# How many of “The Seven Laws of Teaching” are still relevant for teaching maths today?

Wikipedia shows that in 1886 John Milton Gregory outlined his "The Seven Laws of Teaching"; asserting that a teacher should:

• Know thoroughly and familiarly the lesson you wish to teach; or, in other words, teach from a full mind and a clear understanding.
• Gain and keep the attention and interest of the pupils upon the lesson. Refuse to teach without attention.
• Use words understood by both teacher and pupil in the same sense—language clear and vivid alike to both.
• Begin with what is already well known to the pupil in the lesson or upon the subject, and proceed to the unknown by single, easy, and natural steps, letting the known explain the unknown.
• Use the pupil's own mind, exciting his self-activities. keep his thoughts as much as possible ahead of your expression, making him a discoverer of truth.
• Require the pupil to reproduce in thought the lesson he is learning—thinking it out in its parts, proofs, connections, and applications til he can express it in his own language.
• Review, review, REVIEW, reproducing correctly the old, deepening its impression with new thought, correcting false views, and completing the true.

My question is in today's on-line high stakes testing environment, how many of his "Laws of Teaching" are still relevant; in particular, to the teaching of mathematics at the K-12 grade school level. (a modern reference on How People Learn)

• A hard-to-forget personal anecdote on #3, teaching part of a course on aircraft engine maintenance, where the international language is English, but of course that is not necessarily the students' first language. Once, when checking if the description of something as a "bifurcated pipe" was understood (by a class of USAF engineers - who certainly spoke a language they called English) there was silence, until somebody volunteered "does it mean a pipe that has been furcated twice?" – alephzero Apr 17 at 23:37
• @user178758 Welcome to Mathematics Educators SE. I see you cross-posted basically the same question, but asking about computer science education instead, on the Computer Science Educators site at How many of "The Seven Laws of Teaching" are still relevant for teaching computer science today?. Note that such cross-posting of relatively similar questions, even though they each ask it from a somewhat different perspective, is generally discouraged here. However, if you do this, please at least included a link to the other related question. Thanks. – John Omielan Apr 18 at 0:44
• "Gain and keep the attention and interest of the pupils upon the lesson. Refuse to teach without attention." In many less-privileged schools, this might result in the teacher never teaching anything... – nick012000 Apr 18 at 2:31
• You mean, what is relevant in addition to that these days we teach girls, too? ;-) – Peter - Reinstate Monica Apr 18 at 7:31
• @nick012000: That was also my thought. I've run into the issue at the community college where for a confluence of policies, that is something I absolutely am not allowed to require. – Daniel R. Collins Apr 19 at 4:08

OP: "Refuse to teach without attention."

In my role as chair, I attended an instructor's class where he really refused to advance until he was certain the students were all with him, via detailed verbal feedback. The students responded, stopped the presentations and asked questions.

I've changed my own teaching as a result of watching how this can work.

In response to questions:

• The topic was confidence intervals via bootstrap sampling.
• $$37$$ students. $$75$$-minute class.
• He kept asking "Does that make sense?" He asked many simple questions, almost like call & response in jazz. For more complicated questions, he formed in-place groups for them to discuss and report out.
• I didn't notice any student totally zoned-out. But my presence may have altered the atmosphere a bit.
• The magic was not so much his techniques in the observed class. Rather, it was the rapport he earlier established so that the students felt safe in acknowledging they didn't get it, to assume they all can get it, to ask without embarrassment for an explanation to be repeated.
• Can you go into more detail about how this was achieved? – Sue VanHattum Apr 18 at 1:37
• I would genuinely LOVE to see this in practice, but unfortunately this is usually not feasible in classes with ~70 students and one instructor, especially when the classes are not in-person. @Joseph what was the student/teacher ratio for this class? – Kartik Soneji Apr 18 at 8:44
• @KartikSoneji: 37 students the day I attended. – Joseph O'Rourke Apr 18 at 11:23
• In some cases this is absolutely not possible. E,g. at my community college in lower-level classes I routinely have students sleeping during the course session (and also on tests). One time I woke up a person in that state, they physically threatened me, security was involved; and I was advised not to ever do that again, but just let people sleep in place. – Daniel R. Collins Apr 19 at 4:10
• @DanielR.Collins: Yes, this very much depends on the school environment and the motivation of the students. – Joseph O'Rourke Apr 19 at 12:24

They are basic, friendly pieces of pedagogical advice. Most pre-college teaching is very much STILL in this mold.

Where we fall down is in high-end universities and graduate schools, where pedagogy is less emphasized in the paradoxical belief that harder material should be learned with worse training methods. Or that smart students don't need/benefit from gradual training methods. In contrast, for high stakes professional training (like jet pilots and NFL football players) very careful attention is paid to using efficient methods of pedagogy.

I remember irking the hell out of the dad of a shipmate of mine (he was a CSU-level chem prof) when I told him that what was great about the Naval Academy was that it was "taught like high school". But I meant it. And I saw how it was way more efficient than what a relative got at a top named school. (Probably more expensive for the school to deliver though.)

• I often joke "if knowing this was really important we'd be teaching it differently" (cue astronauts and jet pilots...). – user2705196 Apr 18 at 15:59
• Are you making your conclusion based on your own experience from a single high school? "High stakes professional training like jet pilots" - like the 737 MAX pilots who weren't told about MCAS, and who were told the ineffective procedures to regain control of the aircraft? Maybe since you had graduated high school education went south across the board, including jet pilots' training? – Rusty Core Apr 18 at 16:11
• @RustyCore Given the mention of the Naval Academy, I’d be more inclined to interpret ‘jet pilots’ as ‘fighter jet pilots’ than ‘comercial transport pilots’.While I’ve not been in the military myself, most of my friends who have generally make similar observations about the training being handled much more like high school education than collegiate education. – Austin Hemmelgarn Apr 18 at 16:32

Good list.

One of the above comments is true. In the modern classroom, if you wait to get everyone’s attention, you won’t be able to teach anything. That is especially true in today’s math classrooms. Many students are not motivated to learn. Math is objective and cumulative. It is easy to see where someone is on the spectrum of math understanding. However, with social promotion, many students move on (are moved on) with D- grades, and the teacher is blamed for their lack of motivation and understanding. Picture 11th graders coming into a Trig class on Day One and declaring to the teacher that they don’t do fractions.

We should have two math options for high school graduation: Applied Math (calculators and spreadsheets), and Theoretical Math (axioms and proofs, development of the math systems of algebra, geometry, trig, calculus, stats, etc.). The first group would go straight into the work force or attend community college; the 2nd to 4-year colleges and universities.

Oh, and in elementary school, teach the basic arithmetic facts: addition, subtraction, multiplication, and division of natural numbers, integers, and rational numbers (fractions, decimals, and per cents). Worksheet based. You must get this before you move on. Don’t fear the Drill and Kill method of teaching elementary facts.

• Tom
• Much of this seems more like a blog of what you think should be done differently from one specific curriculum, and not at all related to whether the particular ideas mentioned in the question have relevance to teaching maths now. – Nij Apr 19 at 7:23
• Although I think people who don't get "Theoretical Math" are missing something important in science, forcibly sending them "straight into the work force" and baring them from "4-year colleges and universities" seems too radical to me. Also, when some elementary math notion does not get to someone, I try to approach it from another angle, rather than by "drill and kill". More often than not (but still far from most of the time), it succeeds. And I think it tends to make the acquired knowledge more useful and durable than D&K achieves. – fgrieu Apr 19 at 13:26
• This answer seems like a mixed bag to me. Yes, not everyone needs calculus, but basic understanding about trig is important in, say, construction business. There should be no "applied math" with calculator. Worksheets are evil. Instead, just let kids drop off after 9th grade with "basic" education, good enough for flipping burgers. Everyone who wants to continue, should learn functions and proofs and geometry and trig (not to mention history of the world, geography, great literature...). No point keeping them inside the school walls for 13 years offering only fractions, decimals and percents. – Rusty Core Apr 19 at 17:51