Extract 8.1
Context: Four pairs of female students are taught in four consecutive tutorials about the span of a set:
Tutorial 1: Abidul and Eleanor
Tutorial 2: Patricia and Frances
Tutorial 3: Camille and Cleo
Tutorial 4: Beth and Cary
The mathematical structure in the four sessions is as follows:
definition of the span of a set
constructing the span of
{v1},
{v1, v2},
{v1, v2, v3},
where vi are vectors in a vector space V.
In the following I cite extracts from Tutorials 1, 2 and 4, which were almost identical in structure as well as a summary of Tutorial 3 which was structurally different because of one of the students' interventions.
Discusssion of the concept of Span starts with the students' questions about its meaning in all four sessions. In Tutorial 3 the tutor, who in the meantime was surprised by the unanimous unease with the concept, finds out that the definition of Span of a set given in the lectures was
the span of S={v1, v2,...,vk}ÍV is the smallest subspace of V which contains S.
The definition used in these tutorials is
the span of S={v1, v2,...,vk}ÍV is the subspace of V that contains the linear combinations of v1, v2,...,vk
and I emphasise that none of the eight students had heard of it before. The tutor explains that in order to construct the span of S={v1, v2,...,vk} we take all the multiple scalars of vi and the vectors coming out of adding those. She also warns the students that there is a 'change of gear' in the course which now becomes more abstract.
The Episode:
Tutorials 1, 2 and 4
Constructing the span of {v1}. The tutor draws fig.1a and asks the students to find the span of S={v1}. She distinguishes between the cases where v1=0 and v1¹0. The students all point out that if v1=0 then <S>={0}. In Tutorial 4 however Beth claims that <S> is...
B1: ...the empty set.
T: What do you mean by 'empty set'?
B2: Zero.
The tutor repeats clearly to Beth that if v1=0 then <S>={0}. Then she explains that if v1¹0 then <S>=<v1>={av1: aÎÂ} and asks the students to demonstrate on paper (fig1a) what is the role of a: students Abidul, Frances and Beth respond as follows:
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A1: You multiply by the multitude of v1 and [in case a<0] you're going that way [opposite direction to v1]. |
F1: It will be the distance a times v1 for av1 and [in case a<0] you have to turn round beyond zero. |
B3: [shows with her fingers the positive and negative directions of av1] and says that all av1 will be 'on a line'. |
Constructing the span of {v1, v2}. For <v1, v2> the tutor adds a vector v2 to fig.1a (fig. 1b) and distinguishes between v2Ï<v1> and v2Î<v1>. In the case where v2Ï<v1> discussion in the three tutorials is as follows:
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A2: So it's going to be the plane defined by v1 and v2. E1: See how far it goes to this direction and how far to this...[shows with her fingers on the paper something like a translation in order to show 'how far' she wants to go] The tutor draws the parallels but the girls do not know how to continue. Eleanor whispers something that sounds like 'finding b1 and b2', the coefficients of v1 and v2. Abidul adds: A3: It's b1 times the multitude of v1... and b2 times the multitude of v2. |
P1: The sums of the form a1v1+a2v2 will belong to <v1, v2>. Tutor: Pictorially? P2: You complete the parallelogram... The tutor explains that <v1, v2> will be the whole plane and asks 'Can I get to the whole plane this way? By using v1 and v2?'. And 'if I had a vector w here how would you find what scalar multiples of v1 and v2 I should use to write it in terms of v1 and v2?'. Silence. P3: That would be a minus av2... [asked to show on paper Frances and Patricia bend towards the drawing and point with their fingers]. |
B4. Beth, while saying that the span of v1 is one line and the span of v2 is 'another line', is very surprised to hear that <v1, v2> is the whole plane: 'Is it the whole thing?' she asks. The tutor draws a vector w and asks for the coefficients b1 and b2 in the linear combination w=b1v1+b2v2. C1: We just have to take components. Tutor: [makes a wry face at hearing the word 'components'] Although I've drawn this on the plane, I might have matrices or whatever else. With the tutor's closed questioning Cary outlines the Parallelogram Rule. |
In Tutorials 1 and 2 the tutor introduces the Parallelogram Rule.
Asked about the case v2Î<v1> students Abidul, Patricia and Beth respond as follows:
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A4: You're just gonna get the same line again. It's... the sum will be a times v1 plus again... I mean it's gonna be the same line. |
P4: v2 is the spanning of v1... The tutor corrects: the line on which v1 lies is the span of v1. Then Patricia asks: P5: Would it just be around v1? You wouldn't have the plane... |
B4: It would be along the line. |
Since the students are hesitant about how to prove their claims the tutor then, using the linear dependence of v1 and v2, proves that <v1, v2>=<v1>.
Constructing the span of {v1, v2, v3}. Asked about <v1, v2, v3> (fig.1c) in Tutorials 1 and 2 the students distinguish between the cases where v3Î<v1, v2> and v3Ï<v1, v2>. In the latter case 'It's going to give us the three dimensional space' and in the former 'v3 wouldn't matter' point out students Abidul and Frances in tutorials 1 and 2 respectively. In tutorial 4 Beth refers to the former exclusively and only when the tutor draws her attention to the latter she notices that then there is another dimension coming in.
Tutorial 3
Student Camille starts the conversation on Span by confusing the term Span with the word 'scope'. Given the 'more practical definition' of Span of a set used in these tutorials, Camille notes that 0Î<S> because <S> is 'the sum of these vectors, with any coefficients and the coefficients can be zero'. Also that '<S> is bigger than S'. So, she concludes, <S> 'is all the combinations of directions you can get from those directions'. The tutor agrees that 'this is a way of looking at it' and notes that <S> is a subspace of V. Then Camille asks:
C1: How do you prove that this is the smallest subspace... I understood how it contains S but how do you prove it's the smallest?
The tutor explains how <S> can be constructed by 'building it up' from the elements of S, then taking all their multiples and adding these up. Cleo then asks: 'then <S> is a subset of V?'. Camille explains:
C2: It's something more than that, it's a subspace. What's the diff... a subspace is something smaller that has the same properties like the properties on addition is that it... The difference between a subset and a subspace is that a subset can be for instance the empty set and a subspace has to have enough elements to keep all that...
In response to Camille's questions the tutor talks about <v1> and draws it as a line (fig.1a) on paper. She then asks about <v1, v2>. Camille responds: 'it's the whole... plan [in French this means the 'plane']... what-you-call-it provided that v2 is not on that line'. The tutor then points out that if we pick a point W on the plane we can write w=av1+bv2 (fig.1d). How can we then find a and b? she asks. The dialogue is as follows:
C3: By taking the projection of the point to the axes...
T1: What do you mean by the projection?
C4:..the perpendicular, by dropping...
T2: Drawing perpendiculars, is that right?
C5: Mm.[meaning yes] Then measuring...
The tutor then turns to Cleo: does she agree? Cleo asks the tutor to repeat the question as well as what they are trying to do. The tutor repeats the question and concludes by reminding Cleo that Camille 'suggested dropping perpendiculars'. 'Don't you say "take projections" ?' asks Camille. The tutor (due to a phonecall) leaves them for a few minutes to discuss among themselves and when she returns Camille says that they want to take the projections and measure the distance between O and the projected points. Asked by the tutor whether she agrees Cleo nods and the tutor draws two vectors (fig.1b) and asks them how we add these up. They both know the Parallelogram Rule. The tutor asks them if now they 'revise what they suggested'. They are silent and the tutor decides to apply the suggestion and 'drop perpendiculars' (fig. 1e). At the sight of fig.1e Camille changes her mind:
C6: I made a mistake. I thought w was a point in the space. I didn't think...
Everybody is silent for a few seconds. Then the tutor asks how we add v1 and v2. Silence continues. 'Draw the parallel', replies Camille. She then suggests they remove the parallels to the two lines until they meet W and when one parallel meets the first line it is av1 and the other it's bv2 (fig.1f). To the tutor's request Camille points at where av1 and bv2 are on the drawing. For the latter she says 'minus bv2' which she explains with 'because it's on the left'. When the tutor asks her to show 'plus bv2' Camille changes her mind and dismisses 'minus'.
Subsequently the case for v2Î<v1> is discussed and then the tutor asks about <v1, v2, v3>. Camille notes:
C7: So when you take k then you have k dimensions.
T3 Yes, once you make sure that the one you add up is not in the span of the others.
C8: After 3 I can not imagine it. It's getting confusing.
This part of Tutorial 3 closes with the tutor explaining the convenience of working in n dimensions. In the end Camille repeats the definition of <S> used in this tutorial (as shown in the Context) in order to 'make sure she understood it correctly'.