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Suggesting Rudin to an engineering or a programming student not already mathematically inclined is like offering a text in metallurgy or forest management to an apprentice lumberjack. They will learn nothing and give up in frustration almost immediately. Stewart calculus is plenty, and in some ways overkill.
I disagree. There is no point in doing anything if you're not trying to do it right.
Rudin is fundamental, which I find to be the only important thing. He is indeed difficult, but requires no prior knowledge of advanced math.
Maybe I'm wrong, but I think it's worth trying.
In response to
Rewiring my Brain: (gentle) Help Appreciated
Check out Tricki.
It's a repository of useful mathematical techniques. From my experience, many skills can be developed through practice.
For calculus I strongly recommend Rudin. Reading the book (~ half of it) line by line and doing the great exercises was very difficult but gave me a real insight into calculus and mathematical thinking in general.
In response to
Estimation as a game
I have questions. How can one get feedback on the accuracy of the estimates? Can one get feedback on the accuracy of the estimates? Is there value in the practice without feedback?
That's a good question.
Many estimates can be easily checked when you have access to a data source (encyclopedia or the Internet), e.g. object heights, distances, populations etc.
Other estimates are more complicated to check (e.g. probabilities). In that case you can attempt to estimate the same thing using different techniques. This is useful for debugging and may give a general idea of your accuracy (if 3 independent estimates are close to one another, you are likely not mistaken by too much).
Also, its easier when a few people independently estimate the same thing. You can compare your results, discuss the intermediate steps and find errors. This is a great feedback, from my experience.
Is there value in the practice without feedback?
I believe there is. It's valuable as a game and simply as training. Also, sometimes any estimate is better than nothing.
In response to
Estimation as a game
This is a game I play often when it comes to estimating time - probably the most frequent estimation that I conduct in day-to-day life. When on a New York City subway, for instance, I'll make a 50% confidence range guess on how long it will take the subway to get to my stop. The game works equally well when waiting for a light to change, a lecture to end, an elevator to arrive, etc.
I started doing this at a fairly young age when - in response to asking "are we there yet," - my parents told me to guess how long it would take to reach a travel destination.
Great examples. Next step is calibrating the confidence range based on multiple experiments.
In response to
Estimation as a game
I think it's easy and natural for children to grasp if this method is introduced through everyday experiences. By making this into a game children can gain intuitive understanding of quantitative techniques. I suspect many children can enjoy this kind of games.
You can certainly introduce it, and it's a good idea to try, and some children will be interested, but others will find it boring and go back to playing with their dolls or trucks. In my limited experience, when you say "many children can enjoy this kind of games", "many" refers to less than half (and fewer girls than boys, for whatever reasons).
Even 10% of all the children is many. I wonder what percentage was familiarized with numbers in that context. My guess is < 2%.
In response to
Estimation as a game
There was a phase where my 7 year old asked me how many X there are (X being like your examples) and after he knew that things are made of particles (German: 'Teilchen' which also build nicely on 'part'/'Teil') aka atoms he asked lots of questions like "how many water particles are there in a cloud" (or in a water drop).
See also http://lesswrong.com/lw/iha/raising_numerate_children/ esp. the comment http://lesswrong.com/lw/iha/raising_numerate_children/9o8z
I remember that I counted and then calculated lots of things when I was in primary school (windows in a building, squares on the floor), but these were fairly concrete items where you did lots of multiplication but cound still count them in theory.
It looks like you are doing a good job with your kids.
There is also a whole set of questions dealing with probabilities. For example: "what is the chance I'll meet someone I know when going on a weekend trip?". These kind of questions often require more than one step.
In response to
Estimation as a game
Such estimation problems are called http://en.wikipedia.org/wiki/Fermi_problem and they are recommended as real life class room exercises. In Germany you can buy such exercises as a box: http://www.vpmonline.de/index.php/fermi-box_produktdetail/items/1935
Thanks. I wonder if there are more games like this.
In response to
Estimation as a game
From an early age I've been constantly calculating various kinds of estimates - e.g. "how many people live in this building", "how long will it take to cross the US on foot", "what's the height of that tower", "how many BMWs are manufactured annually" and so on.
Did you do this spontaneously, or did your parents or teachers encourage you to?
Do you practice anything similar with your children?
Nope, but I will (once he's a bit older; at two and a half his grasp of "counting" is still limited).
Did you do this spontaneously, or did your parents or teachers encourage you to?
My parents definitely encouraged me, although some inner disposition was probably there as well.
but I will
Glad to hear that.
Estimation as a game
I was inspired to write this post after reading Gunnar_Zarncke's Raising numerate children.
Developing rational patterns of thought in children is very important and I'm glad Gunnar brought that issue up.
I wanted to share with you some thoughts I have regarding estimation games.
From an early age I've been constantly calculating various kinds of estimates - e.g. "how many people live in this building", "how long will it take to cross the US on foot", "what's the height of that tower", "how many BMWs are manufactured annually" and so on.
I believe that practising this technique is not only fun but also helpful. Sometimes one has no way or time to acquire accurate information regarding something and even a rough estimate can be very valuable.
People are often surprised when they see me do it whereas for me it is completely natural. I think the reason is that I do it from a very early age.
I think it's easy and natural for children to grasp if this method is introduced through everyday experiences. By making this into a game children can gain intuitive understanding of quantitative techniques. I suspect many children can enjoy this kind of games.
I'd like to hear your thoughts on the subject.
Do you remember yourself doing something like this? From what age? Do you practice anything similar with your children?
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I suspect it might work quite well in collaborative mathematics. Publish (internally, to your collaborators) when you get a new idea, have a rough sketch of a proof, find a plausible conjecture, think of a good way of presenting something, etc. Eventually, of course, you'd need to clean it all up and get the bugs out, but frequent informal known-buggy "releases" might be an excellent way to get from zero to a high-quality research paper.
That's an interesting idea, although the discussion was about studying. I assume it can also be applied as a studying technique: Read small chunks of material and solve a lot of problems for feedback.
My point was that you cannot overlook the fundamentals.