Teaching and Learning Philosophy

“Best professor, knows students’ names, tries to get students involved, makes math fun, enthusiastic, knows the material and even more important HOW to teach.”

The transition from high-school mathematics to college/university mathematics is a difficult one for most students

Once upon a time, I was teaching first-year calculus at a university. A bright young woman whom I’ll call L. came to my office in tears, and her visit has haunted me ever since. She had scraped through Calculus I, was failing Calculus II, and was bewildered about how this could be. “I got an A in math every year in high school; I wanted to be a math teacher … .” Her dream was shattered.

I have worked closely with many students over the past years, and similar stories has been repeated frequently, in some variation and to some degree. It appears to be getting worse over time. The predominant emotions experienced by the vast majority of first-year university students are anxiety, frustration, exhaustion, depression, fear, and anger at course instructors for perceived unfairness. They feel inadequate, experience constant stress, lack confidence in their abilities, are constantly behind in their work, and are constantly rushed. The pressure is enormous.

Some of these students drop out, some switch majors, some leave their chosen programs; I assert that very few reach their potential. None of this is the fault of the students, most of whom are willing to work hard, and are sufficiently intelligent to do well. Individual course instructors are also not to blame; the vast majority of them are competent, caring teachers (quite a number of them are excellent), work hard on behalf of their students, and do everything in their power (within the constraints imposed upon them) to help their students succeed. University departments are also trying hard, and devoting as many resources as they can afford to supporting students, particularly first-year students.

The problem is systemic, and is typical of most higher-education institutions. Unless students are superbly well-prepared, know how to study effectively, and are aware of how hard one has to work and can organize their time sufficiently well to do so, they struggle to make the transition from high school mathematics to university mathematics. Going from a high-school system where the material is presented at a manageable pace, and they have plenty of personal contact with their teacher, to a university system where they attend three or four hours of very fast-paced lectures per week, in classes of hundreds of students, with minimal opportunities for personal attention from their instructors, is quite an adjustment for students to make.

Students typically get behind almost immediately, which is deadly in mathematics and science courses. Unable to ever really catch up, they scramble inefficiently to complete their assignments and prepare for tests and exams, not having enough time to do all of the activities that would lead them to a deep understanding of their course material (working through many, many practice exercises, deeply engaging with significant problems, “playing” with new concepts, pursuing explorations, making connections, reading the textbook several times, reading around the subject, and so on). Not surprisingly, even those who achieve relatively high grades have gaps in their understanding, which makes them ill-prepared for subsequent courses, and the vicious cycle continues in the following semester.

Because few students can keep up with the work load, by the end of the first few weeks of classes lectures are not very helpful, even in cases where lecturers are excellent; most students are so far behind that they cannot get much out of the lectures, so they just take notes with little understanding and hope to sort it out later — but later never comes. Most students are poorly prepared coming out of high school; they lack basic technical skills, conceptual understanding, reasoning ability, problem-solving skill, and have little idea for how to learn mathematics effectively. Thus, the sincere efforts of fine instructors and intelligent students is not very effective.

Why is the transition difficult?

One reason that students do not perform all of the tasks essential to deep understanding is that they just don’t know what to do. In high school, many students were told exactly what to do and when to do it, but they did not internalize the reasons for their activities. In college/university, they are expected to know what to do, and when to do it, but many are not sure what to do. They may be willing to work hard, but they just don’t know what to do. One of the major goals of this web site is to help students, especially mathematics and science students, to understand how to approach their studies, to know what to do, and to understand why they should do it.

Another major reason that students do not perform all of the tasks essential for deep understanding is that the pressure on them to obtain high grades is enormous. All of the important decisions about their advancement in academia depends on high grades. Graduating from high school, getting accepted into college/university, getting scholarships, getting accepted into professional schools or graduate schools, getting a good job — all depend on getting the highest grades possible. This enormous pressure to obtain high grades is not appreciated by some professors, who complain that students care only about grades instead of caring about learning for its own sake. But one can’t really blame students; after all, they did not create the system, they are just doing their best to succeed and attain their goals within it. And we shouldn’t lose sight of the constant stress that students thereby face.

Because most students are ill-prepared for college/university, and because the pressure to squeeze every last mark out of assignments is so great, they end up taking much longer to complete assignments than they would if they were well-prepared, leaving very little time for all of the other essential tasks. The reward for this kind of behaviour is continual stress, anxiety, and lack of confidence.

Some students who take first-year math “service” courses have extremely poor background, and their problems are compounded. Typical in my experience are D., who freely admitted to sleeping through high school classes, and N., who returned to school as a mature student after a ten-year absence. D. was quite light-hearted about his situation in first year, failed his first-year mathematics service course with a smile on his face, and scraped through the following Spring on a second attempt. D. came by to my office a couple of years later for help with Lagrange multipliers, which he needed to understand for one of his senior economics courses. He was no longer light-hearted; the smile was still there, but it was a grim one, and he mentioned that his classmates all thought he was dumb. D. said that he understood the economics just fine, but that he really wanted to get good at mathematics, which he now recognized was important. I saw N. very frequently one Spring while he was struggling through a first-year mathematics service course. He had forgotten nearly everything about high-school mathematics in the years since he had graduated; he could barely solve a simple linear equation in one unknown, and simplifying simple numerical expressions involving fractions was a nightmare for him. But he was gamely trying to get a credit, which was clearly an enormous obstacle for him. In spite of his time commitments to his family and his full-time job, he was putting an enormous amount of time into his heroic attempt to get his math credit, including spending many hours with me, and also hiring a private tutor. His attitude towards mathematics was clearly one of hatred and fear; he was looking forward to getting rid of this onerous burden so that he would never have to think about mathematics again. However, as a business student, he had not seen the last of mathematics.

The stories are similar, over and over again, year after year. Even many of our most successful students have a harrowing experience. Some of the people I speak to about L., D., N., and numerous students like them, are unmoved. “I had it tough, too,” they say, “and I didn’t even know the language. Where I came from, it was really tough, and only the best students and the hardest workers made it. Kids here have it way too easy.” The people who feel this way are clearly high achievers, and would have succeeded no matter which system was in place. But considering that parents and all taxpayers ultimately pay the bills for our education system, don’t we all deserve a system that makes more sense, and gives more value for our investment?

Ultimately the education system must be accountable to citizens.

Why should we taxpayers invest such an enormous amount of money into a system that does a poor job of producing competent graduates who have mastered something, no matter what that is, and who are highly motivated to contribute their unique gifts for the betterment of our society? To me it certainly does not make sense to admit students into courses for which they are not well-prepared. But colleges and universities are so strapped for cash, and many of them receive government funding based on enrollment, that they willingly accept as many students as possible, whether they are prepared or not.

Most parents are surprised to find out (if they ever do) that professors are not hired to teach their students. The vast majority of professors devote (nominally) 40% of their time, or less, to teaching. For most of them, another 40% (at least) is devoted to research, and the remaining 20% to committee work, and other service to their institution. But don’t blame professors for this situation: The pressure on them to do publishable research and to obtain research funding is enormous, so much so that most of them lead the unbalanced lives typical of workaholics. Contrary to popular belief, college and university professors work extremely hard.

It’s popular nowadays to blame elementary and high-school teachers for the ills of the education system in North America. Teachers are criticized for having cushy jobs, with benefit packages that are far too generous, and vacation time that is disgracefully long.

The reality is just the opposite.

Teachers are run ragged, and typically spend their evenings and weekends working at teaching in one way or another (preparing classes, grading, etc.), and many would collapse from physical and mental exhaustion if they did not get the time off that they do. They are being asked to undertake an increasingly wide range of tasks, from psychological and social work, to health care and logistics, while the demands on their teaching increases dramatically. The amount of official paper work that they are required to do is breathtaking. And they do all this because they genuinely like young people and the idea of helping them develop, in an increasingly hostile environment in which they are blamed for problems that they did not create. In some jurisdictions, teachers’ authority over their own classrooms continues to be undermined in an insane effort that is branded as a way to hold them accountable (the rise of standardized testing, and teacher evaluation formulas that include students’ results on standardized tests), but in reality is intended to increase corporate control on the education system, and to allow corporations access to the education system so that they can siphon off the tax dollars of hard-working citizens into their own already deep pockets.

So teachers are not to blame, professors are not to blame, and students are not to blame. And yet, many students graduate from high school ill-prepared for college/university, and institutions of higher learning are poorly structured to help students to flourish when they arrive. What is one to do?

What we need is to transform our education system with a few key changes that will enable our hard-working teachers, professors, and students to work more effectively at achieving all of their goals. I’ll begin outlining what I think are the necessary changes below, but such changes may take many years to realize; in the meantime, our children need help now.

Which brings us to the point of this site: To provide students with a friendly, supportive environment where they can be guided to master enough mathematics (and physics, if they need it) to succeed in the program of their choice. Our desire is to help students to prepare themselves superbly for college/university before they leave high school, and to continue to support them once they are attending college/university.

Let us work together to improve the education system; but in the meantime, let us give our children the support that they need to succeed NOW.

What can we do to improve mathematics education?

When an infant learns to walk, we do not place the child on a schedule. We do not scold the parents when the child does not meet some artificial schedule for achieving the ability to walk. We do not grade the child. We do not pressure the child (or her parents) to learn to walk by a certain deadline or face severe consequences.

We don’t criticize children who are learning to walk for not walking the way we want them to, nor do we shame them when they do not meet standards imposed by others.

What we do, ideally, is provide a loving, supportive environment for children in which to learn to walk, with encouragement and instruction when needed.

Watch children engaged in learning something that they are interested in, and you will see learning excellence. Children are willing to fail over and over again, at a skateboarding trick, at hitting a tennis ball over a net, at riding a bicycle, at learning to play the guitar, at just about any activity that interests them. They fail over and over again, and it doesn’t hinder them. They just pick themselves up, not wasting time to dust themselves off, and keep at it.

Dr. Tae, the physics prof who skateboards: What learning skateboarding tricks tells us about the learning process in general

Until they begin school.

At school, most students slowly, gradually have their spirits squashed. Now they are graded on their results, and they soon realize how all-important their grades are. Their parents are anxious about their grades, their teachers focus a lot of attention on their grades, and soon they also learn to become anxious about their grades. This anxiety permeates their entire attitude to school, and soon their approach to learning. For most children, their natural, free, engagement with learning is replaced at some point in their school career by an anxious, stressful hunt for grades.

Sir Ken Robinson, Creativity is vital, and how we squander it

Many teachers lament that students nowadays are more concerned with grades than with the true spirit of learning, but students are not to blame. The structure of schooling (and in particular, of assessment) is at fault.

Few students reach high school with their enthusiasm for learning intact and unencumbered by anxiety. If your child is one of these few, then you (and your child) are indeed blessed. The majority of students are burdened by anxiety, stress, and the hollowness that comes with the pursuit of the almighty grades. Some students turn away from school, some no longer enjoy learning, some are paralyzed with fear (procrastination), some rebel; most struggle onwards, as if hindered by a huge weight on their backs.

Sir Ken Robinson, Changing education paradigms

Some parents take their children out of the system to be home-schooled, or even un-schooled. If you are one of these parents, your child still may have some need to consult with an expert mathematician who cares deeply about teaching and learning; if so, you are very welcome at this site, and we shall do our best to support your children. Other parents, wish their children to remain in the system, despite its drawbacks, and instead seek additional support for their children. If so, you are also very welcome at this site, and we shall do our best to support your children.

Changes in the structure of the education system in North America are surely needed. And there are some great models out there from which we can adapt ideas; the Finnish school system is an exemplary one, both in terms of raw results, and in terms of the cost-effectiveness of their excellence.

The Finland Phenomenon, a documentary by Robert A. Compton

But there are powerful forces against positive change in North America, with well-funded campaigns to dramatically increase destructive standardized testing and attack the authority and autonomy of teachers. Tax money that could strengthen public schools is siphoned off by charter schools, who then cherry-pick their students, pocket outrageous profits, and then manipulate data to pretend that they are far more effective than they really are.

It is daunting to work for positive change in this environment, but we must do so, and we will continue to do so. However, in the meantime, our children and grandchildren need support now.

The purpose of this site is to provide a warm, caring, community environment where students can obtain support in reaching goals of their own choosing.

How the FoMaP helps students with mathematics and science

We endeavour to support students in reaching their goals by creating in this site an environment that we wish schools would have (and indeed, some rare schools do have such environments). We believe that such hallmarks of a good learning environment go a long way to recreating the conditions of learning that most of us had naturally for at least some part of our childhood:

  • Students work at their own pace, encouraged by a caring guide.

At FoMaP, students ask questions when they need assistance, and we provide guidance promptly, just when it is needed. If you sign up for a special, personally tailored program, then we guide you through the program at your own pace. Our guidance and feedback is clear, gentle, and supportive, because we genuinely care about your development, and we know from experience that building a strong foundation will assist you in realizing your long-term goals.

  • Students are part of a warm, friendly, supportive community of learners and teachers, helping each other to reach their goals.

Besides getting assistance with their own difficulties, and answers to their own specific questions, it is helpful for learners to see the questions of other students, and to read the discussions and ultimate resolutions of those difficulties. In this way, the learning of each student is multiplied.

The entire database of all questions and discussions is available to all students, so they can browse the questions and discussions and learn a great deal.

A very important beneficial side effect is that students see that mistakes and misconceptions are a normal part of the learning process. When they see that every student has questions and misunderstandings, and that they are resolved gently and without any stigma, they relax and realize that we are all in this together.

“Make as many mistakes as you can, as fast as you can,” was John A. Wheeler’s advice to his physics graduate students, and the same advice applies in any field of learning. (Except perhaps for the people who are learning to defuse bombs, but there is always an exception, isn’t there? But in such cases, simulators allow students to make mistakes non-fatally, accelerating their learning; for example, consider the use of flight simulators in training student pilots.) Fear of making a mistake prevents us from trying things, which prevents us from experiencing life to its fullest. The inventor and the engineer would make no progress if they feared mistakes; rather they must try many different things, constantly learning, modifying approaches, making course corrections, until finally they realize their goals. Artists are the same; when Modigliani was asked how he decided which project to work on next, he replied that he moved in the direction of his greatest fear.

The regular school system unknowingly sows the seeds of anxiety and fear by the standard grading methods that they use. Students are afraid to make mistakes because they fear that it will cost them grades, and grades are made all-important in our current education system. This fear of making mistakes is absolutely counter-productive to learning, as it inhibits students from trying many different approaches, and even inhibits them from studying certain subjects, for fear of getting low grades. Many students have told me that they shied away from studying mathematics and sciences in high school, even though they were interested, because they needed good grades to get accepted to a university, and so chose easier subjects. This hurts individual students, but it also hurts society, as we are not training enough technically adept graduates.

At FoMaP, students are supported to make mistakes in a safe environment, where there will be no shame or ridicule, and where there are no grades, so there can be no damage to their grades. We hope that our caring, supportive environment will help students to lose their inhibitions about making mistakes, and free them to rediscover the creative, purposeful learners that they really are.

  • Students are encouraged to achieve mastery in some fraction of the subject; enough to support them in reaching their goals.

If all mathematical knowledge could be collected into a geometrical shape, then it might be an inverted pyramid, with the point at the bottom. The image is meant to convey the idea that there are only few foundations for this building, but as you advance in your knowledge, you must learn more and more to keep progressing up to higher levels. The other important point implicit in this image is that the foundations must be very strong indeed in order to hold up the building, thanks to its unusual shape compared to actual buildings.

In other words, it is important to master elementary mathematics if you hope to advance to higher levels of learning.

“A little learning is a dangerous thing,” says Alexander Pope (1688–1744) in his poem:

A little learning is a dangerous thing;
Drink deep, or taste not the Pierian spring:
There shallow draughts intoxicate the brain,
And drinking largely sobers us again.
Fired at first sight with what the Muse imparts,
In fearless youth we tempt the heights of Arts;
While from the bounded level of our mind
Short views we take, nor see the lengths behind,
But, more advanced, behold with strange surprise
New distant scenes of endless science rise!
So pleased at first the towering Alps we try,
Mount o’er the vales, and seem to tread the sky;
The eternal snows appear already past,
And the first clouds and mountains seem the last;
But those attained, we tremble to survey
The growing labours of the lengthened way;
The increasing prospect tires our wandering eyes,
Hill peep o’er hills, and Alps on Alps arise!

In learning mathematics, having a shaky grasp on one level is not of much use in advancing to a higher level. Each level builds on the one below, particularly in our current education system. Unfortunately, because of the common practices of assigning “partial credit” (part marks) and testing-by-sampling, it’s common to achieve good grades (even A) without having a deep understanding of the mathematics one was supposed to have just learned. This has the unfortunate consequence of giving students a false sense of security (well, I got a good grade, so I must know what I’m doing), as well as setting them up for enormous stress, inefficiency, and often failure when they tackle the next level.

Work takes much longer to do in Grade 11 mathematics if Grade 10 mathematics has not been mastered, and it is not as well understood. The effect is cumulative, with each subsequent grade becoming harder to handle. First-year university is the hardest step of all, because the material comes so much faster, there is so little personal attention from professors, the number of graded items is far fewer, and the amount of time per semester is less. The fast pace means falling behind is deadly, but falling behind is almost inevitable for the ill-prepared. This is a prescription for misery, and the inverted pyramid often comes tumbling down, crushing the dreams of the poor student in the process.

But it doesn’t have to be this way. The remedy is to heed Pope’s poem.

Master each level, and the next level becomes attainable with less stress. The hard work that a student does will be more purposeful, more effective, and achieve better results. Joy in learning becomes possible again.

University mathematics courses hurl an enormous amount of material at students (relative to what they have been accustomed to in high school), and do so at a much greater pace. Hard work is required to flourish in such an environment, but hard work alone will not help if a student is not well-prepared. Spending twenty hours banging your head against an assignment only to get nowhere is not a recipe for success; but spending ten hours on an assignment, mastering the lessons it holds, and having enough time to tackle your other courses is a recipe for success. This is absolutely possible for a student who is well-prepared for university, both in knowing what to do, and in having a firm grasp on all of the necessary prerequisite skills and knowledge.

Individual learning programs at FoMaP focus on mastery of relevant mathematical skills and processes, including technical excellence, deep conceptual understanding, development of critical thinking and problem-solving skills, and learning how to apply mathematics to understand the beautiful world we live in. Best of all, students work at their own pace to ensure that they integrate their understanding deeply, and that they master all that they need for success in college/university.

  • Experienced, expert, personal guidance and feedback from a caring individual is available.

Whether it has been through teaching, working on textbooks, my extensive experience working with students one-on-one over many years, or furthering my own understanding with deep study, my entire life has been devoted to learning and teaching. I have experienced many difficulties in my own learning, fallen into many pitfalls, made many errors that I have recognized only upon later reflection. And I have witnessed the misunderstandings and learning difficulties of many of my students, thanks to my close rapport with them.

Whatever problems you are facing with learning mathematics or science, it’s almost certain that I have already faced the problem, either personally in my own learning or while helping other students through their difficulties. This vast experience with navigating obstacles on the learning pathways has helped me to become a good guide for others on their own path.

The consistent theme in feedback from students about my teaching is that I care deeply about them, and that I excel at explaining mathematics and science. Students have consistently said over the years that my classrooms have been fun and enjoyable, where I have treated all students gently and with respect, and students have had a very positive learning experience.

The bottom line is that I know my subjects, and I have helped many students; I can help your child, too.

  • Clearly written resource materials are available.

There are many resources on this site, many of which are made available to you for free.

The Online Mathematics Textbook, a multi-volume resource which is in the process of being written, will provide an online repository of clear, step-by-step explanations of fundamental mathematics, intended to help students make the challenging transition from high school mathematics to college/university mathematics. You’ll find this material at the “Textbooks” page.

  • Support is available year-round.

With the semester system in place in many high schools, some students end up with a gap of six months or more between the end of one mathematics course and the beginning of the next one.

This does not support good learning and retention of mathematics.

Learning mathematics is a lot like learning a musical instrument, learning a sport, learning a martial art. Imagine that you are working towards learning the piano or the guitar. Would taking a six-month break help you master the instrument? Would taking a six-month break help you become stronger in your weight-training program?

The summer break in learning typical in almost all schools is already bad enough. Put in place in a bygone era, when children were needed to help on the farm during its busiest period, taking an extended break from learning is not good for students.

At FoMaP we are available year-round to support your students in their learning.

For professional basketball players, the off-seasons (which they call their summers) are the most important times for them to work on their individual skills. Serious players who wish to achieve their greatest potential engage personal trainers, skills trainers, and other professionals to set up personal programs to help them develop themselves. They don’t have time to do this during their regular seasons, because of the exhausting grind of game after game, practice after practice, airport after airport. They certainly work on their games during the regular seasons, but they are busy working on teamwork, preparing for opponents, and so on, and so they do not have the same luxury of time during the regular season that they do in the off season.

Similarly, it’s totally appropriate for students to work on their mathematics while school is in session. However, they may not be able to devote the amount of time that they would like on strengthening their foundations in preparation for college/university. Although they may need a break from the grind of chasing after grades, students don’t need a break from learning when the learning is fun and enjoyable, and makes them feel good about developing themselves.

Nowadays, the vast majority of North American children are not harvesting crops in the summer, so we shouldn’t handicap them by interrupting their learning process; however, even if there is a break in school, summer is the perfect time to help students prepare for the following year of school; far from being an onerous burden, summer preparation will make your child more efficient and effective during the school year, and so will make their studies more enjoyable, and their results will improve.

“Thank you Dr. D’Agostino, this class was a joy to be a part of. I liked the format, it forced me to work independently, think hard, and learn more about my problem-solving abilities. I have definitely gained a ton from this class.”

For more information about tutoring for your child, click on “Tutoring“.