# Emphasizing the discrete in early undergraduate education?

From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's Fantasy Math curriculum (update: see also his material on discrete calculus) and Kelemen et al's article Has Our Curriculum Become Math-Phobic?. For instance, Saveliev refers to discrete functions and data coming in digital format (no formula, just function), leading to a discrete calculus based on discrete differential forms. Kelemen et al discuss early discrete mathematics for computer science students.

Questions: What are the pros and cons of promoting the discrete and, to some extent, de-emphasizing or postponing the continuous in the first year or two of undergraduate education? What are your experiences, if any, of teaching in such an environment?

See also Why does undergraduate discrete math require calculus? on Math Overflow Stack Exchange back in 2010.

Speculatively, it might be possible, albeit unconventional, to teach an early undergraduate course using graph theory as a springboard to introduce a range of concepts, such as linear algebra and network analysis (e.g. Kirchhoff's circuit laws), algorithms, mathematical induction, simplicial complexes, elementary combinatorial topology, relevant aspects of discrete/computational geometry, and possibly some elements of discrete calculus. The topics could be kept as low-dimensional, visual and concrete as feasible. Graph theory is fairly rich in applications these days, so the course could potentially appeal to those with both pure and applied interests.

• My impression is that a too early emphasis on the discrete is a mistake. This is because it is hard to learn the continuous. For example getting into differentials in ODE course (which is hard enough). My counsel would be not to push discrete earlier and screw up continuous. Sep 26 '18 at 19:30

Analyzing data (probably by pushing a button in a statistics programme) instead of learning mathematics is not the same thing as "emphasizing the discrete" as in discrete mathematics.

As far as the discrete vs. continuous goes: I do think that it is a good idea to teach more discrete mathematics early in the curriculum because it is kind of strange if students in the middle of their studies are comfortable with integrals yet wary of sums, understand functions, but are uncomfortable with recursive definitions.

However, it is not true that discrete is generally easier to understand than continuous. Human beings have a good intuition for continuous processes from a continuous concept of space and from movement in space. Almost everyone accepts the intermediate value theorem as eminently reasonable.

Therefore, from me a clear "no" to eliminating continuous content from the beginning of the curriculum.

As far as "mathematics" vs. "data manipulation" goes: I think that it can be harmful to manipulate data without understanding, because it encourages the mindset of "let us push the button for the linear approximation, if this does not work well, push other buttons" among people who are actually studying important data.

• Thank you for your thoughts. I am certainly not advocating button pushing behavior, especially of the mindless variety. What I see from Saveliev's site (e.g. inperc.com/wiki/…) is that the discrete approach to calculus can be quite sophisticated. I agree that some aspects of the continuous can be easier to understand than the discrete. On the other hand, limits can prove troublesome.
– J W
May 2 '14 at 8:15
• @JW I did not mean to apply that you are in favour of button pushing (I actually could not even tell from your question if you are generally in favour of changing the standard approach), it just seems to be one extreme aspect of the "discrete data"-aspect that is too often seen in math courses for non-math majors (who then go on to analyze data on economics and psychology). May 2 '14 at 9:08
• Thanks for clarifying. My question is intentionally fairly neutrally phrased: I am curious about both sides of the coin.
– J W
May 2 '14 at 9:28

Discrete mathematics (some combinatorics, at least) is a requisite for probability and statistics. So it should be covered.

For computer science, combinatorics, techniques to evaluate sums, recurrences, and some graph theory are a must, and should be covered early in some detail.