The study was designed to improve teachers' didactic analysis competence by means of problem-posing tasks. For this purpose, a problemposing strategy has been implemented which sample consisted in in-service mathematics teachers. This strategy involves a reflection stage that is very close to mathematical practices and it encourages to develop didactic analysis competence. Some findings of our research are related to this competence and it means that the posers could formulate better problems with educational purposes.
Problem posing has long been recognized as a critically important intellectual activity
in scientific investigation [
Problem-posing tasks demand a person to expose his mathematical knowledge.
However, if the posed problem is aimed at contributing to the student’s knowledge –
or more specifically, to understanding and solving other more complex problems –
then the didactic-mathematical knowledge of the teacher must also intervene. This
aspect is closely related to the teachers’ didactic analysis competence, which has been
broadly studied within the onto-semiotic approach of cognition and mathematics
instruction (OSA) [
Moreover, [
In this study, we consider two theoretical frameworks, which gave us some tools to analyze the data.
We adopt the OSA as framework because we are interested in teachers’ competences when analyzing the mathematical activities that they develop. Likewise, we believe it is relevant to use an approach that provides us with categories to analyze both teachers’ mathematical knowledge and didactic knowledge. In the OSA, didacticmathematical knowledge is understood as the deepest knowledge of mathematics and its teaching, which a mathematics teacher must have to design, implement and assess the complex processes of mathematics teaching. Also, an important OSA theoretical construct for the analysis of mathematical objects, such as concepts, procedures, propositions and arguments, are the epistemic and cognitive configurations, which we will explain next.
According to Godino, Batanero and Font [
Additionally, Malaspina, Mallart and Font [
In this research, we used a multiple case study with 16 in-service high school
mathematics teachers who participated in a problem-posing workshop. Our study is
exploratory, descriptive and analytical, taking as unit of analysis the problems posed
by the teachers participating in the workshop. We analyse these problems using OSA
tools, it means EC and CC for solving and posing practices. The use of EC and CC is
a methodology previously used in some researches done in the OSA framework [
In our study, we implemented the ERPP strategy in the Problem-Posing Workshop on Quadratic Functions that purpose goes on to stimulate the development of the ability to pose problems by varying a given problem. We focused our attention on preproblem posing, since it requires didactic criteria from the person proposing the problem, so it should have the characteristic to facilitate the comprehension and resolution of a previously given problem.
In the next lines, we summarize the dynamics of the problem-posing workshop.
First session: a test on quadratic functions was applied which purpose was to deep
into participants’ mathematical competence. In addition, a class episode on affine
function that we designed in a previous research [
Considering teachers’ didactic experiences in teaching functions in high school, the research team selected the following episode, in order to present it to the teachers participating in the workshop. This episode includes some comments from students whom aged between 14 and 15 years old and they were exposed to the episode problem:
Mr. Pérez proposed the following problem to eighth-grade students in a mathematics class on functions: Find a pair of numbers whose sum is 43 and their product is the maximum possible. Solve the problem and explain your procedure in detail.
After a few minutes, some students commented: Pedro: The numbers are 21 and 22.
Isabel: You cannot know the maximum product.
Santiago: What good does it do for me to solve this problem?
An expert solution was adopted for the problem, and the EC of such solution was made in order to have it as a reference to analyse and compare it to the CC of the participants’ solutions (CCPp1).
The expert solution to this problem implies defining a function that allows us to obtain a pair of numbers which sum is known and which product must be maximum. In this way, the function f(x)=x(43-x) is defined, where “x” and “43-x” are numbers which product must be maximum. Therefore, both numbers are equal to 43/2 and their product is 1849/4. On the other hand, this answer can also be found by associating the number to the value of the abscissa that maximizes the function f. Another strategy for solving this problem, entails relating the vertex of the parabola that represents f, so the vertex would be the coordinates of the point (43/2, 1849/4). This last strategy makes use of the graphic representation of the function.
While elaborating the EC of the solution to the episode problem, we could recognize different mathematical objects which area involved in the mathematical practices. They are the languages used (verbal, symbolic and graphic representations); the information, requirement, context and mathematical environment; the concepts involved (quadratic function, linear equation, the maximum of a quadratic function, vertex, parabola, graphs of functions). Also, the emerging proposition (the function given by f(x) = x(S–x) will have a maximum for = /2, where S is the sum of the two numbers), the procedure which follows to the solution and the arguments explained to tell the truth about the given proposition, which derives in the conclusion. All of them are explicitly stated.
In relation to the solutions of EP, it was observed that most of the teachers consider the quadratic function as an object associated with the problem and this is closer to what was posed in the expert solution. From the sample, only 12 of the participants solved the problem correctly. In addition, it is significant that, even though 10 participants define a variable for solving the problem, 6 of them make explicit the function to be maximized. On the other hand, in order to find the maximum value of the function, 5 use the completing square strategy, while 5 do it by using the algorithm to find the vertex of the quadratic function and, as a result, to study its corresponding maximum. Three participants solved the problem with the support of a table of values. However, a teacher used the table partially. It gave us and idea to state that he or she recognized the numeric sequence related to the problem. Precisely, the use of the tables led some teachers (4) to make their analysis in the set of natural numbers and to fail the correct answer.
By analyzing in a qualitative way the CCPe made by the teachers, it was observed that most of them recognized the mathematical objects, at least partially. However, only some of the participants were able to elaborate with greater certainty the objects so-called propositions and arguments. In the same way the lack of robustness in their propositions and/or arguments, correspond to a lack of practice in the analysis of their mathematical chore, and by extension we can say that they lack or they do not show the competence of the didactic analysis.
On the other hand, the P1 were categorized by using the analysis of content and the
methodology of expert triangulation. From this categorization, based on the ECPp1
and CCPp1, we can say that most of the participants have an idea of function typified
by an epistemic configuration focused in formalist approach instead of empiricist
approach [
Because of space limitations, in this paper we only present the case of a teacher that hereinafter we will call T11 and analyze ECPe and ECPp1.
For the analysis of the configurations and the reflection on the mathematical practices of solving and posing for this case, we used a scheme (Fig. 2) according to the strategy ERPP. In this scheme it is shown the interaction among different configurations, whether epistemic or cognitive, in order to study the mathematical competence (M) or didactic analysis competence (D) of the teacher. In this context, looking forward to our interests, we focus on studying the interactions involving the competence in didactic analysis (D).
As an example of the pre-problem posing task, we present the first pre-problem posed by T11 (associated to CCPp1):
Determine the pair of numbers whose sum will be, respectively, 1,2, 3,…,10; but in such a way that the product of that pair’s components will be the maximum possible. (a) On the basis of what was observed, could you indicate which are the features that, in each case, the pair of numbers must meet? (b) If you must formulate each product as a mathematical function, express it.
Along with the problem posed, T11 showed a possible situation that will allow us to move into not very explicit aspects of the problem (Fig. 3). Likewise, the teacher elaborated the CCPp1. Finally, he answered the questionnaire about the mathematical practice of posing.
Translation: Let us call “x” and “y” the possible numbers which sum and product are S and P respectively, so… a) The pair of numbers should be two equal integers numbers (if S is an even number) and equal decimal numbers (if S is an odd number). b) In each case, x+y=S and y=S-x. Then, P(x)=x(S-x), where S is the sum of the numbers.
From the analysis of ECPe and ECPp1, it was observed that the information is similar and that it was only modified quantitatively. Even though it is true that the amount of pairs of numbers is greater, these are more manageable numbers for a high school student. In the same way, the requirement suffered a change, since in the P1 there are two questions that invite the student to reflect on his/her procedures and solution.
Also, the problems keep up the intra-mathematical context. In order to answer the requirement of the EP it becomes necessary to use algebraic expressions, however, in P1 posed by T11, it is not necessary to use this resource, except in part “b” in which the notion of generalization using functions is a request. This would be an advantage from the didactic point of view in order to solve the EP, without using the quadratic function.
Based on an expert solution to P1 and the solution proposed by T11, we state the following: for the language, in general terms, the use of graphical representations is highlighted: the parabola for EP and the table of values for P1. Likewise, in EP and P1 the requirement of maximum is evident. Thus it becomes explicit the use of the quadratic function in P1 regarding EP. Unlike P1, in EP it is necessary to use linear equations to formalize and give rigor to the problem solution. For the considered concepts in both configurations, it is easy to see the coincidence in many of them, like the case of the concepts of function, product of function, the idea of maximum of a function, completing squares, among others.
The propositions and arguments in the EP are more formal and rigorous. In P1, the fundamental feature of the propositions and arguments has a lower level of formality, in such way that allows using the inductive and deductive reasoning easily. As a sample to highlight this reasoning, T1 suggests a table of values in order to recognize a pattern that will allow solving the EP easily.
Talking about the procedures, the use of similar strategies is emphasized, in the sense that both problems require the identification of the main information and requirements. However, in P1, the situation becomes more intuitive, since it is not required to formulate a correspondence rule to solve it, except for the explicit requirement. Certainly, in EP there is no need of this, but due to reasons of effectiveness.
The arguments of the EP are more rigorous and formal, since they use concepts that are closer to the quadratic function, for example: concavity, vertex of a quadratic function. Moreover, in P1 the arguments are closer to an inductive reasoning, since the plan is to elaborate a strategy that makes the solution to the EP. For this last case, the use of a value table will permit to guess the practical rule that emerges as consequence of the analysis of the given values and that is one of the purposes of the problem.
The teacher T11 was asked to reformulate his pre-problem (P1), considering a didactic analysis of the problem posing process through the comparison between CCPe and CCPp1. Thus, T11 posed a second pre-problem (P2) that was analyzed by expert triangulation. Next comes the problem P2:
Determine the pair of numbers whose sum will be, respectively, 5 y 6, in such way that the product of the pair’s components will be the maximum as possible. Elaborate a table for the next cases: (a) For the case whose sum is 6, which are the components of the pair of numbers whose product is maximum? (b) For the case whose sum is 5, is there just one pair of numbers whose product is maximum? Explain your answer. (c) If we consider that there is just one pair of non-natural numbers whose sum is 5 and whose product is maximum and it is not a natural number, which are those numbers? (d) Elaborate a strategy in order to obtain the maximum product, in case of the sum of the pairs’ components will be an even number. Is this strategy different in case of the sum will be an odd number? Explain your answer. (d) Formulate the product of the numbers whose sum is 5 as a mathematical function.
From the expert triangulation, there is evidence to say that the P2 posed by T11 has the features of a pre-problem, since it makes easier to conjecture a pattern that allows to solve the EP. It is also observed that it was well conceived and detailed in order to achieve subsequently a solution of the EP easily. Thus part “c” fosters an intuitive solution, against the part “d” that invites to the generalization. The section (e) would be a simple exercise that helps a lot to think about how to solve the EP.
At this time, in the mathematical education field, there are a several theories and
theoretical approaches for researching. A reason for the existence of different theories
and theoretical approaches is the complexity of the topic of research itself [
To sum up, our study proposes a new problem-posing strategy that includes EC
and CC tools taken from the OSA framework to analyze the teacher’s mathematical
practices. Because of this implementation, we have evidence to state that in-service
teachers’ didactic analysis competence shows to be incipient and urge to develop it.
Certainly, our position about the conception of the didactic analysis complements that
proposal of Tichá and Hošpesová [