# Teaching logic through “high school algebra”?

I am going to be teaching a discrete math class in the fall. One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of "and", "or", "not", "implies", "if and only if", "there exists", and "for each" and how to logically manipulate statements involving these.

Often Discrete Math tries to illustrate all of these concepts in the context of set theory, graph theory, combinatorics, elementary number theory, etc. This has the disadvantage that the student is learning new content at the same time they are trying to master the basic logical ideas.

Logic is ubiquitous in mathematics, so I figure that I should be able to illustrate all of these concepts in a familiar mathematical setting: namely "high school algebra".

An example: does the first line imply the second line, does the second line imply the first line, or both ?

\begin{align*} x^2 &= 2x \tag 1\\ x &= 2\tag 2 \end{align*}

I am sure that I could come up with "algebra problems" which illustrate all of the concepts I am trying to convey. For instance, partial fraction decomposition is a fairly logically complex idea (you want to find the only values of some variables ($$A$$, $$B$$, $$C$$ , etc) which make an equation true for all $$x$$).

My questions:

1. Does anyone have any textbooks which would fit well with what I am trying to do? This could either be a book on logic or discrete math which already does what I am proposing, or a book on high school algebra which pays careful attention to logic.

2. Has this idea been tried before? In particular, is there any relevant research on this approach?

• Is your discrete math class an upper division course with a prerequisite, or is it a lower division course, more akin to “finite math?” Also, who is the main audience, perhaps computer science majors? – user52817 Jul 4 '18 at 14:05
• @user52817 This is a sophomore level course for math and computer science majors. This course is being taught in 4 or 5 radically different ways by the instructors who teach it (we are trying to work on that...). I have not taught discrete yet, but I have taught courses which require it as a prerequisite, like abstract algebra. I have not found that my students are being well prepared by it. For instance, hardly any of them know that to prove an "if and only if" statement, that they have to prove both directions. – Steven Gubkin Jul 4 '18 at 14:22
• To be clear, I would still be covering set theory, graph theory, etc. I just think maybe the first 3 weeks or so out of 15 I could devote to learning logic in the context of high school algebra. – Steven Gubkin Jul 4 '18 at 14:23
• Coincidentally, I am also teaching Discrete Math for sophomore CS & math majors for the first time this fall. Also being head of assessment, the one red-flag that consistently pops up for the last 4 years is that students are very weak at proving anything at the end of this course (likely the first having asked for proof work). I think one option is to really make proof-writing a consistent top goal of the course. Another: I've had a proof-writing workshop for math majors added to the curriculum (starting Fall 2019). – Daniel R. Collins Jul 4 '18 at 15:15
• Closer to the stated question: Somewhere around here I recall someone stating they tried to have "rigorous college algebra", including logic and proofs, as the first course for math majors. That warms my heart, but it didn't sound very successful, as the institutional expectation was that students should be beyond algebra at that point, etc. Here we're using Rosen's Discrete Mathematics, and many of the initial examples are usually algebraic (say, the first out of any block of 6 examples or so). Keep in mind that the set, graph theory, etc., is itself a core part of the curriculum. – Daniel R. Collins Jul 4 '18 at 15:25

Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search through.

Regarding the papers listed below, I looked through a lot of papers that I've collected over the years on elementary logic issues (far more than what’s below) and I’ve singled out some papers that might be worth looking at. I’ve divided these papers into two categories: (A) $7$ papers in which I saw some explicit connection with high school algebra topics being made (identified by italic bold numbers --- [9], [16], [18], [19], [21], [23], [29]), and (B) the remaining $22$ papers, which seem to have something of possible pedagogical interest for what you’re dealing with, even though they do not appear to make any explicit connection with high school algebra topics. I googled the titles to find the digital versions I’ve given links to, but nearly all of them are behind paywalls. There might be some papers that I’ve given a paywall link to that can be found elsewhere not behind a paywall, but I did not put any effort into looking for this.

[1] Oliver D. Anderson, How many Venn diagrams are there?, International Journal of Mathematical Education in Science and Technology 19 #2 (1988), 299-305.

[2] Oliver D. Anderson, A universal Venn diagram, Mathematics and Computer Education 22 #2 (Spring 1988), 78-80.

[3] Keith Austin, Set algebra: word proofs and Venn diagrams, International Journal of Mathematical Education in Science and Technology 18 #3 (1987), 482-484.

[4] Christopher Baltus, A truth table on the island of truthtellers and liars, Mathematics Teacher 94 #9 (December 2001), 730-732.

[5] Margaret E. Baron, A note on the historical development of logic diagrams: Leibniz, Euler and Venn, Mathematical Gazette 53 #383 (May 1969), 113-125.

[6] John D. Baum, An arithmetic method in symbolic logic, Mathematical Gazette 56 #396 (May 1972), 91-95.

[7] Javad Behboodian, Set identities and set equations, International Journal of Mathematical Education in Science and Technology 22 #1 (1991), 123-126.

[8] Max Black, The relevance of mathematical philosophy to the teaching of mathematics, Mathematical Gazette 22 #249 (May 1938), 149-163.

[9] Brother T. Brendan, A popular fallacy, School Science and Mathematics 59 #7 (October 1959), 509-513.

[10] Arthur H. Copeland, The algebra of logic, Pi Mu Epsilon Journal 2 #7 (Fall 1957), 317-323.

[11] Carmine DeSanto, A classroom note on Venn diagrams for four and five sets, Mathematics and Computer Education 18 #2 (Spring 1984), 107-110.

[12] Melvin Fitting, Propositional logic using elementary algebra, Mathematics and Computer Education 16 #3 (Fall 1982), 204-207. [related to Goodstein’s paper]

[13] Reuben Louis Goodstein, Solving equations in the algebra of classes, Mathematical Spectrum 2 #1 (1969-1970), 25-28. [related to Fitting’s paper]

[14] David W. Henderson, Venn diagrams for more than four classes, American Mathematical Monthly 70 #4 (April 1963), 424-426.

[15] Clive Kelly, Set algebra via Boolean functions, International Journal of Mathematical Education in Science and Technology 20 #3 (1989), 478-480.

[16] Ernest Albert Kuehls, The truth-value of $\{\forall, \; \exists, \; P(x,y)\}:$ a graphical approach, Mathematics Magazine 43 #5 (November 1970), 260-261.

[17] B. Meltzer, Mathematics, logic and undecidability, Mathematics Gazette 51 #375 (February 1967), 16-25. [See Corrigendum in MG 51 #376, May 1967, p. 134.]

[18] Karl Menger, On necessary and on sufficient conditions in elementary mathematics, School Science and Mathematics 39 #7 (October 1939), 631-642.

[19] Kenneth A. Retzer, Logic, tradition, truth in algebra, and inferences for geometry, School Science and Mathematics 84 #3 (March 1984), 181-188.

[20] Kenneth A. Retzer, Proofs with visible inference schemes, School Science and Mathematics 84 #5 (May-June 1984), 367-376.

[21] Joseph V. Roberti, The indirect method, Mathematics Teacher 80 #1 (January 1987), 41-43.

[22] Mariano Rodrigues, Coding Venn diagrams, Mathematics and Computer Education 15 #1 (Winter 1981), 46-49.

[23] Myron Frederick Rosskopf and Robert M. Exner, Some concepts of logic and their application in elementary mathematics, Mathematics Teacher 48 #5 (May 1955), 290-298.

[24] M. Stephanie, Venn diagrams, Mathematics Teacher 56 #2 (February 1963), 98-101.

[25] Ian Stewart, The truth about Venn diagrams, Mathematical Gazette 60 #411 (March 1976), 47-54.

[26] Charles W. Tryon, The logic of implication, School Science and Mathematics 73 #1 (January 1973), 56-60.

[27] C. H. Wild, The "characteristic function" and set theory, Mathematical Gazette 54 #389 (October 1970), 296-297.

[28] Margaret Wiscamb, Graphing true-false statements, Mathematics Teacher 62 #7 (November 1969), 553-556.

[29] John D. Wiseman, Complex contrapositives, Mathematics Teacher 58 #4 (April 1965), 323-326.

• Thanks! This looks amazing! It will take a while to read all of these... – Steven Gubkin Jul 5 '18 at 0:41

My advice is to teach basic Boolean stuff, with an EE slant.

FWIW, I have never had a course in this stuff, but at least have the very very basics from Ne Math and from EE survey course (for general engineers). I was working at McKinsey, where in theory we do "issue trees" but rarely do. I was told to sketch out a problem once in an issue tree before a partner discussion, did it with and and or gates (abnormal for McK and often causing issues in the though process). Partner liked it. So maybe there is use in places other than strict logic or EE or the like.