I take pains to break down and explain my lessons in simple vernacular (of course,while using correct mathematical terminology)and model problems in step by step fashion. I am fascinated by the students who leave the lesson with complete understanding vs. those with little or none of the same.What did I say that resonated with some students? And didn't with others? Would a different approach to the lesson have made a difference? so I am vexed by this thought exercise: -let x represent the % of students who leave the lesson with understanding and perhaps a reasonable mastery of the skill and/or standard covered that day. -let y (or 1-x) be the % of students who leave with little or no understanding of the skill and/or standard being taught. -obviously, x + y = 1.00 or 100% Can you envision a scenario wherein 2 teachers would have 100% completly different results? i.e, for teacher A and teacher B, the kids who fall into the X group for teacher A become the Y group for teacher B...and vice versa? this question has perplexed me for some time and I am curious as to the opinions of my colleagues.

Yes. (And in my short career so far, I've seen dozens of cases) This is exactly what we, as educators, always have to reflect on, and why collaboration with colleagues who teach the same material can be so effective in helping each other improve!

It might depend on constant and consistent checkingin during and following the lesson, in the minute self reflection, differentiation, etc. but yes, teachers can, will and do achieve different results.

GTB4GT, do you mean that teacher A reaches only the students that teacher B doesn't, and vice versa? That would surprise me. In any given group of students above the age of gotta-please-Teacher, there may, alas, be a few who have learned that they can't learn the content no matter what - and there are almost certain to be at least one or two who will learn the content no matter what.

yes, for the purpose of this discussion. I understand that teachers will generate different results with the same set of students but I am intrigued by the thought of someone reaching the students that I didn't. what would they say/do differently?

Say rather that someone might reach SOME of the students that you didn't. But without having much, much more information as to the students (and their issues!) and the content, I could hazard only the vaguest and most unhelpful of speculations as to what could have been done differently.

It could be a myriad of different things. Sometimes it's as simple as the fact that it's a better teacher/student fit. Some students will simply feel more comfortable and confident of a learner around some teachers than others.

I think that a positive connection between a student and a teacher could make the difference, but it couldn't possibly work to explain the flip in percents for a whole group.

the sense that I am getting from this and other responses above is that (of course) different teachers may shift the %'s a bit in the groups X and Y but it is highly improbable that ANY teacher would have a completely opposite effect such that the members in group X and Y exchange places. Is my inference correct?

I believe that is the sentiment of everyone so far. Think about it, if it was that simple that the exact students would succeed in one class but not the other and vice versa, within the first two weeks of school you could just move some kids around and everyone would be successful. Statistically what you are asking is near impossible. Common sense and experience shows it is near impossible.

a2z...I agree the exercise here was a bit of a stretch. I think the underlying premise here is as follows: you give a lesson for which you have planned and are well prepared for. At the end of said lesson, a hypothetical 17 students leave the classroom with mastery of the material and the other (hypothetical) 10 students did not. Assume another teacher gave the same lesson (with his/her material) and the numbers changed to 23 (achieved mastery) and 4 did not. As a teacher, how can we know what the "better" teacher did/said that we ourselves did or did not do in our lesson? We never get to witness or observe this. Bottom line question is: how does one improve one's teaching in a vacuum?

If you're meaning 1 teacher, with no colleagues in that subject and no other way to "compare" approaches...then you reflect yourself: was there something I could have changed about my approach, materials, or interactions, that would have better helped those 10 students? In that scenario, there is no "some other teacher that got 23 at mastery and 4 not at mastery". Just like we ask kids to think about how they can improve based on where they are at - not where others are at - we must do the same.

mathmagic, maybe it's the mathmatician in me...but I do not ask my kids how they can improve based on where they are at. They simply cannot answer that question. What works is to show them why they made their mistakes (i.e "you are forgetting to take the square root at the end of the quadratic problem" or "you must add exponents when multiplying variables rather than multiply them"). Imo, specific feedback is necessary for improvement, not simply reflecting on how one can do better. The math student cannot reflect (self correct) about forgetting to take the square root. If they had known that, they would have done it to begin with. JMO fwiw. (of course, this assumes that the student has given full effort in his/her attempt at mastery. Most, if not all, students are aware when the issue is self effort)

I'd argue that that holds true in some, but nowhere near all situations. There are definitely times where they'll be at a point where they need a crucial piece that they have never had. In my opinion, many mistakes that are made though are not from missing that crucial piece, but an error in the process that they could catch (i.e. the good ol' "I keep multiplying instead of adding because I wasn't careful enough looking at the symbol") or something that they could figure out. Take coding, for example - which while not exactly math, is similar. Mistakes happen bazillions of times, but most of those mistakes are something that could be caught if they analyze what they have done (oh, computer science class...how torturous you were! ). Some initial guidance or push, sometimes, may be necessary to get them to go back and realize it...but not always. Regardless, the statement about us as teachers still holds -- and really, my prior post had nothing to do with math student reflection with regards to your scenario, but rather just teacher reflection: we can't compare ourselves to a hypothetical teacher, we must reflect on what we can personally try differently to hopefully better reach more students next time (which comes both with just reflection using our own personal understandings/expertise, as well as sometimes with collaboration with colleagues, if applicable).

You try to get out of the vacuum if possible. If not, there are many things you can do. You can look on-line at a number of resource available. Some are teachers giving examples of classroom interaction. You can read about other methods of interaction with students. You can video record yourself and see what you are really doing compared to how you see your lesson in your mind. They are very often very different things.

Great thoughts I must say. Then, I wonder whether teaching is calculating 'x's and 'y' s and so on. I would like to share my experience with my students. The class which I taught was grade XII and the common perception was that if the students do well then it is because they are availing of extra help. The teacher had no role in it. On the other hand, those who did not do well was all because the teacher ( in this case, I ) was not competent. It demotivated me immensely. I lost interest and would not take any initiative to reach out to them. So much so I thought of quitting my job. Somehow good sense prevailed on me and I reasoned that my job was to do my best for my students. They are coming to school with the expectation that I will guide them in the best possible way. I worked for them as best as I could and gradually the scene changed. What I feel teaching means interacting in a natural way without bothering so much comparing with others and dwelling on percentages. It makes one feel as if we are treating the students like mathematical numbers.