I know some teachers are better than others, but I wonder if there are some teachers that are just much better than others.
Has there been any studies that try to quantify this? Like having different teachers teach students at the same initial level for the same amount of time and then having them take a test afterwards?
What made me think of it was this video: https://youtu.be/MwVBzE7Z5gw
I found this on google: Qualities of Effective Math Teachers
It’s a research brief from a doctoral student at the University of Nebraska from 2018. It references quite a few other studies.
As a former teacher (computing and chess), I think that the quality of the students matters a lot.
So it would be difficult to compare teachers without knowing the ability (and potential) of the students.
Based on personal experience as a student and a teacher, there were teachers who had a great reputation within the school (and who achieved excellent results too.)
I expect that there will also be fine teachers of various kinds:
For example, in one (typical) year my school (for 11-18) only sent 4 pupils to University.
Our (superb) Economics teacher had a set of seven18 year-olds. We scored 5 ‘A’ grades and 2 ‘B’ grades. 
Our Physics teacher was useless - aged 15 in the top set, I scored 17% in the yearly exam … and came 4th out of 25. 
Yes, teaching honors students is a wildly different skillset than teaching remedial ones.
But within similar cohorts of children, of course there’s variance. There are effective teachers and ineffective teachers, just like at every other job in the world.
I was one of those students who needed/wanted to know how a particular mathematics concept was useful. I had one teacher who had no idea of how to explain what things like slope were useful for, and I lost all interest in math. Years later I took calculus and suddenly I was excited because the teacher explained that the tangent of a point on a curve of distance vs time was velocity and the tangent of a curve of velocity vs time was acceleration (I hope I got that right).
Some students are excited by the math itself and don’t care about the application and would have been fine with the first teacher.
And schools would love to be able to quantify the difference, and make many efforts to do so, but there are so many variables that it’s almost impossible to tell. You never actually get an instance of the same students from the same background. Even if you have a school where there are multiple teachers teaching sections of the exact same course (i.e., not one regular and one honors or the like), there are still likely to be differences in the quality of students between the classes because one teacher’s class period conflicts with the honors history class and the other doesn’t, or the like.
My soon-to-be-ex wife was a public high school math teacher for many years between stints as an aerospace engineer. She’s a very smart & capable woman in so many ways. Her undergrad degree was in math education, then she later backed into engineering. So attitudinally she was a teacher first and foremost.
She worked a few years at a high school in the poor almost entirely Hispanic part of town where lots of students had parole officers and everybody spoke Spanish at home. She later moved for another few years to the fancy-pants richy rich neighborhood public high school where every student had a new car and went on to college.
She said that very different approaches and skills were needed, but the key is to inspire the kids to stretch from where they were to where they could go. She enjoyed teaching the poor kids more than the richies, despite being of that flavor herself. She was / is extremely proud of living in the exclusive zip code. Sigh.
Here’s an example of a rubbish teacher:
Our class went on a trip to the seaside and walked along the cliffs.
The teacher gave us nothing to do.
So I decided to measure the height of the cliff by timing the drop of a small stone.
My friend and I made sure there was nobody below, then he timed the fall.
As I was using Newton’s equation d = it + 0.5att (where d = distance; i = initial velocity; a = acceleration; t = time), the teacher rushed up and shouted at me to stop.
I tried to explain what I was doing.
He called me a liar (“you can’t possibly know how high you are”) and said I was throwing stones and he would report me.
Nothing came of it - but you see why I thought he was appalling.
In pretty much every field of human endeavor, the ratio of talent & achievement between ‘outstanding’ and ‘ordinary’ exceeds 1000:1.
As others have alluded to, the research shows the most effective math teachers are those that adapt their teaching style to their students’ learning style.
Most folks who have taken a few years of math in school know the answer to this question first hand.
I spent my last 15 years before retirement as a high school and middle school math teacher. The range of effectiveness among my colleagues was always very wide. We need way more math teachers than we have good ones so I expect this situation will continue. Plus, as with any skill that involves human interaction, a lot of the important elements tend to be innate and very difficult to acquire through training.
As noted above too, there are so many kinds of students that no one is likely to be effective for everyone.
Efforts to measure math teacher effectiveness generally get bogged down by this aspect of diversity of student learning styles - the most likely “ideal” outcome would require a large enough faculty to offer specialized teaching approaches to match this range of student needs and some method of channeling students towards the best fit. It happens, but it’s rare.
But of course, you can’t evaluate teachers just by surveying their students, either, because students are likely to answer based on whether they liked the teacher. Which isn’t at all the same thing as them being a good teacher.
One thing that hasn’t been mentioned, maybe because it isn’t especially relevant at first glance, but I can only imagine the difficulty of recruiting. The pool of Individuals proficient at mathematics is not necessarily very large, the number of individuals in this subset who are also proficient at instructing is considerably reduced. From this group that they have to draw from there are going to be even fewer individuals yet, who are actually willing to put up with all the BS.
Are there any studies about how a teacher and student match up?
By which I mean, can we say there is a “best way” to teach for all students?
Anecdotally I know I clicked with some teachers and would rate them very high as excellent teachers. But, others in my class may not have “clicked” with the teacher and would rank them low as a “good” teacher.
My algebra teacher was awesome. My geometry teacher sucked (IMO). Am I right?
Bingo.
Unfortunately, any sort of measurement of the effectiveness of teachers will be horribly warped and subjective. If you measure them by standardized test scores of their students, then that penalizes teachers who work with low-IQ or special-needs or other-issue students - and also encourages gaming of the system. If you measure them by student reviews, then that’s also highly warped and subjective. It’s one of the biggest pitfalls of the education field.
It is pretty hard to get a universal good.
From my teaching days it was pretty apparent that no matter how highly rated someone might be, there was always someone who really didn’t like their style. Student ratings don’t always track effectiveness. But it is often all you see.
As a school student, I rated the majority of mathematics teachers I had as not all that great. The good ones for me were usually those who had a wider enjoyment of the subject. Teachers for whom the concept of recreational mathematics had meaning. They could bring things to life in a way I could engage in.
At university level there was a huge variation, from inspiring to quite dreadful. I do have some appreciation for how hard the jobs is. Even a classroom filled with engineers is never all that engaged with the the subject. And they know they need the knowledge and choose to be there. But a certain subclass of academics resent teaching and make sure you know it. This is something I do feel disappointed about. I remember my complex analysis lecturer making it clear they really didn’t want to be in the lecture theatre with us. So we all did poorly, and I never appreciated the subject until much later in life.
Tons. Of . . . mixed quality.
That’s not math-specific, but it links to some. As mentioned, it’s hard to get large, random samples, and there are many confounding variables. And we’re always talking about averages. Assuming you can measure outcomes, you can’t say a teacher with better average outcomes will have a better outcome with any specific student.
Agree.
I had some good math teachers from 1st grade through 12 grade. I had some bad ones, too. I noticed a common attribute in the latter: they seemed to dislike math, and was teaching the subject because they had to. I mean, they didn’t come out and say it. But you could tell they had absolutely no passion for it. My theory is that their dislike for math rubs off onto (some of) the students. Who can blame these students for hating math when many of their teachers hate math?
The best math teachers (not surprisingly) tend to be the ones who genuinely enjoy, and have a passion for, the subject. Unfortunately they seem to be a rare lot. A student who is decent at math will pick up on their teacher’s positive attitude, resulting in them pursuing more math.
There is always ongoing research into better teaching methods.
I was a teacher for ten years, though not math. My master’s thesis was a comparison of two teaching methods for a certain task. As with all such studies (and mine was a very modest one) there are many, many limitations and they are usually laid out in the write-up. It’s really hard to separate the “treatment” effect from other effects. One of the most insidious is that over time some learning may occur regardless of the teaching method, which can be related to maturation in the subjects too.
Every so often a particular technique gains traction and is adopted by a school district or a training university. Unfortunately, this is often a short lived trend. Several times I saw my school system aggressively push teachers to use a certain new pedagogical method, only to abandon it a year or two later. That’s really frustrating for teachers, who are sometimes forced to use methods they don’t care for. And I imagine it’s not great for students either over time.
This is not to say there aren’t some empirically good teaching methods. But the fact is we are dealing with human learners who have tremendous variability and are sometimes motivated to thwart all attempts to teach them.
I became frustrated with teaching and eventually left the field partly because of this. I think schools try their best to forget that teaching is a human endeavor, and they overestimate how this can be overcome through teaching technique and methodology. A better model IMHO is one that encourages “best practices”. Teachers who are engaged try to update their knowledge and methods by attending conferences and learning from colleagues. It’s usually a mistake to force methodology on them the way it’s currently done.