I've been recently thinking a lot about what would be the "best possible instruction method", and in regards to that, I've been drawn the question of what would be possible if a student had unlimited access to a knowledgeable teacher as their main method of instruction (as opposed to supplementary tutoring). In particular, I'm thinking about how this would enable the student to take a very active role in their education, while having just-in-time answers/feedback from an expert, and maybe aspects of pull-based learning. This train of thought has led me to articles on home schooling, special education and music education, but nothing of a more academic nature and almost nothing quantitative.

If I were to try to study this myself, I'd choose high-school or undergrad mathematics (e.g. calculus) as something where assessment is relatively straightforward. I'd then possibly take a cohort of reasonably competent and motivated students and attempt to measure the outcomes of a course of studies where each student in the experimental group has unlimited (e.g. up to 8 hours a day) access to one-on-one instruction from a knowledgeable and patient teacher. I'd then try to measure various aspects of the instruction and attempt to correlate them with learner outcomes. In particular, I'd be very interested in the effectiveness of different ways of structuring that learning time.

As a side note, I'm well aware that this educational approach is not scalable, but would like to just get some reference measurement as to what is possible.

If you are familiar with any studies like that, either in mathematics or any other field, I would very much appreciate your insights and/or links to relevant literature.

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    $\begingroup$ Home schooling often does not consist in access to a " knowledgeable and patient teacher". On the other hand, all doctoral education is based on this model. $\endgroup$
    – Dan Fox
    Nov 27, 2020 at 17:05
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    $\begingroup$ @DanFox I think I know a few students who would disagree with at least one of your criteria to describe their advisors. In contrast, my experience aligns perfectly with your characterization of my own doctoral education. Furthermore, the public schools have plenty of inpatient ignorant teachers. I know because I talk to teachers. $\endgroup$ Nov 28, 2020 at 0:33

2 Answers 2


Coming back to my question a year later, I'm happy to say that I recently had a breakthrough, by stumbling upon Bloom's 2 Sigma Problem. In his 1984 paper, Bloom discusses findings replicated across subjects and age groups showing that when an "average student" was taught by a "good tutor" via Mastery Learning, over a period of time, they perform approximately 2 full standard deviations better than those in the control group, who received conventional classroom instruction, putting the tutored students at the 98th percentile in their class.

I also discovered a more recent 2011 paper by VanLehn which found human tutoring to only provide an effect size of 0.79 sigma, but it's locked behind a paywall and I haven't had a chance to go over it yet.

So I have a tangible starting point here, which I hope would be useful to others as well, and I'd appreciate any comments regarding additional studies or further detail in these studies that would better answer the question.


My answer is anecdotal, certainly not a reference to a study or anything of the sort.

Background - I work in a high school, as an in-house tutor. I've also subbed, and co-taught a sophomore algebra class.

Our sophomore math is offered at 4 different levels, presumably so an honors student on track for AP calculus (AP = Advanced placement, the student is able to take an exam that might earn them college credit. Some high school students graduate with as many as 5 classes of college credit) isn't in the same class as one struggling with middle school math they haven't mastered.

The classes I am co-teaching this year have a total of 48 students in 2 sections. They are at the second level from the highest. In theory, in the great bell curve of understanding, they should seem to be sliced out rather sharply. In reality, they seem to be a normal distribution of their own. The top 1/6, often completing the week's assignment, (which I load Saturday morning, and mean for it to be done over the course of the entire week) before we meet for class Monday. The bottom 1/6, not looking at the assignments at all, or claiming to do them 'on paper' but never offering proof. Yet, still complaining the quiz was either too tough, or its formatting difficult to follow. (the week's assignment was exactly the same material with different numbers.) Note, my 1/6 is a reference to the bell curve, and my observation that teachers target their lessons to the 1st standard deviation, i.e. the middle 2/3. Yes, I believe that individual (or really, small groups) attention, can keep those students from falling behind.

The above is my attempt to paint a picture showing that even with the attempted stratification, the range from low to high is still too wide. That top 1/6 should have been placed higher, the bottom, lower. Either way, my experience with students that visit me for tutoring, is that the one on one lets me focus on exactly where they are in their understanding, and help get them to the next level. I hear, on a daily basis, "Now I understand it, why doesn't my teacher do it this way?" In return, I offer that the teacher has so much time to get through so much material. There are ways of explaining any math topic a second or third way, if the time were available.

I'd welcome a better system with what we have available. i.e. in my case, 500 students in their sophomore year, divided by level, into 20 classes. In the perfect world scenario, the range of understanding from low to high wouldn't feel so wide. The benefit of one on one or smaller groups than the 20+ we have comes from being able to focus at a certain level. And, while I agree that one on one might be very effective, it's impractical. Separating 500 students into 20 levels? A pipedream, perhaps, but the system we have now is broken, and any move towards this would be an improvement.

Op asked - "do you have any experience/familiarity with this one-on-one approach used as the main method of instruction?"

Again, my experience is anecdotal - 4 years ago, a sophomore came into my tutoring room. I knew him from the class I frequently subbed. He didn't need any help. He told me he wanted to skip the next year's math, and planned to visit me for my help. (One class session twice per week). He was very motivated, needed very little actual attention, except when hitting a bit of trouble. In the end, he passed the exam to skip Junior math. He gave me credit for helping. I think he'd have passed without me.

Second anecdote - This June, after school ended, I heard from 3 students who wanted to summer study to go up a level (not quite a jump of a full year, but the result is to be able to take calculus as a senior vs the precalc course.) This was the summer we would mostly avoid going out due to Covid, and I welcomed the challenge. In this case, I gave them 8 weeks of 2 hours, twice per week. This material was what they'd have taken as seniors. At the end of the summer all three were accepted into the calculus course.

Both anecdotes support the benefit of one on one, or very small groups, but in both cases, these were advanced students to begin with. I have no experience of offering one on one to a student as a main class.

  • $\begingroup$ Thanks, I would love to hear more about the one on one lets me focus on exactly where they are in their understanding, and help get them to the next level - does that imply that the one-on-one instruction brings students to a higher 1/6 in their class? Putting aside the impracticality of doing this for everyone, do you have any experience/familiarity with this one-on-one approach used as the main method of instruction? $\endgroup$
    – yoniLavi
    Nov 29, 2020 at 4:37
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    $\begingroup$ Yes to the first question. Note, my 1/6 is a reference to the bell curve, and my observation that teachers target their lessons to the 1st standard deviation, i.e. the middle 2/3. I'll edit to put this and the second one into my answer. $\endgroup$ Nov 29, 2020 at 12:49

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