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Wondering how to help high school students as they enter the world of upper-level mathematics? If so, today’s episode is for you!

Join us as we discuss math with a man whose name is widely renowned in the world of mathematics textbooks, Paul Foerster. We’ll explore the impact of tackling math in a hands-on way, the importance of a growth mindset when approaching mathematics, and how to coach students as they practice complex math work.

Episode Transcription

Note: This transcription may vary from the words used in the original episode.

Marlin Detweiler:
Hello again and welcome to another episode of Veritas Vox, the voice of classical Christian education. Today we have with us Paul Foerster, a math instructor for his career and an author of math textbooks. Paul, welcome.

Paul Foerster:
Thank you.

Marlin Detweiler:
For a long time, maybe as much as two decades, we have been using Paul Forester's work for algebra two and Pre-calc. And in recent years, I've gotten to know Paul. And it's been a really exciting road as we have talked to him and started working with some of his copyrights to bring them into the Veritas fold. But before we jump in there, Paul, maybe you can tell us a little bit about yourself.

Paul Foerster:
Okay. Well, I was born in England of American parents. My dad worked for Cincinnati Milling Machine Company and was sent to England to open their British branch in the 1930s. And I was born there. 91. 90 years from tomorrow.

Marlin Detweiler:
I heard that tomorrow is your birthday now. Tomorrow will be yesterday or prior to yesterday when people hear this. But as we record.

Paul Foerster:
Okay. Anyway, we lived there before and after World War two. During World War two, my mother, sister and I came back to the United States, lived in Cincinnati. Well, my dad stayed there with the Milling Machine Company and eventually with the Lend-Lease program where we would lend them destroyers and they would lease us air bases. We came back from England when I was 15 years old, 1940, in 1950, moved to Texas, which is my mother's home state and to Sutherland Springs, which is the little town that had 3,300 people while in 1950 it received national recognition or note when the man went in and shot people in the Baptist church. But anyway, the child, one of my former colleagues at the Floral High school, was the one that eventually put that to an end. Anyway, I went to school in Floresville where it's where I should have been a sophomore and said, well, what did you do in your English school?

I said, well, I have four years of algebra. Four years of geometry, two years of Trig, two years of chemistry or physics. And so forth. And I said, well, we're embarrassed to say that we have nothing to offer that you haven't already had.

Marlin Detweiler:
Oh, my word.

Paul Foerster:
For Texas history, which they didn't teach in my English school and Spanish, so they said, we'll put you in the senior class. We'll let you take the geometry, which was their top course at the time, geometry over again. And you can take English and Spanish and well, if you can do the work, we'll let you graduate.

Well, it was easy. And the geometry class is the teacher didn't know what to do. I would tell him, well. Anyway, I did graduate the week before my 16th birthday and spent a year at Trinity University majoring in chemistry, then moved to the University of Texas to major in chemical engineering because I thought that that would be a better option as the Korean War was going on at that time. I figured it would be better to be in the military as an officer than to be recruited to be a foot soldier or something.

So I enrolled in Navy ROTC and got my commission and my chemical engineering degree in a February of 1957. Fortunately, I got interviewed by Admiral Rickover for the nuclear propulsion program and was accepted and spent four years in Washington, DC designing and constructing nuclear submarines. Well, I had only two days of sea duty during that time. Both of them and builders trials of nuclear submarines. So.

Marlin Detweiler:
Apparently, you survived those tests.

Paul Foerster:
Oh, yeah. And I thought that was a very interesting experience. I did get to go to six months of Reactor School in Oak Ridge, Tennessee. That turned out to be a life changing event. Also, it was there that I first understood what definite integrals really were. In my university experience, they were formulas which you used.

And you did this with under certain problems. I had no conceptual idea, but the reactor analysis person who was the sort of person that thought the gutsy course there said, when you find out the power level being generated by a nuclear reactor, you imagine the reactor divided into small parts. You find the amount of power in each part and then add them up and take the limit. And that's called definite integrating. I said, oh, so that's what it is. So that there.

Marlin Detweiler:
Is nothing more exciting to a guy that like math, like you and like me, than a real application of what we learned in a book.

Paul Foerster:
Yeah. So I came to realize that in order to make intelligent applications, you must understand the theory. There were no formulas for building nuclear reactors while I was there. It was all new. You had to go back to basic principles. So that was a milestone in my life and convinced me that it is important to learn the theory, not just the applications.

Toward the end of my Navy career, I thought I'd like, get out of here. I want to get workshop tools because I had gone to Fort Myer and the Washington area, built the furniture and model ships and things like that. I thought, I need to have a workshop when I get out of here.

So to earn money to get workshop tools, I started tutoring high school students in the evenings. What I found there was the thing that I really liked was the. When they would call it on,

Marlin Detweiler:
For that is a teacher is hard at work.

Paul Foerster:
It and so I decided rather than staying with that and work over who wanted me to stay as a civilian, I would leave the nuclear power business and the reactors and engineering and become a teacher. I found that Texas A&M, the rival school of University of Texas, had the best program for getting engineers into the classroom.

So I went there for a spring in the summer, took courses for the education courses that I did for certification and started my chemistry master's degree. But my wife of five years got pregnant at that time and we were anticipating the birth of our first child. I thought, well, I better go out and get some experience teaching that make some money so that I can afford the child.

So I started interviewing, interviewed a number of places and finally wound up at Alamo Heights where the superintendent said, well, we don't have a chemistry job available right now. Why don't you come teach math for a year or two? And when the chemistry opens up, we'll let you move you into that. Well, here I am, 50 years later, still teaching math.

I did get an academic year institute from the National Science Foundation to take off a year and get my master's degree in mathematics, so I have done that. Yeah.

Marlin Detweiler:
But it's notable. I want to make sure the people that are listening didn't miss that. You taught at the same high school, Alamo Heights High School in San Antonio, from 1961 to 2011, 50 years in one spot. In 1983, somewhere in the middle of that, you received a national award for the Texas, for excellence, Presidential Award for excellence in Mathematics. The Texas application of that. That's remarkable.

Paul Foerster:
Under President Reagan, that was the program was conceived to show the public that there really are good teachers in the high schools. So each state was allowed to select one science teacher and one math teacher to receive the award. I was the first one from Texas.

Marlin Detweiler:
That's incredible. Well, I want to jump in here and ask you some questions of things that you've learned from this really incredible career in teaching math in the context of a very difficult point in America with regard to students learning math. What have you learned about secondary school math students? What how would you describe what you experienced as you taught them?

Paul Foerster:
You have to take your students seriously. You take yourself with a grain of salt but you take your students seriously. And you do things that you think will prepare them for what they're going to face when they get out in the big wide world. There are several things that have stood out in my mind.

One of them is that in order to understand what the mathematics is all about, you've got to do it. You've got to do the actually do the mathematics. You can't just hear about it or read about it. You gotta do it yourself. So I started writing what I call explorations, which is a long boring name for worksheets.

I wrote my first woman, my son, which was my second to, was on the way and I had to take paternity leave. I wrote my first one for my trigonometry class and it was pretty much a disaster because I didn't really understand what it was doing but eventually I got to writing explorations for my students, which would be problems that involved variables that really vary.

Now this means that a variable is not just some number you're looking for. It's not X is not an unknown number. X is a number that varies. It's kind of like a the memory location on a computer. It's a place where you store numbers one at a time. And since you can store variable various numbers in that place, it's the variable is always called x.

But it has different numbers at different times. So I wrote problems in which the variable really varied. And I got so many problems like that that I had to write some materials to go along with them to lead students into them. And I got more and more things that I was retyping on blue purple zeros every year.

Marlin Detweiler:
Still remember how good they smelled?

Paul Foerster:
Hey, you don't want to inhale? There's so much there that I finally put them on mimeograph stencils that could be run every year. This is my second year algebra course.

Marlin Detweiler:
Yeah. That one that we use is key to learning. As you look at students, you have dealt with tens of thousands of them, almost certainly over the years. What is, maybe that's a bit of an exaggeration, but certainly a lot. What is key to students learning complex math? You start with algebra one and go the whole way through calculus and the books that you've written. Clearly you understand the material. But tell us about the students.

Paul Foerster:
To the students, you have to convince them that what they're learning is not necessarily difficult, but there are certain things that you have to know in order to fit things into what you are learning. Like the variables really vary is a concept that they need to know. But the main thing is to convince them that you were in on their side.

You are trying to do things for them that will make their of their their lives better. My first idea when I went into teaching was I want to teach the academically talented and teach them critical thinking skills.

Marlin Detweiler:
You enjoy the gifted more than the ones that are harder to bring along. It's an interesting question, and I've observed it. I don't I calculate that I've interviewed more than a thousand teachers for what we do. And one of the questions I've realized is important for me to understand about a math teacher, a math teaching candidate is which side of that continuum they tend to favor and enjoy.

Paul Foerster:
Well, we had a very insightful math department chairman who said each teacher in the department is going to teach at least one lower level course for academically not talented students. So I had a year teaching what the Texas called Fundamentals of Mathematics, which was a course designed for freshmen who were at least two years behind in mathematics.

And so I got this class of unruly kids. And I started working on things with them to show them that there really were things that they could learn, develop various techniques. I developed flow chart idea, and I noticed that the students would respond better to things that were on separate sheets and even better if they were handwritten than not typed.

Interesting. And so anyway, I had, when to control the unruly students, I did various things showing them that I was on their side, like, either who was bigger than all the rest of the kids and loved and all the rest of the kids, a matter of the class cheerleader.

And whenever there was something that they had to memorize, I would have her get up there and lead a cheer. Her means divided by cent means a hundred. So percent means divided by 100. Yeah. And you do, over and over in the class would repeat that. And I found at the end of the year, although it was an ordeal for me to do that, I found that I really had made a difference for these kids, even told me on the last day of class, they said, Mr. Foerster, you were nice to us.

Oh. I made my year. Yeah, I bet it did. I learned how to deal with those who thought they were not good at mathematics. And convince them that the actually, the buzzwords that's going around the mathematics community now is growth mindset. Have you heard or.

Marlin Detweiler:
I'm not familiar with that term?

Paul Foerster:
No mindset. If you want to look up, have your people look up something, look for Carol Dweck, Dweck professor at Stanford. She's one of the leaders in this. And the idea is you may not be good at mathematics, but you can get better. You can grow the growth mindset.

Marlin Detweiler:
Right? I don't believe that everyone is equally talented, but I believe that everyone is capable of learning.

Paul Foerster:
If you believe learning, that's exactly what it is. So that's what I found with my kiddos that were two years behind as freshmen in mathematics. So I could do that. And I enjoyed that kid. Kid comes in. Oh, by the way, I was one of those people myself, in elementary school, I was second worst in arithmetic. Is that right? Only my best friend Eddie was worse, though.

Marlin Detweiler:
It sounds to me like you guys were having too much fun in the back of the class, and it had nothing to do with ability.

Paul Foerster:
When a kid comes in, says Mr. Foerster, I don't understand, as I said, tell me about it. I've been there. I know where you're coming from. Yeah. And they would tell me what they were having difficulty with, and I would not tell them answers. I would ask them questions.

Marlin Detweiler:
I had reasoning works in math, too.

Paul Foerster:
One of my fellow teachers was terrified of having to teach the slope fields, which were. Which is what kind of like what you see on the weather forecast, right? Where it shows you which direction the wind is blowing at every point. And she said, I don't know how to do the slope fields. It's, take your pencil and put it somewhere on the paper.

She did. And I said, now draw a line. She said, where? I said, anywhere. And she drew a line. I said, okay, it can't cross the slope lines. It has to go along the slope lines. So erase it and try again once you try again. And she says, and then I said, oh, you're aiming for the points.

You're not following the path of the slope. You're looking for the points. It's not connect to the dots. That's the iterations. Start over again. And in the ten minutes she knew what the slope field was. And yeah, I'm afraid of it.

Marlin Detweiler:
I have seen that in math curricula where the authors have been very good at understanding what a student will understand. And I'm sure that your 50 years of experience, I understand you wrote the books that you've written before you finished teaching, but nothing helps the teacher like experience with a variety of students.

Paul Foerster:
That's right. It was good for me.

Marlin Detweiler:
How would you describe. Is there anything but we agree. We talked about this before we got, we turned the recording on to that. Students kind of come in along a continuum where you have the slower students or the students to find math difficult, and then you have the curve in the middle, the bell curve, you've got most kids.

And then on the other end of the curve you had the kids that find math comes easy. Have you, any theories that would help us understand how that happens? What causes a student to find math more challenging or come more easily?

Paul Foerster:
I'm not quite sure how the brain works with regard to mathematical ability, but I do know that when you actually do things, when you have trouble with things and then overcome your trouble, it makes structural changes in the brain. Interesting. There are actually synapses develop because you had trouble. Sometimes a student who gets it very easily and breezes through has no change in their brain structure.

Whereas the one who had trouble and learned from the mistakes does have a change in the brain structure. So I like to challenge my more able students and get them to make mistakes and then be able to correct their mistakes just as well as I like to challenge those who are less able and rejoice in their finding out that they really can do some things.

Yeah. That's really there's a lot of individual work where you have to kind of sound out your students and find out what they're doing. Well, there was Derek in my calculus class who would sit there on his desk like that staring at the board, and I say, Derek, aren't you going to take notes? No. And on the test, Derek was the one who got them right.

He was the one who was able to because he was a very visual person. Yeah. So if he had taken time to take notes, it would have been to his detriment. Yeah.

Marlin Detweiler:
So learning styles are applied to math as well?

Paul Foerster:
Absolutely. Yeah. Yeah.

Marlin Detweiler:
How do let me change the subject here? Because I think people will value what you had to say about this. America has shown itself to be by international accomplishment, well, down the list of countries. How does America recover from its current international ranking?

Paul Foerster:
I wish I knew. I know that we have to encourage talented people to go into teaching, okay? And the atmosphere is not that way right now in Texas. We have many teachers who are teaching who are uncertified, because the certified teachers can get positions doing other things besides teaching and are able to afford their families.

I was fortunate to have a source of extra income that allowed me to be able to teach without jeopardizing my three kids. Education and so forth. Yeah. So yeah, getting teachers in the classroom.

Marlin Detweiler:
So quality of teachers is a key ingredient that has caused the system to be on what I would now call an educational dumbing down. Where does that come from? My sense, one of the correctives that I would answer with would be to expect more of students at younger grades, and they will build on that in math. One of the disciplines that builds on what's been learned previously, unlike some others like, say, history or literature, what you did five years ago, in fact, and four years ago and keep going into effects today. Is that part of the problem? Where did the dumbing down come from?

Paul Foerster:
Part of it is the way we test. Okay. In my English school, we never did multiple choice tests. You never did true false tests. And there was a national exam you took at age around 13 or so called the College of Preceptors exam, in which you actually worked the problems, you worked the geometry problems, and they were read by real people who graded you on how you did it.

So there were no test taking skills. There were just mathematical skills. You had to know what you were doing. The exams were. And all subjects, not just math. First there was science, there was religion and that sort of thing. And you had to understand what you were doing. So the way we test and the way we hold people responsible for the test scores is a detriment to actual learning because if you're concentrating on

Oh, yes, always pick option C because that's what it usually is. Then you haven't learned anything. Test taking skills will not get you anywhere in the real world. Yeah. So anyway, it's getting good teachers in and letting them do what they need to do without forcing them to do other things.

Marlin Detweiler:
In your career, you've seen the development of the personal computer and all kinds of technological aspects. What difference should artificial intelligence, AI and technology make in the teaching and learning process?

Paul Foerster:
I'm just scratching the surface of AI. I know that when the calculators came along, people thought, oh, they won't have to memorize their facts anymore. Just use the calculator. But you can use the calculator and particularly the graphing calculator to convert concepts also. And what we found in calculus was you used to find derivatives, which is the rate of change of a function to find the to draw the graph, you say it's a derivative is positive, the graph is going up.

So you make your graph go up because the derivative is zero. It's reached a high point or a low point. So we used derivatives to draw graphs. Now that we have graphing calculators we can use calculators and graphs to understand derivatives rather than the other way around. Okay. You got to adapt your teaching methods to take advantage of the way things are.

Sure not to say, oh, you can't do that. I have heard somebody said, back in the ancient days when they invented writing, everybody said, oh, the brains of the kids brains will rot. Now, they won't have to memorize everything. Whether that's true or not, I don't know, because I was much younger then. Like a thousand years younger.

We need to adapt. And I've seen some interesting work that they're doing in algebra one and actually using my book as input for some of the interesting. Yeah. And it's looking pretty good. You know, I have not used it myself. It's occurred since the 14 years ago when I retired. But I have seen some results of it that are promising.

Marlin Detweiler:
Yeah. When I got involved in classical education, which was around 1992, we were faced with the question which was the big question of the day, how do you use calculators? And we adopted a mindset that calculators should be tools and not crutches, things that came to resonate with many people in classical education and more broadly.

And it was the idea that we still need to learn the math tables. But that doesn't mean that we don't get to use a calculator at some point. Master that and it makes sense. But the idea you mentioned the graphing calculator is an example. If I understand the algebraic equation by having solved it a few times, it becomes a bit of a waste of time to not use technology to see what that formula will produce in the form of a graph.

That is correct. Yeah. And so that was a good example of it being used as a tool and not a crutch. And that seemed to resonate with a lot of people, as I've done speaking over the years related to that. Does that resonate with you? Does that make sense? How would you, how would you further embellish that?

Paul Foerster:
Well, you've got to realize that students can get in a rut. I had, when the computer algebra systems calculator came, that would do calculus. I had a student that had one, and he never learned to integrate x squared in his head. He always used his calculator, and that was a shame.

And so I developed this saying, you know, that your mind has so much creative energy at any one time, you cannot afford to use all your creative energy on tasks that should be routine. So you got to learn some things. You've got to learn to integrate x squared, for example. Yeah. Know what that means?

Without the technology, the simple things you should be able to do in your head, like arithmetic. Yeah. So,

Marlin Detweiler:
We had an experience over the weekend in the grocery store where my wife was faced with buying a vegetable. I'm trying to think what it was. I think it might have been a lettuce. And then smaller packets and larger ones. And they were two different locations. And it was kind of fun to watch her, because I am the math guy between the two of us, and she's not.

But she knew unit costs and those sort of things, and she had to quickly say this was cheaper. I'm buying this one. Okay. We think we don't use math and we use it every day.

Paul Foerster:
Anyway, that's something that you can't afford to do. You can't afford to use up all your creative energy doing two plus three.

Marlin Detweiler:
Yeah. Well Paul, it has been so good to gotten to know you a bit over the last couple of years. Having had many of our students appreciate your writing and your teaching through your books. And now having had a chance to really talk to you a bit more here today. It's been wonderful. Thank you.

Paul Foerster:
Well, I appreciate what you're doing in keeping those books alive.

Marlin Detweiler:
Well, that's very kind of you. It's a blessing to both of us then.

Paul Foerster:
Thank you so much.

Marlin Detweiler:
Thank you. And, folks, thank you for joining us on this episode of Veritas Vox. We hope to see you next time.