## Monday, June 27, 2011

### A Blog From the Math Side...

I am happy to report that John Benson and P.J. Karafiol,two of the great math educators in the country, have a blog up and running at http://www.anglesofreflection.blogspot.com/. Check it out for numerous bits of advice, anecdotes, and interesting problems to work on. I highly recommend it for all teachers and those who enjoy mathematics!

Labels:
Benson and Karafiol,
education,
good teaching,
math blog,
math education

## Wednesday, June 22, 2011

### Trying to Make Some Sense About Quantum Mechanics

Gaining any level of understanding of quantum mechanics is one of the great intellectual challenges in science. In a quantum world of indeterminism and probability, uncertainty and fuzziness, phenomena completely unseen in our everyday lives are the norm for atoms and particles.

At the center of the strangeness is particle-wave duality, the notion that particles can at times act like ‘solid’ balls, but in different circumstances can behave like a wave. Likewise, something we normally think of as a wave, such as light, can certainly act like a wave under certain conditions, but in quantum mechanics light can also behave like particles we refer to as photons. In fact, a favorite question I pose to students is, ‘When light is traveling from a light bulb to your eye, is it a particle or wave?’ Ultimately, someone will offer the answer, ‘It is both!’ That is an acceptable answer; but what does this mean? How can an ‘object’ be two things simultaneously, which is what the answer ‘both’ implies.

No one is comfortable with this answer, and yet it fits in with the foundational principles of quantum mechanics. The reason is, in the mathematics of quantum mechanics, objects are described with a wave function. This is a mathematical function that encompasses possible states the object can take. So a photon that is moving through space can be thought of as a combination of two states, something like

But I think most of us still come back to the same questions: How do we interpret this mathematical nonsense? What does this mean for the object? This is where an analogy comes in handy, that will perhaps put this probabilistic concept into a more understandable context.

If I am talking about this in a class, I ask students to look around at each other and identify the personality snapshot of each of their classmates. This means to identify who is happy, sad, confused, angry, sarcastic, sleepy, bored, or anything else. So while there are numerous possible ‘personality states’ any person can have, while observing a person we can select one personality state at that time because we are interacting with them. However, what do we do when the bell rings and everyone goes on their way? If I ask someone to identify which personality state a specific person is in when they are no longer available for observation or interaction, what is the answer? The best we can do is to effectively guess…but to do this mathematically, we would acknowledge that at any given moment when a person is not being observed in any way, we cannot be certain about the personality state and can only try to identify the probability of that person being in each state. Perhaps there is a 20% chance she is happy, and 25% chance of being sad, and so on for each possible personality state.

This is the way we think about particles and waves when those entities are not being observed. When we do observe the entity, the act of observing selects out the personality from the mix of possible personalities. Another way of saying it is the experiment we do selects out a single observable state that we then identify. For a person, maybe it is the ‘happy’ state that becomes crystallized out of the ‘personality state’ function that includes all the possible personality states. For an electron, if we put it through a diffraction grating the wave personality is selected, whereas if we shoot it at an atom and it is deflected, the particle personality was selected instead.

Thinking this way is not necessarily normal, obvious or instinctive, but it is something we can try to understand the way the quantum world works. Of course, in real quantum mechanical problems, the mathematics becomes very hard very fast, but trying to find more concrete ways of thinking about the consequences of probabilistic concepts can only help the student to whom this is all new.

At the center of the strangeness is particle-wave duality, the notion that particles can at times act like ‘solid’ balls, but in different circumstances can behave like a wave. Likewise, something we normally think of as a wave, such as light, can certainly act like a wave under certain conditions, but in quantum mechanics light can also behave like particles we refer to as photons. In fact, a favorite question I pose to students is, ‘When light is traveling from a light bulb to your eye, is it a particle or wave?’ Ultimately, someone will offer the answer, ‘It is both!’ That is an acceptable answer; but what does this mean? How can an ‘object’ be two things simultaneously, which is what the answer ‘both’ implies.

No one is comfortable with this answer, and yet it fits in with the foundational principles of quantum mechanics. The reason is, in the mathematics of quantum mechanics, objects are described with a wave function. This is a mathematical function that encompasses possible states the object can take. So a photon that is moving through space can be thought of as a combination of two states, something like

*Photon = [particle state] + [wave state]*. More specifically, this function can be used to determine the probability of finding the photon in a particle or wave state.But I think most of us still come back to the same questions: How do we interpret this mathematical nonsense? What does this mean for the object? This is where an analogy comes in handy, that will perhaps put this probabilistic concept into a more understandable context.

If I am talking about this in a class, I ask students to look around at each other and identify the personality snapshot of each of their classmates. This means to identify who is happy, sad, confused, angry, sarcastic, sleepy, bored, or anything else. So while there are numerous possible ‘personality states’ any person can have, while observing a person we can select one personality state at that time because we are interacting with them. However, what do we do when the bell rings and everyone goes on their way? If I ask someone to identify which personality state a specific person is in when they are no longer available for observation or interaction, what is the answer? The best we can do is to effectively guess…but to do this mathematically, we would acknowledge that at any given moment when a person is not being observed in any way, we cannot be certain about the personality state and can only try to identify the probability of that person being in each state. Perhaps there is a 20% chance she is happy, and 25% chance of being sad, and so on for each possible personality state.

This is the way we think about particles and waves when those entities are not being observed. When we do observe the entity, the act of observing selects out the personality from the mix of possible personalities. Another way of saying it is the experiment we do selects out a single observable state that we then identify. For a person, maybe it is the ‘happy’ state that becomes crystallized out of the ‘personality state’ function that includes all the possible personality states. For an electron, if we put it through a diffraction grating the wave personality is selected, whereas if we shoot it at an atom and it is deflected, the particle personality was selected instead.

Thinking this way is not necessarily normal, obvious or instinctive, but it is something we can try to understand the way the quantum world works. Of course, in real quantum mechanical problems, the mathematics becomes very hard very fast, but trying to find more concrete ways of thinking about the consequences of probabilistic concepts can only help the student to whom this is all new.

## Thursday, June 9, 2011

### Deep Thoughts by Dr. Tae - Can Skateboarding Save Education?

I wanted to share this TEDx talk by Dr. Tae Kim, after he sent me the link to it. Dr. Tae is likely the type of educator and scientist most people would label as one who 'thinks outside the box.' I still remember the fun we had when he came over to ETHS one day, while he was at Northwestern, to talk about school, education, physics, and whatever else that would come up in the conversation. We shared many of the same thoughts and ideas about where schools and education should go, but just said it in different ways, it turns out.

In this video, Tae uses skateboarding, which he, seriously, is addicted to and is a master, and more precisely how one learns when trying to do a new trick, to get into how schools set up its environment for learning. The

In this video, Tae uses skateboarding, which he, seriously, is addicted to and is a master, and more precisely how one learns when trying to do a new trick, to get into how schools set up its environment for learning. The

*process*of learning skateboarding on the street or in the park looks very different from the process of learning in school. The*environment*for learning skateboarding is very different from the environment of learning in school. The reason(s) for learning are different, as well, in these realms. I won't say much more here, but rather let you watch Dr. Tae in action and form your own opinions about his points on education and school and skateboarding.
Labels:
Dr. Tae,
education,
how should we do school?,
schools,
skateboarding

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