IIIc.iv The Novice's Difficulties With Linear and Abstract Algebra
In Chapters 8 and 9 an opportunity is given to study the novice's cognition of a wide variety of algebraic topics: most predominantly related to Vector Spaces and Groups. Research interest in these topics is relatively new and a large number of the studies reviewed here are observational studies in which the teaching and learning of these topics merge into a whole of theorising practice: one or a few algebraic problems are given to undergraduates or a new teaching technique is implemented (usually related to current educational technology). Observations are made about the novices' learning and their difficulties. Finally empirical explanations are attempted. Despite their theoretical frailty these studies provide relevant and valuable illumination on the cognition of Algebra. In the following I cite some of these studies.
Orit Hazzan (1994), accounting for the students' repeatedly observed belief that in a group
x*x-1=eÞ x=e,
notes that students 'borrow properties' from  . This possibly reflects their need of a familiar metaphor to associate with the axiomatic definition of operation *. Linguistically calling a*b the 'product' of elements a and b in a group reinforces the illegitimate use of the  metaphor (by an analogous linguistic token she also points at the similar sounding of the Hebrew words for 'keep' and 'preserve'. Students then reinterpret 'an isomorphism preserves the group operation' as keeping the properties of usual operations). Moreover representing elements a and b of a group with different letters may evoke the idea that a and b are two distinct objects. Finally Hazzan claims that students tend to confuse a theorem and its converse. So, because if x=e then x2=e, students also may think that if x2=e then x=e.
In a similar empirical vein Carducci (1993), benefiting from the visualising powers of Mathematica, attempted to question some of her students' persistent intuitions regarding properties of determinants. In the process she observed that, even when required to work in abstract environments, students demonstrate a partiality for some matrix operations (matrix inverse and scalar multiplication). In another IT experiment with ISETL, inspired by the students' frustration with lectures and the low retaining of knowledge even shortly after the lectures, Leron and Dubinsky (1995) note that the construction of meaning 'is at the heart of students' difficulties in abstract algebra' and that the students' difficulty with 'fairly simple' relationships between certain mathematical objects (such as the Homomorphism Theorem or Lagrange's Theorem) should not be attributed to the complexity of the theorems but to the abstract nature of the mathematical objects involved. They then give the example of Lagrange's Theorem which is about 'easy' things such as 'one number dividing the other, two sets having the same cardinality or being disjoint, etc.':
The objects, on the other hand are 'complicated' in the sense of the many levels of abstraction and the great time and effort needed to 'construct' them in the mind of the student. For us the simplicity of the proof lies in our ability to have a clear image of the group as being partitioned into a disjoint union of cosets. But in order for the students to have such an image, they need to 'construct' in their mind not only 'group', 'subgroup' and 'coset' but also 'the set of all cosets of H in G'.
So, they conclude,
the students' difficulty with understanding Lagrange's theorem may be largely due to their confusion about the nature of cosets. We find that they can understand the process of forming a coset, but often cannot take the next step of seeing these cosets as objects to be measured, counted and compared.
Their didactical strategy then revolves around helping students to construct cosets and learn to manipulate them as objects. Harel (1989) a few years earlier also proposed that the teaching of Linear Algebra is based on false assumptions with regard to the students' ability to deal with abstract structures without extensive preparation or to appreciate the economy of thought characterising abstract representations. He then cites examples from school mathematics where the domains of algebraic applications are specific and embedded either in a numerical or a visually accessible geometric context. Like Dubinsky and Leron above, he also recognises the abstract nature of the mathematical objects involved in Linear Algebra as well as the not-always-emphasised multiplicity of its domains of application, as the sources of the students' difficulty. Even when introduced through a few examples, Harel concludes, the Linear Algebra abstract concepts rarely acquire a meaningful reason of existence in the students' minds.
Similarly to Harel, Robert and Schwarzenberger point at the concept of vector space and the concept of group as examples of new mental objects the construction of which causes fundamental difficulties in the transition from elementary to advanced mathematics. In tune with Tucker's historical account of Linear Algebra (1993), they note that 'the problems from which these concepts arose in an essential manner are not accessible to students who are beginning to study (and expected to understand) the concepts today': this historically decontextualised presentation deprives the novices of a potentially meaningful construction of these concepts.
Dorier et al (1994) elaborate on the above, recognising first that, because students do 'not really have to bring the concepts themselves into play', the results of final exams may very well hide the poverty of the students' concept image construction. They then suggest the following explanations:
• the specific epistemological nature of the concepts of Linear Algebra:
These concepts [vector space, linear operators, image, kernel, basis, dependence, dimensions, rank, etc.] took their formal shape only after numerous uses of linear methods in specific contexts without much unification. These concepts, in their present form, are the cause and results of unification and generalisation. Their relevance appears only a posteriori, as they first renewed in an economical manner the solving of old problems and only afterwards allowed new approaches.
So the learners have difficulties learning these concepts because
they only have access to the final phase of the historical process: definition of the concept and systematic use in the solving of problems. Yet the problems they are asked to solve may often be solved with specific tools and methods, more familiar which do not necessarily imply the use of the new concepts. In this case the simplification and improvement induced, in the solving of the problems, by a change in the point of view cannot be foreseen by the students. They may only do this because they are asked to; this is an effect of the teaching contract.
On a more theoretical basis, this implies that these problems, which can be solved in various settings, cannot yet be chosen as 'good' problems in the perspective of the tool/object dialectic [...], as the concepts to be taught are not indispensable for the solving. The only problems which would necessitate an absolute use of the formal concepts of linear algebra are all too complicated for our students (they involve non-countable infinite dimension vector space)...
In other words for the students the concepts of Linear Algebra are above all objects before they can be used as tools; students are therefore deprived of the long progression, which brought mathematicians to express these concepts, step by step.
...[Students have] elements of knowledge which initially are not well enough distributed in different settings (geometrical, analytical, logical, formal settings mainly). [some analyses of students' practices] reveal the lack of ability to change settings and points of view.
Dorier et al's views can be generalised to other mathematical contexts. An example is Nicholson's paper on the historical development of the concept of quotient group which appears implicitly in the works of many 19th century mathematicians, including Galois and Jordan, but only as a context-embedded tool, not formally defined. Despite Dedekind's early understanding of the power and potential of the formalisation of the concept since the 1850s, it is not until the end of the century that Hölder systematically and explicitly defines the quotient group. It is noteworthy that Dedekind's lectures were lukewarmly received (members of the mathematically strong audience admitted they understood little); that is it seems that from a phylogenetic point of view the concept had not completed its cycle of fermentation. As a result professional mathematicians of the time resisted its significance and its acceptance through formalisation.
Harel and Kaput (1991) embed some of the difficulties cited above by Harel in their theory of Object-Valued Operators (an Object-Valued Operator is, for instance, the correspondence between parameters and functions in f(x)=sin(ax)):
...students usually had difficulty dealing with such a correspondence, unless they were able to tag the outputs of the correspondence with familiar geometric figures, such as lines or planes [...]. These geometric figures, which were manipulable objects for the students, apparently helped the students to construct such a correspondence as an object-valued operator.
Another common example involves the construction in abstract algebra of the quotient object associated with a 'normal' sub-object, e.g. in the case of groups. The cosets must be conceived as objects if they are to participate as elements of a group. However the existence of a 'representative element' for a coset, where the operation defined on cosets can be given in terms of an operation on their representatives, makes it possible to deal successfully with many aspects of the quotient group on a symbol manipulation level without treating the subsets of a group as objects, or even as subsets. Students' inadequate conceptions are revealed when one asks them to attempt to create a group using a non-normal subgroup's cosets - they often cannot understand why the subsets 'fall apart' when they attempt to multiply them together as 'sets', or by using representatives.
(Harel & Kaput 1991, p87)
Hillel and Sierpinska (1994) attribute the students' difficulties with learning Linear Algebra partly to their 'inexperience with proofs and proof-based theories. Moreover espousing Piaget's and Garcia's notion of intra-, inter- and trans- level of knowing something, they note that students learning Linear Algebra 'need to operate at the trans- level'. So generality and abstraction coupled with the students inter-level elementary experiences yields problematic cognitive behaviour. The researchers also include in their list of sources of difficulty, things that are not generic to Linear Algebra but relate to advanced mathematics in general, such as 'not understanding the need for proofs nor the various proof techniques, not being able to deal with the often implicit quantifiers, confusing necessary and sufficient conditions'. Below I present Selden and Selden's (1987) list of 'errors and misconceptions' regarding theorem proving in Abstract Algebra which reinforce Hillel's and Sierpinska's observation about the relevance of more general difficulties in advanced mathematical reasoning.
Selden and Selden's taxonomy probably fits better alongside some of the observational studies previously cited in this section, since they are based on their experience from a Moore-type Abstract Algebra course but I mention it here since it helps in placing the learning difficulties with Abstract Algebra in the wider advanced mathematical context. The major 'misconceptions' observed in their students include:
M1. Beginning with the conclusion.
M2. Names confer existence ('failure to distinguish between symbols for things whose existence is established and symbols for things whose existence is not').
M3. Apparent differences are real (failure to see things with different names are not necessarily different).
M4. Using the converse of a theorem.
M5. Real number laws are universal (mentioned also before).
M6. Conservation of relationships.
M7. Element - Set interchanges (difficulty in understanding statements involving sets results in substituting them with statements about the elements of these sets).
Selden and Selden also mention: overextended symbols (the use of the same symbol for two distinct things), weakening the theorem, notational inflexibility, misuse of theorems, circularity, locally unintelligible proofs, substituting with abandon, holes in the implication that links two statements and using information out of context. They then recognise that their taxonomy addresses a diversity of cognitive issues which they sum up as the learner's difficulty with: generalisation, use of theorems, notation and symbols, nature of proofs and quantification.
Back to Hillel and Sierpinska, specifically to Linear Algebra these authors add to Dorier et al's interpretations, cited previously, the 'shuffling back and forth' between different levels of description: within the discourse of Linear Algebra three languages coexist,
the language of the general theory (vector spaces, subspaces, dimensions etc.)
the language of  n (n-tuples, matrices, rank etc.)
the geometric language of  2 and  3 (orthogonality etc.),
that 'are often interchangeable but are certainly not equivalent'. Most emphatically Sierpinska and Hillel raise the problematic issue of 'representing a linear operator in a basis and moving from one such representation to another'. They contend that understanding this representation operation requires that the learner sees the language of Linear Algebra as a 'network of languages and rules of translation between them'. Following Foucault's distinction between language as part of the world it describes and language as a representative system of signs, they suggest that
the problem of understanding the language of linear algebra as a representation rather than as being part of the world lies in the fact that the world of linear algebra is indeed the world of simultaneously used systems of representation. How does one make sure that two externally different representations indeed represent one and the same 'thing'?
In that sense when students encounter for the first time vectors in  n, 'strings of numbers' become to them the 'primary representation', the 'thing'. This view of a vector as identical with one of its representations is profoundly shaken when the student reaches the realisation that this 'string of numbers' represents different vectors in different bases and that the same vector is represented by different 'strings of numbers' in different bases:
Strings of numbers, so familiar, so palpable, suddenly feel like 'ghosts of departed quantities'.
Recurrent mistakes, such as taking the first column of the matrix of a linear operator as the image of the first basis vector under the operator (correct in the canonical basis), must be attributed, contend these authors, to a more deeply ingrained conceptual difficulty than simply to the Piagetian difficulty with internalising an activity as an operation. To this purpose they offer the linguistic scheme briefly outlined above, namely the need to distinguish vectors from their representations as well as reach the 'trans-level' of seeing linear operators as 'objects of inquiry themselves'.
Leron, Hazzan and Zazkis (1994), in a study preceding the first major effort of systematising learning difficulties with regard to Abstract Algebra (Dubinsky et al 1994), interviewed a small number of students on their conceptions of Group Isomorphism. As Harel and others above, they stress that students, accustomed to the affectively secure routine algorithmic behaviour of school mathematics, find it difficult to engage in an existential process of constructing an abstract object. Moreover the existence of more than one possibility also clashes with their previous school experience of mathematical problems having one and only one solution. Decomposing the notion of an isomorphism, they note that it involves two especially complex notions: function and an existential quantifier. They identify three levels of internalisation of the existential quantifier $ that seem to influence their students' conceptions of isomorphisms: the personalised action level, the process level and the object level. The more objectified the conception of a quantifier is, the more appropriated the students' existential activity turns out to be. A similar tripartite levelling of the students' conception of function seems to influence the students' attitudes towards isomorphisms. In addition the authors note the importance of talking about isomorphisms necessarily in terms of their domain and range (which again clashes with the students' previous experiences of talking about functions mostly as procedures or formulas regardless of domain and range).
Finally they state that
the very concept of isomorphism is but a formal expression of many general ideas about similarity and difference, most notably the idea that two things which are different, may be viewed as similar under an appropriate act of abstraction.
Remarkably the notion of function emerges as dominant in advanced mathematical cognition and indeed in much more abstract contexts than it has been mostly studied (see IIIc.ii of this Chapter). In a similar vein in Zazkis' experiment (1992), students, while finding the inverse of a given compound element, repeatedly claimed that (XoY)-1 is X-1oY-1 instead of Y-1oX-1. Zazkis draws on misapplication of linearity and specifically over-generalisation of distributivity as the dominant sources of the error and cites classical algebraic and trigonometric examples of this behaviour. Given that her experiment was carried out in a computer environment, Zazkis also notes that learners do not always 'perceive the [computer] environment as mathematical and therefore they use their 'naive' knowledge and extrapolation techniques, rather than 'formal mathematical knowledge''. She finally employs a proceptual argument in order to indicate the flexibility in shifting from the process to the object aspects of a concept as a sign of mathematical maturity: this flexibility is a requirement for successful handling of tasks like finding the inverse of a compound element.
I close the section on Linear and Abstract Algebra with a reference to the above mentioned first major attempt at a systematic presentation of observations regarding learning difficulties with regard to the notions of group, subgroup, coset, normality and quotient group. As several researchers referred to in this section, Dubinsky et al (1994) also present their observations in terms of the action - process - object framework. In the following I summarise their findings.
Their didactical interest in Abstract Algebra, and in particular Group Theory, was triggered off by the reported failure of the novices to achieve good understanding of the concepts involved in introductory courses. In most cases the novice's formal encounter with the notion of group is the first one in a long series of abstract concepts. Accompanying the problematic novelty of the experience of abstraction is the historical antecedent of the epistemological complexity of the concepts, examples of which were given above in the reference to Nicholson's paper.
Developing the Concepts of Group and Subgroup. In sum the developmental view of these concepts presented in their paper is the following:
Our observations are consistent with a progression in understanding that moves through various intermediate (and incomplete) ways of understanding groups and subgroups... from seeing [them] as primarily sets of discrete elements, to a stage where the operations as well as the group elements are incorporated into the necessary definition [to]...a thorough understanding of a group as an object to which actions can be applied.
In their study the participants seemed to develop the concepts of group and subgroup in parallel. The 'psychogenesis' of these concepts appeared to be 'linear' but also at many points 'in concert with the others'. Often progress with regard to one concept awaits developments in the others. Also other mathematical concepts, most notably Set and Function, are fundamental in these developments. More specifically: at first the most primitive conception of a group is 'based entirely on the student's conception of a set'. Progressively properties of this set are included in this conception; a binary operation maybe one of them and it is crucial when this property is singled out. At this moment the notion of function, and in particular of the two-variable function is needed in order to accommodate the notion of a binary operation. Ideally this process is completed with the encapsulation of the set and the binary operation into the notion of group and the reification of the pair so that the construction of the notion of an isomorphism is possible. It is noteworthy that the procedural aspects of the binary operation seem to attract more of the learner's attention than probably allowed in the above developmental path.
Moreover the notion of function appears strongly in the concept formation of subgroup as a restriction of a function to a subset of its domain. If the role of the binary operation has not been clearly understood it is likely that the learner confuses subgroup with subset.
Developing the Concepts of Coset and Normality. In general the construction of cosets in simple groups appears as a simple task. What seems to be extremely difficult is the conceptualisation of quotient group and normality; also understanding that the quotient group is isomorphic with another familiar group. Difficulty varies in the context of different groups but the overall impression is one of deep confusion. Normality is also confused with commutativity and the students' ignoring the condition of normality for the existence of the quotient group influences heavily their attempts to construct the cosets.
Probably the most problematic concept is the Quotient Group. In terms of the action - process - object schema, a major issue in the psychogenesis of the concept of coset and coset operation appears to be
the encapsulation of the process of forming cosets into objects which are to be the elements of the quotient group. Again the student's conception of function appears to be needed, but confusion seems to have occurred when he or she made an attempt to construct a binary operation on cosets before being able to manipulate these cosets as objects. An interesting point is that in some cases, formation of a coset did not guarantee that the student could not deal with this as an object. This suggests that the student's conception of sets may not have been adequate.
In this last section Linear and Abstract Algebra were presented as two exceptionally difficult introductory topics for novices in which difficulties related to a wide range of basic concepts seems to be condensed. Issues of rigour and intuition, visualisation, notation and language were also raised. The findings of this study, as demonstrated in Chapters 6-10, complement or at some instances extend the findings reported in this Part of the Chapter. In Chapter 2 the theoretical background of the study's methodology is presented, so that in the first two chapters the study is embedded within the current thematic and methodological developments in the field.