.4 * | 0 | 100 * |

2 | In the class there must be an environment where it is possible to explore problems both individually and in groups, communicating all the multiple ways of solving them | 100 * | 0 | 100 * | 0 |

3 | You must first learn a mathematical concept and then apply it to solve problems | 48.2 | 51.8 * | 40.9 | 59.1 * |

4 | The focus of discussion and attention should be on the process | 87.5 * | 12.5 | 92.5 * | 7.5 |

5 | The teacher must show and exemplify, step by step, how problems are solved | 82.1 | 17.9 * | 77.4 | 22.6 * |

6 | Phase and strategies should be taught directly and explicitly | 50.0 * | 50.0 | 38.7 * | 61.3 |

7 | Class discussions should focus on unpacking mathematical concepts involved in PS | 58.9 * | 41.1 | 64.5 * | 35.5 |

8 | The class should start with a problem, then let the students explore it and discover the mathematics involve in it; while the teacher guides the process | 85.7 * | 14.3 | 97.8 * | 2.2 |

9 | The teacher should explain in detail how to solve the problems and the students should listen and then apply | 32.1 | 67.9 * | 30.1 | 69.9 * |

10 | The teacher should teach mathematical concepts first, and then apply them to solve problems | 62.5 * | 37.5 | 50.5 * | 49.5 |

11 | The teacher should teach general aspects of PS (i.e., PS strategies or PS phases), which enhance students’ PS proficiency | 92.9 * | 7.1 | 100 * | 0 |

12 | The teacher should teach a mathematical concept from solving problems | 82.1 * | 17.9 | 89.2 * | 10.8 |

- More GW than GWo correctly agreed with 10 of the 12 factors, suggesting further exposure to formal knowledge on PS made a difference to their learning.
- One factor was particularly challenging for both groups regarding whether the teachers should demonstrate steps to a solution.
- Both groups did best on the set of items dealing with teaching through PS with a majority agreeing with three of the four items, followed by teaching about PS with a majority agreeing with two of the four items, and then teaching for PS with a range of responses for three of the four items.

#### 5.2.2. PTs’ Knowledge of Discourse in PS Teaching

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

13 | Fostering the use of different solution strategies | 100 * | 0 | 100 * | 0 |

14 | To make available to the students a solution-book with all the right answers | 30.4 | 69.6 * | 22.6 | 77.4 * |

15 | Discussing PS strategies used by the students | 100 * | 0 | 98.9 * | 1.1 |

16 | Asking for argumentation and reflection on answers and the mathematics concepts involved in the problem | 100 * | 0 | 100 * | 0 |

17 | Finishing the PS process once the answer is found | 5.4 | 94.6 * | 10.8 | 89.3 * |

18 | Guiding the discussion on how the problem was solved or what procedure was used | 98.2 * | 1.8 | 100 * | 0 |

19 | Asking for problems to be solved quickly | 0 | 100 * | 1.1 | 98.9 * |

20 | Propose problems of easy resolution | 21.4 | 78.6 * | 28.0 | 72.0 * |

21 | Explaining explicitly to students the ways each problem is solved | 80.4 | 19.6 * | 75.3 | 24.7 * |

22 | Encourage indicating agreement or disagreement with classmates’ solutions, giving justified reasons | 98.2 * | 1.8 | 98.9 * | 1.1 |

- The PTs of both groups demonstrated similar level of knowledge of discourse, about the same agreeing with six factors, GWo agreeing with two more than GW and GW agreeing with two others than GWo, suggesting little difference in the knowledge between them despite differences in exposure to formal knowledge in PS.
- However a key issue for both groups was most PTs indicating that the teacher should explain explicitly to students the ways to solve each problem and a few indicating posing problems of easy solution.

#### 5.2.3. PTs’ Knowledge of Stuck State in PS

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

23 | If the student made a mistake in an arithmetic calculation, the teacher should ask to read the problem again until he understands it. | 75.0 | 25.0 * | 72.0 | 28.0 * |

24 | The teacher should identify if the error is related to understanding the problem or strategy’s execution | 98.2 * | 1.8 | 100 * | 0 |

25 | For struggles with problem’ understanding, the teacher should suggest alternative representations. | 96.4 * | 3.6 | 96.8 * | 3.2 |

26 | For struggles with plan’ execution, the teacher should suggest alternative strategies. | 92.9 * | 7.1 | 94.6 * | 5.4 |

27 | If the student struggle with PS, the teacher should give the answer so that the student does not get frustrated. | 14.3 | 85.7 * | 16.1 | 83.9 * |

28 | The teacher should ask the student to represent the problem’s data in a different way ^{+} | 78.6 | 21.4 * | 66.7 | 33.3 * |

29 | The teacher should suggest to the student to change his strategy ^{+} | 53.6 | 46.4 * | 38.7 | 61.3 * |

30 | The teacher should ask the student questions about how he carries out the arithmetic calculations ^{+} | 91.1 * | 8.9 | 97.8 * | 2.2 |

^{+}Participants were asked to respond to a specific stuck situation of a student who understands an arithmetic problem but makes a mistake in one of the calculations. * Intended response.

- The PTs of both groups demonstrated similar level of knowledge of teaching intervention when a student is stuck, suggesting little difference in the knowledge between them despite differences in exposure to formal knowledge on PS.
- They had difficulty identifying the same items including treating an error in arithmetic calculation as a lack of understanding of a problem.

#### 5.2.4. PTs’ Knowledge of Assessment in PS Teaching

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

31 | The student’s understanding of the problem, for example, asking them to explain it with their own words | 94.6 * | 5.4 | 95.7 * | 4.3 |

32 | The organization and representation of problem data by the student | 92.9 * | 7.1 | 95.7 * | 4.3 |

33 | The student’s planning to get the solution | 91.1 * | 8.9 | 93.5 * | 6.5 |

34 | The student’s control of his own PS process, that is, if he is able to notice that if the plan does not allow to find the answer, it must be back and look for a new one | 94.6 * | 5.4 | 93.5 * | 6.5 |

35 | The student’s ability to select and use strategies | 100 * | 0 | 97.8 * | 2.2 |

36 | The existence of appropriate attitudes and beliefs in the student to solve problems | 89.3 * | 10.7 | 86 * | 14.0 |

37 | The student communication, his response and his justifications about what has been done | 94.6 * | 5.4 | 96.8 * | 3.2 |

38 | The student’s ability to use related mathematical knowledge into his PS process | 98.2 * | 1.8 | 92.5 * | 7.5 |

39 | The student’s ability to find the correct answer | 80.4 * | 19.6 | 78.5 * | 21.5 |

40 | The student’s ability to find the answer quickly | 14.3 | 85.7 * | 14 | 86.0 * |

41 | The student’s ability to give tidy and cleanliness work | 82.1 | 17.9 * | 81.7 | 18.3 * |

42 | The student’s ability to identify keywords (give away, lost, etc.) | 87.5 | 12.5 * | 79.6 | 20.4 * |

43 | The student’s ability to make sense of the answer according to the conditions of the problem | 94.6 * | 5.4 | 95.7 * | 4.3 |

44 | The student’s ability to represent ideas and answers only with symbols and numbers | 25.0 | 75.0 * | 17.2 | 82.8 * |

45 | The student’s perseverance to continue working despite not finding the right answer | 98.2 * | 1.8 | 95.7 * | 4.3 |

46 | The student’s confidence and security when facing PS | 78.6 * | 21.4 | 94.6 * | 5.4 |

47A | Observe students | 98.2 * | 1.8 | 82.8 * | 17.2 |

47B | Personal interviews | 75.0 * | 25.0 | 63.4 * | 36.6 |

47C | Self-reports | 83.9 * | 16.1 | 79.6 * | 20.4 |

47D | Problem posing | 82.1 * | 17.9 | 91.4 * | 8.6 |

47E | Written PS responses | 92.9 * | 7.1 | 75.3 * | 24.7 |

47F | Multiple-choice tests | 51.8 | 48.2 * | 52.7 | 47.3 * |

47G | Fill-in-the-blank tests | 30.4 | 69.6 * | 37.6 | 62.4 * |

- There was little difference between the percentage of WGo and WG with appropriate knowledge of assessing PS but more WGo than WG agreed with six of the seven ways of assessing PS with a majority of WG agreeing with problem posing, suggesting little difference or improvement in the knowledge between them despite differences in further exposure to formal knowledge on PS.
- For both groups, there was inconsistency in their knowledge by indicating they would evaluate students’ ability to identify keywords in word problems.

#### 5.2.5. PTs’ Knowledge of Resources in PS Teaching

Item | GWo (N = 56) | GW (N = 93) | |||
---|---|---|---|---|---|

YES | NO | YES | NO | ||

48A | It will help students to systematically write their calculations while solving the problem and respond in a tidy manner | 69.6 | 30.4 * | 73.1 | 26.9 * |

48B | It will allow students to visualize and manipulate relationships and ideas and then generalize some aspects related to the structure of the problem | 98.2 * | 1.8 | 98.9 * | 1.1 |

48C | It is not necessary for students to use resources; it would be better to teach them the mathematical symbols | 10.7 | 89.3 * | 1.1 | 98.9 * |

49 | Promote on the student the use of a single type of representation to avoid confusion | 10.7 | 89.3 * | 4.3 | 95.7 * |

50 | The teacher should use only formal or symbolic representations | 10.7 | 89.3 * | 16.1 | 83.9 * |

51 | The teacher should encourage the use of representations to communicate the problems’ results | 87.5 * | 12.5 | 95.7 * | 4.3 |

52 | The teacher should promote the use of representations because they are the ideas that the student has about the problem | 98.2 * | 1.8 | 96.8 * | 3.2 |

53 | The teacher should foster the use of multiple representations only with younger students or with the student who does not understand | 41.1 | 58.9 * | 31.2 | 68.8 * |

54 | The teacher should encourage its use at the stage of understanding the problem | 67.9 * | 32.1 | 61.3 * | 38.7 |

55 | The teacher should encourage the use of representations throughout across the resolution process | 91.1 * | 8.9 | 95.7 * | 4.3 |

56 | The teacher should encourage the use of personal and spontaneous representations because they prompt the transition to mathematical or formal representations | 100 * | 0 | 97.8 * | 2.2 |

57 | The teacher should encourage the use of the student’s own representation | 92.9 * | 7.1 | 92.5 * | 7.5 |

58 | The teacher should encourage the use of more than one representation in one PS process | 98.2 * | 1.8 | 97.8 * | 2.2 |

- The PTs of both groups demonstrated similar level of knowledge of resources, suggesting little difference in the knowledge between them despite differences in exposure to formal knowledge on PS.
- Many of both groups could not correctly identify whether representations will help students to systematically and clearly write their calculations while solving the problem.

## 6. Discussion and Implications

#### 6.1. Knowledge of PS Learning

#### 6.2. Knowledge of PS Teaching

## 7. Conclusions on PTs’ Pedagogical PS Knowledge

#### Implications for Teacher Education

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Piñeiro, J.L.; Chapman, O.; Castro-Rodríguez, E.; Castro, E.
Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving. *Mathematics* **2021**, *9*, 1811.
https://doi.org/10.3390/math9151811

**AMA Style**

Piñeiro JL, Chapman O, Castro-Rodríguez E, Castro E.
Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving. *Mathematics*. 2021; 9(15):1811.
https://doi.org/10.3390/math9151811

**Chicago/Turabian Style**

Piñeiro, Juan Luis, Olive Chapman, Elena Castro-Rodríguez, and Enrique Castro.
2021. "Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving" *Mathematics* 9, no. 15: 1811.
https://doi.org/10.3390/math9151811