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Category Archives: teaching

Fixing a dull ‘Personal Solar Telescope’

12 Friday Aug 2022

Posted by gfbrandenburg in astronomy, astrophysics, monochromatic, optical flat, Optics, science, teaching, Telescope Making

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ATM, Coronado, filter, Hydorgen-Alpha, Meade, Optics, PST, solar filter, Solar telescope, Telescope

A decade or so ago, I bought a brand-new Personal Solar Telescope from Hands On Optics. It was great! Not only could you see sunspots safely, but you could also make out prominences around the circumference of the sun, and if sky conditions were OK, you could make out plages, striations, and all sorts of other features on the Sun’s surface. If you were patient, you could tune the filters so that with the Doppler effect and the fact that many of the filaments and prominences are moving very quickly, you could make them appear and disappear as you changed the H-alpha frequency ever so slightly to one end of the spectrum to the other.

However, as the years went on, the Sun’s image got harder and harder to see. Finally I couldn’t see anything at all. And the Sun got quiet, so my PST just sat in its case, unused, for over a year. I was hoping it wasn’t my eyes!

I later found some information at Starry Nights on fixing the problem: one of the several filters (a ‘blocking’ or ‘ITF’ filter) not far in front of the eyepiece tends to get oxidized, and hence, opaque. I ordered a replacement from Meier at about $80, but was frankly rather apprehensive about figuring out how to do the actual deed. (Unfortunately they are now out of stock: https://maierphotonics.com/656bandpassfilter-1.aspx )

I finally found some threads on Starry Nights that explained more clearly what one was supposed to do ( https://www.cloudynights.com/topic/530890-newbie-trouble-with-coronado-pst/page-4 ) and with a pair of taped-up channel lock pliers and an old 3/4″ chisel that I ground down so that it would turn the threads on the retaining ring, I was able to remove the old filter and put in the new one. Here is a photo of the old filter (to the right, yellowish – blue) and the new one, which is so reflective you can see my red-and-blue cell phone with a fuzzy shiny Apple logo in the middle.

This afternoon, since for a change it wasn’t raining, I got to take it out and use it.

Verdict?

It works great again!

A Navigation / Geometry Problem

24 Sunday Oct 2021

Posted by gfbrandenburg in astronomy, History, Math, science, teaching

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Benjamin Banneker, District of Columbia, Pierre l"Enfant, Washington

I had the pleasure of helping lead a field trip for 9th grade Geometry students at School Without Walls SHS that we call ‘Math on the Mall’ assisting with two colleagues from the SWW math faculty.

One of our goals is for the students to see how beautifully and geometrically this city was laid out by Pierre l’Enfant, Andrew Ellicott, and Benjamin Banneker about 230 years ago.

While there are lots of myths written and repeated about Banneker’s actual contribution, the fact is that he was the astronomer, who was responsible for determining due north, exactly, and the exact latitude and longitude of the southern tip of the original 10-mile-square piece of land. With no Internet or SatNav or even a telegraph or steam engine, but with a very nice refractor and highly accurate clock that he was entrusted with, but with no landmarks to measure from, he was able to do so, in 1790.

I was sad to see that exactly none of the students know which way was north – in a city where the numbered streets near the Mall and the rest of DC’s historic downtown were almost all laid out perfectly north-south, and the streets whose names begin with letters or words like ‘Newark’, and the streets along the Mall, are all laid out perfectly east-west. Very few of them had ever seen the Milky Way, though most had heard of Polaris or the North star.

Hopefully they will remember that in the future as they do more navigation on their own in this great city.

I challenged them to try to figure out why the angle of elevation of the North Star is the same as their latitude. Here is a diagram illustrating the problem:

The Earth, Polaris, and You.

This diagram is intended to help you understand why the North Star’s elevation above your horizon always gives you your  latitude (if you live north of the Equator.

The big circle represents the Earth. The center of the earth is at E. The equator is AD.

YOU, the observer, are standing outside on a clear night. You see Polaris in the direction of ray BG. Line HCE is the Earth’s axis, and it also points at Polaris – which is so far away, and seems so tiny, but yet is also so large, that yes, parallel rays BG and CH do, for all practical purposes, point at the same point in the sky. Ray ED starts at the center of the Earth, passes through you at B, and goes on to the zenith (the part of the sky that is directly overhead). The horizon (BF) and the zenith (ray EB) are perpendicular. Also, line HCE (the earth’s axis) is perpendicular to its equator (segment AED).

Using some sort of angle measuring device, if you are out on the National Mall at night, you can very carefully measure the angle of elevation of the North Star above the local horizon, and you should ideally find that angle, FBG, is about 38.9 degrees, but we could also call it X degrees.

Prove (i.e. explain) why your latitude (which is angle AEB) measures the same as angle FBG.

What are the givens?

=========================================================

Full disclosure: My daughter graduated from SWW two decades ago, and I taught there as well for a year and for 10 years at a school that is now associated with it: Francis (then JHS now a middle school).

The kids were nice back then, and they still are. I thought the teachers did a great job.

This is a DC public high school that you have to apply to.

Benjamin Banneker was an amazing person. There are a lot of myths that have been attached to his work and accomplishments, which I am guessing might be because those people didn’t actually understand the math and astronomy that he did accomplish. The best book on him is by Silvio Bedini.

‘Math on the Mall’ was originated by Florence Fasanelli, Richard Thorington, and V. Frederick Rickey around 1990. I participated as a math teacher in a couple of those tours led by FF. Later, I wanted to take my students on a similar tour that would include a trip to see a number of the works of the geometer and artist Maurice C. Escher, and couldn’t find my copy of their work, so I made up my own, and added to it using the work of FF, RT, and VFR and suggestions from teachers and students. Later on, the Mathematical Association of America made something similar, which you can find here.

My version was on the website of the Carnegie Institution for Science for a number of years. See page 56 on this link. I need to find someone to cut out some of my excess verbiage and then trot it out to a publisher.

Can Mathematicians be Replaced by Computers?

30 Sunday Aug 2020

Posted by gfbrandenburg in education, Math, teaching

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artificial intelligence, computers, conjecture, induction, logic, mathematicians, mathematics, proof

The short answer is, certainly not yet.

Can they ever be? From reading this article and my own experience with the geometry-proving-and-construction software called Geometrix, written by my friend and colleague Jacques Gressier, I am not sure it’s possible at all.

Here is an interesting article that I’m copying and pasting from Jerry Becker at SIU, who got it from Quanta:

From Quanta, Thursday, August 27, 2020. SEE
https://www.quantamagazine.org/how-close-are-computers-to-automating-mathematical-reasoning-20200827/
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How Close Are Computers to Automating Mathematical Reasoning?
AI tools are shaping next-generation theorem provers, and with them the relationship between math and machine.
By Stephen Ornes
In the 1970s, the late mathematician Paul Cohen, the only person to ever win a Fields Medal for work in mathematical logic, reportedly made a sweeping prediction that continues to excite and irritate mathematicians – that “at some unspecified future time, mathematicians would be replaced by computers.” Cohen, legendary for his daring methods in set theory, predicted that all of mathematics could be automated, including the writing of proofs.

A proof is a step-by-step logical argument that verifies the truth of a conjecture, or a mathematical proposition. (Once it’s proved, a conjecture becomes a theorem.) It both establishes the validity of a statement and explains why it’s true. A proof is strange, though. It’s abstract and untethered to material experience. “They’re this crazy contact between an imaginary, nonphysical world and biologically evolved creatures,” said the cognitive scientist Simon DeDeo of Carnegie Mellon University, who studies mathematical certainty by analyzing the structure of proofs. “We did not evolve to do this.”

Computers are useful for big calculations, but proofs require something different. Conjectures arise from inductive reasoning – a kind of intuition about an interesting problem – and proofs generally follow deductive, step-by-step logic. They often require complicated creative thinking as well as the more laborious work of filling in the gaps, and machines can’t achieve this combination.

Computerized theorem provers can be broken down into two categories. Automated theorem provers, or ATPs, typically use brute-force methods to crunch through big calculations. Interactive theorem provers, or ITPs, act as proof assistants that can verify the accuracy of an argument and check existing proofs for errors. But these two strategies, even when combined (as is the case with newer theorem provers), don’t add up to automated reasoning.

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SIDEBAR PHOTO:  Simon DeDeo of Carnegie Mellon helped show that people and machines seem to construct mathematical proofs in similar ways. Courtesy of Simon DeDeo
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Plus, the tools haven’t been met with open arms, and the majority of mathematicians don’t use or welcome them. “They’re very controversial for mathematicians,” DeDeo said. “Most of them don’t like the idea.”

A formidable open challenge in the field asks how much proof-making can actually be automated: Can a system generate an interesting conjecture and prove it in a way that people understand? A slew of recent advances from labs around the world suggests ways that artificial intelligence tools may answer that question. Josef Urban at the Czech Institute of Informatics, Robotics and Cybernetics in Prague is exploring a variety of approaches that use machine learning to boost the efficiency and performance of existing provers. In July, his group reported a set of original conjectures and proofs generated and verified by machines. And in June, a group at Google Research led by Christian Szegedy posted recent results from efforts to harness the strengths of natural language processing to make computer proofs more human-seeming in structure and explanation.

Some mathematicians see theorem provers as a potentially game-changing tool for training undergraduates in proof writing. Others say that getting computers to write proofs is unnecessary for advancing mathematics and probably impossible. But a system that can predict a useful conjecture and prove a new theorem will achieve something new –  some machine version of understanding, Szegedy said. And that suggests the possibility of automating reason itself.
Useful Machines

Mathematicians, logicians and philosophers have long argued over what part of creating proofs is fundamentally human, and debates about mechanized mathematics continue today, especially in the deep valleys connecting computer science and pure mathematics.

For computer scientists, theorem provers are not controversial. They offer a rigorous way to verify that a program works, and arguments about intuition and creativity are less important than finding an efficient way to solve a problem. At the Massachusetts Institute of Technology, for example, the computer scientist Adam Chlipala has designed theorem-proving tools that generate cryptographic algorithms – traditionally written by humans – to safeguard internet transactions. Already, his group’s code is used for the majority of the communication on Google’s Chrome browser.

——————-
SIDEBAR PHOTO: Emily Riehl of Johns Hopkins University uses theorem provers in teaching students and proof assistants in her own work. “Using a proof assistant has changed the way I think about writing proofs,” she said. Will Kirk/Johns Hopkins University
———————-
“You can take any kind of mathematical argument and code it with one tool, and connect your arguments together to create proofs of security,” Chlipala said.

In math, theorem provers have helped produce complicated, calculation-heavy proofs that otherwise would have occupied hundreds of years of mathematicians’ lives. The Kepler conjecture, which describes the best way to stack spheres (or, historically, oranges or cannonballs), offers a telling example. In 1998, Thomas Hales, together with his student Sam Ferguson, completed a proof using a variety of computerized math techniques. The result was so cumbersome – the results took up 3 gigabytes – that 12 mathematicians analyzed it for years before announcing they were 99% certain it was correct.

The Kepler conjecture isn’t the only famous question to be solved by machines. The four-color theorem, which says you only need four hues to color any two-dimensional map so that no two adjoining regions share a color, was settled in 1977 by mathematicians using a computer program that churned through five-colored maps to show they could all be reduced to four. And in 2016, a trio of mathematicians used a computer program to prove a longstanding open challenge called the Boolean Pythagorean triples problem, but the initial version of the proof was 200 terabytes in size. With a high-speed internet connection, a person could download it in a little over three weeks.

Complicated Feelings
These examples are often trumpeted as successes, but they’ve also added to the debate. The computer code proving the four-color theorem, which was settled more than 40 years ago, was impossible for humans to check on their own. “Mathematicians have been arguing ever since whether or not it’s a proof,” said the mathematician Michael Harris of Columbia University.
———————-
SIDEBAR PHOTO:  Many mathematicians, like Columbia University’s Michael Harris, disagree with the idea that computerized theorem provers are necessary – or that they’ll make human mathematicians obsolete. Béatrice Antolin
———————–
Another gripe is that if they want to use theorem provers, mathematicians must first learn to code and then figure out how to express their problem in computer-friendly language – activities that detract from the act of doing math. “By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself,” Harris said.

Many just don’t see a need for theorem solvers in their work. “They have a system, and it’s pencil and paper, and it works,” said Kevin Buzzard, a mathematician at Imperial College London who three years ago pivoted his work from pure math to focus on theorem provers and formal proofs. “Computers have done amazing calculations for us, but they have never solved a hard problem on their own,” he said. “Until they do, mathematicians aren’t going to be buying into this stuff.”

But Buzzard and others think maybe they should. For one thing, “computer proofs may not be as alien as we think,” DeDeo said. Recently, together with Scott Viteri, a computer scientist now at Stanford University, he reverse-engineered a handful of famous canonical proofs (including one from Euclid’s Elements) and dozens of machine-generated proofs, written using a theorem prover called Coq, to look for commonalities. They found that the networked structure of machine proofs was remarkably similar to the structure of proofs made by people. That shared trait, he said, may help researchers find a way to get proof assistants to, in some sense, explain themselves.
“Machine proofs may not be as mysterious as they appear,” DeDeo said.

Others say theorem provers can be useful teaching tools, in both computer science and mathematics. At Johns Hopkins University, the mathematician Emily Riehl has developed courses in which students write proofs using a theorem prover. “It forces you to be very organized and think clearly,” she said. “Students who write proofs for the first time can have trouble knowing what they need and understanding the logical structure.”

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SIDEBAR:  By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself.  Michael Harris, Columbia University
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Riehl also says that she’s been increasingly using theorem provers in her own work. “It’s not necessarily something you have to use all the time, and will never substitute for scribbling on a piece of paper,” she said, “but using a proof assistant has changed the way I think about writing proofs.”

Theorem provers also offer a way to keep the field honest. In 1999, the Russian American mathematician Vladimir Voevodsky discovered an error in one of his proofs. From then until his death in 2017, he was a vocal proponent of using computers to check proofs. Hales said that he and Ferguson found hundreds of errors in their original proof when they checked it with computers. Even the very first proposition in Euclid’s Elements isn’t perfect. If a machine can help mathematicians avoid such mistakes, why not take advantage of it? (The practical objection, justified or not, is the one suggested by Harris: If mathematicians have to spend their time formalizing math to be understood by a computer, that’s time they’re not spending doing new math.)

But Timothy Gowers, a mathematician and Fields medalist at the University of Cambridge, wants to go even further: He envisions a future in which theorem provers replace human referees at major journals. “I can see it becoming standard practice that if you want your paper to be accepted, you have to get it past an automatic checker,” he said.

————-
But before computers can universally check or even devise proofs, researchers first have to clear a significant hurdle: the communication barrier between the language of humans and the language of computers.

Today’s theorem provers weren’t designed to be mathematician-friendly. ATPs, the first type, are generally used to check if a statement is correct, often by testing possible cases. Ask an ATP to verify that a person can drive from Miami to Seattle, for example, and it might search all cities connected by roads leading away from Miami and eventually finding a city with a road leading into Seattle.

———————–
SIDEBAR PHOTO: Not every mathematician hates theorem provers. Timothy Gowers, of the University of Cambridge, thinks they may one day replace human reviewers at mathematical journals. The Abel Prize
———————-
With an ATP, a programmer can code in all the rules, or axioms, and then ask if a particular conjecture follows those rules. The computer then does all the work. “You just type in the conjecture you want to prove, and you hope you get an answer,” said Daniel Huang, a computer scientist who recently left the University of California, Berkeley, to work at a startup.

But here’s the rub: What an ATP doesn’t do is explain its work. All that calculating happens within the machine, and to human eyes it would look like a long string of 0s and 1s. Huang said it’s impossible to scan the proof and follow the reasoning, because it looks like a pile of random data. “No human will ever look at that proof and be able to say, ‘I get it,'” he said.

ITPs, the second category, have vast data sets containing up to tens of thousands of theorems and proofs, which they can scan to verify that a proof is accurate. Unlike ATPs, which operate in a kind of black box and just spit out an answer, ITPs require human interaction and even guidance along the way, so they’re not as inaccessible. “A human could sit down and understand what the proof-level techniques are,” said Huang. (These are the kinds of machine proofs DeDeo and Viteri studied.)
ITPs have become increasingly popular in recent years. In 2017, the trio behind the Boolean Pythagorean triples problem used Coq, an ITP, to create and verify a formal version of their proof; in 2005 Georges Gonthier at Microsoft Research Cambridge used Coq to formalize the four-color theorem. Hales also used ITPs called HOL Light and Isabelle on the formal proof of the Kepler conjecture. (“HOL” stands for “higher-order logic.”)

Efforts at the forefront of the field today aim to blend learning with reasoning. They often combine ATPs with ITPs and also integrate machine learning tools to improve the efficiency of both. They envision ATP/ITP programs that can use deductive reasoning – and even communicate mathematical ideas – the same way people do, or at least in similar ways.

The Limits of Reason

Josef Urban thinks that the marriage of deductive and inductive reasoning required for proofs can be achieved through this kind of combined approach. His group has built theorem provers guided by machine learning tools, which allow computers to learn on their own through experience. Over the last few years, they’ve explored the use of neural networks – layers of computations that help machines process information through a rough approximation of our brain’s neuronal activity. In July, his group reported on new conjectures generated by a neural network trained on theorem-proving data.

Urban was partially inspired by Andrej Karpathy, who a few years ago trained a neural network to generate mathematical-looking nonsense that looked legitimate to nonexperts. Urban didn’t want nonsense, though – he and his group instead designed their own tool to find new proofs after training on millions of theorems. Then they used the network to generate new conjectures and checked the validity of those conjectures using an ATP called E.

The network proposed more than 50,000 new formulas, though tens of thousands were duplicates. “It seems that we are not yet capable of proving the more interesting conjectures,” Urban said.
————————–

SIDEBAR: [Machines] will learn how to do their own prompts.  Timothy Gowers, University of Cambridge
—————————-
Szegedy at Google Research sees the challenge of automating reasoning in computer proofs as a subset of a much bigger field: natural language processing, which involves pattern recognition in the usage of words and sentences. (Pattern recognition is also the driving idea behind computer vision, the object of Szegedy’s previous project at Google.) Like other groups, his team wants theorem provers that can find and explain useful proofs.

Inspired by the rapid development of AI tools like AlphaZero – the DeepMind program that can defeat humans at chess, Go and shogi – Szegedy’s group wants to capitalize on recent advances in language recognition to write proofs. Language models, he said, can demonstrate surprisingly solid mathematical reasoning.

His group at Google Research recently described a way to use language models – which often use neural networks – to generate new proofs. After training the model to recognize a kind of treelike structure in theorems that are known to be true, they ran a kind of free-form experiment, simply asking the network to generate and prove a theorem without any further guidance. Of the thousands of generated conjectures, about 13% were both provable and new (meaning they didn’t duplicate other theorems in the database). The experiment, he said, suggests that the neural net could teach itself a kind of understanding of what a proof looks like.

“Neural networks are able to develop an artificial style of intuition,” Szegedy said.

Of course, it’s still unclear whether these efforts will fulfill Cohen’s prophecy from over 40 years ago. Gowers has said that he thinks computers will be able to out-reason mathematicians by 2099. At first, he predicts, mathematicians will enjoy a kind of golden age, “when mathematicians do all the fun parts and computers do all the boring parts. But I think it will last a very short time.”
———————–
Related:
.  Machines Beat Humans on a Reading Test. But Do They Understand?  —  https://www.quantamagazine.org/machines-beat-humans-on-a-reading-test-but-do-they-understand-20191017/
.  Will Computers Redefine the Roots of Math?  —  https://www.quantamagazine.org/univalent-foundations-redefines-mathematics-20150519/
.  Symbolic Mathematics Finally Yields to Neural Networks  —  https://www.quantamagazine.org/symbolic-mathematics-finally-yields-to-neural-networks-20200520/
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After all, if the machines continue to improve, and they have access to vast amounts of data, they should become very good at doing the fun parts, too. “They will learn how to do their own prompts,” Gowers said.

Harris disagrees. He doesn’t think computer provers are necessary, or that they will inevitably “make human mathematicians obsolete.” If computer scientists are ever able to program a kind of synthetic intuition, he says, it still won’t rival that of humans. “Even if computers understand, they don’t understand in a human way.”

*********************************************
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A neat geometry lesson! And a rant…

13 Thursday Feb 2020

Posted by gfbrandenburg in education, flat, History, Math, Optics, teaching, Telescope Making, Uncategorized

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apps, computer, computer-managed instruction, geometry, kaleidoscope, Math, Mirror, programs, reflection, school

Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!

Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?

This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)

I am reproducing a couple of the images and text that Angel Gilding provides on their website (which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon)).

At 72º you see 4 complete reflections.

When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.

 

https://angelgilding.com/multiple-reflections/

===========

Rant, in the form of a long footnote:

* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.

Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.

I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill  to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.

Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).

By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.

I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.

Silvering Mirrors, and More, at Stellafane

05 Monday Aug 2019

Posted by gfbrandenburg in astronomy, flat, History, Math, monochromatic, optical flat, Optics, science, teaching, Telescope Making, Uncategorized

≈ 3 Comments

For me, these were the two most significant demos at the 2019 Stellafane Convention in Springfield, Vermont:

(1) Silvering large mirrors, no vacuum needed

We had a demonstration by Peter Pekurar on how to apply a layer of Silver (metallic Ag, not aluminum) onto a telescope mirror, accurately, with a protective, non-tarnishing overcoat, that works well. I looked through such a scope; the view was quite good, and I was told that interferograms are great also.

What’s more, the process involves overcoating a mirror with spray bottles of the reagents, without any vacuum apparatus needed at all. Note: Silver coated, not aluminum coated. This is big for me because the upper limit at our club’s aluminizer is 12.5″, but some of us are working on larger mirrors than that; commercial coaters currently charge many hundreds of dollars to coat them.

You can find information on some of these materials at Angel Gilding. Peter P said he will have an article out in not too long. Here are a few photos and videos of the process:

IMG_4972

Finished mirror; notice it’s a little blotchy

 

 

IMG_4985IMG_4987

(2) Demo and links for Bath Interferometer (see http://gr5.org/bath )

How to set up and use a Bath interferometer to produce highly accurate interferograms of any mirror for many orders of magnitude less cash than a Zygo interferometer. As I wrote earlier, Alan Tarica had taken the lead on fabricating one at the CCCC – NCA ATM workshop, and we eventually got it to work, but found it rather frustrating and fiddly to use.

The presenter is a HS teacher, and it shows: he explains things very clearly! On his website ( http://gr5.org/bath ) you can get plans for 3-D printing the parts for the Bath device, if you have any access to a 3-D printer, so you can print the parts out for yourself. He also has links to vendors that are selling parts for it, such as certain small lenses, mirrors and beam splitters. He shows you where you can get them for very little money from Surplus Shed and such places. Or you can purchase his really inexpensive kits that he’s already 3-D printed for you. Plus parts for an XYZ stage, which you will need for fine focus. The whole setup (not counting mirror stand and two tripods, which he assumes you have access to already) is under $130.

I will need to look carefully at our setup as built almost completely by Alan, and see how it differs and what we would need to do to make it better. The problem is that there are lots of little, tiny parts, and many of them need to be adjustable. We saw him doing LOTS of little adjustments!

Before his talk, I had absolutely no idea how this (or similar interformeters) really worked. Now I understand: the interference fringes that we see are really contour lines – like we see on on a USGS topo map, only with the mirror tilted in one direction or the other. A big difference with the USGS topo map is that there, the contour lines (isohypses – a new word for me today) are often 10 feet to 100 meters apart. In interferometry, the contour intervals are either one or one-half lambda (wavelength of light) apart – a really tiny amount! We need that level of accuracy because the surface we are studying is sooooooo flat that no other measuring system can work. His explanation of this whole thing now makes perfect sense to me. And the purpose of the software (free!) is to un-slant the mirror and re-draw it using the countour-line information.

Beautifully clear explanation!

Caution: a friend who works professionally in optics told me his team had made three Bath interferometers, using cheap but good quality ebay xyz stages, and found that they were just too much trouble; so they borrowed a very expensive commercial interferometer (costing many tens of kilobucks) from another department and are using that instead. I’m not selling my house to get a Zygo interferometer!!! But I will try the Bath interferometer instead.

 

 

Why Not Show Students the Beauty of Math?

16 Tuesday Oct 2018

Posted by gfbrandenburg in education, Math, teaching

≈ Leave a comment

Tags

algebra 2, algebra two, beauty, benoit mandelbrot, complex numbers, education, imaginary numbers, julia set, mandelbrot set, Math, strange, weird

When I taught math, I tried to get students to see both the usefulness and beauty of whatever topic we were discussing. The most beautiful mathematical objects I know of are the Mandelbrot and Julia sets, which in my opinion should be brought up whenever one is studying imaginary and complex numbers.

To illustrate what I mean, here are some blown up pieces of the Mandelbrot set. Below,  I’ll explain the very simple algebra that goes into making it.

 

I made these images using an app called FastFractal on my iPhone. The math goes like this:

Normally, you can’t take the square root of a negative number. But let’s pretend that you can, and that the square root of negative one is the imaginary number i. So the square root of -16 is 4i. Furthermore, we can invent complex numbers that have a real part like 2, or 3.1416, or -25/17, or anything else, and an imaginary part like 3i or -0.25i. So 2-3i is a complex number.

Ok so far?

We can add, subtract, multiply and divide real, imaginary and complex numbers if we want, just remembering that we need to add and subtract like terms, so 4+3i cannot be simplified to 7i; it’s already as simple as it gets. Remember that i multiplied by i gives you negative one!

Interesting fact: if you multiply a complex number (say, 4+3i) by its conjugate (namely 4-3i) you get a strictly REAL answer: 25! (Try it, using FOIL if you need to, and remember that i*i=-1!)

Furthermore, let us now pretend that we can place complex numbers on something that looks just like the familiar x-y coordinate plane, only now the x-axis becomes the real axis and the y-axis becomes the imaginary axis. So our complex number 4+3i is located where the Cartesian point (4, 3) would be.

Ok — but what’s the connection to those pretty pictures?

It’s coming, I promise!

Here’s the connection: take any point on the complex plane, in other words, any complex number you wish. Call it z. Then:

(1) Square it.

(2) Add the original complex number z to that result.

(3) See how far the result is from the origin.

(4) Repeat steps 1 – 3 a whole lot of times, always adding the original z.

One of two things will happen:

(A) your result stays close to the origin, OR

(B) it will go far, far away from the origin.

If it stays close to the origin, color the original point black.

If it gets far away, pick some other color.

Then repeat steps 1-4 for the point “right next” to your original complex point z. (Obviously, the phrase “right next to” depends on the scale you are using for your graph, but you probably want fine coverage.)

When you are done, print your picture!

If we start with 4+3i, after one round I get 11+27i. After two rounds I get -604 + 597i, which is very far from the origin, so I’m going to stop here and color it blue. I’ll also decide that every time a result gets into the hundreds after merely two rounds, that point will also be blue.

Now let’s try a complex point much closer to the origin: how about 0.2+0.4i? I tried that a bunch of times and the result seems to converge on about 0.024+0.420i — so I’ll color that point black.

This whole process would of course be very, very tedious to do by hand, but it’s pretty easy to program a graphing calculator to do this for you.

When Benoit Mandelbrot and others first did this set of computations in 1978-1980, and printed the results, they were amazed at its complexity and strange beauty: the border between the points we color black and those we color otherwise is unbelievably complicated, even when you zoom in really, really close. Who woulda thunk that a simple operation with complex numbers, that any high school student in Algebra 2 can do and perform, could produce something so beautiful and weird?

So, why not take a little time in Algebra 2 and have students explore the Mandelbrot set and it’s sister the Julia set? They might just get the idea that math is beautiful!!!

IMG_1735

Math – How Come We Forget So Much of What We Learned in School?

27 Monday Aug 2018

Posted by gfbrandenburg in astrophysics, education, History, Math, science, teaching, Telescope Making, Uncategorized

≈ Leave a comment

Tags

education, engineering, forgetting, France, mathematics, scientists, USA

This was a question on  Quora. Here is an answer I wrote:

In the US, judging strictly on what I’ve seen from my time in the classroom as both a student, a teacher, and a visiting mentor of other math teachers, I find that math and science was very often taught as sort of cookbook recipes without any real depth of understanding. The recent National Council of Teachers of Mathematics prescriptions have attempted to correct that, but results have been mixed, and the Common Core has ironically fostered a weird mix of conceptual math marred by teachers being *OBLIGATED* to follow a script, word-for-word, if they want to remain employed. Obviously, if students are really trying to understand WHY a certain mathematical or scientific thing/fact/theorem/theory/law is true, they are going to have questions, and it’s obviously the teacher’s job to figure out how best to answer said questions — which are not likely to have pre-formulated scripts to follow in case they come up — and which are going to take time.

Another thing that is true is that not everything in mathematics has real-world applications in every single person’s life. I taught a good bit of computer programming (aka ‘coding’ today), geometry, arithmetic, probability, algebra, statistics, and conic sections, and in fact I use a LOT of that every week fabricating telescope mirrors to amazing levels of precision, by hand, not for a living, but because I find telescope-making to be a lot of fun and good mental, aesthetic, manual, and physical exercise. But I’m a pretty rare exception!

Most people obviously don’t dabble in math and physics and optics like I do, nor should they!

In fact, I have made it a point to ask professional scientists and engineers that I meet if they actually use, on their jobs, all the calculus that they learned back in HS and college. So far, I think my count is several dozen “Noes” and only one definite “Yes” – and the latter was an actual rocket scientist / engineer and MIT grad and pro-am astronomer (and wonderful, funny, smart person) who deals/dealt with orbital rocket trajectories. (IIRC).

In France, when I went to school there 50 years ago and in my experience tutoring some kids at the fully-French Lycee Rochambeau near Washington, DC, is that they go very deeply into various topics in math, and the sequence of topics is very carefully thought out for each year for each kid in the entire nation (with varying levels of depth depending on what sort of track that the students elected to go into (say, languages/literature, pure math, or applied sciences, etc), but the kids were essentially obligated to accept certain ideas as factual givens and then work out more and more difficult problems that dealt with those particular givens. No questions allowed on where the givens came from, except to note the name of the long-dead classical Greek, French, Italian or German savant whose name is associated with it.

As an American kid who was mostly taught in American schools, but who also took 2 full years of the French system (half a year each of neuvieme, septieme, premiere, terminale, and then passed the baccalaureat in what they called at the time mathematiques elementaires, I found the choice of topics [eg ‘casting out nines’ and barycenters and non-orthogonal coordinate systems] in France rather strange. Interesting topics perhaps, but strange. And not necessarily any more related to the real world than what we teach here in the US.

Over in France, however, intellectuals are (mostly) respected, even revered, and of all the various academic strands, pure math has the highest level of respect. So people over there tend to be proud of however far they got in mathematics, and what they remember. Discourse in French tends to be extremely logical and clear in a way that I cannot imagine happening here in the public sphere.

So to sum up:

(a) most people never learned all that much math better than what was required to pass the test;

(b) only a very few geeky students like myself were motivated to ask ‘why’;

(c) most people don’t use all that much math in their real lives in the first place.

 

 

Different Ways to Teach Math

16 Friday Sep 2016

Posted by gfbrandenburg in education, Math, teaching

≈ Leave a comment

Tags

discovery, education, Math, school, teaching

I recommend looking at different ways to teach mathematics. Here is one take on the topic, from our friends up north. I reprinted this on my mostly-education blog, here.

https://gfbrandenburg.wordpress.com/2016/09/15/discovery-math-is-weird-but-a-good-idea-nonetheless/

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