Assessing the didactic and mathematical knowledge of prospective mathematics teachers in Namibe, Angola

In this work the results of applying a questionnaire to assess didactic-mathematical knowledge of prospective secondary school teachers are analyzed. The sample consists of 31 students, prospective teachers, from the School of Teacher Training of Namibe in Angola. The questionnaire is made of 20 items that assess algebraic content and didactic content knowledge in secondary education. The quantitative analysis of the results show a low level of algebraic and didactic content knowledge and we conclude that it is necessary to improve the teacher training program to enable these teachers to develop the mathematics and didactic competences they need.


Introduction
The research presented, which is the beginning part of an ongoing study, is designed to assess the didactic-mathematical knowledge of prospective secondary teachers on algebraic reasoning. The question we want to answer is how to characterize the didactic-mathematical knowledge of future high school teachers in Angola on algebraic reasoning? 1 3 Mathematical competence is a fundamental support for a good quality teaching of mathematics and a mathematics teacher should know the mathematics at the educational level he/she teaches, but he/she must also be able to articulate this knowledge with those corresponding to some higher-level. This knowledge constitutes the knowledge of the mathematical content per se (Godino et al. 2017) which, according to these authors, constitute common knowledge (corresponding to the level at which it is taught) and expanded knowledge (relative to higher levels). The mathematics teacher must recognize that the factors influencing practices of teaching are complex and that it is necessary to have didacticmathematical knowledge.
In previous publications [3][4][5] we proposed a characterization of algebraic reasoning in primary and secondary education. We propose the following six levels of algebraic thinking in primary and secondary education (along with level 0, indicating absence of algebraization): level 0: operations with particular objects using natural, numerical, iconic, gestural languages, are carried out. Level 1: use of intensive objects (generic entities), properties of algebraic structures and algebraic equality (equivalence). Level 2: use of symbolic-alphanumeric representations to refer to the intensive objects recognized, although linked to the spatial, temporal and contextual information; solving equations of the form Ax ± B = C. Level 3: symbols are used analytically, without referring to contextual information. Operations with indeterminate quantities or variables are carried out. Level 4: studying families of equations and functions using parameters and coefficients. Level 5: analytical (syntactic) calculations are carried out involving one or more parameters. Level 6: study of algebraic structures themselves, their definitions and structural properties. The last three levels (level 4, level 5 and level 6) are based on the following considerations: (1) using and processing parameters to represent families of equations and functions, (2) the study of algebraic structures themselves, their definitions and properties. Succeeding this general introduction, there is a description of the theoretical framework supporting the development of the questionnaire. Ensuing there is a brief characterisation of the context and student sample answering the questionnaire. Then, there is a presentation of global and partial results distinguishing algebraic and didactical knowledge and a discussion of the results. Finally, the results and their discussion are presented.

Theoretical framework
To progress in clarifying the nature of algebraic reasoning is a complex, but necessary topic, from an educational point of view and developing a comprehensive model of algebraic reasoning may facilitate the design of instructional activities supporting the emergence and progressive consolidation of algebraic reasoning.
Shulman [8] stated that among multiple knowledge domains for teaching (e.g., content knowledge, general pedagogical knowledge, curriculum knowledge, knowledge of learners, etc.), it is the pedagogical content knowledge, that is the category most likely to distinguish the understanding of the content specialist from that of the pedagogue. Godino [2] presents an analysis of Shulman's teacher knowledge model and some adaptations and interpretations developed by different authors in the field of teacher education in mathematics. After identifying some limitations in these models, he proposed a model that includes more detailed categories of analysis, for the knowledge of teachers of mathematics and didactics, based on the application of a semiotic approach to knowledge and mathematical instruction.
The research is based on two theoretical components: (1) nature of algebraic reasoning in primary and secondary education [1,[3][4][5][6][7]; and (2) didactic-mathematical knowledge of the mathematics teacher [2]. These components base the evaluation instrument used in this research.
Godino et al. [4,5] presents a model of algebraic reasoning structured into six levels (as mentioned in the introduction). This model extends the model of elemental algebraic reasoning [3]. The use and treatment of parameters is a criterion for defining higher levels of algebraization, as it is linked to the presence of equations and functions families, and, therefore it implies new approaches or levels of generality. The intervention of parameters is linked to the fourth and fifth algebraization levels, while the study of specific algebraic structures determines a sixth level.
Regarding the model of didactic mathematical knowledge, two types of variables were considered: algebraic content and didactical content. For the variable algebraic content, three values or categories are considered, in which different subcategories can be distinguished: Structures (equivalence relation, properties of operations, equations,…); Functions (arithmetic patterns, geometric patterns, linear function, affine, quadratic,…); Modelling (context problems solved by using equations or functional relationships). For the variable didactical content, referred to algebraic contents, the following categories are considered: • Epistemic facet, recognition of objects and algebraic processes (representations, concepts, procedures, properties, generalization, modelling), recognition of algebrization levels; • Cognitive facet, personal meanings of the students (knowledge, understanding and competence on elementary and secondary algebraic contents), learning conflicts about objects and algebraic processes; • Instructional facet, resources for the teaching of algebra in basic and secondary school (situations-problem, technical means), and their adaptation to the curriculum school.
Is a fact that mathematical knowledge and didactical knowledge strongly intertwine, and for this reason, this intertwining has been taken into consideration in the methodological design of the study, by combining mathematical and didactical questions in the tasks proposed to the teachers.

Methodology
Participants of this study were 31 students of the Mathematics and Physics course, finishing the first semester of their last academic of their last academic year. They are future teachers of the first cycle of secondary education in Angola. The instrument used for data collection was a questionnaire (see "Appendix") composed by 8 tasks, each of which consisting of items that evaluate different aspects of didactical mathematical knowledge of algebraic reasoning. The description of the questionnaire development process and the discussion of its validity and reliability are available in Godino et al. [5]. This instrument is designed to obtain the information provided by the students, prospective secondary school teachers, in written form within a limited period of time (2 h approximately).
To illustrate the classification of the different items of the questionnaire within the categories (for the variables "algebraic content" and "didactic content"), we present two examples of task analysis.
Example 1 Task 2 of questionnaire ("Appendix"): A pupil made the following conjecture: "I add three consecutive natural numbers. If I divide the result by three I always get the second number".
(a) Is the statement valid for all-natural numbers? Why? (b) In your opinion, what kind of justification could a primary school pupil give to this conjecture?
Expected solution It is expected that the future teacher indicates that the conjecture formulated by the student is correct and valid for all natural numbers, elaborating a justification based on the use of a variable to express "any natural number": n + (n + 1) + (n + 2) = 3n + 3 = 3 (n + 1); 3 (n + 1)/3 = n + 1, or (n − 1) + n + (n + 1) = 3n. The justification of the conjecture that a primary school pupil can give will be based very possibly on the verification of some particular cases.
Objects and processes The expected solution requires the representation with a variable of any natural number, and the next two numbers, operation with these expressions and simplification of the result, identifying it with the intermediate number.
Algebraic level An algebrization level 3 is put into play.
Categories of didactical knowledge Item (a) requires a more advanced level of knowledge; representation of a natural number by a letter that can takes any natural number as value (variable n as a generalized number); translation of natural language to symbolic-literal language; transformation of algebraic expressions (variable n as syntactic inscription); enunciation and proof of a proposition (the arithmetic mean of three consecutive natural numbers is equal to the intermediate number), item (b) refers to the possible reasoning of a student of primary school.  Table 1.
And conjecturing that the expression describing the relationship is: f(n) = (n + 1) 2 − 1. Visually, we can also see that each figure has quadrangular shape with side (n + 1), where n is the order number of the figure, eliminating a point from the square. For item (b) the student is expected to refer to the use of an inductive process as indicated in the table, or a deductive reasoning based on the visual characteristics of the task. In item (c), basic school students are expected to use an inductive procedure and be able to describe a pattern forming successive figures.
Objects and processes In this solution, a table is proposed for registering the number of points according to the position of the figure. The notion of function and variable (independent variable, serial number of the figure, and dependent, the number of points) is involved; besides the general rule of the pattern is recognized and expressed in a symbolicliteral language; in addition the variable is used as generalized number (expression of the general criteria depending on n) and changing value (expression of the functional relationship) and placeholder (when the function is particularized to find the number of points in a certain figure).
Algebraic level Level 2 of algebraization. When the student is able to perform transformations on the expression, that is, transforming the expression f(n) = (n + 1) 2 − 1 into f(n) = n 2 − 2n, then the activity should be considered of level 3.
Categories of didactical knowledge Item (a) involves knowledge of the mathematical content of the functional type of the secondary school; item (b) implies an epistemic reflection (recognition of different procedures to perform the task), and item (c) mobilizes cognitive knowledge (skills of basic school students). Table 2 summarizes the categories that we take into account in the design of the tasks, in order to evaluate the didactic-mathematical knowledge in the scope of the algebraic reasoning, according to the model described in the theoretical reference. The categories of algebraic content and didactic content are not disjoint or exclusive. The same item can be in more than one category. For example, item 7a," Design two problems that can be proposed to students of the 8th class involving the solution of a system of linear equations with two unknowns" includes content about "equations", but also "modelling". Moreover, the prospective teacher must mobilize knowledge about an "instructional resource." In addition, the item also includes a specialized knowledge of mathematical content in itself, that is, the epistemic facet in its situational and regulative components (concept of equation and its representations). For that reason, the item 7a is in different cells of Table 2. The codes used to classify responses were the following: Code 2-correct solution-Code 1: partially correct solution; Code 0-wrong solution; Code x-without solution; Code n: answered correct but only numerically; Code c: answered correctly without displaying the calculations. Table 3 presents the results of the questionnaire, where (Fr) represents the absolute frequency of the various response possibilities and we give also the percentual value (approximate).

Results and discussion
The processes of generalization and their corresponding symbolic representation, constituted a great challenge for the students (e.g., the items 2a, 3a, 8a). For example, in item 3a, students do not recognize the rule of the pattern, in a symbolic literal language, not permitting the determination of the number of points in the twenty-fifth position (code  ; code X-51.6%). The items 8a, 4b and 4c, also reflect great difficulties of the students, because they present high percentages of non-responses (code X). Another example is item 2a (45.2% correctly answered but only numerically-code n; 12.9% answered correctly without displaying the calculations-code c). For example, in item 8a students were expected to draw the following conclusions: (1) the sign of the parameter a determines if the parabola is facing up or down; (2) the absolute value of a influences the more or less opening of the parabola; the greater the absolute value of a is, the lower the opening of the parabola is; (3) all parabolas have their vertex at the point (0, 0) and the symmetry axis is the line with equation x = 0, from which it follows that they it is independent of a. Let's look at an example of a student conclusion: The answer presented in Fig. 1 does not involve any generalization, it only involves the graphical representation of some elements of the family of quadratic functions (with the resource of tables).
We find deficiencies in the recognition of algebraic objects and processes, as revealed in the analysis of the answers to items (e.g., items 6b, 8b). In Table 3, and related to item 8b, we can see that 70% of students did not answer. The following answer was expected: the activity involves a family of quadratic functions, were x and y stand for the independent and dependent real variables of the functions, and a is a parameter with real variable value; all parabolas have the y-axis as axis of symmetry. The activity requires interpreting the role of the parameter for identifying properties of the function family, level 4 of algebraization. In item 6b, it is expected to identify the linear function, graph C. But this has not happened.
Tasks designed in order to develop algebraic thinking were also challenging, as revealed by items (5b, 7a).
For instance, we present a student's response to the item 5b (Fig. 2). This student designs the following tasks: (1) "I have x loaves. The quadruple of my loaves plus five units results in 25. How many loaves do I have? "; (2) graph the function: y = 2x + 1; (3) calculate the perimeter of rectangle of length 6 cm and width 2 cm ". The task described in the manuscript is challenging for students because involve concepts such as equations and functions are involved in a context of modeling (they must design problems) involving, the letter symbols (variables) play different roles in different Fig. 1 A example of a student answer expressions, unknown and changing quantities. The equal sign is used as conditional equation (4x + 5 = 25), and a specification or definition (y = 2x + 1; P = 2c + 2l). In the student answer we observe the use of symbolic-alphanumeric representations. About algebraic thinking, operations with the algebraic expression are not carried out, so the level of algebraization is 2.
The items 5a, was where the students had the best results (67.7% presents correct solution). He involves the interpretation of functional relations (4x + 5 = 25, it is a linear equation with one unknown; y = 2x + 1, it is an affine function that allows us to find the value of a variable y based on another variable x; P = 2c + 2l, it can be interpreted as a formula to calculate the value of a variable P depending to other two, c and l).
The application of the questionnaire allowed to characterize relevant aspects of didacticmathematical knowledge on algebraic reasoning of prospective secondary school teachers. The results of this study reflected a lack of consolidation of algebrization levels, difficulties in recognising algebraic objects and processes (representations, concepts, procedures, properties, generalisation, modelling). These difficulties have manifested themselves in the form of a large percentage of incorrect answers or missing answers.
Given the lows indicators of algebraic and didactic knowledge, shown in Table 3, it is important to design a programme that promotes the competencies of prospective teachers of mathematics in Angola. The model presented by Godino et al. [4], brings about a new research and development programme focused on the design, experimentation and evaluation of training interventions that enhance the knowledge of mathematics among teachers which considering the different categories of knowledge and didactic competencies.
(a) How many glasses have to put in the third balance to make it balanced? (b) What interpretation of "balance" is associated with the situation? How would you explain to your pupils? 2. A pupil made the following conjecture: "I add three consecutive natural numbers. If I divide the result by three I always get the second number". (a) Determine the number of dots that the shape placed on the twenty-fifth (25th) position of this sequence will have, assuming it continues with the same rule of formation. Justify your answer. (b) Indicate the techniques or different ways in which the problem can be solved. (c) Do you consider that this task could be proposed to pupils in the basic school?
How could they reach a solution?
4. An "Economic" taxi company launch a fare campaign during the summer holidays. For a taxi ride, a flat fee is charged 530 (Kwanza) and more 170 (Kwanza) by kilometer up to a distance of 5 km; and 70 (Kwanza) by kilometers after 5 km of distance. A taxi driver is willing to wait if a passenger wants to stop before continuing his journey and charges 350 (Kwanza) for 10 min of waiting.
(a) What is the amount to pay on a trip of 4 km? And 8 km? (b) What is the distance of the trip, if a taxi driver receives from 2600 (Kwanza) with 20 min of waiting? (c) Do you consider that this task can be proposed to pupils of the 8th Class? How could you approach the resolution with these pupils?
(a) Describe your interpretation of each of the above expressions.
(b) Come up with three problems that may be proposed to pupils and whose solution leads to these expressions. 6. To fill a container with a maximum capacity of 90 L with water a faucet whose flow is constant and equal to 18 L per minute is used. (a) Indicate which of the three graphical representations corresponds to the situation described above, being that the X axis represents time in minutes and the Y axis the volume of water in liters.
Answer: _____; Justification: (b) What mathematical knowledge or other type of knowledge is used to solve this task? (c) Do you consider that this task could be given to basic school pupils? If so, what level? Justify your answers. 7. Consider the general form of a system of linear equations with two unknowns.
(a) Design two problems that can be proposed to students of the 8th Class involving the solution of a system of linear equations with two unknowns.
(b) Solve them by the method that is considered most appropriate for the 8th Class.
8. Consider the family of quadratic functions defined by the expression y = ax 2 .