Introduction to Differential Calculus for Algebra Students

My earlier blog about math education generated some offline discussion. In it, I suggested that students who are stuck on fundamental concepts should not continue forward until they understand those concepts. I also suggested that students could study principles of calculus while studying algebra. These two points seemed to some readers to generate a contradiction. I disagree. All that is required to study an advanced math topic is the set of specific fundamentals foundational to that topic. Beyond that, the traditional schedule followed in K-12 curriculum is arbitrary. An algebra student who understands how to plot a polynomial and how to find the slope of a line is prepared for differential calculus in its most fundamental form. Thus, for example, geometry and trigonometry are not fundamental to basic calculus. On the other hand, a student who does not understand the concept of slope should master that concept before attempting basic calculus.

As an illustration of how an algebra student could learn basic calculus, I include a slide show (below). Students working through the example should not simply read the slides, but should reproduce the ideas on paper and work out the algebra for themselves. Upon completion, they should understand how to find the slope of a line tangent to a continuous function, which represents the derivative of that function at that point.

Note the missing signs on slide 19: Students should take advantage of this error by filling them in themselves, then checking their result against the next slide.

Derivativedemo from Paul Roundy

Science and the Culture of Education

I love the approach of science to learning. As a professor with a research focus in weather and climate science, my objective is to better understand aspects of the natural world. Science is a way of interrogating that world through observation, developing models of what we observe, and testing them against observations and other knowledge. This testing is necessary because we cannot simply accept our hypothetical models as fact. Fundamental to the whole scientific process is skepticism about our explanations. Successful scientists doubt their own explanations and those proposed by other scientists. Every idea is open to being discredited and discarded, even those with deep consensus, and confidence in an idea depends on the evidence presented in favor and the lack of verifiable evidence against.

In the end, the “truth” of a scientific notion or model depends not on how many scientists accept it, but on whether observed facts remain consistent with its predictions. The more evidence that piles up over years or even centuries of testing and experience raises confidence in an explanation or ultimately discredits it. Some models have been so thoroughly tested without refutation that confidence levels of most experts exceed 99%. Some aspects of quantum mechanics, the germ theory of disease, the general theory of evolution of living organisms by natural selection, and even the safety and efficacy of some algorithms of genetic modification of food crops have reached that level of confidence in the scientific enterprise, even if some political or religious movements resist or ignore the evidence. At its most fundamental level, the theory of climate change, including global warming in response to burning of fossil fuels is at a similarly high level of confidence. It is a fact, tested repeatedly in the laboratory, that CO₂ absorbs infrared radiation and re emits it. In the atmosphere, it is demonstrably true that CO₂ reduces the rate at which the atmosphere emits radiation into space, thereby increasing the heat retained at the surface of the earth. The direct effects of changes in CO₂ concentration due to human activities is fairly small, but several known feedbacks amplify or reduce the resulting warming. Some gaps in our understanding of the size of these competing feedbacks remain. Thus the details of climate change induced by human activities, including how warm the planet will ultimately get given doubling of CO₂, are not as firmly grounded as our simple knowledge that increases in CO₂ concentration would raise the average temperature of the surface of the earth. In that context, the most appropriate public policy decisions with respect to climate change should be informed by our understanding of risks and the likelihoods of negative and positive outcomes, even though we are not absolutely certain at what level the altered climate will ultimately verify.

Although scientists do exist in a culture that motivates asking certain questions over others, the conclusions of science are not forced upon scientists in advance by some authority or by pre conceived notion. Instead, scientists are free to engage the data, the models of other scientists, and their own ideas, and they are free to present their ideas. Healthy scientific culture welcomes new ideas. In general, no formally submitted scientific proposition is shouted down or mocked, but new ideas or claims are brutally scrutinized through peer review. Those ideas that survive the test of time become broadly accepted.

Several aspects of the general processes of science seem natural to our youngest children. Young children are always asking questions. They observe nature and poke and prod at it, developing explanations for what they see, and they test those explanations. Of course they do not always come to conclusions that are consistent with prevailing scientific views, but most children genuinely want to understand nature. Although a small fraction of children maintain that healthy curiosity into adulthood, many of them lose it while very young, as adults squash their curiosity by suppressing their questions. They also lose interest as they begin to see science and math from the perspective enforced in public school. Many great teachers and teaching algorithms inspire interest in scientific thinking, but the system as a whole destroys that thinking, reducing it to sets of facts, rote exercises with clear pathways to a single right answer, and bubbles on standardized tests. The curriculum has become so standardized that the next steps and the next questions are almost entirely specified by the system. Thus students rarely maintain the vision of how to craft questions themselves and how to follow the evidence where it leads. The curriculum quells curiosity and excitement as a heavy wet blanket kills fire, and I am convinced that loss of these motivations for deep learning is one of the biggest tragedies of the modern world. Many heroes in education are working from within to improve the effectiveness of science curriculum, and I admire their efforts. 

I love and respect my students at the university, and nothing thrills me more than to see them ultimately buck the shackles of the system by taking back responsibility for their own learning. I teach applied mathematics, statistics, and computer programming as tools to better understand natural systems. I find that my greatest challenge is to help students learn to craft questions and algorithms to solve broadly stated problems. The education system has trained students to constantly ask authorities what they should do next. Students want the exercises set up for them. In my view, this represents the greatest failure of public education. It emerges because students are almost constantly directed by autocracy in the classroom that tells them when to do something, how to do it, and what to conclude from it.  

The real world, in contrast, requires that they learn how to pose questions, and then work out pathways to answering them when there is both too little and too much information available. They need to learn to make appropriate assumptions and how to test those assumptions. This type of learning cannot be reduced to a blank space on a worksheet or a bubble on a test form.

My own children are not perfect students, but they are curious contemplators. They left the public school system for home education years ago, and I don't recall them ever asking to go back. They spend hours each day playing, reading, computer programming, drawing, craft making, and in other pursuits. They are largely self-directed. My wife and I provide them with the topical expectations of the state, and we help them generate learning plans. We expose them to people and ideas that they might not necessarily have discovered by themselves, but only rarely do we impose our wills firmly on them. They connect with other children and adults at the park and at Yacon Village, a community-learning center in the Albany New York area. They occasionally participate in classes taught by other students and adults in the community. As one might expect, they end up behind their peers in public school in some areas and well ahead of them in others. 

We do support limited testing and assessment, by offering our children one standardized test each year. We use that test as a tool to help guide their future learning activities. Curriculum dominated by testing does not yield vibrant curiosity-driven learning. When student motivations for learning are internal, their outcomes nearly always exceed expectations. They wake up excited about what the new day will bring. They thrive on a healthy level of autonomy. Students around the world cry out for such autonomy. Let's help them achieve it.  

You may contact the author at proundy at Albany dot edu

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.

The Daily Agenda of Typical Advanced Academic Track Teens

On the average school day, the typical middle-class public-schooled teenager on an advanced academic track spends

•          6-7 hours in school
•          1 hour waiting for and riding the school bus
•          3 hours on homework

Many students also participate in 1-2 hours of structured extracurricular activities. Most of them probably spend 1-2 hours preparing to go to school in the morning and eating breakfast and dinner. At the end of it all, most of them would benefit from more than 8 hours of sleep.

That leaves roughly 1-4 waking hours for their other pursuits, except that their drive for such pursuits naturally leads many of them to swipe sleep time. Most students spend their personal time networking with friends, playing online games, or participating in other entertainment.

This schedule model denies many academically motivated students what they really need most to become creative critical thinkers: Time to contemplate, to play with math, to investigate science, and to practice in the arts. Given that time for personal study, for unstructured play, for structured cooperative activities like sports, and for hobbies is so important to personal development, what can students trim to gain their best advantage possible?

My proposed solution might seem radical, but it is becoming natural for increasing numbers of students: Cut out school and the school bus!  Replace that time with educational and personal pursuits driven by the student’s actual needs, crafted by the students themselves and their parents. Academically motivated students nearly always learn better driving their own educations than following the script of public school curriculum, because they are not arbitrarily confined to continue work on topics they already understand, and they can move forward at their own paces, not slowed down by the system’s schedule. The system conditions students to think that they cannot learn certain topics without a trained teacher, but this is simply not true, especially in today’s world with so much information and free educational resources right at our fingertips.

The school culture also has deceived many students that healthy social development depends on public school, but opportunities to build social skills can be much better outside in the community than inside a school—It just takes open eyes and some determination and planning.

For those who cannot ditch the school for home education or cooperative groups, work within the system to increase the amount of time for student-driven activities at school, and for greater focus on the individual.

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You may contact the author at proundy at Albany dot edu

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.

How K-12 Math Education is Broken, and How to Fix It

As a university professor in natural sciences, I have come to a generalization about how I think K-12 math education too often fails in reaching its primary goals. My focus is broad, extending from basic arithmetic through calculus. Not all teaching and learning follows this generalization, and although I think the system is not completely broken, I am convinced that understanding the context of my generalization would illuminate errors in our approaches and would suggest directions we can move so that more students become proficient in math.

Consider how we usually teach and learn multiplication. Right at first, students might be introduced to the concept of adding up the members of some groups of equal size (like, 4+4+4 is the same as 4 times 3), using counting as the fundamental proof. After that, training is heavily dominated by memorization of tables and repeatedly writing the answer to a mix of simple exercises in recall. Now, don’t get me wrong: Repetition can be great at reinforcing a skill, making it a necessary component of good math education. Yet, by weighting too strongly on repetition and drills, the concept gets buried under a mass of scripted rubble, as the system collapses in fulfilling the needs of a student who loses grasp of the central point.

In more advanced mathematics, such as trigonometry, K-12 education typically follows the same general pathway. Teachers and textbooks give token time to the fundamentals that lead to some math fact, including providing proof. Then, they highlight the conclusion that comes from that proof. Students memorize that conclusion as a rule, and then as with multiplication, they drill-drill-drill. Homework is nearly always dominated by carrying out the tasks of repetition, and exams, for the most part, assess whether students can carry out the same assigned tasks. With insufficient emphasis on the fundamental reasoning in the practice and assessment chain, students themselves naturally focus on learning the rules and carrying out the tasks, glossing over the motivations behind it all. The supporting proof and fundamental understanding gets buried in the quest to complete long lists of exercises, while the student and the instructor often see illusions of understanding based on success in the exercises, or students fall behind with no clear path to understanding. Over the whole trigonometry course, students memorize chains of rules that they then usually forget within a few weeks as they move to other topics. Yet, deep understanding of trigonometry, like most branches of mathematics, requires remembering only a handful of basic fundamentals, from which nearly all rules can be easily derived, often by simply drawing a triangle. Students who learn by following the public school paradigm eventually find themselves lost in the sea of rules, frequently looking up the rules again and again in order to carry out future tasks. The core of the students’ understanding of the math itself is therefore shallow. Many students who are perceived as successful complete assigned tasks and get excellent grades on exams, but really understand little of the subject.

An appalling fraction of former high school students who have completed even our best courses in Advanced Placement calculus, who then take calculus again in college, fail to attain a C or higher there. Calculus instructors in college frequently question why students who seem so well prepared on paper are actually poorly prepared for the intuitive thinking of a good college calculus course. In contrast, students who succeed typically have gained fundamental understandings of the roots of the algorithms applied: They are not simply satisfied by completing the drill or memorizing the rule. Understanding the motivation for the rule is far more important than the rule itself. Flaws driven by shallow understandings accumulate over time, eventually generating an overwhelming feeling of being lost, often leading students to think that they are incapable of success in math. In most cases what is really wrong is how they go about learning it. The most common errors students make in college calculus involve poor application of the fundamentals of algebra, which then only compound their errors derived from poor understanding of the fundamentals of calculus.

http://dimacs.rutgers.edu/~joer/2005-prep-for-college.htm

I personally can contribute little to the mountains of research on effective learning in mathematics. Yet, my experience as a college professor, whose courses involve applications of mathematics, provides me some insights on how students, parents, instructors, and education planners might improve outcomes:

1.  Place more emphasis on understanding the fundamentals that motivate a rule.
2.  Do not memorize rules beyond the small handful that is most fundamental to a subject (For example, in trigonometry, memorize the Pythagorean theorem and the definitions of sine, cosine, and tangent in terms of the lengths of the sides of a right triangle, and memorize a few of the most fundamental values of these ratios).
3. Know how to recreate the rule from the fundamentals.
4. Practice proofs and exercises as needed, playing with the math to consider different correct pathways to the same answer, making mistakes and learning from them along the way. This kind of “play” with math is what actually led to most of the proofs found in textbooks and in the math literature. Most students who ultimately succeed in college math do so because they paid a price of individual effort, sometimes fraught with frustration, which led to better fundamental understanding.
5. Students who do not understand a concept should not simply move on to the next one and forget about it, but should be given the freedom to work at it until proficiency is attained. Being slower than a group of other students during K-12 years should not carry a stigma: Students need to start from somewhere.
6. Students need to learn to find their own mistakes by asking themselves whether their answers make sense, and they should confirm answers by working through exercises in different ways. If students are assigned exercise sets, the assignments should be sufficiently long to provide enough practice, but sufficiently short that students can check themselves and play with the math a little, and they should be able to move on without penalty, leaving assignments incomplete, if they demonstrate understanding.
7. When a student makes a mistake, the mistake should provide a teaching and learning opportunity that can be used to drive better understanding of the fundamentals.

These rules might seem obvious on paper, but the whole system of a one-size-fits-all curriculum makes their application extraordinarily difficult in most public school environments. A truly successful math education that leaves nearly no one far behind must treat each student as an individual. Responsibility for success rests on the whole chain of those involved, beginning with the students themselves, their teachers, their parents, and their peers. Students must be allowed to take ownership of their own learning. In order to allow such ownership to develop, the system must not force each individual to move at exactly the same pace: The upcoming steps in a learning plan must adjust to the individual abilities and needs of the student. Those who do not yet understand a concept need to work at it personally, with assistance from a good mentor.

I invite discussion on these and alternative solutions, in the comment section below.

You may contact the author at proundy at Albany dot edu

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.

Be Part of a Revolution in Education
For K-12 students most interested in math and science, and their parents.
A blog by Paul E. Roundy, Associate Professor, University at Albany

I thought I liked public school while I was in it—it was all I knew. I loved science, math, geography, and reading great classics. Yet, at the time, I didn’t realize just how much the curriculum was built around an attempt by the system to provide every student in a class with the same learning outcomes. In retrospect, in terms of the way it fulfilled my own needs, I think it was too shallow and insufficiently challenging.  My teachers were great, but they were constrained by the system they served. Sometimes I found myself excited about a topic of discussion only to have my excitement interrupted by the bell at the end of class, when I was forced to move on to something else. Other times, I found myself slogging through long homework assignments that I had to complete for a reasonable grade on topics that I already understood. The time I spent on such assignments was almost entirely wasted when I could have been learning something new.  Had I been given the opportunity, I think I would have excelled at going straight to college at age 16. In the very least, I would have benefitted from being given the opportunity to design part of my own learning plan, or to have my learning plan change according to my progress in fulfilling it.

With so much information at our fingertips and systems for social networking, the rigid schedule of K-12 education is rapidly becoming antiquated. It has been evolving toward a more scripted one-size-fits-all curriculum, retaining too much rote memorization and repetition and insufficient time for students to learn to critically assess evidence and how to pose probing questions themselves. Tragically, the system has moved toward a test-centered atmosphere that crushes the vibrance of curiosity-driven learning, a type of learning that must focus on individuals rather than the student population as a whole. Many great teachers themselves question how we got to this point.

Children and teens need time for free play and loosely supervised self-directed learning in order to develop important cognitive pathways. Public schools typically offer little flexibility for students to personalize their experience beyond selection of an elective course or choice in the topic of a research project. Options to personalize learning plans for academically advanced students could be endless. For example, the school system constrains students to wait to learn basic calculus until after they have completed all of two courses in algebra, a course in geometry, and a course in trigonometry. Yet, some basic calculus fits naturally into topics of algebra and geometry before even starting a course in trigonometry. Helping students learn the most basic calculus much earlier, when they show interest and readiness, would expose them to the concepts, so that the whole subject might feel more accessible to them. The rigid structure across the public school curriculum is motivated more by tradition than by logic and evidence.

One answer to a better life might surprise you. Rapidly expanding communities of parents and students, together with easily accessible information on the Internet, are making home education a viable alternative to students of all types who wish to diversify their experience beyond that which public school can provide. Although these alternatives do not always compete well with the physical and financial resources available to public schools because they are financed through the tax base, a wide range of benefits and freedoms supersedes such shortcomings.

Home-schooled students do not generally sit at home hiding from commitment and social experience. They create their own experiences. They can design their own learning plans, in the context of state requirements and under review of their parents and the school districts. When students and their parents so choose, they can network across a community, travel to museums, tour businesses, or study nature. They can join sports or gaming teams or take part time employment or internships to build experience. They can take free online courses from expert college professors on topics ranging from astrophysics to horticulture to composing poetry, or take formal college classes for credit, learning a field at an advanced level the first time instead of taking a shallow version of the class in high school followed by a more advanced version in college. They can work through textbooks or exercise sets at their own paces and move forward when they are prepared to do so. They can create their own YouTube videos on topics of science, the arts, or sports. They can spend hours at a time studying topics of their own choosing. They can also build meaningful service portfolios. They are not confined to working just with students their own ages. They do not need to ask for permission to use the bathroom or get a drink. Should they so desire, they can continue to study and network online if they become ill.  Parents do not need to be teaching experts to provide good learning environments for their children. Many home-educated students find that deep learning of a subject does not depend as much on following detailed step-by-step instructions of a teacher as it depends on their own grit and determination.

I am a professor of atmospheric sciences at the University at Albany. I teach courses in applications of mathematics and statistics to natural systems, such as the atmosphere and ocean. I respect and admire my students. Most of them came through the public school system. I find that their biggest challenges in tackling advanced problems in science and mathematics involve learning to pose questions about systems they observe, then using the languages of mathematics, physics, and statistics to develop and test solutions to those questions. Although the technical abilities that the curricula of public schools provide students are usually helpful, the schools typically fail to help students learn to craft appropriate questions and to set up logical pathways to the answers. Instead, schools more often teach and test scripted pathways to solutions to pre written questions. Research suggests that more time to play and to design their own pathways through problems, with wise mentoring, can better prepare students for the modern world. Practically any information a student can want is available at the push of a button, so that it is more important today to learn how to assimilate information critically and how to ask the best questions and seek solutions relevant to real situations. Unfortunately, most math and science exercises in the school system provide exactly the right pieces of information along with examples showing how to solve that kind of exercise. Problems in the real world, in contrast, typically provide both too little and too much information, and require making assumptions that need to be carefully tested for applicability under the conditions of the problem. Students need to learn how to disregard useless information and how to use what is most relevant. Learning to pose the question can be as important as learning to find answers. Public schools typically do not often facilitate this type of learning, probably because it is difficult to manage in classrooms with many students and because it is difficult to assign grades objectively.

I suggest that, as a student yourself, your greatest advantage will be to learn to study deeply, think critically, and question everything about the world around you. Do not fill your entire schedule with pre scripted programs and activities, but leave yourself some free time to go where your interests lead. Work hard, play hard, and enjoy positive healthy relationships. Learn to discuss controversial topics with those with whom you disagree. Learn to give up your opinions when you find they are not supported by experience and evidence.

Home education is not for everyone, but many people dismiss it outright simply because working in the style of the public school system has become a habit. Yet the most popular point of view is not necessarily the best one. If after real contemplation and investigation of local home education communities, you decide to try this option, my own children and many others would welcome you among their ranks.  If home education is not an option for you, voice your opinions about modern educational methods to public school leaders and legislators, to pressure the system to accommodate more student individuality and real world thinking. Together, we can reform the education system from within and without.

You may contact the author at proundy at Albany dot edu

The opinions presented here are those of the author alone and do not necessarily reflect the perspectives of my employer or the State of New York.