6 Opportunities to learn
6.3 Mathematical tasks
Mathematical tasks or exercises are important in all mathematics class- rooms. In the TIMSS 1999 video study, classrooms in seven countries around the world13 were videotaped. It was found that at least 80 % of les-
son time was commonly devoted to mathematical tasks (Hiebert, et al., 2003). Although this study concerned mathematics in grade 8, the results from TIMSS advanced (Grønmo, et al., 2010) showed that the situation in grade 13 did not differ that much. Students in upper secondary school par- ticipating in mathematics courses, and their teachers answered in ques- tionnaires that the most common activity in the mathematics classes was to solve tasks; tasks similar to the examples provided in the textbook.
The considerable time spent on mathematical tasks in mathematics classrooms is one reason for focusing on them. Another reason is that the tasks students work on, influence their view of the nature of mathematics and what they learn. Stein, Remillard and Smith (2007) claim:
Tasks that ask students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that demand engage- ment with concepts and that stimulate students to make connections lead to a dif- ferent set of opportunities for student thinking. (p. 347).
Studies done on tasks implemented in the mathematics classrooms con- clude that tasks which are cognitively challenging, and which are not tak- en over by teachers or higher achieving peers, promote students’ concep- tual understanding (Henningsen & Stein, 1997; Stein, Grover, &
Henningsen, 1996; Stein, et al., 2007). In the work on mathematical tasks and students’ learning (Stein, et al., 1996; Stein & Smith, 1998) Stein and colleagues developed a framework for analysing mathematical tasks ac- cording to the potential levels of cognitive demand. They categorised the
13 The countries were Australia, Czech Republic, Hong Kong SAR, the Netherlands, Switzer-
tasks into two levels ‘the low level of cognitive demand’ and the ‘high level of cognitive demand’, which in turn were divided into two sub- categories each. For the first category, the two subcategories were: ‘mem- orisation tasks’ and ‘procedures without connections to meaning’. The latter main category was divided into the ‘procedures with connections to meaning tasks’ and ‘doing mathematics tasks’ (for a thorough description of each category see appendix 4).
This categorisation was done for assigned tasks. During classroom ob- servation, however, it was clear that although a task was categorised to be a ‘high level demand task’, it was difficult to keep the high level of chal- lenge (ibid), and often high level tasks were experienced declining into lower cognitive levels. The reason was mostly that students pressed for help, and thus they avoided the mental struggling period, which according to Hiebert and Grows (2007) was one prerequisite for the development of conceptual understanding (section 6.1).
There are several definitions for what is a task. Doyle claims that the concept of task:
calls attention to four aspects of work in class (a) a goal state or end product to be achieved; (b) a problem space or set of conditions and resources available to ac- complish the task, (c) the operations involved in assembling and using resources to reach the goal state or generate the product, and (d) the importance of the task in the overall system in the class (Doyle, 1988, p. 169).
A task he said, is defined “by the answers students are required to produce and the routes that can be used to obtain these answers” (Doyle, 1983 p. 161). This means that for him a task (he used the notion ‘academic task’) includes the products students are to produce, the operations they have to execute in order to generate the expected products, and the resources available for students in this production process. Stein et al. (Henningsen & Stein, 1997; Stein, et al., 1996; Stein, et al., 2007) based their definition on Doyle’s work. The difference lies in the duration or length of a task. In their work, a mathematical task is defined to be a classroom activity, whose purpose is to focus students' attention on a particular mathematical idea. An activity is not classified as a new task unless the underlying mathematical idea of the activity changes.
Niss (1993) defines a task to be “an oriented activity, i.e. a set of ac- tions oriented towards undertaking certain missions such as orders, pro- posals, or challenges” (ibid p.17). He further asserts that a task can be formulated orally or in written form, often by using the imperatives: com- pute, solve, draw, construct, determine, describe and so on. They might also be formulated as questions. In all cases the ‘mission’ is to carry out the task and to come up with a solution or an answer. He listed three cate- gories of tasks: questionnaires, exercises, and problems. The first he de-
fines to be “a collection of questions that usually concern facts” (p.18). There are normally only one or very few options for correct answers to each question. The next category, exercises, are tasks which involve pri- marily routine considerations or operations to be executed in straightfor- ward combinations, and which are set within a well-defined part of a giv- en mathematical topic. The last category, problems, Niss (ibid) are tasks involving non-routine considerations, or operations, or they are set within a context not related to a specific topic in the syllabus.
Problems and exercises are emphasised as not being absolute concepts. The reason for this is that an exercise might be a problem to one person, while it might be a routine task – an exercise - to another. Types of tasks which have been problems for a student might later in the development be perceived as an exercise for the same person.
In the report from the TIMSS Video Study 1999 (Hiebert, 2003) all mathematical tasks are termed problems, no matter if they are problems to the persons trying to solve them or not. According to the coding manual from the TIMSS Video study a mathematical problem is defined to be a an explicit or implicit problem statement that includes an unknown aspect, something that must be determined by applying a mathematical operation. Included in the mathematical problem is the solution or answer, as well as the checking of the answer. This means, in the TIMSS terminology, that as long as there is an expected answer to be found by mathematical opera- tions, there is a mathematical problem.
In this thesis a mathematical task will be defined according to the definition of a mathematical problem in the TIMSS Video Study, and it will include questionnaires, exercises and word problems (Niss, 1993). This means that as long as there is an expected answer to be found by mathematical operations, there is rather a mathematical task than a math- ematical problem as in TIMSS.
The reason for the choice of the notion task, instead of problem, is that as Niss (ibid) emphasises, a problem is a problem in relation to the indi- vidual student. Another reason is that the assigned tasks were mostly of short duration. This might be due to the mathematical content and to the case that only a short time period was allocated to basic algebra.
However, the investigation has to go beyond the content and the quali- ty of tasks and examples. Stein, Grover, and Henningsen (1996) provided a model in which a task is seen to pass 3 phases. The first phase is the task in the textbook or in other written materials, the second phase is the task set up by the teacher, and the last phase is the task implemented and worked upon by the students in the classroom. This means that the task as it appears in the textbook or in other materials might be altered from phase to phase and even in the classroom situation.
In this thesis the tasks will be categorised in line with former studies about number and algebra, and analysed according to levels of cognitive demand. Then the implementation of the tasks will be focused upon.
6.4 Mathematics syllabus and tasks
According to the perspective outlined above it is important to provide op- portunities to learn mathematics. The ultimate aim of mathematics teach- ing is to enhance students’ mathematical proficiency (Kilpatrick, et al., 2001) or mathematical competence (Niss, 2004). According to both
frameworks one cannot hold isolated strands of proficiencies or individual mathematical competencies, and at the same time claim to be mathemati- cally competent or proficient. However, in school, different strands or competencies are emphasised at different times, although the aim is to promote connections and an overview of mathematical strands.
The conventional curriculum model as Stein et al. (2007) describe it, is focused on carefully sequenced theory and tasks. This is still the case for most commercial textbooks. Although the goal is that students are to reach a level where mathematics is seen as connected and meaningful, one has also to practice skills in order to be fluent.
One concern has been that the focus on conceptual learning of mathe- matics would result in students having problems executing procedures and computations. Studies, however, show that students who have followed a reform based curriculum with emphasis on conceptual learning tend to perform as well as students following a conventional program on stand- ardised tests (Swafford, 2003). In addition, those students outperformed students following a conventional program, when it came to tasks testing what had been emphasised in the reform curriculum.
In this chapter, curriculum material, textbooks and tasks have been discussed and related to students’ opportunities to learn. It is problematic to make direct links between curriculum material and learning although there are indications that the different features of curriculum material might influence students’ learning in different ways (Remillard, Harris, & Agodini, 2014). One result from earlier studies is clear though. Topics covered in the material are more likely to be learned than topics not cov- ered (Stein, et al., 2007). It is also clear that cognitive challenging tasks promote another learning than tasks of a low level of cognitive demand (ibid). Grouws and Hiebert (2007) in a meta-study found that different teaching promoted different learning, and came to a conclusion about what promoted conceptual understanding and what promoted procedural fluency (see section 6.1). This will be drawn upon in the further work with the data in this thesis.