In the previous post, I mentioned a forceful article by Paul Lockhart illustrating the problem with teaching math in school as facts to memorize. He is exactly right. But I feel that given our current system, it would be nearly impossible to turn math education into a "exploratory" subject - that is, a form of art. The barriers include:
- our cultural perception. Until someone is powerful enough to convince everyone that math, currently viewed as the most rigid and systematic of the sciences, actually exists with an artistic side that rivals the fine arts themselves, we are not going to go anywhere. Parents, principals, and even teachers would be confused as to why children should learn math as an extension to their creativity (which is what should ultimately happen if the right method is employed throughout). I believe that a good first step is to publicize articles like Lockhart's so that not only the math-people lament; they already know that math is supposed to be more of an art.
- the standard curriculum. There is no flexibility, and if the standards stay the way they are, any math reform attempt is doomed. The number of concepts to be covered in a year is designed to have a teacher introduce something new almost constantly, resulting in no time to reflect. "Exploring" takes a lot of time - sometimes days to find out a theorem that could be presented as fact in 30 seconds, but the result of 3 days of careful thinking would produce more benefits than 30 days of concept regurgitation. Learning all the formulas for area is building knowledge. Knowing how to derive one is building skill.
There are two benefits to the approach of improving creativity and problem solving skills. The first is that with a more solid understanding of where formulas come from, students retain them better. This helps them to learn advanced formulas faster because they already have a firm grasp of the fundamental ideas, which actually may speed up learning in future math classes! A good example is in trigonometry; there are about 10-15 "identities" (i.e. equations) that students are expected to memorize, some of which involve square roots or other fancy symbols, and all of which contain a mess of "sin" and "cos" and "tan". However, all these identities can be derived just one of them. Thus, if a student has one, he has all; memorizing one is a little easier than 15, but teachers never suggest this, leaving helpless souls with a list of a disconnected symbols.
The second is that in the end, most people will cherish a skill much more than knowledge. Only about 10% of kids will actually need the math they learn in school (so the students are justified when they ask "what do I need this for?"), but 100% of them will need to problem-solve. Obviously it would be better to give the students an ability they can use in all parts of life. This should be what a curriculum is designed to do.
- assessment problems. Encouraging "creativity" is hard work, as Lockhart says. Teachers need to carefully plan their lessons, patiently work with students on problems (not exercises), and at the same time deal with those who are less motivated. Furthermore, there is no way to measure how students progress without setting "deadlines", which may force teachers to give ideas away to improve their own performance, leaving the student with no time to think on his own. Voila the problem we have right now. However, without guidelines for performance, teachers would not teach at all. There are those few teachers who genuinely care about students' learning, but that's definitely not the majority. As masterminded the ideas for a new curriculum may be, the instructors are the front line in enforcing new methods. This will certainly require some experimentation in the classroom before we go any further in planning any further.
Saturday, October 17, 2009
Friday, October 16, 2009
Mathematics Education in Schools
An excellent article on mathematics education, which I recommend you read fully (it is a rather quick read, despite its length):
"A Mathematician's lament"
It's sad that just 1% of students in school ever get to enjoy the full range of creativity that mathematics has to offer by participating in interesting math competitions or exploring the problem-solving aspect. I believe that the math curriculum is supposed to improve quantitative and logical reasoning skills as well as creativity, but because of our "standards-based" approach, teaching math has become reduced to presenting facts without any useful context and regurgitating definitions without the slightest idea of the point of such technical terms.
Students, for the most part, never retain any of the concepts they learned in school (unless they go into a math-related field), AND they never truly learn the critical thinking skills that a correct math education will instill. So, they are definitely wasting their time in class; to quote Lockhart: "They say, 'math class is stupid and boring,' and they are right".
"A Mathematician's lament"
It's sad that just 1% of students in school ever get to enjoy the full range of creativity that mathematics has to offer by participating in interesting math competitions or exploring the problem-solving aspect. I believe that the math curriculum is supposed to improve quantitative and logical reasoning skills as well as creativity, but because of our "standards-based" approach, teaching math has become reduced to presenting facts without any useful context and regurgitating definitions without the slightest idea of the point of such technical terms.
Students, for the most part, never retain any of the concepts they learned in school (unless they go into a math-related field), AND they never truly learn the critical thinking skills that a correct math education will instill. So, they are definitely wasting their time in class; to quote Lockhart: "They say, 'math class is stupid and boring,' and they are right".
Thursday, October 15, 2009
Speed limits
I was quite shocked to read a short while ago that a "speed limit" on a road represents the 85th percentile speed of drivers, meaning that 85% of cars should be going at or below the limit.
To be honest, I think that in some roads it better represents the 5th percentile; for 95% of people, it is an inconvenience to drive at the limit. On small roads within residential neighborhoods, the posted limit is 30mph. Normally, people drive at 30 mph on if they:
- are new drivers
- are highly distracted
- are elderly people (slower reaction times)
- need to slow down for weather or night driving
- are purposefully trying to anger the cars behind
- operate an over-sized or especially heavy vehicle
Highways are typically not as notorious for slow speed postings, but still noncompliance is seen on almost every roadway. In fact, on average, the true "85th percentile" is 10-15 mph higher than the posted maximum. However, attempts to increase limits have received opposition from insurance and political standpoints, arguing that encouraging faster driving will increase accidents.
Well, a Montana highway study showed that when speed limits were imposed, accident rates actually doubled. Whoa. It's not clear what exactly happened, but the hypothesis was something along the lines of more courteous driving - when certain people decide to strictly adhere to the limit, aggressive drivers try to pass them as quickly as possible, and this can cause problems, especially in one-lane roads. Besides, most people have comfortable speeds that they drive at regardless of the limit, so removing the speed limit will have no effect on their driving. The Autobahn in Germany also had lower fatality rates after a "reasonable and prudent speed" incentive was entertained.
I'm not advocating for a removal of speed limits because teenagers can have crazy ideas and driving at 70 mph on a rural road can't go unpunished. But I do wish that the laws were a bit more reasonable. I have a safe speed that I like to travel at in my neighborhood, but 30 mph doesn't always cut it.
To be honest, I think that in some roads it better represents the 5th percentile; for 95% of people, it is an inconvenience to drive at the limit. On small roads within residential neighborhoods, the posted limit is 30mph. Normally, people drive at 30 mph on if they:
- are new drivers
- are highly distracted
- are elderly people (slower reaction times)
- need to slow down for weather or night driving
- are purposefully trying to anger the cars behind
- operate an over-sized or especially heavy vehicle
Highways are typically not as notorious for slow speed postings, but still noncompliance is seen on almost every roadway. In fact, on average, the true "85th percentile" is 10-15 mph higher than the posted maximum. However, attempts to increase limits have received opposition from insurance and political standpoints, arguing that encouraging faster driving will increase accidents.
Well, a Montana highway study showed that when speed limits were imposed, accident rates actually doubled. Whoa. It's not clear what exactly happened, but the hypothesis was something along the lines of more courteous driving - when certain people decide to strictly adhere to the limit, aggressive drivers try to pass them as quickly as possible, and this can cause problems, especially in one-lane roads. Besides, most people have comfortable speeds that they drive at regardless of the limit, so removing the speed limit will have no effect on their driving. The Autobahn in Germany also had lower fatality rates after a "reasonable and prudent speed" incentive was entertained.
I'm not advocating for a removal of speed limits because teenagers can have crazy ideas and driving at 70 mph on a rural road can't go unpunished. But I do wish that the laws were a bit more reasonable. I have a safe speed that I like to travel at in my neighborhood, but 30 mph doesn't always cut it.
Wednesday, October 14, 2009
Top chemistry mistakes that people make
These drive me crazy. Even chemistry students can't get this stuff straight sometimes because of culture conditioning.
1. If you can't read it, it's bad. Why? What's dihydrogen monoxide? Maltodextrin? The first is just a technical name for water and the second is a relatively safe food additive. Yet there are always people falling for the "dihydrogen monoxide hoax" and some of the top searches on google for maltodextrin include "maltodextrin side effects" and "maltodextrin dangers" - people search for the bad effects before they know what it actually is. Technical names say nothing about the substance itself. We are too developed a society to still have "chemophobia".
2. Atoms are round, hard objects. This was sort of like what the Greeks thought of, but we are not in 500 BC. Atoms actually have no boundaries; they are basically composed of nuclei (containing positively charged protons and neutral neutrons) surrounded by negatively-charged electrons that move around really fast. A very close (but incorrect) simplification is that the atom is a "solar system" - the nucleus is the "sun" and the electrons are the "planets". If this works for you, believe this, but just don't buy into the hard sphere concept.
3. Reactions: mix chemicals, wait, isolate product. If everything were this easy then people could be chemists in 30 minutes. Lab work is MUCH harder than it seems. Sometimes the compounds can't be exposed to air. Other times, you finish your reaction and can't find your products. Maybe they will decompose if they are not purified so you have to purify them immediately (which usually is a difficult process in itself, requiring hours of manual work). Often you will use other substances to speed up the reaction; how do you remove them? How do you know if your reaction is done (it's not always a color change!)? How do you know that you actually make the right thing? There are extensive methods to answer each question that take years of experience to master. Even more interesting is when things go wrong - then you really have to know what's going on to fix it.
4. Water conducts electricity. It doesn't. It is a molecular substance, and doesn't exist as charged species (aka ions), so an electrical charge cannot travel through. Tap water, however, does conduct electricity because it contains dissolved ions, like F- and Ca2+, which creates paths for charges.
5. Assuming that the chemical definitions of "metals", "sugar", and "salt" are the same as the common ones. This is a more obscure difference that most people are not expected to know. Metals are not necessarily shiny solids. Mercury is not a solid, lead is not shiny. The concept of metals has more to do with the bonding, i.e. how the atoms are held together. Sugar is not just the white powder you put in cake. Sugar is a rather vague term that refers to a compound with a formula that is a multiple of CH2O or any related compound. Salt is more than the thing you put as pranks in pixie sticks. When you combine a cation with an anion, you get a salt (cation=positively charged ion, anion=negatively charged ion). NaCl is our familiar salt, but MgCl2 and Al2O3 are all salts. Basically, you combine a metal and a nonmetal to get a salt.
6. Boiling, sublimation, and dissolving are chemical changes. In a chemical change, the composition of the substance must change. When water boils, it is still water. When table salt dissolves, it is still NaCl. Phase changes are called physical changes, which don't involve any new substances formed.
Warning: not all color changes are chemical reactions: heating glass until it is red is a physical change (it's still the glass), and iodine sublimination gives off a purple gas that is still iodine.
7. Atoms of solids don't move. Particles are always in motion. In solids, atoms are actually vibrating rather quickly. Temperature is related to the particle velocities, so if nothing moved in the solid, it would be very, very cold (in fact, it would be "absolute zero", the absolute lowest temperature possible). Remember that the visible properties of the solid itself doesn't tell anything about the atoms contained.
1. If you can't read it, it's bad. Why? What's dihydrogen monoxide? Maltodextrin? The first is just a technical name for water and the second is a relatively safe food additive. Yet there are always people falling for the "dihydrogen monoxide hoax" and some of the top searches on google for maltodextrin include "maltodextrin side effects" and "maltodextrin dangers" - people search for the bad effects before they know what it actually is. Technical names say nothing about the substance itself. We are too developed a society to still have "chemophobia".
2. Atoms are round, hard objects. This was sort of like what the Greeks thought of, but we are not in 500 BC. Atoms actually have no boundaries; they are basically composed of nuclei (containing positively charged protons and neutral neutrons) surrounded by negatively-charged electrons that move around really fast. A very close (but incorrect) simplification is that the atom is a "solar system" - the nucleus is the "sun" and the electrons are the "planets". If this works for you, believe this, but just don't buy into the hard sphere concept.
3. Reactions: mix chemicals, wait, isolate product. If everything were this easy then people could be chemists in 30 minutes. Lab work is MUCH harder than it seems. Sometimes the compounds can't be exposed to air. Other times, you finish your reaction and can't find your products. Maybe they will decompose if they are not purified so you have to purify them immediately (which usually is a difficult process in itself, requiring hours of manual work). Often you will use other substances to speed up the reaction; how do you remove them? How do you know if your reaction is done (it's not always a color change!)? How do you know that you actually make the right thing? There are extensive methods to answer each question that take years of experience to master. Even more interesting is when things go wrong - then you really have to know what's going on to fix it.
4. Water conducts electricity. It doesn't. It is a molecular substance, and doesn't exist as charged species (aka ions), so an electrical charge cannot travel through. Tap water, however, does conduct electricity because it contains dissolved ions, like F- and Ca2+, which creates paths for charges.
5. Assuming that the chemical definitions of "metals", "sugar", and "salt" are the same as the common ones. This is a more obscure difference that most people are not expected to know. Metals are not necessarily shiny solids. Mercury is not a solid, lead is not shiny. The concept of metals has more to do with the bonding, i.e. how the atoms are held together. Sugar is not just the white powder you put in cake. Sugar is a rather vague term that refers to a compound with a formula that is a multiple of CH2O or any related compound. Salt is more than the thing you put as pranks in pixie sticks. When you combine a cation with an anion, you get a salt (cation=positively charged ion, anion=negatively charged ion). NaCl is our familiar salt, but MgCl2 and Al2O3 are all salts. Basically, you combine a metal and a nonmetal to get a salt.
6. Boiling, sublimation, and dissolving are chemical changes. In a chemical change, the composition of the substance must change. When water boils, it is still water. When table salt dissolves, it is still NaCl. Phase changes are called physical changes, which don't involve any new substances formed.
Warning: not all color changes are chemical reactions: heating glass until it is red is a physical change (it's still the glass), and iodine sublimination gives off a purple gas that is still iodine.
7. Atoms of solids don't move. Particles are always in motion. In solids, atoms are actually vibrating rather quickly. Temperature is related to the particle velocities, so if nothing moved in the solid, it would be very, very cold (in fact, it would be "absolute zero", the absolute lowest temperature possible). Remember that the visible properties of the solid itself doesn't tell anything about the atoms contained.
Sunday, October 11, 2009
The (harder) part of the Common Application
Continued from the previous post:
Now we will discuss the part of the application that has no right answers: here you can really show yourself to the committee.
Part IV: "Short answer" and "Personal Essay"
The first bit isn't too intimidating. Pick something that meant a lot to you and describe why it meant so much.
The "Personal Essay" - oh no, I can hear your heart beating faster. "250 words minimum" (usually 400-700 is a good range). Yes, it's a major piece of work, but the college people aren't expecting literature. Here is actually an opportunity to describe yourself in a way that would be impossible to see otherwise in the application. Do you have a special talent? An experience that had a profound effect on your character? A teacher who taught you new perspectives to view the world? All of these are possible topics. There are six options and the last choice, "topic of your choice", makes this section as open-ended as it could get.
What the essay is not: the essay should NOT be a highlight of your achievements, because that was written on the activities/honors chart. It is NOT necessarily a 5-paragraph essay with intro, conclusion, and three body paragraphs. That was for middle school students to be able to structure their thoughts. In fact, the best essays often dive right into the narrative. The writing should NOT be over-formal or wordy; saying "the perplexity of all the intricate wonders in the exhilarating field of biology corroborates my tenacious interest in the science of organisms and their physiology" is far too presumptous and would be better written as "I enjoy the complexity biology has to offer".
Part V: "Disciplinary History", "Additional Information", "Signature"
I probably only have to describe the middle form. If you have taken any college courses, have had any impacting family situations, speak several other languages, moved extensively in a way that academics were altered, have a disability, or have experienced any abnormal events that had an effect on your life, they should go here. It is important that you take the time to elaborate on these additional facts, if applicable. To the admissions people, it may also clear up some other parts of an application, such as lower grades for a period of time.
Part VI: "Teacher Evaluation", "Secondary School Report", "Midyear Report", "Final report"
Teacher and counselor recommendations form the bulk of this part. The teachers you pick evaluate your character, grades, and class behavior here, through a form and a letter. You just have to trust that the recommenders do their job. If you've been a good person and an active class participant, this will definitely not hurt you. However, if you have shown superior traits in any area, they will glow; you may want to emphasize these characteristics when you give the recommendation to your teachers. The mid-year and final reports are mysterious entities that are almost identical to the secondary school report. I am not entirely clear on their use but I would guess that they track your performance throughout the school year, just to make sure that the "senioritis" isn't kicking in too far. Trust me, I have heard of students who were dropped from their accpetance because of poor performance senior year. Not the way to end your high school years.
Part VII: Extracurricular supplements
The Arts supplement are for those who really have the talent. Don't send a portfolio or CD if you're not that good. Be honest. Would you be confident presenting your skills in a competition? If not, then forget about it. Some of these applicants are REALLY GOOD at what they do, and you don't want to be judged against them if you're mediocre. Most people should not be sending anything for this section.
With Athletic supplements, I would go and contact the coach directly. Unless you are pretty sure about Varsity-level training, don't bother with this. It is never good ethically to be admitted for your athletic strength and then decide not to pursue the activity that helped your chances.
Part VIII: College-specific supplements
Some more selective colleges have another form that you have to fill out. Usually they include additional essays, stating your academic interests, and family involvement in the university.
End of tour! It is truly an ardous process that will exhaust any senior. For those who are going to send more than 12 applications, let me wish you best of luck. Be prepared to write at least 20 essays or short answer responses. And, get a lot of stamps ready.
In any case, if you are a senior, enjoy writing, and good luck! If you are younger, then I hope you found this helpful for future reference. If you have graduated already, then maybe you can show this to your children. You may have been amazed at how much this application has evolved over the years - it reflects the ever-increasing competitiveness over the nation's universities.
Now we will discuss the part of the application that has no right answers: here you can really show yourself to the committee.
Part IV: "Short answer" and "Personal Essay"
The first bit isn't too intimidating. Pick something that meant a lot to you and describe why it meant so much.
The "Personal Essay" - oh no, I can hear your heart beating faster. "250 words minimum" (usually 400-700 is a good range). Yes, it's a major piece of work, but the college people aren't expecting literature. Here is actually an opportunity to describe yourself in a way that would be impossible to see otherwise in the application. Do you have a special talent? An experience that had a profound effect on your character? A teacher who taught you new perspectives to view the world? All of these are possible topics. There are six options and the last choice, "topic of your choice", makes this section as open-ended as it could get.
What the essay is not: the essay should NOT be a highlight of your achievements, because that was written on the activities/honors chart. It is NOT necessarily a 5-paragraph essay with intro, conclusion, and three body paragraphs. That was for middle school students to be able to structure their thoughts. In fact, the best essays often dive right into the narrative. The writing should NOT be over-formal or wordy; saying "the perplexity of all the intricate wonders in the exhilarating field of biology corroborates my tenacious interest in the science of organisms and their physiology" is far too presumptous and would be better written as "I enjoy the complexity biology has to offer".
Part V: "Disciplinary History", "Additional Information", "Signature"
I probably only have to describe the middle form. If you have taken any college courses, have had any impacting family situations, speak several other languages, moved extensively in a way that academics were altered, have a disability, or have experienced any abnormal events that had an effect on your life, they should go here. It is important that you take the time to elaborate on these additional facts, if applicable. To the admissions people, it may also clear up some other parts of an application, such as lower grades for a period of time.
Part VI: "Teacher Evaluation", "Secondary School Report", "Midyear Report", "Final report"
Teacher and counselor recommendations form the bulk of this part. The teachers you pick evaluate your character, grades, and class behavior here, through a form and a letter. You just have to trust that the recommenders do their job. If you've been a good person and an active class participant, this will definitely not hurt you. However, if you have shown superior traits in any area, they will glow; you may want to emphasize these characteristics when you give the recommendation to your teachers. The mid-year and final reports are mysterious entities that are almost identical to the secondary school report. I am not entirely clear on their use but I would guess that they track your performance throughout the school year, just to make sure that the "senioritis" isn't kicking in too far. Trust me, I have heard of students who were dropped from their accpetance because of poor performance senior year. Not the way to end your high school years.
Part VII: Extracurricular supplements
The Arts supplement are for those who really have the talent. Don't send a portfolio or CD if you're not that good. Be honest. Would you be confident presenting your skills in a competition? If not, then forget about it. Some of these applicants are REALLY GOOD at what they do, and you don't want to be judged against them if you're mediocre. Most people should not be sending anything for this section.
With Athletic supplements, I would go and contact the coach directly. Unless you are pretty sure about Varsity-level training, don't bother with this. It is never good ethically to be admitted for your athletic strength and then decide not to pursue the activity that helped your chances.
Part VIII: College-specific supplements
Some more selective colleges have another form that you have to fill out. Usually they include additional essays, stating your academic interests, and family involvement in the university.
End of tour! It is truly an ardous process that will exhaust any senior. For those who are going to send more than 12 applications, let me wish you best of luck. Be prepared to write at least 20 essays or short answer responses. And, get a lot of stamps ready.
In any case, if you are a senior, enjoy writing, and good luck! If you are younger, then I hope you found this helpful for future reference. If you have graduated already, then maybe you can show this to your children. You may have been amazed at how much this application has evolved over the years - it reflects the ever-increasing competitiveness over the nation's universities.
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