I have spent a lot of time researching the difference between teaching and training in working toward a master's degree. I'd like to clarify that I feel that both are important, but teachers most of all should know the limitations and applications of both in everyday life. The structure of our modern schools allows for little authentic teaching to occur outside of the realm of training. I hope that teachers who read this to connect their own experiences with brain theory.
Training uses pleasure and pain to achieve a desired response to a particular set of environmental conditions. The purpose of training is to get the trainee to perform an action or set of actions with as little thought as possible. Although it is impossible to eliminate all thought from "doing", it is possible to almost entirely eliminate thinking from the "doing" process in a limited environment. Any major change in the environment or situation could result in paralyzing stress on the trainee. A new situation demands thinking which the pure trainee is ill-equipped to handle. Training by it's nature takes thought out of the equation. If the environment is sufficiently different, the trainee will forget their training entirely and look to their own nature and experience for answers. Little thought needs to be given to the trainee's needs by the trainer because every trainer has but one goal. Get the trainee to always do what they say well and without question through practice/repetition. The ultimate goal of training is impossible because the trainee will always have free will, but training is such that a partial goal will reap varying degrees of success. A well trained basketball team that trusts their coach 95% does better than the one that doesn't trust their coach at all. Training has applications in nearly all sports, the military, animal obedience, the safety of young children, police forces, firefighting, emergency medicine, etc. It is impossible to train anyone who doesn't agree (at least reluctantly) to the training. People trained against their will learn to frustrate and antagonize the system and take every opportunity to do so.
Training does have drawbacks that are rooted in the brain and the methods used in training. The biggest one being the ability to think for one's self and problem solve. Training is about following. A good trainee does what he or she has been told without question. Problem solving and thinking for yourself are processes that require some degree of leadership over self and thus no amount of training will develop these skills.
Animal training is a good example. A lion is trained with meat and the whip. Lions have a need for survival; eating and avoiding bodily harm are high on their survival list so they will "agree" to be trained in this way. The trainer stands tall, looks intimidating, and uses his posture and nerves of steel to establish trust with the animals. When the lions begin to trust him the training begins. No trainer starts with the difficult tricks first. In order for the lions to be ready for those they must first learn the basics: How to stand on a pedestal and wait for instruction, how to follow the trainer properly, how to go where the trainer points on command, and so on. As the lions' brains become accustom to these behaviors they begin to do them without much thought.
The repetition of behavior is very important to training because of the way the brain works. The nerve cells in our brains have small gaps between them called synapses. These gaps are needed because they act like switches that control the flow of electricity in the brain. Nerves communicate with each other via chemicals called neurotransmitters. A nerve on one side of the synapse releases neurotransmitters which then travel the gap to the other nerve and the message is communicated. The longer this chemical exchange takes, the more delay we experience in producing behavior, giving us more time to think about other possibilities. Repetition of a behavior physically alters the brain so that the nerve cells involved in that behavior move closer together thus shortening the synapse gap and making the behavior happen more quickly, eliminating extra think time. It is a mistake to think that any trained behavior is cemented. No trained behavior is instant, there will always be time to think. A person who has bitten their nails their entire life will almost instinctively put their hand in their mouth when nervous. This is a self-trained behavior, and the key to changing it is to start training the brain to realize when it is happening and to stop. When nervous, the brain is going to want to take the path of least resistance, nail-biting. These relapses occur because the nail-biting neurons are still close together, but eventually with practice the training for realization will become more second nature than the nail-biting behavior. If you have spent years biting your nails it may take years to practice realizing your are biting your nails and stop. Over-eating, hair twirling, shouting in a classroom, etc. are all examples of self-trained behavior.
Teachers and psychologists stress the importance of catching negative behaviors when the child is young because it is harder to retrain than it is to train. To retrain themselves a person must believe their behavior is bad and be willing to replace it with a new one. Self-control is essential to the re-training process. Since self-control requires self-leadership, external training cannot be used alone to re-train an engrained behavior. Can a person who has not practiced self control retrain themselves? It is possible, but only if the person truly wants it enough. Even then it is a struggle, because the brain always wants to take path of least resistance. Over time the neurons involved in the negative behavior may distance themselves a bit from less frequent use resulting in small amounts of un-training but the person will always have to vigilant. Un-trained just means "out of practice" or "rusty". If they do the negative behavior again the brain will speed-learn and the behavior could reassert itself. Re-training is about the brain learning a new positive behavior neural pathway better than the negative behavior pathway it is currently used to taking to enact real long term behavioral change.
In the same way, it is a mistake to think that the lion is not choosing its behavior. Because the repetition has made the behavior easy, the lion chooses the path of least resistance. A new situation will increase the resistance of the trained behavior, perhaps long enough for the lion to think up a different behavior and choose that instead. The behavior the lion chooses will certainly be one that is well ingrained in its nature.
Let's take for example the lion who has been trained to open its mouth on command. This lion knows that opening its mouth in its limited caged environment on the command of a trainer produces pleasure (or did at one time with meat) The trainer may even feel confident enough to stick his head in the lion's mouth, but what happens when someone brings an old fashioned flash-bulb camera and the light bulb explodes with a loud flash and a pop? This unexpected change in the environment gives the lion think time. Lions that are stressed out in the wild do not sit still on pedestals with their mouths open. Their mouths are closed and they are alert. The lion may choose to close his mouth on the trainer. We all know what lions do when they feel something struggling in their jaws. Incidentally, the most common cause of a trainer being mauled is when the trainer trips while walking backwards. Something as simple as a trainer falling is irresistible to the lion and all of the training in its feline head evaporates as it's true nature comes forward.
I acknowledge the animals themselves are the main attraction, but there is also an element of drama. If training were 100% effective terrible accidents would never happen, and the drama and tension would be lost. Many trainers will not work with a lion who has mauled someone because the lion will always know that killing the trainer is an option and the more practice the lion gets at this, the more likely it will attack. A trained Orca named Tilikum mauled and killed 3 trainers before being retired as untrainable.
So here is the million dollar question. Are your students trained into creating the appearance of "good behavior" while inside their nature to be kids and goof around is just below the surface eagerly waiting for an opportunity to express itself in negative and unproductive ways? What do trained kids do when they have a substitute? What do they do when they have a fire drill after a snowstorm? What do they do when you have an activity with rubber bands? What do they do when one person in a lunchroom throws food? What do they do when they go to college? Is it possible to train our students to behave in every conceivable situation? If so, is that the goal of our public education system?
What if students no longer fear the teacher's punishment or accept their bribes? What do we do with these un-trainable students? Appallingly the current solution in the US is to stick them in self-contained classrooms, give them an alphabet soup of labels that tell the kid what's "wrong" with them, and sometimes drug them into proper behavior. Wake up people! Maybe the kid doesn't learn because he thinks its a waste of time and wants to disrupt the class because its and easy way to impress a girl he likes. That is not a label or a syndrome, its common sense! The kid has already figured out he doesn't have to do anything you tell him to do and accepts the consequences. The answer isn't finding bigger consequences! It's finding a way to meet his needs with the goal of him valuing and choosing learning. Maybe the answer is to pair the two because she is an A+ student. He doesn't want to look dumb, does he? The short term goal is to get him to learn for the girl he likes, the long term goal is to get him to connect that the learning itself feels good by asking him how he feels about his learning, listening, and helping so he will learn for himself.
Some teachers will say that training is an important step that must happen if real teaching is to occur. I would take issue with that. In my experience training just begets more training. When students learn that training is the game they tend to apply what they learn from it universally to other aspects of education and begin to expect it. In fact they may be begin to demand it both because it is familiar and because it transfers almost all personal responsibility to the trainer. By high school students are so good at frustrating and defeating the training model that all of school is becomes a game of cat-and-mouse to get a letter (A, B, C, or D) with minimal effort. Students see training and school seem to go so hand-in-hand and they begin to confuse training and authentic learning altogether.
As evidence consider the kid who can do the problem: 2+3x=14 but has no idea how to do 15=7x+1. Why is that? If the kid has truly learned the processes of inverse operations and equality, this should be a slam dunk. I think is is obvious to any math teacher what is going on here. The kid has learned and memorized an order of doing (training), not a learned a way of thinking. Students are applying their behavior training from follow the rules (sit down, take out a notebook, be quite, take notes) to the learning of math itself. Why else would the order and position of the variables matter so much to them? Students who have been trained into math compartmentalize each kind of problem: "this is what I do when the x is first, this is what I do when their is no number in front of the x, this is what I do when the x is on the other side, etc." When given a story problem outside of their training these kids raise their hands and ask teachers to "set this problem up for them so they can solve it." That's a BIG problem! No one is going to be there to set it up for them in the real world, so by clinging to these repetitious training exercises instead of encouraging open-ended and dynamic real-world problems what are we teaching them? If they can't use the math for anything then I guess we are teaching them how to move X's and Y's around to look pretty and NOTHING ELSE!
Why? There are 2 reasons. First is because we were trained in this way. A few of us honed the curiosity inherent in our human nature and explored math on a conceptual level, but most of us did not. I once had difficulty figuring out how and where a student had made a mistake on a problem involving the area of a triangle. A seasoned 10 year math teacher 'corrected' me that the area of a triangle A=.5B*H was one half the base times one half the height! I was once at a professional development where my group was asked to develop an activity for a computer. Since our activity involved a falling object, and I was in charge of the technology side of the lesson I asked if another group member could make up some real world data for an example by taking a parabolic model and making the data "off" by a little bit to simulate human error. "Any parabola will do, it doesn't have to be earth gravity", I told them. Not a single one of them could do it. "I'm not good at physics", one of them said. That means 3 certified math teachers out of 4 had no idea how to apply simple algebra 1.
Second is that training is so much faster and easier than the alternative. The trainer does not have to take into account the trainee's needs. (although ones who do will have more success) The trainer only needs to find out the students 'stops' and use them to 'force' the student to 'learn'. Johnny isn't learning well so I'll talk to his coach and he won't play basketball again until he can do what I tell him. Sheniqua likes photography, so I'll talk to her photography teacher and let her leave class 15 minutes early to take pictures outside, but only if she does what I say.
Seasoned teachers know that training is almost always made easier when they build and cultivate a relationship with their students. (I think this is why elementary teachers have more success than high school teachers. They have more time to develop relationships and if students destroy their relationship with their teacher there is greater impact on their everyday learning) It seems there is something need satisfying for teachers and students to care about one another and this relationship makes even the hardest training seem easier. The essence of that is the core of real teaching and authentic learning and will be discussed in my next post. True teaching is channeling the nature of the student itself to produce positive outcomes, not trying to change the nature of the student using punishment and/or bribes.
Trained kids cannot adapt to even the simplest changes in routine. This is why so much emphasis is on structure in modern classrooms. Structuring the student's life at school becomes the teacher's job under the training model. The essence of personal responsibility is delegated to an authority figure and it is deemed acceptable by pretty much everyone involved!!! It's not acceptable because unless students learn to set positive goals for themselves on their own (which takes practice), they won't set their own goals or structure their own lives because we insist on doing that for them! This essentially leaves them always looking for some external source to control their behavior and structure their lives for them. (perhaps they'll find a controlling abusive husband, an unhealthy spirituality, join the military for the wrong reasons, or lean on mom and dad forever for structure instead of self-reflecting and growing as a person)
The inability of students to adapt to new conditions is shocking! Students fail simple standardized tests because they 'look different' than normal tests. If they can't do that, what happens to them when they leave school and the knowledge they got there was trained into that particular school environment? What happens when a professor doesn't mark them absent and call home? What happens when The instructor doesn't make special interventions for them when they struggle? THE STUDENT LOSES when we take the easy way by training instead of teaching.
Training isn't all negative. Their are many positives of training. In sports the field and rules do not change. When they do change is is relatively minor and the athletes have plenty of time to adjust their training. The grace and beauty of an athlete free in action is a site to behold. Unencumbered by thought they fly effortlessly down the field awing fans everywhere.
The policeman who is in a shootout or the firefighter who runs into a burning building risk life and limb to save others. When asked by a news reporter after the fact about how they manage to stay cool under fire they will almost certainly say "it's all training"
The well trained EMT assesses the situation and applies medicine quickly to save lives.
The soldier trains for combat is many situations. Elite soldiers can learn to tame every conceivable condition: wind, rain, heat, freezing cold water, exhaustion, etc. The training of these brave men and women protects us and the freedoms we enjoy.
It is for those reasons and many more that we should never get rid of training, but...
...Getting rid of thought is not something we should be doing in the classroom! EVER! PERIOD!
Showing posts with label fuzzy math. Show all posts
Showing posts with label fuzzy math. Show all posts
Friday, April 2, 2010
Friday, March 5, 2010
The myth of "the basics"
What's WRONG with math education?
Before I talk about why I want to define what I mean by the phrase "doing mathematics". Stephen Wolfram defines doing mathematics as a 4 step process ad I agree with him. Doing mathematics is...
1) Finding a problem in the real world.
2) translating the real world elements into the language of math using formulas and symbols.
3) Computing the answer.
4) Reinterpreting the answer's value back into the real world to help solve the problem.
I think we have lost sight of what mathematics is all about. We now have machines to free us from the drudgery of step 3. We no longer have to fixate on it as we did in the past. In the days of pencil and paper it was extremely important to focus on step 3. There were no machines to do step 3 and it was very difficult to compute answers. Procedures were made and proven to allow mankind to extend his ability to compute ever more difficult mathematics. Some Mathematicians have built their careers almost exclusively on creating and proving abstract computational procedures for step 3.
Mathematics has developed and grown as a field because it solves problems in the real world, and computing is just a part of that process. From here I will be talking about what I observe in the classroom and how it compares with "doing mathematics". I will also analyze some of the arguments used to perpetuate and justify the status-quo of mathematics: procedural step 3 thinking and technological obsolescence of content.
I want to point out that what I am advocating is that we switch how we handle and do step 3. Instead of using paper and pencil, we use computers to do the computing work. Every other step in doing mathematics will remain the same. That may mean instead of making the assignments like "solve for x" we ask questions with higher cognitive demand like "write a real world story problem that uses the equation below and a person would need to find x". The difference here is that to answer the second question, you really have to know what the symbols of math mean. Students often limp their way though the procedures of math by memorizing processes without attaching meaning to the procedure. A student may memorize the procedure of finding an average as "add all the numbers together then divide by how many numbers there are." That's great, but what good is it if they don't know that the average is the exact center or balancing point of a set of numbers? Faking math by memorizing a bunch of steps superficially is common crutch for students. Asking questions like the second one above ensure that the purpose and concept behind a math idea is learned because in order to answer the question you need to understand math, not just be good at memorization.
"What if the computer is wrong? How would anyone know?" is another excuse that is often given for the "reason" we need to do things by hand. This is a silly argument when you think about it. Why stop at computers? How can we trust your own hand calculations? How can we trust mathematicians? How can you trust the internet? How can you trust your eyes, you could be delusional couldn't you? You can be forever agnostic about everything if you want. The fact is, its easy to point to a time when computers failed while ignoring that they are thousands of times more accurate and faster than humans at the same computational tasks. If some person can find an error with Mathematica, does that mean all computers everywhere are dangerous and will destroy math? It doesn't seem to be true of hand algorithms. When we get those wrong we acknowledge it was a mistake and correct it. We don't abandon hand algorithms and move back to stones and abacuses for counting, so why would computers be any different?
"What if everyone just trusted computers all the time? That's dangerous to trust computers to such a high degree all the time. If we don't teach our students how to do things by hand how will they verify the accuracy of computers? People won't be able to think!" You might think I am crazy, but I have heard this argument more than once so I will address it. Forgive my sarcasm but in society we have books, a primitive form of data storage. Should all of the world fall into the chaos that will "surely" follow by trusting a computer to solve for X and it failing to do so, people could read these books. Some people even learn things like math by reading books alone. In fact, I think some people will want to learn the hand algorithms because they find them interesting.(Wouldn't teaching that class would be a math teacher's dream!?!) Unless all math books are destroyed these hand algorithms will not be lost.
I think its questions and lines of reasoning like the last one that really tell it like it is. People do not know the difference between knowledge that everyone should know and knowledge that only a few people need to know. The current math procedures could be taught in a math history class where the inventive and ingenious people who developed them would get the honor in history they have earned. Now most of their names are hardly mentioned because we are to busy drilling everyone on ALL the algorithms.
"The basics" in math is this undefined hazy ideal that seems to impede progress in math education. No one would EVER put up with in a class that was not so abstract. What would you say if you wanted to take a class to basic computer class and on the first day the teacher gave you some computer chips a soldering iron and began teaching circuit dynamics? In the afternoon you were taught binary and machine code. You wanted to learn how to use Microsoft Windows, but when you tell the teacher this he teacher assures you that in order to REALLY understand Windows you must first master machine code and later MS DOS, then you might be ready to learn Windows NT then Windows 2000, then Windows Vista, then Windows 7, but only if you get your doctorate. Would you pursue such a program or would you look at all the people you know, realize that none of them know machine code but still know enough about computers to use them and then drop out? This is currently what happens in mathematics education. Is it any wonder why our students don't want to learn math? THIS IS UNACCEPTABLE! No one would take a computer class like the one above unless it was a special course in computing history. So long as that was the only program available, those that did would probably make a lot of money, but their success wouldn't be because machine code was the bedrock of the industry or gave them new insight into programming. It would be because few people would have the patience to deal with that kind of program. Fewer people in the program means higher demand for qualified people and thus higher wages, but students learning of machine code and soldering could have been better spend on other more important concepts like object oriented programming, system design, program architecture, etc.
If this program were like math education today students would complete their mostly irrelevant coursework, the professors would see that the few students who completed the program were successful and confirm their beliefs about just how important it was for students to know "the basics" of computer design. Even a few of the students who liked their machine code classes would extol how useful it was that they learned the basics. Since their were no other programs out there that didn't obsess over the basics, there would be no way to tell just how useful the basics really were in creating competent computer operators. If no another program was skipping the basics to teach more useful and practical information would we assume that there was no other way to teach it as we currently do in mathematics education today? Is it really necessary to get good with a soldering iron and circuit boards for you to use a computer? Is this knowledge everyone needs to know or some people need to know?
So why all the confusion? Why is it that math never seems to move forward?
What's wrong with mathematics education is simple: everyone is totally obsessed and hung up on step 3. Since step 3 is taught almost exclusively in the classroom from day 1, often the people who get into math fall in love the the procedural elegance and exact nature of the subject. Even though we now have computers to to the drudgery of calculation, they still extol computation by hand as some kind of virtue, while ignoring the actual application of math to the real world by glossing over steps 1, 2, and 4. Simply put we have not moved math into the age of computers; We must teach all 4 steps of mathematics in balance. Bringing math into the computer age will also make math accessible to the general population, democratizing math and opening exciting possibilities. No longer would mathematical analysis be restricted to the few who learned hand procedures. Anyone who had an understanding of math terminology and relationships could analyze data or use data to make decisions.
The age of computers has made hand calculation obsolete. If you don't believe me go to wolframalpha.com and punch in an equation try "3x+2=8" Magic. Check out the introduction video here.
http://www.wolframalpha.com/screencast/introducingwolframalpha.html
It is amazing what will happen in the future! We could be awestruck by its possibilities or fearful what will math teachers do when free websites like Wolframalpha are available to them 24/7 on smartphones. In 10 years, most US citizens will have a smartphone, are we going to force more algorithms down the throats of people who don't want to learn them or will we strengthen the rigor of our math curricula and make mathematics accessible to people who don't have the patience to learn the computational end of math but thrive on steps 1, 2, and 4. Which is better for the economy: fear and no change or change and possibility?
Math is one of those subjects that people think does not change over time. I would agree that the theoretical foundations of math have not and may never change, but how we teach math, how we learn math, where math is applied, and technology are factors that do change the game of math education. Fortunately, technology has long changed the game for mathematics for scientists and engineers, but it has not for students. Technological advances are being made at an exponential rate and we need more mathematically competent people to operate and understand these complex machines.
Are we producing people fluent in the concepts of math? The answer is a resounding NO! From my experience, most people have no concept of math beyond it is working with numbers or doing problems like "solve for x", "simplify the expression" or "rationalize the denominator". You would think these procedures were all made of gold and absolutely necessary to understand math the way some math teachers and professors talk about them. "They need to know the basics" is appealing rhetoric, but are students able to do the basics?
A simple look at the NAEP, TIMMS or PISA (international math and reasoning tests) we see US students' scores doubled by their rivals in other countries. It's pretty obvious students are not getting the basics (or much else) out of math classrooms today. I don't think foreign born students are twice as smart as us or that some sort of genetic advantage evolved in the last 100 years. There is nothing wrong with the American student!!!! They have every right to be bored to tears with hand algorithms they are forced to learn in exchange for a letter on a sheet of paper. (A,B,C,D,F) Especially after they have been shown that a computer can do it in a second. Nothing highlights our dismal failure to advance math curricula forward like the poor scores we see in the US today.
The students we do produce are can barely compute all the rigorous procedures and algorithms we teach them at all much less explain why and where we would apply those procedures. Basically, we cram computation down their throats, claim is important to know, and punish them when they choke on it. We busily teach them that doing math like solving equations by hand is very important, but then after we spend hours and days in the classroom teaching them these pencil and paper procedures, we then show them how it can be done on a calculator in 20 seconds. After all that, its more than unfair to be expected to be taken seriously by students who rely on technology everyday. Shouldn't technology make these manual calculation units easier and shorter? Shouldn't the questions we pose to students be becoming more conceptual and real-world based as technology frees them from the drudgery of manual calculation? According to most mathematics educators today, the answer is no.
I will end with some good news. We may be behind as a country, but no one is where they need to be in this math debacle. Other countries are busy teaching their students the same "by hand" procedures as we are. Some like Japan do not allow calculators, and I have even seen Japanese students wasting hours working abacuses with lightning speed and precision. (apparently while calculators are cheating abacuses are not)
Why do other countries seem to destroy us in math? Its a good question that demands an answer. This could be a thesis, but I will briefly discuss some of the reasons. Most countries scores benefit from the educational systems they have in place that test students after elementary school and then place children in different educational "tracks". Often only the students who are on an academic track take the tests for the country. Students who are on a good track and like mathematics do not have to contend with the poor behavior of students who don't want to be there. This is a huge advantage. Another factor I feel is not often discussed is why Asian countries do so well on tests is language. Many Asian countries have languages that are made up of thousands of symbols. The order of symbols and their placement in relation to each other acts a lot like an algorithm to change word meaning. Math symbolism may come easier to them because many of the learning techniques they use in language (memorization and word order) happen to be useful in math symbolic manipulation.
Another reason for the decline of education in the US is the abundance of supplies Americans have. Most poor families in the US still own a TV, have enough to eat and drink, and have access to shelter. To them physical survival is not really the issue.
In other countries the poor have much less. It is obvious that education creates opportunity to move up in social standing. People who are rich are educated, and vis-versa. To our uneducated and media brainwashed youth, learning seems a lot less appealing then waiting for your big chance and making it big. Their ignorance of probability aside, it is a matter of values. Whenever our youth have the good life already they take it for granted and don't look at the factors that may have bought their family to their current status. In China the children of the middle class are already refusing to work in school. In Japan there is the increasing problem of Japanese youth not learning the full Japanese language. In Japan this language is what all academic books are written in!
Teachers keep hope, math education in the world is not much better than in the US. We now have a choice: will we move forward by using technology in place of hand algorithms, or will we forbid it until students can do everything by hand? Will we move forward or backward?
Before I talk about why I want to define what I mean by the phrase "doing mathematics". Stephen Wolfram defines doing mathematics as a 4 step process ad I agree with him. Doing mathematics is...
1) Finding a problem in the real world.
2) translating the real world elements into the language of math using formulas and symbols.
3) Computing the answer.
4) Reinterpreting the answer's value back into the real world to help solve the problem.
I think we have lost sight of what mathematics is all about. We now have machines to free us from the drudgery of step 3. We no longer have to fixate on it as we did in the past. In the days of pencil and paper it was extremely important to focus on step 3. There were no machines to do step 3 and it was very difficult to compute answers. Procedures were made and proven to allow mankind to extend his ability to compute ever more difficult mathematics. Some Mathematicians have built their careers almost exclusively on creating and proving abstract computational procedures for step 3.
Mathematics has developed and grown as a field because it solves problems in the real world, and computing is just a part of that process. From here I will be talking about what I observe in the classroom and how it compares with "doing mathematics". I will also analyze some of the arguments used to perpetuate and justify the status-quo of mathematics: procedural step 3 thinking and technological obsolescence of content.
I want to point out that what I am advocating is that we switch how we handle and do step 3. Instead of using paper and pencil, we use computers to do the computing work. Every other step in doing mathematics will remain the same. That may mean instead of making the assignments like "solve for x" we ask questions with higher cognitive demand like "write a real world story problem that uses the equation below and a person would need to find x". The difference here is that to answer the second question, you really have to know what the symbols of math mean. Students often limp their way though the procedures of math by memorizing processes without attaching meaning to the procedure. A student may memorize the procedure of finding an average as "add all the numbers together then divide by how many numbers there are." That's great, but what good is it if they don't know that the average is the exact center or balancing point of a set of numbers? Faking math by memorizing a bunch of steps superficially is common crutch for students. Asking questions like the second one above ensure that the purpose and concept behind a math idea is learned because in order to answer the question you need to understand math, not just be good at memorization.
"What if the computer is wrong? How would anyone know?" is another excuse that is often given for the "reason" we need to do things by hand. This is a silly argument when you think about it. Why stop at computers? How can we trust your own hand calculations? How can we trust mathematicians? How can you trust the internet? How can you trust your eyes, you could be delusional couldn't you? You can be forever agnostic about everything if you want. The fact is, its easy to point to a time when computers failed while ignoring that they are thousands of times more accurate and faster than humans at the same computational tasks. If some person can find an error with Mathematica, does that mean all computers everywhere are dangerous and will destroy math? It doesn't seem to be true of hand algorithms. When we get those wrong we acknowledge it was a mistake and correct it. We don't abandon hand algorithms and move back to stones and abacuses for counting, so why would computers be any different?
"What if everyone just trusted computers all the time? That's dangerous to trust computers to such a high degree all the time. If we don't teach our students how to do things by hand how will they verify the accuracy of computers? People won't be able to think!" You might think I am crazy, but I have heard this argument more than once so I will address it. Forgive my sarcasm but in society we have books, a primitive form of data storage. Should all of the world fall into the chaos that will "surely" follow by trusting a computer to solve for X and it failing to do so, people could read these books. Some people even learn things like math by reading books alone. In fact, I think some people will want to learn the hand algorithms because they find them interesting.(Wouldn't teaching that class would be a math teacher's dream!?!) Unless all math books are destroyed these hand algorithms will not be lost.
I think its questions and lines of reasoning like the last one that really tell it like it is. People do not know the difference between knowledge that everyone should know and knowledge that only a few people need to know. The current math procedures could be taught in a math history class where the inventive and ingenious people who developed them would get the honor in history they have earned. Now most of their names are hardly mentioned because we are to busy drilling everyone on ALL the algorithms.
"The basics" in math is this undefined hazy ideal that seems to impede progress in math education. No one would EVER put up with in a class that was not so abstract. What would you say if you wanted to take a class to basic computer class and on the first day the teacher gave you some computer chips a soldering iron and began teaching circuit dynamics? In the afternoon you were taught binary and machine code. You wanted to learn how to use Microsoft Windows, but when you tell the teacher this he teacher assures you that in order to REALLY understand Windows you must first master machine code and later MS DOS, then you might be ready to learn Windows NT then Windows 2000, then Windows Vista, then Windows 7, but only if you get your doctorate. Would you pursue such a program or would you look at all the people you know, realize that none of them know machine code but still know enough about computers to use them and then drop out? This is currently what happens in mathematics education. Is it any wonder why our students don't want to learn math? THIS IS UNACCEPTABLE! No one would take a computer class like the one above unless it was a special course in computing history. So long as that was the only program available, those that did would probably make a lot of money, but their success wouldn't be because machine code was the bedrock of the industry or gave them new insight into programming. It would be because few people would have the patience to deal with that kind of program. Fewer people in the program means higher demand for qualified people and thus higher wages, but students learning of machine code and soldering could have been better spend on other more important concepts like object oriented programming, system design, program architecture, etc.
If this program were like math education today students would complete their mostly irrelevant coursework, the professors would see that the few students who completed the program were successful and confirm their beliefs about just how important it was for students to know "the basics" of computer design. Even a few of the students who liked their machine code classes would extol how useful it was that they learned the basics. Since their were no other programs out there that didn't obsess over the basics, there would be no way to tell just how useful the basics really were in creating competent computer operators. If no another program was skipping the basics to teach more useful and practical information would we assume that there was no other way to teach it as we currently do in mathematics education today? Is it really necessary to get good with a soldering iron and circuit boards for you to use a computer? Is this knowledge everyone needs to know or some people need to know?
So why all the confusion? Why is it that math never seems to move forward?
What's wrong with mathematics education is simple: everyone is totally obsessed and hung up on step 3. Since step 3 is taught almost exclusively in the classroom from day 1, often the people who get into math fall in love the the procedural elegance and exact nature of the subject. Even though we now have computers to to the drudgery of calculation, they still extol computation by hand as some kind of virtue, while ignoring the actual application of math to the real world by glossing over steps 1, 2, and 4. Simply put we have not moved math into the age of computers; We must teach all 4 steps of mathematics in balance. Bringing math into the computer age will also make math accessible to the general population, democratizing math and opening exciting possibilities. No longer would mathematical analysis be restricted to the few who learned hand procedures. Anyone who had an understanding of math terminology and relationships could analyze data or use data to make decisions.
The age of computers has made hand calculation obsolete. If you don't believe me go to wolframalpha.com and punch in an equation try "3x+2=8" Magic. Check out the introduction video here.
http://www.wolframalpha.com/screencast/introducingwolframalpha.html
It is amazing what will happen in the future! We could be awestruck by its possibilities or fearful what will math teachers do when free websites like Wolframalpha are available to them 24/7 on smartphones. In 10 years, most US citizens will have a smartphone, are we going to force more algorithms down the throats of people who don't want to learn them or will we strengthen the rigor of our math curricula and make mathematics accessible to people who don't have the patience to learn the computational end of math but thrive on steps 1, 2, and 4. Which is better for the economy: fear and no change or change and possibility?
Math is one of those subjects that people think does not change over time. I would agree that the theoretical foundations of math have not and may never change, but how we teach math, how we learn math, where math is applied, and technology are factors that do change the game of math education. Fortunately, technology has long changed the game for mathematics for scientists and engineers, but it has not for students. Technological advances are being made at an exponential rate and we need more mathematically competent people to operate and understand these complex machines.
Are we producing people fluent in the concepts of math? The answer is a resounding NO! From my experience, most people have no concept of math beyond it is working with numbers or doing problems like "solve for x", "simplify the expression" or "rationalize the denominator". You would think these procedures were all made of gold and absolutely necessary to understand math the way some math teachers and professors talk about them. "They need to know the basics" is appealing rhetoric, but are students able to do the basics?
A simple look at the NAEP, TIMMS or PISA (international math and reasoning tests) we see US students' scores doubled by their rivals in other countries. It's pretty obvious students are not getting the basics (or much else) out of math classrooms today. I don't think foreign born students are twice as smart as us or that some sort of genetic advantage evolved in the last 100 years. There is nothing wrong with the American student!!!! They have every right to be bored to tears with hand algorithms they are forced to learn in exchange for a letter on a sheet of paper. (A,B,C,D,F) Especially after they have been shown that a computer can do it in a second. Nothing highlights our dismal failure to advance math curricula forward like the poor scores we see in the US today.
The students we do produce are can barely compute all the rigorous procedures and algorithms we teach them at all much less explain why and where we would apply those procedures. Basically, we cram computation down their throats, claim is important to know, and punish them when they choke on it. We busily teach them that doing math like solving equations by hand is very important, but then after we spend hours and days in the classroom teaching them these pencil and paper procedures, we then show them how it can be done on a calculator in 20 seconds. After all that, its more than unfair to be expected to be taken seriously by students who rely on technology everyday. Shouldn't technology make these manual calculation units easier and shorter? Shouldn't the questions we pose to students be becoming more conceptual and real-world based as technology frees them from the drudgery of manual calculation? According to most mathematics educators today, the answer is no.
I will end with some good news. We may be behind as a country, but no one is where they need to be in this math debacle. Other countries are busy teaching their students the same "by hand" procedures as we are. Some like Japan do not allow calculators, and I have even seen Japanese students wasting hours working abacuses with lightning speed and precision. (apparently while calculators are cheating abacuses are not)
Why do other countries seem to destroy us in math? Its a good question that demands an answer. This could be a thesis, but I will briefly discuss some of the reasons. Most countries scores benefit from the educational systems they have in place that test students after elementary school and then place children in different educational "tracks". Often only the students who are on an academic track take the tests for the country. Students who are on a good track and like mathematics do not have to contend with the poor behavior of students who don't want to be there. This is a huge advantage. Another factor I feel is not often discussed is why Asian countries do so well on tests is language. Many Asian countries have languages that are made up of thousands of symbols. The order of symbols and their placement in relation to each other acts a lot like an algorithm to change word meaning. Math symbolism may come easier to them because many of the learning techniques they use in language (memorization and word order) happen to be useful in math symbolic manipulation.
Another reason for the decline of education in the US is the abundance of supplies Americans have. Most poor families in the US still own a TV, have enough to eat and drink, and have access to shelter. To them physical survival is not really the issue.
In other countries the poor have much less. It is obvious that education creates opportunity to move up in social standing. People who are rich are educated, and vis-versa. To our uneducated and media brainwashed youth, learning seems a lot less appealing then waiting for your big chance and making it big. Their ignorance of probability aside, it is a matter of values. Whenever our youth have the good life already they take it for granted and don't look at the factors that may have bought their family to their current status. In China the children of the middle class are already refusing to work in school. In Japan there is the increasing problem of Japanese youth not learning the full Japanese language. In Japan this language is what all academic books are written in!
Teachers keep hope, math education in the world is not much better than in the US. We now have a choice: will we move forward by using technology in place of hand algorithms, or will we forbid it until students can do everything by hand? Will we move forward or backward?
Wednesday, March 3, 2010
Dedication
I am dedicating this blog to all the hardworking teachers out there who work tirelessly to the bone just to get all that is expected of them done, particularly urban teachers. I am especially proud of those who not only get it all done, but still dedicate time to helping students by being innovative and creating meaningful learning experiences for their students. They work long hours for students who do not seem to have any intrinsic motivation at all. They live in frustration as students bash a subject they love but still push though it to teach it anyway; They want their students to learn. Most are forced by policy to teach some particular topic on a particular day like a robot by a district that 'knows best". All too often teachers work in an environment that feeds them disrespect from all sides: student, administrators and parents and punishes them when they choke on it. In some urban areas teachers have to all but beg a student to learn just one fact. (How good it feels to think you may have accomplished something) They are blamed for what others do or fail to do. They are underpaid, overworked and rarely have the support and materials they need. Public education is a sinking ship! To those who stay in it anyway bailing water to help our children, You are heroes!
I have been a math teacher for 7 years. I will likely keep teaching until this June because I lost faith in the systems ability to right itself from the inside. I don't believe anything short of gutting whats there and restructuring the way we all manage from superintendent down to teachers in the classroom will solve the problems our great nation faces. Education is a mess, until I feel like I can be productive as a teacher again I will not return to the profession. There are also other personal factors involved with my choice to leave like starting a family.
Teaching is the hardest job in the world!!! PERIOD!
One fellow at a bar I was visiting said "Teaching is easy! Once you get started you just have to dust off lesson plans and do the same thing every year until your 3 month vacation" People like him have no knowledge of the reality of any classroom. It kind of reminds of when Rush Limbaugh told a caller that the price of his recent doctor visits were less than the cost of an SUV and were perfectly reasonable. Hey Rush, $15,000 is a lot of money to OTHER people.
Attention teacher haters: I have a challenge for you! Name one other job where you are charged to manage a bunch of workers who don't get paid anything, are forced to go to work everyday, can't usually be fired, aren't usually screened before they are hired, and, worst of all, know if they don't work it will be you who is blamed for it. As a manager you can't use tough language or physical force to get them to work. You are also held accountable to at least three entities who often have conflicting interests. (peers, parents, administrators, district policy, state standards, district standards, etc.) In addition, you will have to pay for all of your training and must take college courses to keep your job.
Good luck!
The purpose of this blog is to create awareness of and spread ideas that educators may find insightful and helpful. To this end, most of my posts will be about or connected to at least one of the following ideas.
1) Choice Theory and the Deming management model must used in education.
2) Math must "grow up"! I mean that as a statement of where math is and also as a challenge to professors and others in the field to be aware of technologies ever-changing role in the process of mathematics. Currently we are teaching things "by hand" that students have no use for. They have calculators. Soon they will all have access to smartphones 24-7 and free websites like Wolframalpha.com (which can solve most math problems like a click of a button.) What will happen when a kid can just punch in "2x+3y=5 and 5x+3y^2=9" into a search bar and get an answer? Maybe we should spend our time on meaningful problems in the REAL world and leave raw, intense computing to... computers. Maybe that's why they call them computers, because they compute for you. Isn't that the point?
I will address these points in posts to come, but I hope you will find this blog insightful, witty and most of all I hope it give you faith to keep moving forward. You are not alone.
Keep the faith.
I have been a math teacher for 7 years. I will likely keep teaching until this June because I lost faith in the systems ability to right itself from the inside. I don't believe anything short of gutting whats there and restructuring the way we all manage from superintendent down to teachers in the classroom will solve the problems our great nation faces. Education is a mess, until I feel like I can be productive as a teacher again I will not return to the profession. There are also other personal factors involved with my choice to leave like starting a family.
Teaching is the hardest job in the world!!! PERIOD!
One fellow at a bar I was visiting said "Teaching is easy! Once you get started you just have to dust off lesson plans and do the same thing every year until your 3 month vacation" People like him have no knowledge of the reality of any classroom. It kind of reminds of when Rush Limbaugh told a caller that the price of his recent doctor visits were less than the cost of an SUV and were perfectly reasonable. Hey Rush, $15,000 is a lot of money to OTHER people.
Attention teacher haters: I have a challenge for you! Name one other job where you are charged to manage a bunch of workers who don't get paid anything, are forced to go to work everyday, can't usually be fired, aren't usually screened before they are hired, and, worst of all, know if they don't work it will be you who is blamed for it. As a manager you can't use tough language or physical force to get them to work. You are also held accountable to at least three entities who often have conflicting interests. (peers, parents, administrators, district policy, state standards, district standards, etc.) In addition, you will have to pay for all of your training and must take college courses to keep your job.
Good luck!
The purpose of this blog is to create awareness of and spread ideas that educators may find insightful and helpful. To this end, most of my posts will be about or connected to at least one of the following ideas.
1) Choice Theory and the Deming management model must used in education.
2) Math must "grow up"! I mean that as a statement of where math is and also as a challenge to professors and others in the field to be aware of technologies ever-changing role in the process of mathematics. Currently we are teaching things "by hand" that students have no use for. They have calculators. Soon they will all have access to smartphones 24-7 and free websites like Wolframalpha.com (which can solve most math problems like a click of a button.) What will happen when a kid can just punch in "2x+3y=5 and 5x+3y^2=9" into a search bar and get an answer? Maybe we should spend our time on meaningful problems in the REAL world and leave raw, intense computing to... computers. Maybe that's why they call them computers, because they compute for you. Isn't that the point?
I will address these points in posts to come, but I hope you will find this blog insightful, witty and most of all I hope it give you faith to keep moving forward. You are not alone.
Keep the faith.
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