Local Cognitive Theory for Coordinate Systems. Components of Understanding:

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1 Local Cognitive Theory for Coordinate Systems Components of Understanding: Student difficulties: Students do not naturally consider any basis beyond the natural (standard) basis. Students do not understand that a basis for a vector space (or subspace) must be comprised of a set of linearly independent vectors that span the vector space (or subspace). Students do not consider a vector, such as p =!2 % $ ', as a descriptor with respect to a # 1 & particular basis such as the natural basis for! 2. That is, they do not consider p =!2 % $ # 1 & ' =!2 1 # $ % 0& ' +1 $ 0% # 1 & '. Students do not consider the change of coordinate matrices, e.g. a matrix Q such that Q [ P] N!! [ P] { u,v}, where N is the natural basis of! 2, from a functional perspective. Students do not carefully attend to the order of matrices and how order impacts ability to functionally navigate from one coordinate space to another.

2 Characteristics of Understanding: An APOS Genetic Decomposition for Coordinate Systems Action: Given a nonstandard basis and a vector coordinatized with respect to the nonstandard basis, one can express that coordinatized vector with respect to the natural (standard) basis, or visa versa. Process: One can describe the process of coordinatizing a vector with respect to different bases (whether they be standard or nonstandard) without specifically referring to particular vectors. The student has begun to transcend the physical manipulation of vectors and consider the manipulations in various forms. Object: Coordinatization of vectors is considered as an object when one can conceptualize of the change of coordinates from one nonstandard basis into another nonstandard basis not in terms of a series of changes but rather as a singular change that transcends the process of moving from P without having to consider going through the intermediary of P coordinatized with respect to the standard basis). to [ P] G [ ] N (the vector Schema: One can see coordinatization as a concept beyond vectors in R n, but as a concept relating to objects in Vector spaces and bases within those abstract vector spaces. Proposed learning paths and instructional activities: Problems 1 and 2 Problems 1 and 2 ask students to find the vector P with respect to the standard (or natural) basis determined by the given coordinate vector [ P] ( u,v) and the given basis {u,v}. In problem #1, the student is presented with the applets starting information and is asked to interpret this. The 0 % teacher should ask the students to conjecture how [ P] ( u,v) can equal with respect to #$!1&' (! { u,v} = #.5$.5% &,! 2 $ + ),. The teacher may wish to suggest to the students to show the corresponding * # '1%& - grid and ask the students how they can use the grid to argue that [ P] ( u,v) = 0 % with respect to #$!1&' (! { u,v} = #.5$.5% &,! 2 $ + ),. In Problem #2, the student is expected to explore a similar situation but has * # '1%& - to utilize the applet to reorient the basis vectors and conduct a similar thought experiment.

3 Problems 3a and 4a Problems 3a and 4a ask students to find the coordinate vector [ P] ( u,v), given the vector P and a basis {u,v}. The goal of these questions is to get the student to consider a variety of representations and the utility of each of them. In problems 1 and 2, the student was drawn into thinking of a coordinatized vector, [ P] ( u,v), such as [ P] ( u,v) =!3 % #$ 2 &' ( [ P ] ( u,v) =!3 2 % #$!1&' + 2!3 % #$.5 &'. A vector expressed in a vector equation. It is the goal of these questions to compel the students to consider alternate forms for solving a problem such as If [ P] = 2 % ( 3 % and u,v #$!6 { } = &' #$!5&', #$!4 % + ), * 6 &' -, then find [ P] ( u,v). In the presence of the applet, students will quickly be able to answer the question but the teacher will need to probe student thinking to draw out how one could avoid the applet and generate the vector [ P] ( u,v) via some computational method. Drawing upon their insights from problems #1 and #2, students natural inclinations will be to explore this problem by expressing the following equation: a 3 % #$!5&' + b!4 % #$ 6 &' = 2 % #$!6&'. However. students may not automatically translate this into the following Matrix Equation form: 3!4% #$!5 6 &' #$ a% b& ' = 2 % #$!6&' or System of Equations form: 3a! 4b = 2!5a + 6b =!6 Students may need to be led toward these alternate forms by asking questions such as: In what other forms can we express this relationship? What is the power of each of these different forms? Which form do you think will be most beneficial to solving the problem? Which form will be most beneficial to describing what is being asked? Which form will be most beneficial to generalizing your results? In other words, if given a slightly different problem, in what form can you express the solution in a generalized way? Once students affix on the various forms, students may need to be reminded that previous knowledge and techniques associated with such forms can be used to establish values for a and b. Students may also need to be reminded that one can consider a Matrix equation such as 3!4% #$!5 6 &' #$ a% b& ' = 2 % #$!6&'

4 from a functional perspective, where A = 3!4 % and we have the question, does there exist an #$!5 6 &' x in! 2, such that T (x) = Ax = 2 %. It is the purpose of moving to either the Matrix Equation #$!6&' or functional perspective, to help the student begin to conceptualize of how to convert from P [ ] u,v to P ( ) as well as review concepts such as domain, codomain, range, span, linear independence, invertibility, etc. [ ] Problems 3b and 4b Problems 3b and 4b ask students to find a change of coordinate matrix which transforms P into [ P] ( u,v) when given the vector P, a basis {u,v} and the coordinate vector [ P] ( u,v). In particular, these two problems are designed to drive the student to finding a matrix Q such that Q [ P] N!! [ P] { u,v}, where N is the natural basis of! 2. In doing so, students may need to 3!4% more fully explore Matrix equations such as, #$!5 6 &' #$ a% b& ' = 2 %, or linear transformations #$!6&' such as, T (x) = Ax = 2 %. Students need to be encouraged to move away from a trial-and-error #$!6&'! methodology and move to systematic ways of isolating the x = # a$ b% & from either 3!4% #$!5 6 &' #$ a% b& ' = 2 % 2 % or T (x) = Ax =. It would be reasonable to encourage students to #$!6&' #$!6&' express the relations from P into P [ ] u,v ( ) as In doing so, students will begin to build the connections necessary to envision a change of coordinate matrix from P Problem 5! [ P] C. Problem 5 pushes students to consider the applicability of two coordinate systems and how to change between them. Using the applet, the students should quickly identify that the [ P] G can be easily found by moving the blue and green vectors to correspond to the basis elements ( identified as B = #$!1 % 3 &', #$ 2% + ) * 5& ', - and G = ( 3 % #$!5&', #$!4 % + ),. Once this is accomplished, move the red * 6 &' -! vector so it is coordinatized as [ P] B = # 3$ 1% & that is with respect to the blue vectors. The

5 coordinates associated with [ P] G can then be read from the applet although students should be encouraged to use the grids to double check the computations. Once students arrive at a correct answer to this problem, they should be pushed to think about how to solve this problem in general if the applet was not available to them. Specifically, students should be asked to consider what they uncovered from their investigations with 3b and 4b. The teacher then needs to help the students conceive of the conversion from P to [ P] G as a composition of functions and a product of matrices. In doing so, students need to begin to recognize the power of the following diagram: as well as the meaning behind the diagram. Problem 6 Problem 6 asks students to consider the existence and uniqueness of a coordinatzing basis when students are presented with a vector coordinatized with respect to a nonstandard basis that is given and another basis that is unknown. In particular, the problem asks students to accomplish this without attempting to coordinatize P with respect to the natural basis for! 2. It is this requirement that frustrates students because their natural reaction is to try to equate Q P = [ P] N = T!1 [ P] G after working with Problem 5. It is good to allow students to take this tactic and see if they can use it to establish the missing basis because students will need to grapple with a variety of conceptual issues linked to invertibility of a matrix and its connection to a basis with ideas of span and linear independence parlaying into the exploration. It should be noted that this particular problem can also be solved with the applet by making judicious choices for one of the green vectors and using the grid to adjust the other green vector so the point P, when oriented with respect to the other nonstandard basis, is coordinatized according to the grid in the appropriate manner. This technique can arrive at a solution to the problem through trial and error; however, the students should be reminded that determining the basis in this manner actually violates the condition of attempting to do this without attempting to coordinatize P with respect to the natural basis for! 2 since in doing this, the red vector has been coordinatized with respect to the standard basis by the applet. However, if a student cannot arrive at any other answer, the teacher should ask the student if there are other bases that might work? The teacher should be prepared to ask the students: If one makes another judicious choice for one of the green vectors, does another basis arise? If another basis does arise, is there any commonality between these bases?

6 If another basis does arise and you can t see a commonality between the two, can you find a third basis to which you can compare? If students get stuck, it might be helpful to suggest that they affix the green v 2 and see if they can see any relationship between this vector and the green u 2 when the red vector satisfies the condition. From that, make a conjecture about the vectors in the basis and test the hypothesis. The students should be informed that proceeding through this process, although arriving at a generalized solution for a class of bases, does not achieve the requirement of assurance that this class is the only class. In particular, students need to understand the difference between a generalized solution that is known to work and assurance that the generalized solution is descriptive of the entire class of solutions. A solution that some students who have potentially transcended the process to conceive of the process as complete and an object descriptive of the completed process will arrive upon! T [ P]!1 Q Q B # [ P] G and [ P]!1 T G # [ P] B where Q = [ u 1 v 1 ] and T = a c $ # b d% & where u =! a 2 # $ b % & and! v 2 = # c$ d% &. The students will need to be directed to consider if { u 1,v 1 } are linearly independent, then what can be said about the matrix Q? Using these pieces, the students can begin to conceptualize that Q!1 T P [ ] G = [ P] B and use this to describe all of the solutions, in parametric vector form, of the non-homogeneous matrix equation. Then using this to translate back to the vectors contained in the basis.