A recent request posted on AERNet that mentions "issues for complicated visual displays" brings up a concern which I think BANA needs to consider. This concern is the growing use of non-traditional algorithms and the representation of both traditional and non-traditional algorithms with visual displays and graphical notations.
There has always been a strong tradition that braille should follow the print. But I'm wondering whether sticking to this tradition is the best strategy in situations where print uses non-traditional algorithms or new ad hoc visual notations to represent algorithms? Of course, these are to some extent separate questions as it is usually possible to represent even a non-traditional algorithm without the use of visual notation.
As an example of this trend, I've recently learned that the combined use of non-traditional algorithms and visual notations is becoming common even in basic arithmetic now that many schools have adopted one of the new mathematics curricula called constructivist or "new-new mathematics." For those not familiar with these new curricula, the next sections describe the new-new approach to multi-digit addition using the partial sums algorithm.
A very readable summary description of the arithmetic algorithms used in the particular new-new textbook titled Everyday Mathematics is available at the link. The summary reports that the teacher materials for Everyday Mathematics include the suggestion that the teacher should explain all of the different algorithms (partial sums, column addition, traditional, and opposite change rule in the case of addition ) but should also
identify one of the alternative algorithms as the focus algorithm ... to provide a common basis for further work. However, students are encouraged to use whatever method they prefer. ["Partial sums" is the suggested focus algorithm for addition.]
The new-new math curricula bring up at least two related questions for the braille teacher and braille transcriber:
With the new curriculum currently used in teaching elementary arithmetic, students often use a visual approach to adding multi-digit numbers. Rather than using a traditional spatial arrangement where the multi-digit numbers are lined up in columns as in the example to the right, students write the numbers that are to be added side-by-side and then sketch graphical notations below the numbers to illustrate the various steps in the "partial sums" addition process.
Even the simple addition of pair of two-digit numbers requires a complicated visual notation to show all of the steps. In the first step, graphical connectives that resemble upside-down vees are sketched under each of the numbers to show how these numbers can be separated into a tens component and a ones component. In the second step another set of connectives, which this time resembles a pair of overlapping ordinary vees, is sketched below the separated numbers in order to show the respective grouping of the tens components and the ones components. Finally, more connectives are added until the result is finally determined.
The next section explains how this new-new math process is applied to the addition of 19 and 19.
The picture to the left was scanned from actual schoolwork done in the fall of 2005 by a sighted second-grader who had recently been taught to use this method. Here the student is using a visual notation to compute the sum of 19 and 19 by partial sums.
The student first shows how the number 19 is decomposed into the two numbers 10 and 9 by writing these numbers below the ends of the legs of an upside-down vee which the student has sketched under the number 19. This vee structure is then duplicated for the second 19.
In the next step, the student appends a second graphical representation, resembling a pair of large overlapping vees, to show how to group the previously decomposed numbers, in this case the 10, 9, 10, 9 sequence. (Here one of these vees groups the two 10's and the other one groups the two 9's.) The student also writes the two partial sums of the grouped numbers, 20 and 18, under the appropriate vertices of the large vees. A final connective is then sketched below the partial sums to indicate that these are added to yield the final result.
The previous sections describe a visual notation used by sighted elementary students to represent the partial sums method for addition of multi-digit numbers. The partial sums method can, of course, be represented as a spatial arrangement without the use of the graphical connectives employed in the visual notation. A suggesed spatial arrangement, which is is suitable for direct translation* to Nemeth braille, is shown below on the left. The standard spatial arrangements for the corresponding traditional algorithm are shown on the right.
Partial Sums Traditional Traditional
(with carried digit) (carried digit not shown)
1
=====
19 19 19
+19 +19 +19
----- ----- -----
20 38 38
+18
-----
38
There is a trend toward using non-traditional algorithms in arithmetic and also of representing algorithms using visual notations. These visual notations are generally not the only way of representing the algorithms. When confronted with this situation, the braille teacher or transcriber needs to be able to identify the particular algorithm and perhaps isolate it from the visual notation in order to provide a consistent method for braille transcription and for the student to use in writing out solutions using the algorithm.
Also, where students are given a choice of different algorithms, careful thought needs to be given to which algorithm is preferable for the braille-using student.
First posted 3/9/2006.
Significantly revised version posted 3/11/2006.