Algebraic notation and the variable

Perhaps the most important concept in algebra is the concept of variable. In school algebra a letter is normally used as a place holder for a number. This number can be a specific unknown like the unknown in an equation, it can be a general number as in the general rules for arithmetic, or it can be a number which varies in relation to other numbers as in the case of functions.

In mathematics, this notion of variable is used for all the above- mentioned situations. Some researchers in mathematics education also used the notion variable for all these cases (Drijvers, 2002 ; Usiskin,

1988). Küchemann (1981) who investigated students’ different interpreta- tions of letters in algebra warned against the “blanket use of the term “var- iable” in generalised arithmetic” (ibid p. 110). He asserted that this use had blurred the meaning of the term itself and also the different meanings

that students assign to the letters. For him it was important to emphasise that the concept of variable includes an interpretation of an unknown, not a specific unknown, but an unknown with a changing value.

Küchemann (ibid) focused on different ways in which students inter- preted letters in algebra. His work was part of a large-scale survey of Concepts in Secondary Mathematics and Science in England, the CSMS- project. One of the aims of the study was to categorise mathematics un- derstanding according to Piagetian developmental stages. This aspect of the research has since been questioned; however, the study is still relevant in that it gives a valuable insight into students’ thinking about mathemati- cal concepts.

In the study the underlying assumption was that algebra was viewed as generalised arithmetic. Küchemann (ibid) identified six categories of how students interpreted literal symbols. In the first two categories letters are ignored or given a numerical value from the outset.

In the third category letters are used as objects. In this category letters are regarded as shorthand for an object or as an object in its own right. When students are introduced to collecting like terms in algebra, as in the example 2a + 5b + a, teachers often use the “fruit salad” method of say- ing for example 2 apples + 5 bananas + one more apple, which can totally mislead students. Another possibility is to see the letter as an object in its own right collecting a’s and b’s and so on. It is fundamental in algebra that students understand that letter symbols are substitutes for numbers. A lack of such understanding might initially lead to success in solving some types of tasks, it will, however, mislead students and cause problems in the further work with algebra. Another problem occurs when students have to distinguish between the objects themselves and the numbers of objects. If students are given the problem: If cakes cost c pence each and buns b pence each, and 4 cakes and 3 buns are bought, what does 4c + 3b stand for? Many students also in higher levels tend to interpret this to be 4 cakes and 3 buns, instead of seeing the expression as an expression for the total price for the 4 cakes and the 3 buns.

The last three categories are as follows: letter used as a specific un- known. Here the letter is regarded as a specific, but unknown number, that can be operated upon directly as in equation solving, or for example in tasks where it is asked for the answer when 4 is added to 3n. The next cat- egory is letter used as a generalised number. This means that the letter is seen as representing, or at least as being able to take, several values rather than just one, for example letters in formulas.

The highest level of understanding, according to Küchemann, is

reached when letters are used as variables. By this he asserts that the letter is seen as representing a range of unspecified values, and a systematic re-

lationship is seen to exist between two such sets of values (Küchemann, 1981a, p.104). One example he uses is the statement 5b + 6r = 90. If the letters here are interpreted as specific unknown numbers, the statement is seen to be true for an unknown, particular pair of numbers. It is also pos- sible to interpret these letters as generalised numbers without having in mind how they change, only that they are different isolated pairs of num- bers making the statement true. For these letters to be interpreted as varia- bles, according to Küchemann’s meaning of a variable, it is important that there is an established relationship between the two letters b and r in the example above, so that it is seen by the students that for example a change in the value of b causes a change in the value of r.

In a follow-up study, Strategies and Errors in Secondary Mathematics (SESM) (Booth, 1984), the tasks and the findings from the algebra part of the CSMS study were used in order to investigate the reasons for the most frequent errors. The researchers interviewed 50 students from the same population as in the CSMS study. In addition, teaching experiments were carried out both in small groups and in whole classes. Booth (ibid) found that some problems students had in interpreting letters as generalised numbers, were due to their cognitive development. Other problems re- ported seemed to have been resolved during the teaching experiment. An emphasis on open expressions led for example students to accept expres- sions as x + 3 as a final result. This was not obvious in the start, but they developed an acceptance of the ‘lack of closure’ (see the next section 4.1).

In a recent intervention study in England, the Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS), the first phase was a large scale survey using tasks from the CSMS study. It was concluded that the tasks were still relevant, 30 years on. The algebra test was administered to ca 5000 children from 12 to 14 years old. It was reported (Brown, Hodgen, & Küchemann, 2012) that the results were the same or slightly worse than the results from the CSMS project, and that less than 10 % interpreted the letter symbol as a variable.

Based on the test results and interviews with students in phase 1, on pedagogical principles, and research literature about algebra, rich tasks were designed, and collections of interlinked lessons were outlined. The research team worked with teachers, and after implementing the lessons and the material, many teachers told that the intervention project made them change their ways of teaching. Students’ responses on tests indicated that the rate of learning had been double over a year for those who had joined the project compared to those who had not (Brown, Hodgen, & Küchemann, 2014)

When investigating students’ conception of variable, Wagner (1981) inquired into students’ ability to conserve equations and functions when

changing the literal symbols. In interviews with 29 students between 10 and 18 years from schools in the USA, she found that fewer than half the students, having studied algebra, could state that the functions and equa- tions presented, were conserved when the letters denoting the variables were changed. They seemed to believe that the value of the variable was changed by the changing of the letter symbol, even though the structure of the functions and equations was kept the same. Some gave values accord- ing to the ordering of the letters in the alphabet.

Usiskin (1988) describes many uses of the letters (he uses the term variable to be synonymous with the word letter) to which secondary stu- dents are exposed. He asserts that these different uses are related to the purposes to which algebra is used, and this again is determined by which approach to algebra is chosen (see section 3.1.1).

MacGregor and Stacey (1997) reported from a longitudinal study in Australia including 22 schools from grade 7 to grade 10. The students were tested at several occasions during three years, and some students were interviewed. They found that students were making the same errors as reported in earlier studies (Booth, 1984; Küchemann, 1981; Wagner, 1981). Also they (Stacey & MacGregor, 1997) found that the difficulties students have in learning to use algebraic notations, appear to have several origins. The authors found evidence that “intuitive assumptions and sensi- ble, pragmatic reasoning about an unfamiliar notation system” caused problems in interpreting and using letter symbols correctly.

‘Interference from new learning in mathematics’, was one such obsta- cle for some students, especially the older ones. The researchers found evidence that students wrote the right expression or equation, however, misused newly learned rules for solving equations. Another example was students’ use of exponential notation to express multiplication, e.g. x³ in- stead of 3x.

‘Poorly-designed and misleading teaching materials’ was another source for problems with letter symbols. In some schools, students’ per- formance was relatively poor compared to the others. In these schools the authors discussed with the teachers and found for example that textbooks used in the first algebra course, explicitly stated that letters could be used as abbreviated words and labels.

In one school, they found that students interpreted letters according to their order in the alphabet. The teachers told that they had been working with puzzles and codes in the mathematics lessons. The authors character- ised this source of mis-interpretations as: “analogies with symbol systems used in everyday life, in other parts of mathematics or in other school sub- jects”.

In this section, it is shown that many students have problems in their interpretation and use of letters (the variable) in algebra, and Küchemann (1981) in the CSMS study categorised these different interpretations in six categories. Other studies reported the same problems as was found in the CSMS study.

It was also shown that many students considered that functions and equations changed by changing only the letter variable (Wagner, (1981).

Booth’s teaching experiment was promising in that a focus on specific problems helped some students to resolve some of their problems. Also the more recent intervention project, ICCAMS, reported a positive devel- opment in students’ conception of algebra (Brown, et al., 2014).

Later in this thesis, when referring to students’ work, the notion ‘vari- able’ will be used only in accordance with Küchemann’s definition of the variable. The reason for this is that the notion ‘variable’ was only used in class in relation to linear function, and that observation done when stu- dents were working with functions created a question if all students had the conception of the letter symbols as variables even in that context.

4.2 Lack of closure

One important difference from arithmetic to algebra is that expressions can be the final result in the transformational activity. Collis denoted this as the ‘acceptance of lack of closure’ (Collis, 1972). In the literature it is reported that many students show a reluctance to accept this lack of clo- sure.

Tall and Thomas (2001) use the example of 7 + 4 in arithmetic and the expression 7 + x in algebra to describe this difference between arith- metic and algebra. Although 7 + 4 is an open operation or a procept in the vocabulary of Gray and Tall (1994) (see section 3.3), it has a built-in pro- cess of computing the numbers to reach to a number result. When it

comes to the algebraic expression this is not possible without knowing the value of x. The last expression therefore has to be left open. For students it is not easy to accept this lack of closure. For many this tends to lead stu- dents to conjoin the terms, which means that they for example regard

3 4n to be equal to 7n, or that they alternatively might assign values to the letter symbol (Küchemann, 1981). Several researchers have reported these phenomena (Booth, 1984; Brekke, 2005; Chalouh & Herscovics, 1988; Küchemann, 1981).

The reluctance among students to accept open expressions is more likely in the beginning of the algebra learning.