Dutch vs. Americans on Education – Implications for the Gifted

I have come to believe over the past six months that Dutch and American values have influenced and shaped each country’s educational policies and attitudes much more than research or lessons learned over the past fifty years.  This raises important questions about the implications for each nation’s future and for gifted children in particular.  Which system is better in the long run (e.g. 30 to 50 years), and which is better for gifted children?  Unsurprisingly, I think the answers are not as clear cut as it may seem at first, and I don’t claim to know all the answers.  What follows are my “somewhat” informed opinions based on anecdotal observations, devoid of any serious study.

I think the key questions before us are the following:

  1. What national values are most likely reflected in the national approaches to education?
  2. What are some of the key differences between the two educational systems?
  3. What system seems more likely to support the long-run economic development and happiness of a nation?
  4. Which system is best for gifted students?

Unfortunately, it is impossible to answer any of the above questions independently of the others, but I will give it my best shot.  Moreover, I believe it is IMPOSSIBLE to segregate them from the far more important socioeconomic and political considerations, but it is instructive to think of the answers in isolation.

I believe that American and Dutch education policies have been strongly shaped by the key traits of each nation’s aggregate personality.  Capitalism and individual freedom are at the core of America’s history (true for most of America’s history, although now changing for the worse in my opinion), and Americans believe that you get what you work for.  This is the cornerstone of the “American dream.”  With few exceptions, talent has been historically rewarded.  As a result, Americans are fascinated by prodigies, and parents want to believe their children are special.  Many states have enacted laws to address the needs of the gifted, although most of these programs are being curtailed due to the political cost of allocating scarce resources that benefit a small cross section of the population.  On the other hand, the Dutch strive for societal uniformity.  This is no accident because the country owes much to its ability to tackle difficult problems (e.g. land reclamation) through consensus.  The Dutch strongly believe in lifting the entire population, even if this means shortchanging the potential of the exceptional few.  This prompts me to ask which system is better.  Clearly, “better” is an ambiguous term requiring context (e.g. in what sense and for whom).

The Dutch and American education systems differ in some important ways.  The first is the percent of adults post-secondary education.  Some statistics peg the percent of adults over 25 with a college degree at nearly 30% (source: United States Census Bureau).  Education at a Glance 2011, OECD Indicators states that approximately 45% of US adults over 25 have attained tertiary education.  In the Netherlands, the level is below 30% for the 55-64 year old cohort and approximately 40% for 25-34 year olds.  On the other hand,  anecdotal evidence I have collected suggests that the Dutch are much more likely to get a vocational or specialized degree such as electrician.  The interesting thing is that these statistics suggest the Dutch have just started valuing college degrees as much as Americans.  The second difference is that grade skipping is extremely rare in the Netherlands.  Administrators oppose it vehemently based on social development concerns, with little regard to research or evidence from the trenches.  The third major difference is the focus on and the role of homework.  It is my impression (based on my daughter’s own experience, from reading numerous articles on the subject, and from talking to many parents in both countries) that American kids (specially in the large cities) are overloaded with “busy work” (e.g. not necessarily a bad thing for the average student).  Dutch children, on the other other hand, are expected to do some homework and spend the rest of the time playing.  The American approach is to force learning via a heavy homework load.  On the other hand, the Dutch believe that learning should be done mainly at school.

I firmly believe that the American system is theoretically better for long-term economic development.  Unfortunately, it has deteriorated markedly in the past thirty to forty years, and it is now failing to deliver the education required for the US to keep the economic, scientific, and  academic leadership it has hitherto enjoyed.  The Dutch (who have their own distinct challenges) are doing a better job than Americans at providing the majority of its population with a decent education.  The problem is that the Dutch system provides virtually no structural way to address the fundamental right of the gifted to reach their potential.  I believe this results in an immeasurable loss of human capital that far exceeds the modest capital investment require to remedy the situation.  Unfortunately, US gifted education programs are failing.  Many states (e.g. California) have laws mandating gifted education programs, but the recent economic crisis, political issues, and general the incompetence of school administrators keeps most gifted children from getting an honest shot at developing their rare talents.

My review of the research and my own personal experience shows that there is no better country than the US for gifted children, provided they come from well-to-do families.  While there are some unique public programs (e.g. the Davidson Academy being the most spectacular), setting the right educational environment requires financial resources and time commitments beyond the reach of most parents.  I believe this to be the case because the US leads the rest of the world in terms of gifted programs (both public and private) and because homeschooling is legal in many states.  On the other hand, there are only a few gifted programs in the Netherlands, grade skipping is virtually non-existent, and home-schooling is essentially illegal under anything but the most extreme religious justifications.  To put it bluntly, there is little parents can do in the Netherlands to accelerate the academic career or to provide an appropriately stimulating intellectual environment for an exceptionally talented child.

My wife and I took Paulina to a birthday party this Sunday.  As we waited to pick her up, we sat in a bar discussing how well Paulina is doing in algebra and in elementary physics (which I teach her three to four times a week after school).  Then, we remembered that despite being very happy at the British School in Amsterdam Paulina is taking 4th grade arithmetic for the third time (year five in the British system) and finds it extremely mind numbing.  We also recalled that Paulina talks often about how much she enjoyed being home-schooled last year because she learned a lot more than she ever did in a traditional classroom.  We then made the decision to seriously consider Stanford University’s OHS in two years when Paulina will be old enough to apply for admission to middle school.  We know this could mean confronting the Dutch education bureaucracy, but we have a moral obligation to provide the right environment for Paulina.  We will not let the “system” tell us what it thinks is best because the system never redresses its errors.

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Math is About Solving Problems, Not Doing Arithmetic

Note: If you do not have Wolfram’s free CDF player installed, you will need to visit http://www.wolfram.com/cdf-player/ to install the browser CDF plugin.  This blog post contains two interactive examples that use the plugin.

It has bothered me for as long as I can remember.  “You must be so good at doing arithmetic in your head because you have a doctorate in math.”  This is one of the first things people tell me when they find out I studied math.  However, nothing could be further from the truth.  I can estimate reasonably quickly, but my mental, arithmetic skills are decidedly normal.  I usually tell people that math is not about arithmetic and that others at the table might be faster at calculating how much to tip our waiter.  People react almost invariably with an expression of confusion.  The real issue is that most people do not understand what mathematics is, and the real tragedy is that this shapes the way our society approaches its teaching.

My approach is to give my daughter what she is unlikely to get at school.  Why do countless arithmetic problems when they are not fun, and she will get plenty of them from her teacher?  I choose to focus on problem solving instead of computation.  Paulina loves everything about space.  So, I crafted a whole lesson about it learning about scientific notation and how to use it to estimate stellar distances and how long it would take to get from one place to another.  I asked her questions such as how long it would take to get to Proxima Centauri if we travelled, on average, as fast as Apollo 11.

I think it is important to push questions without a “correct” answer because they force kids to think creatively.  I once asked Paulina how she would estimate the mass of the earth.  A few weeks before, she had learned the concept of density.  So, her idea was to take a bunch of cubic meter samples and use their average.  Since she knew the diameter of the earth (e.g. from Wolfram Alpha at http://www.wolframalpha.com/) and the formula for the volume of a sphere, she was able to use her estimate of the Earth’s density to compute its mass.  The point here is that there is no “correct” method.  There are better and worse approaches.  However, problems of this sort are far better than those typically found at end of a chapter because they require not only recalling formulas and facts but also true understanding of the key concepts.

I am also a big proponent of using advanced computer algebra systems like Mathematica to focus on problem solving instead of computation.  Say your child is learning about the slope/intercept equation of a line and already knows how to plot lines.  Why not focus on how the line responds to changes in the slope and intercept?  Wouldn’t this approach develop more intuition than a hundred exercises?   I built the following gadget in ten seconds using Wolfram’s Mathematica (affordable to most people at $300 for home use).

[WolframCDF source=”http://localhost/wp-content/PostCDFs/LineSlopeIntersect.cdf” CDFwidth=”600″ CDFheight=”400″ altimage=”file”]

My daughter can vary \(m\) and \(b\) to get a sense of what happens to the line.  I can do even better by giving her a similar gadget with two lines so she may set one while altering the other to compare changes.

[WolframCDF source=”http://localhost/wp-content/PostCDFs/TwoLinesSlopeIntersect.cdf” CDFwidth=”600″ CDFheight=”400″ altimage=”file”]

Once again, these gadgets helped her get an intuitive understanding of how a line responds to changes in the slope and intercept, and she can do this much faster with my gadgets than if she plotted one hundred different lines.

The key point of this post is that math is about solving problems, not doing arithmetic.  Gifted kids should spend most of their time thinking about solutions, not computing.  Open problems are far better than the ones typically found at the end of a book, and computer algebra systems can be integrated into everyday lessons to accelerate the development of intuition and to focus on problem solving.

Finally, I would recommend that parents and educator watch Conrad Wolfram’s TED talk on how to fix math education.  Find his talk here.

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Site Restored!

Our move to Europe went relatively smoothly, but I fried the server holding this blog.  Fortunately, I had backed up everything in an external drive, and I just found it!

I plan to start posting soon. Please, visit here regularly.  The next few posts should be quite interesting given our experience in The Netherlands.

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Physical Models in Elementary School Science

I have been paying very close attention lately to my daughter’s science curriculum, and I have reached the conclusion that there is something terribly wrong with it. I am relatively happy with the exposition of earth science, but I am completely dumbfounded by the approach to the physical sciences. Paulina has done countless experiments this year using the scientific method to study physical phenomena, despite having virtually no mental models of the underlying processes. This makes no sense. Bear in mind that hers is the gifted track, which suggests it is utterly unnecessary to revisit the scientific method ad infinitum simply for the sake of teaching the ideas behind it.  My issue today is that I believe it is of the utmost importance at this stage to teach simple, physical reasoning grounded on models and that this is being sacrificed to allot plenty of time to a few concepts like the scientific method.

I came to the conclusion that our elementary school science curriculum is inadequate after listening to my daughter’s explanations for the physical drivers of several experiments.  All of them involved how temperature affects gases, and it was truly amusing to hear her explanations of why the experiments turned out they way they did.  Yes, she had been taught that a gas expands as temperature rises, but she had no idea why.  Some people may argue that doing these experiments develops intuition, and I do not disagree.  However, much more could be achieved by teaching physical models and asking students to do experiments requiring logical deduction from them.

My thoughts here are backed by academic research on gifted education. Andrew Grevatt, John K. Gilbert, and Matthew Newberry argue in chapter seven of Science Education for Gifted Learners that gifted students learn better by forming mental models and making inferences. This process is self-reinforcing because as new models are internalized, pupils derive ever more intricate relationships.

Let me focus on a simple example. What is an ideal gas is. The basic characteristics are:

  1. An ideal gas is a collection of free moving particles of negligible size.
  2. Gas particles move randomly in all directions.

Clearly, understanding the simple model I have described above requires rudimentary knowledge of the nature of energy. In particular, a moving object has energy. We call it kinetic. Intuitively, one knows this because effort is required to stop a moving object. In addition, it exerts force on any object it hits. It is logical that the faster something moves, the more kinetic energy it has. Finally, we measure the kinetic energy of a gas using temperature. In the case of a gas, the higher the temperature the higher the kinetic energy of its particles. Explain to your kid that rising the temperature of a gas is like giving lots of sugar to a group of kids, they bounce off the walls! This is a funny analogy I am sure will prompt some laughs. The point here is that the above facts are all that is needed to explain a wide range of physical phenomena involving gases.

Let’s discuss two recent experiments from Paulina’s science class:

  • Impact of air temperature on a 2-litter, plastic bottle – Fill a two-litter soda bottle with hot water (not so hot that the bottle melts). Let it sit for a minute of so. Empty the bottle and put the cap back up. After a short while, as the bottle cools down, you should see the bottle collapsing on itself. Specifically, the sides will come in towards the central axis of symmetry. Now, open the bottle and watch it regain its original shape.
  • Moving a Paper Spiral By Heating Air – Make a paper spiral from a standard 8.5 inch \(\times\) 11 inch piece of paper. Hang it over a light bulb using a string. Turn on and wait for the light bulb to get hot. Shortly afterwards, the spiral should start moving.

These are two examples of of experiments about weather and temperature that Paulina was required to do in the past month. While she was asked to hypothesize about their outcomes, she was never asked to explained them. This is baffling to me. What is the point of doing these two experiments in the context of weather if the student is not taught any of the underlying physical principles? Bear in mind that both were done to explain that temperature has an impact on weather. The only relationship between them and the class material was the fact that they dealt with temperature. Exactly how was the student supposed to extrapolate knowledge from these experiments about the behavior of weather? The answer is poorly. Paulina’s explanations ranged from the fantastical to the comical.

We laughed a lot as she explained these experiments because an educated adult would be incapable of such flights of fancy. Fortunately, the good mood set the tone for discussing the physical world. We talked about the ideal gas model, the concept of pressure exerted by a gas, and how changes in pressure are induced by temperature variations. More precisely, I showed Paulina how the ideal gas model implies the concept of gas pressure as well as the movement from high to low pressure regions. Once she understood this, it was an easy leap for her to explain the above experiments and even why a hot air balloon floats. This makes me happy because Paulina is now ready for fundamental principles like that of Archimedes’.

I will continue to use the school’s curriculum because it would be a monumental task to create my own in every subject, but I will never again let a school decide what it is important for my daughter to learn. It is ultimately my responsibility to remain vigilant and correct errors and omissions. Science is, in its essence, about discovering knowledge to explain the world around us. Science boils down to explanations through verifiable models that are simple enough for us to understand and then making deductions about the world (subject themselves to experimental verification). Shouldn’t our kids learn the same way?

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Using Programming to Teach Algebra

One of the hardest things for home schooling parents is to learn effective teaching techniques. In math and science, this means finding good ways to explain key concepts and their applications. As Paulina plunges herself into algebra, I realize that I must find ways to anchor the subject on concrete applications. This has prompted me to remember how I learned the subject. Before algebra, I did fine in arithmetic, but I did not excel. Looking back, I think I hated the subject because I found it boring. Fortunately, my mother bought me a TRS-80 Model III (Google it for a flashback into retro personal computers) when I was 12 and told me that if I wanted to play any games I would have to write them. In retrospect, my mom’s decision was a pivotal moment in my education. First, I had little choice but to learn technical English. Second, my interest in computers put me in the hands of Mr. Richard Murphy, head of the computer department at the National Astronomy and Ionosphere Center (i.e. the famed Arecibo Observatory). I learned more computer science from him in four years than most people in college. Finally, programming taught me algebra before I knew what it was, and this is the genesis of my idea.

I think I should introduce Paulina to the world of programming concurrently with algebra. My first experience with computers happened before the era of the Internet, graphical user interfaces, and ubiquitous computing. My only digital connection to the outside world was a 300Kbps modem connected to the Arecibo Observatory mainframe. This led me to spent countless hours writing programs to do everything from balancing my parents checkbook to play tic-tac-toe and checkers. By the time I started algebra in the summer of 7th grade, I already knew how to program in Basic, Pascal, and z80 assembly. This is whn things got interesting. While my friends struggled with abstract concepts like variables, I already knew about them as well as operator precedence, expressions, formulas, and functions. While they could not read simple linear expressions, I could work with complex formulas full of parenthesis, exponents, and functions. The bottom line is that algebra turned out to be the easiest math class I had ever taken. The interesting thing is that there were other smart students in my class, but I had a much easier time in algebra than they did. Why did I do so much better? Perhaps, I was smart enough and worked harder than others, but I firmly believe that my knowledge of computer programming gave me an unexpected edge.

The question is then how to approach computers today when all kids want to do is surf the web, watch YouTube videos, and play computer games. I was fortunate that to grow up with computers but few of today’s informational distractions. This forced me to learn how to program because hacking was the cool thing to do. This suggests a few strategies:

  • Restrict non-creative, non-educational computer time. We don’t let Paulina spend any time on the computer unless it is for school work or to research a topic she is interested in. We are big fans of Wikipedia, Google Earth, and the Google Art Project. We own a Wii and an Xbox 360 with Kinect, but we typically limit playtime to weekends. The point is that we want to create a clear demarcation between toys and real tools.
  • Foster a Do-It-Yourself Environment – Encourage your kids to learn how computers work and how to program. Tell them to learn how to write their own games. You may not know how to do it, but they will learn if you get them a computer and some software. I started Paulina on Alice (from Carnegie Mellon University) last year, and she loves to create imaginary worlds. Sometimes she surprises me. She has learned quite a lot by just tinkering with the system and exploring the objects. I have also introduced her to Scilab (an open source Matlab clone) to do computations.
  • Introduce Real Computer Programming – I am not sure what is an appropriate age to do this. Paulina has started writing very simple programs. I think this is okay in our case because Paulina has been doing pre-algebraic concepts for a little while. I get a feeling that kids are ready for basic programming when they start to understand 6th/7th grade math. It may be different for other kids, but this has worked for Paulina. I think the trick is to observe your children to get a sense of how mature they are and trust that they probably are a little more capable than you think.

There are computer languages I would recommend:

  • Python – This is a very easy computer language, but it is incredibly powerful. The best part is that it is completely portable. Write once and run it on PCs, Macs, Linux, and even mobile phones. Python is a free, open-source project with a large user base. Finally, there are tons of tutorials online, so it is quite easy to learn the basics.
  • Scilab – This is a free version of Matlab. It is mostly compatible with it. The package offers great numerical capabilities, but it lacks symbolic support (i.e. manipulating algebraic expressions). Nonetheless, it is a powerful development environment which is simple enough to learn quickly.

I prefer Python. It is super easy to learn the basics, but it is complex enough to develop industrial-scale applications. Don’t be intimidated by programming. Give your child the gift of programming. You will never regret it.

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Is Our Science Curriculum Properly Focused?

I love the scientific method as much as any self-respecting, educated person. However, is the public school’s obsession with it getting in the way of teaching fundamental principles? Is it more important for a 4th grader to learn fundamental principles or do twenty simplistic experiments as an excuse to memorize the high-level steps of the scientific method. This is the issue I want to consider today because I believe our science curriculum needs to focus much more than it does on fundamental physical, biological, and chemical principles.

Let me give you a specific example. Last week, my daughter’s science course asked her to theorize what would happened if she filled a bottle with hot water, emptied it after a while and closed it immediately before it cooled down. The instructions were to come up with a hypothesis, conduct the experiment, collect data, and determine if this supported or rejected the hypothesis. The student was to decide on the control, the variables, etc. This is not a bad experiment. It is simple. It gets the children to design the experimental protocol. Finally, the student needs to figure out if the experiment supports the hypothesis. My issue is that Paulina has done a ton of these mini experiments this year, but she still has not been taught the rudimentary theory of ideal gases. Hence, she cannot possibly rationalize the outcome of her experiment. Let’s be honest. The scientific method is simple. The difficult things are proper experimental design and the painstaking work required to draw meaningful conclusions. Mindless repetition of the scientific method is pointless. Fourth grade kids have little knowledge of statistics, experimental protocols, etc. However, they are mature enough to understand the concepts of temperature and the ideal gas model.

The question is then how I would have structured the lessons relating to the experiment. First, I would have explained that matter is composed of atoms, which group together into molecules which exhibit the material properties we perceived. I would have given a simplistic explanation of the atomic differences between solids, liquids, and gases. At this point, I would have introduced the concept of kinetic energy for point-like (or small spherical objects) particles and how temperature is a measure of the average kinetic energy of the molecules of the gas. Finally, I would have explained by the kinetic energy of gases gives rise to pressure and how volume, pressure, and temperature are all related (Boyle’s and Charles’ Law. Finally, I would have asked students to think about the pressure exerted by air molecules on the outer surface of the bottle and how that interacts with the inner pressure of the air trapped inside the bottle. My point this program is to get students to develop mental models of matter so they may make prediction to compare with the experimental results. Kids are not discovering new science with this experiments. They are learning how to construct experiments to test their theories. However, the whole thing is pointless, in my opinion, if they are incapable of explaining what they see.

It may be apparent from my tone that I am a bit enraged. I suffered from poor scientific exposition throughout my entire primary education. I was forced to teach myself because there was nobody able to help me. I cannot let this happen to my daughter. It would be unforgivable.

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What is Different About Gifted Education in Practice?

I had a rare, quiet moment today to reflect on whether or not there are any real differences between teaching the normal and gifted children. I asked myself if I could classify there differences in simple, general terms? I believe it is possible and much simpler than most people think. Finally, I asked myself if I could draw applicable conclusions from this classification.

I have not had the luxury of conducting exhaustive research on the subject, and I am limited by my own observations and my experience dealing with schools in California (both public and private settings). However, I have read plenty of research as well, so I am well aware of the general academic views on the subject. Bear this in mind as you read this short blog post.

Here are the practical differences I have observed:

  • Pace – This is the single biggest difference between traditional and gifted education. Gifted students learn material faster, and this drives the need for acceleration. The challenge is that giftedness tends to be subject-specific or subject-group-specific. By this I mean that students are unlikely to be gifted in every area. Many are gifted in math, languages, and the arts. Some are only gifted in mathematics, while others are only gifted in languages. Finally, while some students are gifted in many academic subjects, the pace of development is almost always non-uniform. This is precisely the biggest challenge with gifted kids. One has to work very hard to craft the right personalized learning plan.
  • Complexity - This is an interesting issue. It is clear, in my opinion, that gifted students can handle higher-than-average complexity. However, I struggle constantly trying to figure out if acceleration is an appropriate substitute for complexity. I don’t have a satisfactory answer to this question. Allow me to elaborate. Acceleration is definitely important, and it is particularly advantageous when going through elementary school math. However, one can create very challenging problem sets for high school students, and the importance of learning how to tackle difficult problems is undeniable. Let’s not forget that the best universities anchor mathematics and science around difficult problem sets. I have a feeling that there is a real benefit in slowing down the pace of acceleration after algebra to focus on solving tough problems and understanding connections with other areas of knowledge (i.e. applications).

I think most other differences are cosmetic. Of course, I will almost certainly adjust this list as my knowledge grows, but this is what I see today. I am ignoring emotional issues here, although they can have a measurable impact on the child, because I am focusing on the composition of the curriculum and not on how the curriculum is delivered.

What do my observations imply? I think the answer is simple. Gifted children can and should be accelerated, and their curriculum should include plenty of complex, challenging work. How this is incorporated into any particular study plan is a complicated issue that I do not want to deal with here. The bottom line is that I have seen too many gifted children withering away in traditional schools because neither the principals nor the teachers are equipped to handle extreme deviations from the mean. I cannot let this happen to my girl.

Here is my advice. If your child needs an intellectual challenge (please, make sure it is your child and not you who wants it), try to work with your school. If that does not work, check other schools. If you cannot find a school willing to accommodate your child’s needs, consider supplementing your child’s education through tutoring (with college or graduate students), distance learning (like EPGY and CTY), and plenty of books. If all these alternatives fail, try homeschooling (but only if you have the energy and understand what it entails). I don’t think homeschooling is for everyone, but it is the only solution sometimes. My biggest piece of advice is that you be actively involved in your child’s education. Don’t just say you care. Be your child’s primary educator.

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A New Dutch Approach to National Gifted Education

I have been researching for a few weeks how European countries approach gifted education, and I just bumped into a fairly comprehensive review of the Leonardo schools in the Netherlands. I have linked to the review here. Enjoy.

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A Call to Teach Formulas with “Proofs”

I have seen it countless times, and I just saw it once more: formulas without proofs. I love Capistrano Connections. It works for my family. However, no school is perfect, and Paulina’s 6th grade math class just annoyed me so much that I must write about it. For reasons that escape me, educators believe it is okay to teach formulas in elementary school, but that it is unnecessary or counterproductive to devote any time to their proofs. I find this appalling because proofs are critical to understanding. It can take longer than necessary to learn applications and be next to impossible to modify formulas when proofs are not understood. Formulas for areas are a concrete example of this. Most plane geometric areas can be derived from that of a right triangle. In fact, learning how to derive the formula for the area of a triangle and parallelogram teaches the basic technique for dealing with complex, plane figures. The bottom line is that proofs SHOULD NOT be avoided.

Let me clarify what I mean by proofs. On the one hand, I refer to formal, mathematical constructs. On the other, I refer to informal applications of the algorithms described by proofs. The fact is that most proofs for elementary school formulas are constructive. This means that they offer a algorithm (i.e. recipe) for solving problems. For instance, instead of teaching \(A = \frac{1}{2} b \times h\) for a right triangle, it might better to teach young students to double right triangles and then compute \(\frac{1}{2}\) of the area of the resulting rectangle. In fact, this could be done using numbers instead of variables with little loss of generality. The point here is that once a student is ready for algebra, he or she will quickly “remember” the correct formula because he used it before countless times. Finally, if the student knows algebra, there is absolutely no reason to avoid teaching the derivation of area formulas.

The point of this article is to refresh the reader’s memory about plane area formulas and their proofs. It is critical to establish a few basic facts and then derive everything from them.

  • Basic Area Element – We define the area \(A\) of a rectangle with sides \(L\) and \(W\) to be \(A = L \times W\). Think of \(L\) and \(W\) as the length and the width.
  • Areas Are Additive – This means that you can add the areas of non-overlapping regions to compute the area of a complex shape. Overlapping exclusive along the boundary (i.e. a shared boundary) does not invalidate this additive property. Put otherwise, it is okay to split a complex figure into simpler ones and then add the areas of these smaller pieces. In fact, the area of a complex figure IS DEFINED as the addition of the areas of its elementary building blocks.

We will follow the following program:

  • Start by assuming the formula for the area of a rectangle.
  • Derive the formula for the area of a right triangle.
  • Derive the formula for the area of an arbitrary triangle.
  • Derive the formula for the area of a parallelogram.

We will stop with the parallelogram because the areas of most other regular polygons require trigonometry. However, this short program teaches the most fundamental ideas.

Let’s start with a right triangle.

The only thing we know at this point is how to compute the area of a rectangle. Hence, we need to turn this triangle into a rectangle.

This rectangle has twice the area of the original triangle. Assuming the length and width are \(L\) and \(W\) respectively, the area of the original triangle is

$$A = \frac{1}{2} L \times W.$$

The key observation here is that we double the triangle, compute the area of the resulting rectangle, and then take half of the area of the rectangle.

We now know how to compute the areas of rectangles and right triangles. Let’s figure out how to compute the area of an arbitrary triangle.

We took the triangle above and split it into two right triangles by drawing the height from the top vertex down to the base. We then have two right triangles whose areas we have learned how to compute. Because the triangle may not be isosceles, one cannot assume that the two bases are congruent. So, we introduce an auxiliary variable \(l\).

The base of the triangle on the right is \(l\), and the base of the triangle on the right is \(b-l\). Let’s compute the areas of the two triangles:

  • Left Triangle – \(A_{\text{left}} = \frac{1}{2} h \times (b-l)\)
  • Right Triangle – \(A_{\text{right}} = \frac{1}{2} h \times l\)

We can add the areas of these two triangles to get the area of the big one.

$$A = A_{\text{left}} + A_{\text{right}} = \frac{1}{2} h \times (b-l) + \frac{1}{2} h \times l.$$

The next step is to simplify this expression by cancelling terms after using the distributive law.

$$
\begin{align*}
A &= \frac{h \times (b-l)}{2} h \times (b-l) + \frac{1}{2} h \times l \\
&= \frac{h \times (b-l)}{2} + \frac{h \times l}{2} \\
&= \frac{hb – hl}{2} + \frac{hl}{2} \\
&= \frac{hb – hl + hl}{2} \\
&= \frac{hb}{2}
\end{align*}
$$

Make sure the student understands what we have done here. We split the figure into two right triangles. We then added the two areas to get the formula

$$A = h \times b.$$

It is also important to make sure the student understands that this formula agrees with the one for a right triangle.

It is time to derive the formula for the area of a parallelogram. Start by drawing an arbitrary parallelogram. By arbitrary we mean that we do not know the exact dimensions and angles involved.

We need to divide this parallelogram into shapes whose areas we know how to compute. The easiest way is to divide the parallelogram into a rectangle and two congruent triangles.

Using this diagram, we can write

$$
\begin{align*}
A &= A_{\text{left triangle}} + A_{\text{rectangle}} + A_{\text{right triangle}} \\
&= \frac{h \times l}{2} + (b-l) h + \frac{h \times l}{2} \\
&= \frac{hl + 2(b-l) h + hl}{2} \\
&= \frac{hl + 2bh -2 lh + hl}{2} \\
&= \frac{2bh}{2}
\end{align*}
$$

Notice above that we had to double \((b-l)h\) in the numerator when we wrote the term as a fraction \(\frac{2(b-l)h}{2}\). This is because

$$(b-l)h = \frac{2}{2} (b-l)h = \frac{2(b-l)h}{2}.$$

This concludes this post. We showed how to derive the formulas for the areas of a right triangle, an arbitrary triangle, and a parallelogram starting from the definition of the area of a rectangle

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Introduction to Proofs Using Euclid’s Postulates

There comes a time when every teacher realizes that a particular student  is ready for logical rigor, and I believe that high school geometry is the first, and for many students the last, time when they are required to prove theorems. The problem with doing proofs at an early age is that things can get out of hands very quickly. There are few areas of math where proofs require modest machinery. Fortunately, geometry and graph theory are two wonderful topics that I think are accessible to young, gifted kids. With this in mind, I have started teaching Paulina basic geometry, starting from Euclid’s postulates. My idea is that we are very early in Paulina’s education and can afford to play with the postulates to derive most of the basic theorems of plane geometry. What I really mean is that we can pay very close attention to the logical dependency of the various assumptions without being rushed by an academic schedule. My real goal here is to give her a gentle, enjoyable introduction to art of mathematical proof.

Disclosure: You can teach very basic geometric reasoning without algebra, but some knowledge is required to make the experience worthwhile. For instance, many proofs require understanding that if \(a+b = 180\) and \(a+c=180\) then \(b=c\). Another example of the type of algebra needed is solving for \(a\) in \(a+b+c=180\). Finally, the student needs to be familiar with substitution.

Euclid’s Postulates

The basic postulates are five. The 5th was hotly debate (i.e. Cnn it be derive from the other four?) for a long time, but it is now known to be independent from and consistent with from the first four.

  1. A line can be drawn through any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angle, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

From http://mathworld.wolfram.com/EuclidsPostulates.html

Proving Things from the Postulates

Let’s to prove some basic statements that I think are accessible to young students.

Exercise 1 : Prove that vertical angles are congruent.

Proof: This is a fundamental proof in geometry. It is simple and variants come up repeatedly. First draw two intersecting lines and label the four angles.

We can conclude a few things

\(
\begin{align*}
a+b&=180\\
c+b&=180
\end{align*}
\)

Notice that both equations add the same thing to \(b\) to get 180. Hence, \(a\) and \(c\) must be equal. Hence, we have proven that the vertical angles \(a\) and \(c\) are equal.

To make sure the student understood the proof, ask him or her to prove that \(b\) and \(d\) are equal.

Exercise 2: Prove that alternating interior angles equal.

Proof: This one can be a bit tricky when you first see it, but it is important because it is a relatively simple application of the technique of proof by contradiction. I would advice skipping this proof for now and revisiting it once a good number of simpler proofs have been understood.

The classic proof uses a technique called proof by contradiction. Let me explain in plain English what is meant by proving by contradiction. Accept the assumptions of the theorem but not the conclusion. Then, reach a contradiction. This means reaching a conclusion that contradicts one of the assumptions we started with.

Let’s try to understand what the problem is asking us to prove. We are told to assume as true that:

  • We have two parallel lines.
  • We have a third line intersecting the two parallel lines.

We are then asked to show that alternating interior angles are equal. Hence, our proof by contradiction will assume that while the two bullet items above are true, alternate, interior angles are not equal.

It always helps to draw diagrams when solving geometric problems. Below, lines \(l\) and \(k\) are parallel. Letters \(a\) through \(h\) denote all the angles in the diagram.

What does the term alternating interior angles mean? Look at the intersecting line. Angles \(c\) and \(f\) are alternating. So are \(e\) and \(d\).

I have labeled every angle in the diagram, but we will simplify it in a second. We can use exercise 1 to conclude that \(a = d\), \(c = b\), \(e = h\), and \(g = f\). This simplifies our diagram to

We are going to proceed by contradiction.

$$
\begin{matrix}
l, k \text{ are parallel} & \text{by assumption} \\
b \neq f  & \text{because we are proceeding by contradiction} \\
\end{matrix}
$$

We are assuming that \(b \neq f\). This means that one is bigger than the other. It does not make a difference which one is because the proof is identical in the other case.  Assume that \(b>f\). Then, \(a+f<180\) because \(b+a=180\) since they are supplementary. However, postulate 5 says that if the sum of \(a\) and \(f\) is less than 180, then lines \(l\) and \(k\) eventually intersect. This would mean they are not parallel, which is a contradiction of one of our assumptions. Therefore, it must be true that \(b = f\).

Exercise 3: Proof that the sum of the interior angles of triangle is 180

Solution: We will use the previous exercise to solve this one. Let’s draw a triangle with its interior angles.

Let’s draw a line at the top that is parallel to the base of the triangle.

We can conclude that the angle to the left of \(a\) is \(c\) and the one to the right is \(b\). This last step is where we used the last exercise about alternating, interior angles being equal. We now have three angles covering one side of a straight line:

\(c+a+b=180\)

However, this is exactly what we wanted to prove.

Exercise 4: Show that the sum of the interior angles of any polygon is \((n-2) \times 180\).

Solutions: There are two elementary solutions that I know. The first is the one found in high school text books. Paulina taught me the other one about a year ago. I guess this shows the advantage of asking kids to prove things on their own. Sometimes, they come up with new ways to do things.

For the sake of simplicity, let’s start with a quadrilateral. Split it into two triangles as show below.

Notice that the sum of the interior angles triangles 1 and 2 equals the sum of the interior angles of the quadrilateral. Hence, the sum of the interior angles of a quadrilateral is \(2 \times 180=360\).

Let’s do it for a pentagon.

By the same argument as for the triangle, we compute the sum of the interior angles to be \(3 \times 180 = 540\).

Let’s do it for a hexagon.

Hence, the sum of the interior angles of a hexagon is \(4 \times 180 = 720\).

The pattern should be obvious by know and could be formalized by induction, but the important thing is for the student to realize that a polygon can be “triangulated” with \(n-2\) triangles, where \(n\) is the number of sides in the polygon. Hence, the formula for the sum of the interior angles of an arbitrary polygon is

\(180 (n-2), \text{ where } n \text{ is the number of sides in the polygon}.\)

Paulina’s Alternative Solution

I will do it with a hexagon, but the process is the same for all polygons.

Pick an interior point. Draw a line from this point to each of the hexagon’s vertices. This creates six triangles. If one adds the interior angles of the six triangles, one gets the sum of the interior angles of the hexagon plus 360. Hence, the formula is

\(180 \times n – 360\)

However, this formula is the same as \(180 \times (n-2)\).

Exercise 5: Show that the sum of the exterior angles of any polygon equals 360.

Solution: The way to solve this problem is to invoke the previous exercise. However, I would advice you to ask you student to solve the problem for a triangle, a quadrilateral, and a pentagon first. Then, ask then to give you the general formula. With this in mind, let me show you how to do it for a triangle.

A triangle has three interior angles adding to 180. We know that the sum of any of these plus its corresponding angle is 180. Let \(i_1, i_2, i_3\) be the three interior angles. Let \(e_1, e_2,e_3\) be the three exterior angles. Then,

\((i_1+e_1)+(i_2+e_2)+(i_3+e_3) = 3 \times 180.\)

We can simplify this by regrouping and using the fact that the sum of the interior angles is 180.

\(
\begin{align*}
(e_1+e_2+e_3)+(i_1+i_2+i_3) &= 3 \times 180\\
(e_1+e_2+e_3) + 180 &=3 \times 180 \\
e_1+e_2+e_3 &= 2 \times 180 \\
e_1+e_2+e_3 &=360
\end{align*}
\)

As I wrote above, have the student do this same proof for a quadrilateral and a square. The reasoning and formulas are identical. The point is to have him or her figure out the basic ideas and how to extend the formula to the general case.

For completeness, let me include here the proof of the general case. Assume that you have an \(n\)-sided polygon. The sum of the interior and exterior angles at any vertex equals 180. Hence, we have

$$
\begin{align*}
(e_1 + e_2 +\ldots + e_n)+(i_1 + i_2 + i_n) &= n \times 180\\
(e_1 + e_2 +\ldots + e_n) + (n-2) \times 180 &=n \times 180 \\
e_1 + e_2 +\ldots + e_n &= n \times 180 – (n-2) \times 180\\
e_1 + e_2 +\ldots + e_n &=360
\end{align*}
$$

I will stop this long blog post here. My hope is that you have seen how basic Euclidean geometry could serve as an introduction to mathematical proofs for very young students.

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