First question: What is Vedic mathematics? It is not mathematics from the Vedic period. Rather it consists of many clever mathematical sutras and algorithms that were devised by Swami Bharati Krishna Tirtha (1884-1960) who for a long time was the Shankaracharya of Govardhan Matha in Puri.
The teaching of Vedic mathematics cannot be justified on the grounds that it tells the students something about India's ancient mathematical heritage. This heritage without even counting the invention of zero was brilliant and unique but Swami Bharati Krishna Tirtha's Vedic mathematics has nothing to do with it. Some of the great gifts given by India to world mathematics include the Pythagoras theorem (by Baudhayana in the Shulba Sutras centuries before Pythagoras in Greece), permutations and combinations in Bharata Muni's Natya Shastra, binary numbers by Pingala, abstract computations in the style of a computer programme (as in Panini's celebrated grammar of Sanskrit), number theoretic algorithms of Aryabhata, Bhaskara and others, beginnings of calculus, and the great Kerala school of mathematics whose contributions to infinite series predate that of Newton and Leibnitz.
Since Indians are ignorant of their history, a course on history of Indian sciences should be taught at high school or college. To truly imbibe the scientific method it is not enough to understand current science but also how great scientists of the past - both from India and elsewhere - made their advances. Since a broad history of Western science is indirectly a part of the instructional material, a course on history of Indian sciences would nicely complement the current curriculum.
But to return to the recent controversy, can a case be made to teach Vedic mathematics to motivate students to learn the subject? Some teachers who have used this material argue that it can get a young person interested in mathematics. If there is good pedagogical evidence in support of this argument, then it may be taught as an elective to interested students or perhaps taught in workshops.
The problem of getting students motivated to do mathematics has become quite acute both in India and in developed countries. In mathematical Olympiads, the best performance in recent years has been by China (before its break-up the dominant nation at mathematics was the Soviet Union). American students do not do well at what are termed STEM (science, technology, engineering, and mathematics) courses even though these subjects are central to modern technology. Indeed one reason, why there is such demand for Indians in information technology fields is because of the weakness of American school instruction.
My own take on this problem is that the pedagogy of mathematics has lost the proper balance between mastery of basic mathematical operations and more abstract reasoning. The use of Vedic mathematics, with its emphasis on mental calculations, can be a corrective to this imbalance. In any event this prescription is rather ad hoc. What is needed is a careful analysis of why since 1985, China has on 19 occasions achieved highest team score in the "international math Olympiad," whereas India has not done so even once. Other countries that have done well include Romania (three times), US and Russia (twice), and South Korea and Iran (once each). If the controversy over Vedic mathematics prompts an analysis of why India is lagging in research and quality school education, then something good would come out of it.