Herb Gross was born in Boston in 1928. He studied mathematics at Brandeis University and did graduate work in math at MIT, to which he returned as Senior Lecturer at MIT’s Center for Advanced Engineering Study. During his work there, he created the highly acclaimed lecture series, “Calculus Revisited.” It is through viewing that series online that I became acquainted with his masterful teaching style. I was thrilled to learn that he is still going strong at 87, having turned his attention in recent years to K-12 mathematics in a free series called “Mathematics as a Second Language.” It is with great pleasure that I was able to speak with him for the inaugural interview in this series.
Michael Paul Goldenberg: Herb, you’ve made analogies between teaching math and playing sports, suggesting that it is often the case that a great player does not make an effective coaches, particularly if what he offers his charges is showing how far he can hit the ball (reminds me of some of Ted Williams frustrations as a manager and hitting instructor). You’ve stated,
“It was at the community college that I learned how to actually coach mathematics. Professors who are actually excellent mathematicians) tend to believe that because of that, they are also the greatest math coaches. For example, when I was at MIT I essentially didn’t have to teach. In essence, all I did was tell the students about mathematics. It was at the community college that I learned how to actually coach mathematics. Yet when it comes to remediation the government confers with the professors at the most prestigious universities. In other words, we try to improve coaching by talking to the best players rather than to the coaches themselves.”
I’d like you to comment a bit further on this point in light of the last 30 years or so of the so-called “Math Wars,” where many of the experts who’ve garnered the greatest media attention are professional mathematicians, often prestigious ones, who seem to have a great deal of contempt for “mere teachers” of mathematics, particularly when it comes to K-12 education, where they themselves have little or no experience as instructors.
Herb Gross: Let me first take your question literally and give you an answer based on that. Then I will offer a more detailed answer that is based on a broader picture.
I think part of the reason is caused by some of the “coaches” not being particularly good coaches. More specifically, in terms of our sports analogy, it is true that the coaches don’t have to play the game well but they have to understand the game even better than the great players do. In a sort of humorous way, the great coach should have a T-shirt that reads, “Ain’t it amazing what you can produce with your brains and someone else’s body.” In terms of the wording of your question, I would say that the mathematicians get upset by how many teachers are presenting math topics in ways that are counterproductive to having students internalize the logic behind the algorithm, even though they might have trouble doing a better job because how they see the topic may be beyond the scope/interest of the students.
The broader picture is a version of the sports adage “Great players make great coaches.” It doesn’t mean that if you were a great player you would also be a great coach. Rather it means that if you want to be a great coach, just makes sure you have great players. In other words, great players will make you a great coach. The point is that the professors who excel at “playing” math are often at institutions where the students are excellent. However, in every group there is always a worst student. At an institution such as MIT, the lower half of the class is simply the lower half of the top 2% of high school graduates in the country/world. So the professors at MIT are teaching the brightest students and after a while they visualize the lower half of math students in the image of the lower half of their own students. However, not realizing this, they merely assume that with teachers such as themselves, the lower half of students in a developmental math course would do much better if only they were the teachers.
Again in terms of a sports analogy, it would be like the head football coach talking to a physical education teacher who is teaching a compulsory class for all students and saying “I don’t know why your classes don’t have better morale. We certainly don’t have that problem on the football team.” Or in terms of another analogy, there are summer camps that meet the needs of any student and there are also specialized sports clinics that cater to the better athletes. It can be “tragic” for the kids who should be in summer camps to be thrust into a sports clinic. In this analogy, the students in developmental math courses, say, at the community colleges are the kids who belong in “summer camp”; and the professors at the prestigious university are viewing them as if they were in a “summer clinic.“ So, for example, having a Ph. D. in mathematics says nothing about how the recipient would fare teaching fractions to 40-year- old adult mathephobes.
MPG: Having taught at MIT and community colleges, what are some of the key “problems of pedagogy” that have been central to your teaching practice?
HG: This is an interesting question. Let me once again use a sports analogy. Imagine that you are a successful basketball coach whose team has won the state championship for 10 years in a row. During that time the shortest player on your team was 6’7’’. Then, during tryouts in the 11th year, the tallest player is only 5’10’’. This doesn’t mean that you have to change your approach to the game or your value structure, but it does mean that you might want to reassess using the slam dunk or the alley-oop plays.
In this context, if you compare my arithmetic lectures and my calculus lectures, I use exactly the same approach. Everything is taught logically in a user-friendly way. What is different is that I recognize that my arithmetic class is filled with “5 footers,” but my calculus class is filled with “6 (or 7) footers.” In a sense, I invoke a theory along the lines of “rigor is a function of the rigoree.” So in formulating my approach I use my experience as a math player (for example, during my years at MIT) to determine what the content of the lecture should be, and I invoke my coaching experience at the community college to determine how I will teach it in the way that will make it easier for my students to internalize the content.
MPG: As someone who has viewed with enormous pleasure many videos from your 1970-71 “Calculus Revisited” series (some of them multiple times over the last half dozen years or so), and who has also perused many of the mostly glowing comments other viewers have left for you, I’ve noticed a consistent theme that focuses on how excellent your presentations are of mathematical ideas in contrast to most or all of the other instruction commentators have experienced. To what do you attribute this apparent gap in the ability of many mathematics professors and teachers to effectively communicate key mathematical ideas to what I suspect to be a sizable percentage of their students? In particular, what would you say to the open or implied claims I’ve heard from countless K-12 and college teachers that seem to boil down to “I taught a great lesson/course, but the students just refused to learn the mathematics!”
HG: In my opinion, it is interesting to note that while the Golden Rule is a wonderful guideline in most cases, it is counter-productive when we are trying to teach. More specifically, the wording of the Golden Rule (“Do unto others as you would have had others do unto you”) makes you the focal point. However, when we teach it is the students who should be the focal point. Hence the teachers’ Golden Rule (which I refer to as the Platinum Rule) is, “Do unto others as others would have done unto themselves”. Too often we teach students in the way we would have liked to have been taught and too often we fail to look at things from the students’ point of view. We might not know what the students’ point of view is, but we should at least be trying to think about it from the students’ point of view.
I was once told to never think so much of my subject that I neglected my subjects. While teaching the content is by far the main objective in teaching, it must also be remembered that giving students a sense of some sort of ownership of the course is also very important. In essence, I always tried to make my classes seem like a home away from home for the students. I was motivated by the adage “People don’t care how much you know until they know how much you care.” There are many different ways to show caring, and the choice will often depend on the experience of the instructor. Again, in my desire to use sports’ analogies, too often the professor is the home team, and the students are the visiting team. It is the home team that makes up the ground rules. For example, the professor might invoke something along the lines of “I want all the papers to be turned in by Wednesday, and for every day they are late I will lower the grade by 10 points.” Unfortunately, the captain of the visiting team cannot insert a rule along the lines of “We worked hard to get the papers done in time, and we want to have them back by Monday. So for every day that you are late in getting the papers back to us, you have to add 10 points to the grade.”
MPG: Would you address what has struck me as a failure on the part of many mathematics professors to help students see: a) larger interconnections between what they’re teaching and significant mathematical ideas that run through a range of the main branches of mathematics, and b) how certain key elements of doing and working with specific mathematical ideas and procedures hit notes that will resonate repeatedly throughout their mathematical studies. Perhaps those are simply two ways of saying the same thing.
HG: I’m not sure I know the answer, but what I do know is that the question you raise is even more important now than it was when I was a student. More specifically, in my time there were no calculators and the important thing was to get the correct answer and not to worry too much about why the algorithms worked but rather to learn how to use them correctly. In those days the slogan would not have been “drill and kill” but rather “drill and survive.” Nowadays, however, the Internet, Google, and calculators have made it easy to do computations almost instantly. I do not have to memorize anything about decimal arithmetic to compute 3.14 X 2.7. Indeed, I simply type the information into my calculator and almost instantly see that the product is 8.478. However, this knowledge will not help me find the answer to “What is the circumference of a circular disc if its diameter is 2.7 inches?” if I do not know the formula for computing the circumference of a circle given its diameter. And even then this is not a serious problem because all I have to do if I don’t know the formula is to go to Google and type, “Find the formula for the circumference of a circle.”
However, even when I know that the answer is obtained by the formula 3.14 X 2.7, I might still make a mistake in entering the information. So it is a good idea for me to have enough number sense to know that 3.14 X 2.7 is greater than 3 X 2 but less than 4 X3. So the answer has to be between 6 and 12. So if by mistake I forget to enter the decimal point in 3.14 and obtained 847.8 as the product, I should know that the correct answer has to be between 6 and 12; and that 847.8 isn’t such a number!
MPG: I, too, grew up in the EBC (“era before calculators”) and despite being someone who appreciates all the electronic tools as both a teacher and learner of mathematics, I still often rely on mental math when I teach (as well as when I’m doing my own calculations in various other contexts). I’ve noticed how amazed many of my students are when I do such calculations aloud before putting the results on the board and then explain what I did. Similarly, I’ve often asked whether sqrt(5) + sqrt(11) = sqrt(16). Students immediately grab calculators and I tell them to stop and think about the question without resorting to any technology at all (including paper and pencil). Very rarely does some student realize that while sqrt(16) equals 4, sqrt(5) must be between 2 & 3, and sqrt(11) must be between 3 and 4; hence, the smallest that sqrt(5) + sqrt(11) can be is already greater than 5, and thus not equal to 4.
Does this suggest to you that the technology has undermined students’ ability to estimate and use mental math, or is there something else at work? You are of my parents’ generation, so I wonder what your experiences were along these lines going to school in the 1930s. Was it a “Golden Age” of American K-12 mathematics, as has often been claimed by people who dislike current trends? Is it your sense that the average American high school student in, say, 1945, knew significantly more mathematics than is the case with high school kids today?
HG: Strange as it may appear to be at first glance, my opinion is that modern technology has made it even more important to have a number sense. In the spirit of the National Rifle Association, “Calculators don’t make mistakes. The people who enter the information do!” For example, with respect to something we discussed earlier, suppose that by mistake a person omits the decimal point in 3.14 and because of this the calculator gives the result that 3.14 X 2.7 = 847.8 In other words, we might say that the calculator gave the right answer to the wrong question (which can be just as disastrous as the wrong answer to the right problem!) In this case, it would be crucial that the person understand immediately that the correct answer has to be between 6 and 12 (that is the answer has to be greater than 6 but less than 12 (more specifically, the correct answer has to be between 3 X 2 and 4 X 3). While there are “lots of numbers between 6 and 12,” it is clear that 847.8 isn’t one of them!!!
In my mind, using a calculator is no more brain-deadening than using a paper-and-pencil algorithm by rote. In fact, I feel that if all you are going to use is rote, you are better served by using a calculator. It is faster and most likely error free (barring typos). Moreover, I feel strongly that if a teacher forbids students to use calculators in class, I think the teacher will lose credibility in the sense that the students will find it “crazy” that they can use their calculators at any time and in place they want, except in that teacher’s classroom.
My approach would be to let students use calculators, for example in an arithmetic course, by giving problems that cannot be answered correctly unless one goes beyond simply entering data into a calculator. As an example, don’t ask them a question such as “How much is 2,821 ÷ 13?” Instead, ask them “By what number must we multiply 13 in order to obtain 2,821 as the product?”
This even applies to my calculus video series [in 1970-71]. In those days, it was crucial to understand the role that knowing how the derivatives of a function influenced the graph of the function. However, today if I want to graph, there is absolutely no need for me to know anything at all about calculus. All I have to do is go to Google and type “graph ” and not only will the graph appear almost instantly, but if you drag the cursor along the curve, you can see the coordinates of each point on the curve. So it would seem to me that if my teaching is to be perceived as bring relevant by the students, I should be doing things that go beyond the information that is easily accessed on the Internet.
Finally, with respect to our question concerning 1945 versus 2017 let me simply say that it is two different worlds. My recollection is that in my day students did not have better number sense than their current counterparts, but rather that they had no need to have a better number sense. More specifically, jobs that in 1945 could have been obtained by students who had only a high school diploma now require some evidence of post-secondary achievement by the applicant. In essence, it seems that the associate’s degree has replaced the high school diploma (or the equivalent GED) as the entry level credential for jobs that promise upward mobility and/or a better quality of life.
So today many students come to college not because they have a thirst for higher education, but rather because they more or less have to go. In essence, these students become the ones who don’t ask such questions as “How do you do this?” but rather they ask “Why do I have to know this? I’m never going to use it once this course is over!” And answering “why” takes much greater insight than answering “how”.
MPG: What is your view of using models of various kinds, including alternative algorithms such as lattice multiplication, verbal analogies, hands-on tools like algebra tiles, etc., to teach mathematics to students? Do you think that students gain from these things, or do they ultimately become burdensome, even handicaps?
HG: I think they are fine, provided that they are used as supplemental enrichment for the students. For example, in my liberal arts math courses (as well as in my developmental math courses) I introduced the lattice method from a historical point of view. Lattice multiplication first was introduced to Europe by Fibonacci (Leonardo of Pisa), whose 1202 treatise Liber Abacii (Book of the Abacus) was the most sophisticated work on arithmetic and number theory written in medieval Europe. It was known then as Gelosia (an Italian word meaning “iron grill,” which the format resembled). However, I would never use it as a method for students to use in place of the usual multiplication algorithm.
The same applies to the use of algebra tiles. More generally, my opinion is that any manipulative that is presented in the form of pure rote should never be used for anything other than as supplemental enrichment for the students.
As for verbal analogies, I believe that my stock-in-trade “math as a second language” theme is a very effective teaching device. Let me share one or two examples with you.
- When mathematicians talk about “numbers”, students often see “quantities.” A quantity is a noun phrase in which the number is the adjective, and as an adjective it modifies a noun (usually referred to as a “unit.” So for example, “3 apples” is a quantity in which the adjective is 3, and the noun (unit) is “apples.” In this vein, we have seen 3 apples, 3 people, 3 centimeters, 3 tally marks etc.; but never “threeness” by itself. It is my opinion that a major reason that students have trouble with math is that math is a world in which there are only adjectives. If we were taught to view the numbers as modifying units, it would be easy to see that while as adjectives 1 = 1, as quantities we would see that 1 inch is not equal to 1 foot. But without the nouns how do we know what the 1’s modify?
It is not uncommon for students to confuse a million and a billion because they are both “big numbers.” However, a million seconds is a little less than 12 days, but a billion seconds is a bit more than 31 years. Has anyone ever confused 12 days with 31 years? Can you picture a construction company saying to a client “The job will takes either 12 days or 31 years?” And while as adjectives a million is always less than a billion, when used in quantities that might not be the case. For example, while a billion seconds dwarfs a million seconds, a million days dwarfs a billion seconds. I should mention as an aside that even the most math phobic students experience an “aha” moment when they see that it would be a million days since the birth of Jesus until the 28th century! And at a more elementary level, youngsters know that 7 is greater than 1, but 7 pennies is less than 1 dime, etc.
- Students find it very non-threatening when they see that if we had enough nouns and could use them in math, there would never be a need to talk about fractions. For example, when we say “It took me 7 minutes to walk here from the parking lot” it is easy not to notice that the word “minute” is an abridged way of saying “7 of what it takes 60 of in order to equal the whole unit”. Notice how less threating it is to say “7 minutes” as opposed to saying “7 of what it takes 60 of to equal a minute.” So, when I teach fractions, I do not start, for example, by introducing such symbols as ¾. Rather I introduce it as 3 fourths, where a fourth means “1 of what it takes 4 of, to equal the entire unit”. So with respect to 60 seconds, 1 fourth would be 1 minute ÷ 4 or 15 seconds; whence 3 fourths of a minute would be 3 X 15 (or 45) seconds.
Once the students get comfortable with this, only then do I introduce the notation “ ¾” as an “abbreviation” for “3 fourths.” I think this is very important because it helps students understand fractions rather than having to rely on rote memory. For example, I don’t believe any student would have trouble with seeing, for example, that 3 sevenths + 2 sevenths is equal to 5 sevenths. Yet when written in the form 3/7 + 2/7 they will tend to “add across” and say the 3/7 + 2/ 7 = 5/14 (that is 3/7 + 2/ 7 = (3 +2)/(7 + 7) = 5/14). And when it comes to finding common denominators, we have all done that when the nouns are present. For example we know that 3 dimes + 2 nickels = 40 cents. We found the answer by converting “nickels” and “dimes” into a common noun (usually cents). More specifically,
3 dimes + 2 nickels = 30 cents + 10 cents = 40 cents.
Notice that we could also have said that
3 dimes + 2 nickels = 3 dimes + 1 dime = 4 dimes
3 dimes + 2 nickels = 6 nickels + 2 nickels = 8 nickels.
Notice that while as adjectives 4, 8 and 40 are different, the quantities 40 cents, 8 nickels and 4 dimes are equal.
As an aside, I have found that whenever I can introduce a topic in a way that is accessible to all students, independently of how they view math, the class goes quite smoothly. One such application is to ask the class a question such as “Since “teen” means “plus ten”, why doesn’t the first teen come after ten rather than after twelve?” Eventually we get to a point in our discussion where students learn that ten was not an important number until the invention of place value. Prior to that, people encountered fractional parts and as a result they wanted to choose units that had many divisors. In that respect 12 was a better choice than 10 because 12 has more divisors than 10 has. In a similar way, we chose to have a circle divided into 360 degrees rather than, say, 400 degrees because 360 has many more factors than 400 has; and similarly, we chose to have 5,280 feet in a mile rather than a number like 5,000 (which is easier to remember) because of the large number of factors 5,280 has. In other words many fractional parts of a mile are a whole number of feet.
MPG: You have a great story in your CALCULUS REVISITED lecture series about trigonometry in which you say that your high school teacher told the class when asked where trigonometry was used, “Surveying!” and you thought that while you didn’t know what you wanted to be at 16, you knew that you didn’t want to be a surveyor. At want point did you know you wanted to teach mathematics, and what experiences and influences inspired you to do so?
HG: This is a difficult question for me to answer accurately because of all the years that have transpired since then. However, to the best of my recollection, I found that the math courses at the high school level were very easy for me; I believe there is a strong correlation between how good we are at something and how fulfilling it is to be engaged in trying to become even better at it. On the other hand, many of my classmates were not enjoying the same “happiness” in studying math as I was. And so it was not unnatural for my classmates to seek my help when we were preparing to take a test.
I guess it was at that time that I realized that it was “fun” trying to teach others (which is probably the time when I decided that my professional goal would be to teach math at the high school level).
At the same time I would be sort of depressed when the students I helped still did poorly on the test. That was my first contact with what I now refer to as “The Teachers’ Golden Rule”. To repeat what I have said previously in this interview, as good as the Golden Rule is, it is self-centered. When we teach, it is the students who should be the focal point. In other words, it was the first time I realized that I had to “dig” into the students’ view of math before I could help them. Up to that time I would show students how I would approach a problem. But eventually I realized that I had to watch them as they tried to solve the problem so that I could see where they might be going astray.
In summary, I would say that having to put the students first was a different job than my being able to be “good at” math. In my opinion if we don’t do that (especially in classes that consist of math phobic students) we will not truly educate our students. The students will strive to pass our course and even if they are successful it is possible that shortly thereafter they will have forgotten much of what they had been “taught.” In fact, that’s what led me to define education as the part that is left after we have forgotten everything else that we’ve been taught. And in that sense I worry that many of our math courses do little, if anything, to educate the students who are not planning to enter a STEM-oriented program.
MPG: Finally, if you were offering advice to someone who was considering becoming a mathematics teacher, a mathematician, or what I’ll call a “high-end” user of mathematics, what advice would you offer?
There are two types of people who plan to be mathematicians. One kind is the kind that I was. Namely I thought mathematics was the “stuff” we were being taught in high school. And then there are those who, like my MIT officemates, knew what mathematics really entailed.
To those who are like I was, I would say to take your college math courses seriously to make sure you really want to become a mathematician. If it turns out that you don’t think you would like such a career, think about a field that you would enjoy pursuing and see how what you have already learned in your math courses enhances your chances of being good in any other field that you might choose. In my own case, it turns out that what I really wanted to do was to become a high school math teacher and I had not realized that this was a much different career than being a math-ematician. Nevertheless the advanced math courses I took made me a much more effective teacher in the sense that I could now better realize what parts of the curriculum were the most important for my students to understand, etc. And if you choose to be a math teacher, make sure that you understand that being a good “coach” goes beyond being a good “player.” In essence, make sure you understand and have internalized the math you will be expected to teach; then concentrate on how to transmit this information to students in ways that help the students truly internalize the content.
On the other hand, I think that to those who understand what it means to be a mathematician and still aspire to become a mathematician, I would offer them no further advice than what I said above. Watching my officemates at MIT study, it was clear that they were self-motivated and enthusiastic about achieving their goal. I also noted that all of my officemates did their undergraduate studies at colleges that stressed teaching over research. And I think that’s a subjective piece of advice I would give students who want to be mathematicians. Namely, think about getting your undergraduate degree from a college that values teaching. Then, once you have a good foundation, pick a graduate school that is staffed with professors who are good mathematicians.
MPG: Thank you for sharing your thoughts with our readers, Herb. It’s been a privilege to hear your ideas and learn from your experiences.