We summarize the pros and cons of all the four methods of
the matrix multiplication adaptation.
The first method is useful for
students who begin working with matrices. One of the workshop
participants confirmed that:
"I think that the braille
print is optimal to use during the introductory
lessons about matrices. The teacher could demonstrate the basic
properties of the matrix more easily. Thanks to the braille print
a student has a better idea about the arrangement of values in
the matrix."
We also discussed the question of whether a blind
person is able to read information in the braille print located at
different positions at the same time, and whether he/she prefers the
matrices arranged one beside the other or one below the other. The
workshop participants declared they did not have a problem with
simultaneous reading of two sources of information at different
positions. What is more, they do not distinguish between the two types
of arrangement of matrices mentioned above. Rather, they confirmed the
following fact:
"If values of the matrix are complex, then they cannot be
aligned vertically because of their different length; that’s why it is
not possible to follow them in columns."
Therefore they all agree that it
is suitable for blind students studying matrices for the first time to
use a tactile print as it helps them imagine their structure and
arrangement of values in rows and columns better. For those who work
with complex values in matrices it is necessary to use the editor Lambda
or a spreadsheet.
The workshop participants validated our
assumption concerning the accessibility of objects we work with during
the computation. Manipulation with three matrices in one digital file
(the second method) is too time-consuming and inefficient. One spends
a lot of time moving between both input matrices
A, B
because he has to read through all the values in previous rows and
columns. To speed up the computation he can remember the particular
row’s values of the first matrix which he multiplies with numbers or
expressions located in the given column of the second matrix. At the
same time he/she adds up these semi-results. It is absolutely clear that
although this approach saves time, it also causes overloading of memory
and therefore it can lead to mistakes during the computation.
If blind
students manipulate matrices only on a computer, the third of
fourth method is optimal. They can access the particular row or column
very easily. Additionally, when they leave the matrix located in the
particular file (or sheet) and move to the other one, the cursor stays
on the position where it was the last time. Therefore if they return to
the matrix they do not have to look for the relevant row or column
again.
There was a question whether the workshop participants
distinguish between
- distributing all three matrices to separate files or
- using sheets of only one spreadsheet document.
They were not able
to recognize any substantial difference between these two approaches,
therefore they consider the first option as redundant.
They also
appreciated the Lambda tool for matrix manipulation. Expressions
including complicated mathematical symbols (powers and square roots,
fractions, Greek letters, etc.) are correctly reproduced to the blind
user whether by a screenreader or a refreshable braille
display ()According to the relevant national 8-dot rules.
Standard spreadsheets do not provide users with this kind of
service.
Finally, the workshop participiants suggested the following
idea. "It would be useful if the blind person could ask the
application to read the content of one single cell, row, or column of
the matrix without having to go through all the previous values".
This
is a reasonable requirement. For example, if the student intends to
analyze the fourth column of matrix B,
he/she has to pass first three cells of the first row, which slows
down the process of looking for the information.