The normal distribution is pretty cool. It’s a mathematically determined probability distribution that does a good job of describing the patterns of variability between scores for many variables in the natural world. Sometimes called the Gaussian Distribution, after its creator, German mathematician Carl Friedrich Gauss, the normal distribution has become a centerpiece for many-a-statistics class around the world.

One criticism of the normal distribution is that it is over-applied and over-used. This criticism is partly based on the ideas that (a) fewer naturally occurring variables than one might believe actually conform to the normal distribution and, importantly, (b) forcing groups of humans into bell-curve conceptualizations of variables has the capacity to stifle growth.

I am fortunate to be an educator - having taught at the college level since Fall of 1994. And I absolutely love it. In this job, I get to take several bright young minds and lead them on a journey of discovery - helping them to learn totally new ways to understand the world and their place in it. My students are, almost across the board, bright, hard-working, and appreciative - and they learn a ton in their four(ish) years here at SUNY New Paltz.

To some extent all teachers take some time to think about their philosophy of education - asking, essentially: What are my goals here? What am I trying to teach? What do I want my students to walk away with? And so forth. I’ve been thinking a lot about my philosophy of teaching lately - and it relates strongly to issues related to the bell curve. I thought others might find it provocative and perhaps useful. Here is the story of my philosophy of teaching.

It all started when I was a PhD student in Psychology at the University of New Hampshire in the 1990s. The Psychology PhD program there is really great - largely as it includes a very thoughtful teacher-education element - all about helping mentor young PhD students to learn about the process of college teaching. You see, in most academic departments, graduate students who teach are just given a textbook and are told “good luck!” At UNH, third-year graduate students all take a one-year “teaching practicum course” that runs concurrent with their teaching of Introductory Psychology to undergraduate students. The practicum course is taught by masterful and experienced teachers who have a lot to say about issues of college teaching. I got a ton out of that experience.

And I will never forget one particular lesson - provided by Dr. Peter Fernald, who started class by saying essentially this: It’s bad practice to shoot for a normally distributed grade distribution in the teaching process. Whoa! Never heard that before. For years, prior teachers of mine had literally drawn normal distributions on the board after exams and show students scores in this context. This Dr. Fernald fellow was going fully against the grain. But I knew him to be an extremely strong and experienced teacher - and something about this idea matched naturally with my own take on the world. I was all ears. What was he talking about!? …

So Dr. Fernald painted a portrait of two (highly simplified) teaching philosophies - one that is based on a relativistic approach to understanding student success and the other based on the concept of student mastery. The relativistic approach is very grade-focused. It essentially sees grades as a critical part of the teaching process - with a priority placed on the teacher optimally discriminating students from one another in terms of achievement. From this perspective, the main goal of teaching is to, using highly valid measuring instruments, sort the students in some kind of order - ultimately leading to a distribution of scores (one that maps onto a normal distribution) that accurately reflects who’s at the top, who’s in the middle, and who’s at the bottom. This relativistic approach to teaching is actually quite common at the college level.

Dr. Fernald had some nits to pick with the relativistic approach to teaching - as follows:

From this perspective, grades are not used to motivate or inspire students - they serve little function for the students at all, actually.

From this approach, the ideal outcome is to have most students end up near the middle of the distribution. This is how normally distributed variables come out. Well, let’s dissect that a bit. Suppose that you are teaching a class on statistics, and your distribution of scores is normal - and the mean (average) score is 80 (of a possible 100). So that means that the lion’s share of students got about 80. Well to some extent, a student with a score of 80 missed 20% of the material. That student is, at the end of the class, fully ignorant of ⅕ of the course material. And if this is a normally distributed variable, then get this: MOST students are clueless when it comes to about ⅕ of the material. Further, get this: The teacher, who takes pride in a grade distribution that is a normal distribution, sees this outcome as optimal!

In light of all this, Dr. Fernald went out on a limb and, with his gentle manner, made a very bold statement. If you are teaching in a way that is focused on trying to create a normally distributed grading system, then you, the teacher, must be unclear in your presentation. In fact, you must be effortfully unclear, to some extent. If the goal is to have most students not understand 20% of the material at the end of the semester, then your teaching simply must be unclear about 20% of the time as far as the typical student in the class is concerned.

As you may have guessed, I don’t exactly adhere to the relativistic approach to teaching and grading. Dr. Fernald, a true humanist in every sense, shared, then, an alternative approach to teaching - which we might call the mastery approach to teaching. From this perspective, all students are given the benefit of the doubt - the idea is that if the student got into this college in the first place, he or she should be able to master the material that we have to present to him or her - and the job of the teacher is to facilitate such mastery on behalf of each and every student.

The mastery approach requires an extraordinary amount of work on behalf of both instructor and student. Some material is not easy. For instance, when I teach statistics, I have students compute something called “statistical power” by hand. Wow, do students find it difficult. So I work with them on it - a lot. They read a full chapter on this topic in the textbook. We go over multiple example problems in class. They have an extra lab component in the course - and the lab instructor goes over this content with them multiple times. I have eight hours of office hours a week, and I tell students in the class to come to my office hours to further go over the material more if they like. Several students take me up on that. It’s a ton of work. But you know what? Most of them ultimately get it. They master the material. And my standards for the material are absolute. There is real content - and they only get credit for fully understanding that content - and they are given every opportunity to master that content to meet the absolute standards.

The mastery approach is, clearly, student-focused. The core goal of this approach is education. Facilitating learning. Getting students to connect with and understand the material. It is not focused on showing who gets it and who doesn’t on some variant of a normal distribution .

In using a mastery approach, grading still exists - but it plays out differently. From this perspective, grades aren’t used primarily to discriminate students from one another. Grades are, rather, used to reward students for meeting absolute standards. If a student meets an absolute standard, then he or she should get an A. Otherwise, perhaps an F actually makes most sense.

So a grading distribution from a class of students in a mastery-taught class looks a bit different from a distribution that pertains to a relativistic class. While the mean, or average score, may be similar across the two classes, the pattern of variability will differ. The relativistic class, as we’ve discussed, will have a normal distribution - with few scores that are very high and few that are very low. A mastery-taught class will be more bi-modal - with, ideally, many scores in the A range - and then, as not every student puts in the work to master the material, several scores in the D and F range as well.

I have to say, I agree wholeheartedly with Dr. Fernald’s approach to teaching - teachers are here to inspire - to lead - to work to get all of their students to see how awesome the content is and to take them on a journey of discovery that gives each student the opportunity to fully master the material at hand. Focusing on teaching in a way that is designed to lead to a normally distributed grading distribution, on the other hand, misses this boat - completely.

If you are a teacher, I urge you to spend more time inspiring and working first-hand with your students and less time worrying about whether the students’ grades are normally distributed. The greatest teachers are not the ones whose gradebooks best reflect normally distributed grades -- they are the ones who give each of their students 100% of their time and attention - and who believe in the success of each student they encounter.

The normal distribution is pretty cool. It’s a mathematically determined probability distribution that does a good job of describing the patterns of variability between scores for many variables in the natural world. Sometimes called the Gaussian Distribution, after its creator, German mathematician Carl Friedrich Gauss, the normal distribution has become a centerpiece for many-a-statistics class around the world.

One criticism of the normal distribution is that it is over-applied and over-used. This criticism is partly based on the ideas that (a) fewer naturally occurring variables than one might believe actually conform to the normal distribution and, importantly, (b) forcing groups of humans into bell-curve conceptualizations of variables has the capacity to stifle growth.

I am fortunate to be an educator - having taught at the college level since Fall of 1994. And I absolutely love it. In this job, I get to take several bright young minds and lead them on a journey of discovery - helping them to learn totally new ways to understand the world and their place in it. My students are, almost across the board, bright, hard-working, and appreciative - and they learn a ton in their four(ish) years here at SUNY New Paltz.

To some extent all teachers take some time to think about their philosophy of education - asking, essentially: What are my goals here? What am I trying to teach? What do I want my students to walk away with? And so forth. I’ve been thinking a lot about my philosophy of teaching lately - and it relates strongly to issues related to the bell curve. I thought others might find it provocative and perhaps useful. Here is the story of my philosophy of teaching.

It all started when I was a PhD student in Psychology at the University of New Hampshire in the 1990s. The Psychology PhD program there is really great - largely as it includes a very thoughtful teacher-education element - all about helping mentor young PhD students to learn about the process of college teaching. You see, in most academic departments, graduate students who teach are just given a textbook and are told “good luck!” At UNH, third-year graduate students all take a one-year “teaching practicum course” that runs concurrent with their teaching of Introductory Psychology to undergraduate students. The practicum course is taught by masterful and experienced teachers who have a lot to say about issues of college teaching. I got a ton out of that experience.

And I will never forget one particular lesson - provided by Dr. Peter Fernald, who started class by saying essentially this: It’s bad practice to shoot for a normally distributed grade distribution in the teaching process. Whoa! Never heard that before. For years, prior teachers of mine had literally drawn normal distributions on the board after exams and show students scores in this context. This Dr. Fernald fellow was going fully against the grain. But I knew him to be an extremely strong and experienced teacher - and something about this idea matched naturally with my own take on the world. I was all ears. What was he talking about!? …

So Dr. Fernald painted a portrait of two (highly simplified) teaching philosophies - one that is based on a relativistic approach to understanding student success and the other based on the concept of student mastery. The relativistic approach is very grade-focused. It essentially sees grades as a critical part of the teaching process - with a priority placed on the teacher optimally discriminating students from one another in terms of achievement. From this perspective, the main goal of teaching is to, using highly valid measuring instruments, sort the students in some kind of order - ultimately leading to a distribution of scores (one that maps onto a normal distribution) that accurately reflects who’s at the top, who’s in the middle, and who’s at the bottom. This relativistic approach to teaching is actually quite common at the college level.

Dr. Fernald had some nits to pick with the relativistic approach to teaching - as follows:

From this perspective, grades are not used to motivate or inspire students - they serve little function for the students at all, actually.

From this approach, the ideal outcome is to have most students end up near the middle of the distribution. This is how normally distributed variables come out. Well, let’s dissect that a bit. Suppose that you are teaching a class on statistics, and your distribution of scores is normal - and the mean (average) score is 80 (of a possible 100). So that means that the lion’s share of students got about 80. Well to some extent, a student with a score of 80 missed 20% of the material. That student is, at the end of the class, fully ignorant of ⅕ of the course material. And if this is a normally distributed variable, then get this: MOST students are clueless when it comes to about ⅕ of the material. Further, get this: The teacher, who takes pride in a grade distribution that is a normal distribution, sees this outcome as optimal!

In light of all this, Dr. Fernald went out on a limb and, with his gentle manner, made a very bold statement. If you are teaching in a way that is focused on trying to create a normally distributed grading system, then you, the teacher, must be unclear in your presentation. In fact, you must be

effortfullyunclear, to some extent. If the goal is to have most students not understand 20% of the material at the end of the semester, then your teaching simply must be unclear about 20% of the time as far as the typical student in the class is concerned.As you may have guessed, I don’t exactly adhere to the relativistic approach to teaching and grading. Dr. Fernald, a true humanist in every sense, shared, then, an alternative approach to teaching - which we might call the

mastery approachto teaching. From this perspective, all students are given the benefit of the doubt - the idea is that if the student got into this college in the first place, he or she should be able to master the material that we have to present to him or her - and the job of the teacher is to facilitate such mastery on behalf of each and every student.The mastery approach requires an extraordinary amount of work on behalf of both instructor and student. Some material is not easy. For instance, when I teach statistics, I have students compute something called “statistical power” by hand. Wow, do students find it difficult. So I work with them on it - a lot. They read a full chapter on this topic in the textbook. We go over multiple example problems in class. They have an extra lab component in the course - and the lab instructor goes over this content with them multiple times. I have eight hours of office hours a week, and I tell students in the class to come to my office hours to further go over the material more if they like. Several students take me up on that. It’s a ton of work. But you know what? Most of them ultimately get it. They master the material. And my standards for the material are absolute. There is real content - and they only get credit for fully understanding that content - and they are given every opportunity to master that content to meet the absolute standards.

The mastery approach is, clearly, student-focused. The core goal of this approach is education. Facilitating learning. Getting students to connect with and understand the material. It is not focused on showing who gets it and who doesn’t on some variant of a normal distribution .

In using a mastery approach, grading still exists - but it plays out differently. From this perspective, grades aren’t used primarily to discriminate students from one another. Grades are, rather, used to reward students for meeting absolute standards. If a student meets an absolute standard, then he or she should get an A. Otherwise, perhaps an F actually makes most sense.

So a grading distribution from a class of students in a mastery-taught class looks a bit different from a distribution that pertains to a relativistic class. While the mean, or average score, may be similar across the two classes, the pattern of variability will differ. The relativistic class, as we’ve discussed, will have a normal distribution - with few scores that are very high and few that are very low. A mastery-taught class will be more bi-modal - with, ideally, many scores in the A range - and then, as not every student puts in the work to master the material, several scores in the D and F range as well.

I have to say, I agree wholeheartedly with Dr. Fernald’s approach to teaching - teachers are here to inspire - to lead - to work to get all of their students to see how awesome the content is and to take them on a journey of discovery that gives each student the opportunity to fully master the material at hand. Focusing on teaching in a way that is designed to lead to a normally distributed grading distribution, on the other hand, misses this boat - completely.

If you are a teacher, I urge you to spend more time inspiring and working first-hand with your students and less time worrying about whether the students’ grades are normally distributed. The greatest teachers are not the ones whose gradebooks best reflect normally distributed grades -- they are the ones who give each of their students 100% of their time and attention - and who believe in the success of each student they encounter.