In Kenneth Monks’s Bhāskara’s Approximation to and Mādhava’s Series for Sine, students refine several approximations for the sine function to eventually obtain Bhāskara I’s (ca 600–680 CE) approximation formula for the sine function. They then translate Mādhava’s (1350–1425) sine series into modern notation and compare the accuracy of these methods. The tasks in this PSP require students to write solutions to mathematical problems and to explain their thinking in words and in symbols. Students also construct graphs and other diagrams as appropriate. This PSP was assigned as a course writing project at the end of the semester. Students wrote a full draft, engaged in peer review, and then wrote a final draft for submission.
One of the goals of consistently using RA routines is that students explain their thinking without being asked. In the sample response below, the student’s first draft included such an explanation of his reasoning, using a similar approach to a metacognitive reading log. Previously, students had been provided with the log and asked to respond in that format. In this instance, the student utilized the approach without being prompted.
Task 2 of the PSP prompts the student to translate the following excerpt into modern symbolic algebra.
Figure 10. Primary source excerpt giving Bhāskara's description of his sine approximation [Monks 2020, p. 3].
Figure 11. Same excerpt in original Sanskrit [Gupta 1967, p. 122].
The students’ response to this translation task was as follows.
Table 2. Sample first draft student response to translating Bhāskara’s approximation.
\(180x\)

Subtract the degrees of the bhuja from the degrees of half of a circle.

We know that the degrees of a half circle is 180. With that known all we need to do is subtract the angle \(x\).

\(x(180x)\)

Then multiply the remainder by the degrees of the bhuja and put down the result at two places.

After doing the math above, the rule states that we need to multiply it by the same angle.

\(40500x(180x)\)

At one place subtract the result from 40500.

The rule states that we need to subtract our result from 40500.

\(\frac{4x(180x)}{40500x(180x)}\)

By onefourth of the remainder, divide the result at the other place

When rereading the rules it referred to \(x(180x)\) as remainders. So I took that number and divided it by the whole equation given to use earlier. At first the 4 was in the bottom because it said 1/4th but after simplifying it came to the top.

The student adapted the threecolumn log to draft his initial translation. The first column is his algebraic expression, which addresses the task prompt. The second column is Bhāskara’s text, “chunked” into one mathematical operation at a time. The third column is his explanation of “how I know,” similar to the final column in the threecolumn log from the Gaussian Guesswork project. His explanation of the final step suggests that he was still unsure about the meaning of the final sentence, but he gave his reasoning for translating it as he did.
The student was well prepared to complete the next task in the PSP because of his RA approach to the initial draft. In the next task, students need to algebraically manipulate their expression and convert to radians to obtain Bhāskara’s approximation that is used throughout the first half of the PSP. Students may find that their translation is incorrect and that they are thus unable to derive the given expression in radians. While this is part of the iterative process of problem solving in this PSP, it can lead to frustration if the student is not able to articulate how and why they have given their initial translation (and how it might need to be revised). The sample student’s response gives a clear insight into each step of his thought process and supports thinkaloud problem solving with peers as they continue refining their translations.