Zimbabwean undergraduate mathematics in-service teachers understanding of matrix operations

Transcription

1 Zimbabwean undergraduate mathematics in-service teachers understanding of matrix operations Cathrine Kazunga and Sarah Bansilal University of KwaZulu-Natal

2 Introduction Linear algebra First year course Context of the study Block release In-service teachers

3 Literature Sometimes at the end of the linear algebra course many students do reasonably well in their final examinations, since most mathematical questions require knowledge of certain procedures, rather than conceptual understanding of the concept. (Siyepu, 2013)

4 Literature review cont Most participants were confident applying algorithms but had difficulties in answering the questions requiring them to give reasons. Ndlovu and Brijlall (2015); Kazunga and Bansilal (2015)

5 Research Question The research question that is explored in this study is: How can in-service mathematics teachers understanding of matrix operation concepts be described using APOS theory?

6 Methodology The study comprised a qualitative analysis of the written responses of 116 in-service Zimbabwean teachers to five questions based on addition, subtraction, multiplication and transpose operations on matrices. The teachers were attending an introductory linear algebra content module as part of their three-year in-service program.

7 Methodology cont The design of the program was such that the teachers would complete the work usually done in an undergraduate three- year degree program except that the lectures were offered in two intensive block sessions for each semester. These block release sessions, coinciding with the school and university holidays, were very intensive with classes being held from 8 Am to 6 Pm, every day.

8 Theoretical Framework The APOS framework describes the growth in understanding of mathematics concepts through the hierarchical development of mental constructions, namely, action, process, objects and schema. APOS theory was built on the work of Piaget and constructivist ideas (Arnon, Cottrill, Dubinsky, Oktac, Fuentes, Trigueros & Weller, 2014).

9 Theoretical Framework cont The theory centers on the models of what might be taking place in the mind of a student as s/he engages with mathematical concepts like matrix algebra. It involves general descriptions of the mental structures and mental mechanisms (Arnon et al., 2014). Dubinsky (1991 identified five types of mental mechanisms which are interiorisation, coordination, reversal, encapsulation and generalization. These will lead to the construction of hierarchal mental structures of actions, processes, objects and schemas.

10 Preliminary Genetic decomposition for matrix operations The preliminary genetic decomposition for matrix operations was adopted and modified from Arnon et al. (2014). The specific constructions relating to concepts of scalar matrix multiplication, addition of matrices and matrix multiplications will be described.

11 Addition of matrices Action: The individual performs single additions (resulting in a new entry of the required matrix or row or column) at a time, without thinking beyond the addition of the numbers being added. Process: The individual can imagine what the sums of the corresponding elements will be without carrying out step-by-step procedures. Addition of multiples of matrices can be done in one step, without having to first work out the result of the scalar multiples of the matrices. At this level, the individual is able to predict whether it is possible to add given matrices. Object: The individual can see the effect of the matrix addition as a totality on any given matrix n by m. S/he is able to explain why it possible or not possible to given matrices. The individual will be able to apply processes or further transformations on matrix addition.

12 Scalar Matrix multiplication Action: The individual multiplies each element at a time by k. An individual cannot think beyond the single multiplication being carried out. Process: An individual reflects on the rule and thinks about the effect of the scalar k on all the elements of the row or column or matrix A to form ka, by imagining that each element has been multiplied by the scalar k. The individual has interiorized the scalar multiplication and can carry out operations without doing step-by-step procedures. S/he is able to symbolically express the result of the scalar multiple using algebraic notation. Object: The individual can see the effect of the scalar multiplication as a totality. The individual will be able to apply processes or further transformations on a scalar multiple of a matrix or scalar multiple of a row or column.

13 Matrix Transpose Action: The individual able performs a single transformation of a row to a column, by systematically considering each row and transforming it into a column in a step-by-step manner without thinking beyond the rearrangement of each row. Process: The individual able to imagine the effect of transposing each row into a column and can also see how the reversal of the transpose operation can result in the original matrix. Object: The individual can see the effect of the transpose as a totality on any given matrix. The individual will be able to apply processes or further transformations on matrix addition. The individual can see the A T as an object in its own right and can carry out further actions on A T and recognize that two consecutive transpose operations has the effect of returning to the original matrix

14 Matrix multiplication Action: In working out the product AB=C of two matrices, the individual is able to multiply out one row by one column at a time, by multiplying each element in a row from the first matrix by the corresponding element of a column from B and then adding them up, in the same way as a vector dot product is computed. The individual is able to identify the i th row of matrix A that must be multiplied by the j th column of matrix B that results in the ij th, element c ij of the product C for matrix with numbers as elements Compare two matrices if the number of element in the row of the first matrix is the same as the number of elements in the column of the second matrix

15 Matrix multiplication Process: The individual is able to imagine the effect of finding the dot product of the i th row of the first matrix with the j th column of the second matrix to generate a new specific element c ij. S/he does not necessarily have to go through the pair-wise multiplication of each element of the row with each element of the corresponding column but is able to recognize the corresponding elements of the rows and columns that are paired. Can predict the order of the product of the matrix they want to multiply

16 Matrix multiplication Object: The individual is able to carry out further operations or transformation on a product of matrices. S/he is able to see the result AB as an object separate from the process that produced it. The individual is able to generalize about properties of products of particular matrices.

17 Results The results of the study are reported in terms of operations of matrix addition and scalar multiplication (Question 1); transpose of a matrix (Question 2) and matrix multiplication (Questions 3, 4 and 5). The same set of matrices was provided for questions 1, 2 and 3 and the teachers were asked to carry out some of the operations on these.

18 es of matrices Matrix addition and Scalar multiplication A = B = C = D = E = F = G = Instruction: Compute the following if possible, using the matrices given above Correct responses as percentage of group 1.1 E + G 100% 1.2 A + B 86% 1.3 4G + 3E 94% 1.4 3F 2C 88%

19 Student response

20 Student response

21 Interview response The solution will be the same provided we change this one [pointing to minus sign between 2G and 3E] to plus. When I will be teaching my pupils in class on scalar multiplication I emphasized that they must maintain the sign. I say leave the minus alone just multiply this [pointing to the scalar 3]. If you multiply by -3, what do they say? Otherwise people make a problem if they insert a -3 inside the matrix, they create a different question and end up multiplying. Other pupils make a problem if they insert -3 inside the matrix, they thought there is no sign then they end up multiplying.

22 Transpose of a matrix A = D = E = Instruction: Compute the following if possible. Correct responses as percentage of group 2.1 A T 91% 2.2 D T 91% 2.3 E T 93%

23 Students responses

24 Interview response 3 4 R: If B T = 5 6, is it possible to find B? 3 2 S39: umm not sure. S111: This one oh it has already been transposed finding um (shaking head). I was thinking that we first find the inverse of B T then we multiply it by B T so we will get B.

25 Matrix multiplication Consider the following matrices : A = B = C = D = E = F = G = Instruction: Compute the following if possible. Correct responses as percentage of group The order of the two matrices are given below 3.1 BA 48 % CF 84 % DC 90 % EG 84 % DG 78 %

26 Student responses

27 Matrix multiplication given order 4. The size of matrix A and B are given. Find the size of AB and BA whenever they are defined Percentage with correct answer AB BA 4.1 A is of size 3 4, and B is of size % 67% 4.2. A is of size 2 3, and B is of size % 60% 4.3 A is of size 7 3, and B is of size % 67% 4.4 A is of size 2 3, and B is of size % 54%

28 Student response

29 Matrix multiplication given order 5. Let A be a matrix of size m n and B a matrix of size s t. Find the condition on m, n, s, and t so that both matrix products AB and BA are defined correct conditions for AB correct conditions for BA 48% 48%

30 Conclusion The questions involved operations on matrices, in order to explore their mental constructions of the understanding of matrix addition, scalar multiplication, transpose of a matrix and matrix multiplication using APOS theory. The findings reveal that many teachers were able to answer those items requiring action level engagement with the matrix operations but struggled with those which required higher levels of engagement. For example, some teachers confused the operations of addition and matrix multiplication while others saw no differences between a matrix and its transpose.

31 Conclusion cont The interviews confirmed some of these struggles experienced by the teachers. The interviews also highlighted the constraints of the mode of delivery of the in-service program. The teachers were offered limited opportunities to actually engage with the concepts and consequently were only able to respond to items which required action levels of engagement. It is recommended that programs for upgrading teachers should be carefully designed so that their constraints can be taken into account. If part-time courses are crammed into the same time frames as those for full-time students, it is inevitable that the quality of the learning opportunity will be compromised.