The reasons that lead teachers to use textbooks are diverse and complex. Thus, [15
] explains that the deficient initial and continuing training of teachers does not enable them to face their own teacher practice autonomously; hence, they need to resort to “pre-elaborated curriculum materials” that facilitate their job. In this sense, [16
] (p. 63) claims that for teachers, “textbooks act as a navigation map that reduce the uncertainty and complexity of teaching”. According to other authors such as [17
], there is a technocratic conception of the teaching profession that predominates in school, whose greatest exponent is the textbook. From this rationale, homogeneous curricular designs that are valid for any context and student are advocated; they must be thought out and elaborated by experts, and teachers must apply them routinely and mechanically. Finally, the Spanish educational system’s policies of free textbooks naturalize the presence of this resource in schools as something typical of pedagogical normality, making it the quintessential material. In this regard, the authors of [18
] (p. 266) ask, “Why are textbooks free, and other materials and resources are not?”
In this paper, we address the analysis of the assessment tests presented in the didactic guides of the Spanish mathematics textbooks in two specific aspects: (1) the distribution of the items by type of curriculum contents that the assessment tests include, and (2) the treatment given to the assessment of the problem-solving process.
1.1. Curricular Frameworks for the Study of the Contents of the Mathematics Area
In Spain, the Royal Decree 126/2014 of February 28, which establishes the basic curriculum of primary education [24
], includes five content blocks in the area of mathematics: processes, methods, and attitudes in mathematics; numbers; measure; geometry; and statistics and probability.
According to [24
], the block of processes, methods, and attitudes in mathematics constitutes the cornerstone of the rest of the content blocks. For this reason, this block has been formulated with the aim of being part of the daily classroom tasks in order to work on the rest of the contents. The different steps that constitute the process of solving a problem, which are articulated around a method or model, as well as the development of a positive attitude toward mathematics, are part of this block.
The block of numbers include two conceptual categories: development of number sense or number literacy, and operability. The first category includes reading, writing, and ordering of different types of numbers (natural, fractions, decimals, roman). The second category includes knowledge, use, and automation of the algorithms of addition, subtraction, multiplication, and division with different types of numbers; calculation, taking into account the hierarchy of operations and applying their properties; and initiation in the use of percentages and direct proportionality.
In the measurement block, two conceptual categories are also contemplated: first, the knowledge, the use, and the transformation of the different measurement units related to length, surface area, weight/mass, capacity, time, and monetary systems; second, the use of the most relevant measuring instrument according to the type of measurement.
The block of geometry is organized into a single conceptual category focused on the knowledge, use, classification, reproduction, and representation of objects on the plane and in space. The basic geometric notions are part of this block; flat figures and the calculation of their areas; and polyhedrons, prisms, pyramids, and round figures.
Finally, the statistics and probability block is structured around two conceptual categories. One category comprises contents that allow the processing of information: the collection and recording of quantifiable information; the use of graphic representation resources such as data tables, bar blocks, and line diagrams; and reading and interpreting graphical representations of a data set. As its second category, this block also includes the contents related to the prediction of results and the calculation of probabilities.
At the international level, two frames of reference use the curriculum and its organization as a key element that contributes to the analysis of the level of knowledge and cognitive skills acquired by students in the area of mathematics. We refer to the Program for International Student Assessment (the PISA program) [25
] in secondary education and Trends in International Mathematics and Science Study (TIMSS) [26
] in primary and secondary education. Both stand out for being the most consolidated and having the greatest international follow-up. Although both have a very similar structure, we have focused on the theoretical framework proposed by TIMSS [26
] due to the purposes of our work, because our subject of study is focused on the area of mathematics in primary education.
] aims to assess the level of achievement of students, compare the results between different participating countries, and explain the differences detected according to the different educational systems. These objectives constitute what in this framework is called the attained curriculum. However, to achieve these objectives, TIMSS [26
] starts from a first curricular level (official or intended curriculum) that is compared to a second curricular level (implemented curriculum).
Precisely, one of the aims of this study is to compare the official curriculum of the Spanish educational system with the implemented curriculum by textbooks; hence, the theoretical framework offered by TIMSS [26
] constitutes a relevant reference for our research.
In Spain, Royal Decree 126/2014 [24
] does not prescribe to what extent each content block must be present in the curriculum (beyond the mention that the first block must be the cornerstone of the rest of the blocks). However, the TIMSS [26
] proposes a gradation regarding the distribution of individual blocks of the curriculum, which are called dimensions of knowledge, referring to the mathematical content involved in the task.
Specifically, in the test for the 4th grade of primary education, TIMSS gives a weight of 50% of the total of the test to the dimension of numbers, the dimensions of measurement and geometry are given 15%, respectively, and the statistics block is given the remaining 20% [26
1.2. Problem Solving Models
Solving a problem is a cognitively complex task in which a set of skills, strategies, steps, or stages come into play. When explaining the process that the solver carries out during the PS, two types of theoretical models have been developed: on the one hand, general models based on heuristics; on the other hand, models from the field of cognitive psychology.
Models of the first type try to explain through a series of general or heuristic steps how students solve any type of problem (arithmetic, algebraic, geometric, etc.). Therefore, their main advantages are breadth and flexibility, as they can be adapted to different problem situations.
It is considered that the model based on heuristics proposed by [27
] was the pioneer in the description of the phases of solving problems; however, [28
] had already described in 1933 the stages of thought in the process of PS: (1) Identification of the problematic situation; (2) precise definition of the problem; (3) means-ends analysis and solving plan; (4) execution of the plan; (5) assumption of consequences; and (6) valuation of the solution, supervision, generalization.
According to [29
], the stages proposed by [28
] are a “prelude” or precedent to those proposed by [27
] in 1945, “an action guide” so that the teacher could help the students in the PS process effectively: (1) Understand the problem; (2) design a plan; (3) execute the plan; and (4) verify the solution obtained. In any case, Pólya’s model [27
] was a reference for the later development of heuristic models in which different phases are proposed to solve problems: a first phase of analysis and understanding of the information provided in the problem statement; a second planning phase aimed at finding the right strategy, that is, an action plan; a third phase that is more automatic and closely linked to the previous one, in which the plan is executed by applying the corresponding algorithm; and a fourth, final phase of verification-reflection of the process followed and the meaning of the result obtained.
Another model based on heuristics was the one proposed by [30
], who established four phases, just as [27
], to (1) analyze and understand the problem; (2) design and plan the solution; (3) explore solutions; and (4) verify the solution. The main novelty of this model in comparison with the previous one was the consideration of other dimensions, apart from the heuristics, that are necessary to solve a problem, such as the prior knowledge of the solver, a control that allows the efficient use of available resources, and the belief system that students have about mathematics.
As we stated previously, the advantage of models based on heuristics is that they can be applied to any type of mathematical problem. However, their main limitation is that they make very generic descriptions of the steps that form the PS process. Thus, for example, in the first phase of the model proposed by [27
] (to understand the problem), which type of understanding is necessary? Is it necessary to have a conceptual or mathematical comprehension, or a textual comprehension? It must be remembered that verbal arithmetic problems of additive structure, which we will refer to in this article as “word problems”, have double character: mathematical and textual. In this sense, a mathematical equation with numerical data that must be solved by applying one or more mathematical operations underlies every problem, hence the mathematical nature of the problem. For instance, “Alberto had 54 euros. He got 57 euros more. He has spent a few euros and in the end, he has 13 euros left over. How much money has Alberto spent?” (54 + 57 − X = 13). That said, if the solving of the problem ends up with the execution of one or more algorithms that result in a numerical answer, the first step should be a comprehensive reading of its statement. Hence, different authors consider word problems as short texts, verbal or textual descriptions, authentic discursive entities, specific textual genres or textual units [32
]. In short, against the vagueness (generality) of heuristic models, cognitive models, which are much more specific, do not limit themselves to general descriptions of the PS process, but rather delve into the mental processes involved in the task of solving a problem. Therefore, for these models, solving the problem in the previous example or any other word problem goes beyond mathematical knowledge. A deep comprehension of the problem involves understanding the problem as a mathematical structure and as a text. So much so that for the solving, “it is necessary to build a bridge: a link between the semantics of the language of mathematics and the semantics of natural language is required” [33
] (p. 112). That is, if the students do not understand the problem as a textual statement, they will not be able to extract the underlying mathematical essence.
On another note, the different cognitive models also propose a set of stages or steps to solve problems. All of them agree on the idea that solving a problem is a complex task in which it is necessary to activate sophisticated strategies to understand the statement of the problem. This is the common thread of all cognitive models, even though these models differ from each other depending on the emphasis that each places on one or another aspect of the solving process. One of the most relevant models, which belongs to the latest generation of cognitive models, is the one proposed by [38
]. In the model of these authors, the emphasis is on the importance of both quantitative (mathematical) and qualitative (situational) understanding to solve word problems. Thus, solving the problem requires a numerical answer, but also a realistic answer that the solver acquires by reasoning about the context of the problem and based on the previous knowledge. Therefore, to solve the following problem: “A man needs a rope that is long enough to stretch between two poles that are separated 12 m from each other, but he only has 1.5 m lengths of rope. How many pieces will he need to bind in order to spread the rope between the two poles?” mathematical knowledge (division algorithm) is not enough, because it must be taken into account that, when knotting the rope between the posts, a few centimeters of length are lost. Therefore, it is necessary to apply a reasoning that goes beyond the strictly mathematical (situational context), because if we base the solving of this problem exclusively on an arithmetic procedure, it will lead us to an answer that, being correct from a mathematical point of view, will be meaningless.
Because not all problems involve the same level of difficulty, in the cognitive model of [38
], two modes of solving are distinguished: the superficial mode and the genuine mode, whose main difference lies in the promotion, or not, of reasoning. Solving a problem using the superficial mode can be carried out in three steps: (1) selection of numerical data of the problem; (2) execution of the corresponding operation; and (3) expression of the result. For example, in the problem: “Andrés has 36 candies and Daniel has 24. How many less candies does Daniel have than Andrés?” even though it is possible to reason, the problem can be solved in three steps: (a) select the data from the statement without the need to understand it (36 and 24); (b) deduce the operation using a superficial strategy, specifically, the “keyword” strategy [39
], also called the direct or literal translation strategy [42
], which consists of selecting the data of the problem, “holding onto the numbers”, according to the expression of [43
], and operating with them taking as reference the linguistic terms of the statement (in the example problem: less than = subtract); and (c) report the result without checking if it is plausible from a mathematical or situational point of view. The superficial solving mode allows the solving of the easiest word problems, that is, the so-called consistent problems in the terminology of [44
], in which there is a consistency or coherence between the superficial structure of the problem and the algorithm that must be applied. In this way, terms such as “win”, “collect”, “more”, etc. imply a sum, while other terms such as “lose”, “reduce”, “less”, etc. involve a subtraction. According to these authors, the following word problems are consistent: problems of combination 1; change 1, 2, and 4; comparison and equalization 2, 3, and 4.
Conversely, mathematically difficult or inconsistent problems require a schematic knowledge of the part–whole structure that organizes the relationships between quantities. For example, in the problem “Andrés has 29 candies and his brother Daniel has 44, how many candies does Daniel have more than Andrés?”, the use of the “keyword” strategy would lead the solver to a wrong answer. Therefore, to solve this problem, the student needs more advanced mathematical knowledge; in this case, it is necessary to reason that, if Daniel has more candies than his brother Andrés, to know how many more he has (the difference), it is necessary to subtract, although in the surface structure of the problem statement appears the expression “more ... than”. According to the dichotomous classification established by [44
], inconsistent problems are those in which there is no coherence between the surface structure of the problem and the algorithm necessary for its solving: combination problems 2; change 3, 5, and 6; comparison and equalization 1, 5, and 6.
Moreover, a second factor that determines the difficulty of word problems is the location of the unknown; so that the problem will be more difficult, the further to the left the unknown is located. For instance, change problems have a maximum difficulty when the unknown is the initial set; the difficulty is less when the unknown appears in the change set; and even less when the unknown refers to the final set [45
Finally, without diminishing the importance of the previous classifications, it should be clarified that the difficulty of a problem cannot be considered in absolute terms, because the perceived difficulty of a problem also depends frequently on the knowledge and experiences of individuals [46
]. In this sense, [49
] proposed in 1985 the concept of threshold of problematicity, which will be different for each person. In this regard, [50
] also stated that the existence of difficulties is not only an intrinsic characteristic of the problem, because the ease or difficulty in solving also depends on the prior knowledge of the solver. The author of [50
] places the problem threshold according to the subject who faces the problem; if he masters all the necessary concepts and procedures, he will be in front of an exercise, while if he does not know them, he will have a problem.
In short, solving problems is a fundamental tool typical of advanced societies that all citizens must master in adulthood and that is mainly developed in school. Considering that the teaching guides that complement the textbooks contain the assessment tests of the students, an analysis of these curricular materials will allow us to know the publishers’ treatment of PS. Therefore, given the relevance of these materials as documents that include the students’ assessment tests already prepared, the main contribution of this study is to analyze the treatment of the word problem solving process in the assessment tests of the mathematics area of primary education, published in the teaching guides of six of the main publishing houses in Spain. Specifically, two objectives are proposed:
To analyze which content blocks are given priority in the assessment tests of the mathematics didactic guides.
To test to what extent these tests assess the different steps of the PS process, including reasoning as a key step, and what type of solving modes they propose to assess this process (superficial mode or genuine mode).