IIa. Students' Topical Difficulties

The transition from familiar, concrete to abstract mathematical contexts generates a series of difficulties for the novice. Some of these difficulties were observed in these tutorials. The following examples relate to the use of the concept of function in the context of mappings between vector spaces in Vectorial Analysis and to the introductory concepts of Topology, most notably the notion of compactness.

Example 1: The notion of function in the unfamiliar context of mappings between vector spaces. The mathematical question in the tutorial is about finding the matrix of a mapping between two vector spaces. During the discussion the tutor asks: 'what if the matrix we found was the zero matrix O? What would that mean for the mapping between the two vector spaces?'. The student answers that the mapping would 'map everything to zero'.

A little later the question is repeated for another pair of vector spaces and another mapping. Only this time instead of O, the tutor asks about I, the identity matrix. The same student replies 'everything would go to one'.

Both the tutor's questions here are supposed to be simple. They merely seem to require the student to generalise from

• the notion of the constant real function that maps every real number to zero (f(x)=0) and

• the notion of the identity function, the real function that leaves every real number the same (f(x)=x),

to the notion of mapping a vector of one vector space to another via a matrix that in the first case it is O and in the second it is I. Whereas the first transfer seems to be carried out smoothly, the second is not. A vector that is mapped via the identity matrix remains the same; the student's claim 'everything would go to 1' is a perplexed interpretation possibly of the kind 'since O means everything mapped to 0 then I means everything would go to 1''. It seems that it cannot be taken for granted that the transfer

from thinking in terms of one variable correspondences between numbers

to thinking in terms of mappings between vector spaces

— vectors after all are multidimensional and non-numerical mathematical objects — comes natural to the novice learner.

Example 2. The notion of compactness. Topology seems to be one of the areas in which the difficulties of the transition from the concrete to the abstract are mostly apparent. Most evidently the students seem to make no sense of the definition of compactness. As a flavour of the basic difficulties the students have with their introduction to Topology, I cite two instances relating to the notion of set of sets.

Instance 1:

X is an open set.

Tutor: What is an open cover of X?...first what is an open cover?

Student: ...is it the union of lots of open sets?

Tutor: ...well, it’s a family of open sets.

Instance 2:

{Xi} is a cover for X. This means that X is a subset of ÈXi.

Student: Why then X is not in ÈXi?

Since the issues raised by the students' questions are elaborated upon in the Main Study here I merely cite the instances noting their significance with regard to the kind of evidence of the students' learning difficulties available during tutorial observation. Knowing what a union of sets is and what a subset is does not imply that the notions of a cover {Xi} and of ÈXi will also come naturally.

One of the consequences of the novices' difficulties with the transition from concrete to formal mathematical contexts seems to be that they tend to adhere persistently to the contexts with which they are familiar from earlier mathematical experiences. The following examples relate to the use of the concept of function as a mapping in Vectorial Analysis and to the definition of a vector space.

Example 3. The concept of Function in Vectorial Analysis. The same student as in Example 1, but in another instance, suggests, when asked to find the value of c in f(x) = c = constant: 'let's try a couple of values for x'.

This is a standard practice used when looking for the values of a and b in f(x) =ax+b, where f is a linear function. A familiar practice is activated and employed inappropriately. Incidentally this instance is evidence of the students' persistent image of linearity: linear functions and their properties seem to dominate a large part of the novices' concept image of function. Similar observations were raised in these tutorials in cases where the students demonstrated difficulty with replacing variable x with x+1 in f(x) and often replied that f(x+1) = f(x)+1 or f(x+1) = f(x)+f(1) which are properties held only by some linear functions.

Example 4. The definition of Vector Space. The students seem constantly to assume that a vector space is always defined over Â. When the tutor brings their attention to other possibilities and asks them what the scalars are, responses are like the one below:

Student: ...constants.

Tutor: Constants?

Student: Numbers.

Tutor: How do you know?

Student: Are they real?

The students are usually reluctant to pay special attention to 'details' like this because the majority of vector spaces they have to deal with at this stage is defined over Â. So in this case this avoidance of the general case of a field in favour of the specific and familiar  seems to be convenient.

The students occasionally express a preference for notation at the expense of conciseness and comprehensiveness. For example, and in relation to the use of the concept of absolute value, as well as notation ||, in a number of cases the students appear as if they do know that |a-b|<c translates into -c<a-b<c, but given an inequality of the latter form, they can rarely see it concisely represented by the former.

This becomes more explicit in the cases where in the course of a proof they devise the inequalities themselves. In these tutorials no student came up with a direct algebraic expression involving ||. Their explicit preference was for long inequality algebraic expressions which, despite their length, appear visually as much closer to their mental image of the inequality to be expressed.

'||' is a compressed expression whose interpretation and handling repeatedly seemed to be problematic. It is therefore expected that in a selective process amongst various alternatives the learners will not favour a mathematical tool that, so far, has caused them inconvenience and confusion. Its comprehensiveness and elegance are of little relevance: when a mathematical tool is not embraced smoothly as a personal construct, the learner's response to it is likely to be poor; however highly recommended is its use for epistemological reasons.

In the following I present a sample of observations from the Pilot Study relating to the students' difficulties with mathematical reasoning.

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