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Re: Teaching kids real math with computers

Posted: Mon Feb 13, 2012 3:22 am
I stumbled upon this TED talk a couple days ago. In the video, Conrad Wolfram discusses his feelings towards mathematics education. He essentially states that we need to teach the essence of math itself, the theories and ideas, rather than the computation.

My favorite quote from the talk: ?"Programming is... a great way to engage students much more and to check they understand. If you really want to check you understand math then write a program to do it." I know that the purpose of the Raspberry Pi is to teach kids programming. The Raspberry Pi is the means to an end: a new generation of students and professionals with a much better understanding of computers. I, however, do not think that programming should be the ultimate goal. While programming is a great thing to know, it can itself be the means to an end: a much better understanding of math and problem solving. I myself have used programming to cement my understanding of all kinds of math. If you can make a program that does it, you have a very solid understanding of it.

I think we should consider a somewhat different learning approach when we try to teach students using the Raspberry Pi. Rather than bringing the students through various levels of programming, resulting in complete mastery of the language(s), we should consider teaching students the basics of programming and problem solving skills. From there we can use those concepts to help the students learn math. For example, we could have the students create an algorithm that calculates angles in a triangle using the law of cosines. The student must understand the law of cosines completely in order to create an algorithm that works every time.

The Raspberry Pi is a great device that has the potential to get a lot of kids interesting in programming. I simply think that we should expand the focus of the device to include math education as well. I am planning on bringing this up to my math teacher; maybe I can find some interest in this type of teaching.

Ideas?

Eric.

Re: Teaching kids real math with computers

Posted: Mon Feb 13, 2012 8:07 am
Eric Middleton said:

I know that the purpose of the Raspberry Pi is to teach kids programming. The Raspberry Pi is the means to an end: a new generation of students and professionals with a much better understanding of computers. I, however, do not think that programming should be the ultimate goal.

You are right of course. The "teach programming thing" is just the hook. It's what was exciting in the 8-bit era and what led a lot of people to get into computing in a much wider sense.

The Foundation is well aware that computing is not programming, and in education it's understood that the language(s) learned are largely irrelveant, certanly to start with - it's the underlying concepts and principles of computing that matter. Even computers are redundant in this sense (see e.g. the fantastic CS Unplugged) and there is a big push at the moment in the UK to get computational thinking and computing taught from the early years.

How to do all this is a different matter so your questions are very relevant. But it's not a coincidence that (AFAIK) there is a maths sections in the learn Python bit of the user guide that will come with the full release of the Pi Q3 2012

Re: Teaching kids real math with computers

Posted: Mon Feb 13, 2012 12:41 pm
Doubtless this will be contentious point of view in this audience, and beforehand I would like to say:

I love the RasPi and the foundation;
I love Numerical Methods and had great fun programming them and using them to solve engineering mathematics problems;
I’ve made a good living out of computers;

However, I can hear my 6th form applied maths teacher, and my lecturers, telling me that proper maths has nothing to do with numbers Despite the smiley, they were perfectly serious.

I would always advise that for maths, the computers are the last thing that you reach for when the equations will take you no further, and then use with caution.

Re: Teaching kids real math with computers

Posted: Mon Feb 13, 2012 7:33 pm
Hi,

In primary school education numerical manipulation is important.  During secondary school numbers become less important.  Undergraduate mathematics exams can normally be completed without a calculator, since the problems are arithmetical.  Physics undergraduate problems are similar and may only require a calculate for the final step.  Computers become important for two sets of problems: (i) when many calculations need to be repeated, and (ii) numerical solutions to problems which do not have an arithmetic solution.  The simulation of particles passing through matter is an example of (i), where the equations are known, but the number of calculations is rather high.  Numerical integration is often the only way to solve matrix element calculations of particle physics.

Two of my sons have started to use python to check their numerical maths programs, (after finishing their exercises.)  To practice their addition, I wrote a simple JAVA application with a reward system and increasing difficulty.  For the Raspberry PI, the JAVA application could be re-written with Qt.

Regards,

Will

Re: Teaching kids real math with computers

Posted: Tue Feb 14, 2012 6:36 am
I may have failed to make my point clear - I have nothing against doing math by hand. I actually prefer doing math by hand in most cases, especially trigonometry and calculus. The point that I am trying to make is that by leveraging computers we can involve the student in all aspects of solving the problem:

1) identifying the problem

2) designing an equation

3) solving the equation

4) make sure we completely solved the problem

Right now, most problems that students are given are, at most, covering steps two and three. Another added bonus of using computers is the mighty algorithm: the best way to check you understanding of a math formula.

I am actually working on a lesson plan for a high school Pre-Calculus class (a class that covers trigonometry and basic limits) that involves designing a flow chart that solves the law of cosines. The point is to get students to think about all aspects of the math instead of just the numbers.

Re: Teaching kids real math with computers

Posted: Tue Feb 14, 2012 9:06 pm
Hi Eric,

As you say, computers are good for going to the limit of a particular calculation.  Another area where this can be applied is that of statistical methods.

At the University level we often ask the students to solve equations numerically, which cannot be easily solved arithmetically.  For example, rough projectiles with complex drag modelling, angular momentum calculations of objects with non-unform density and complex electric/magetic field problems.  The flow of liquids or plasmas can be quite interesting too.

Best regards,

Will

raspberry pi and teaching stats using r commander

Posted: Wed Feb 19, 2014 5:27 pm
I'm in the process of writing a book primarily for undergraduate medics/ health sciences students and focusing on using R and R commander - I known nothing really about the raspberry pi but have managed to get both to run on it - is there any experience of using either of these programs in schools for helping explain topics in the AS stats curriculum etc. Any advice pointers welcome.

you can see a draft of the chapter here
https://www.dropbox.com/s/20iwwszpndu9l ... y%20Pi.pdf

Re: Teaching kids real math with computers

Posted: Mon Feb 24, 2014 12:00 am
Computing has been with us long before we even had the abacus, writing, and probably before we had achieved speech beyond animalistic grunts. Our modern-day computers are even described in these terms, aka digital - using our built-in digits ... yes, counting on our fingers! The fact that today's "fingers" number up into the trillions (in the case of rotating disk drives) and can be counted billions of times a second is just a slight improvement over the manual method. The question isn't whether to use computing to help with learning math(s), it's how to do it. Learning to write code is just a form of writing, albeit one that requires a level of focus and syntactic precision way beyond prose, although poetry isn't a bad second to programming in that there can be very specific rules.

We know that computing tools such as Khan Academy hold advantages over traditional methods. It's self-paced and students (of all ages, including kids' parents, BTW) can spend as much time in as many repetitions as are necessary to master each concept. Behind the content in such successful systems is assessment, and the incredible benefit of computing-based assessment is that it can dynamically modify problems in response to the assessed student's abilities and progress. This can help reduce, if not eliminate test stress, which has to be one of the major reasons that many people wind up hating the traditional education process and don't get much from the experience, not proceeding to higher levels where they could eventually excel, sometimes to the point of dropping out prematurely altogether.

I recently attended a Silicon Valley Math Initiative (SVMI, sponsored by the Noyce Foundation formed by the late Intel cofounder/CEO Robert Noyce) workshop that introduced the Common Core Curriculum for Mathematics (CCCM) which is required to be implemented in classrooms in 34 states by 2015 (10 other states will be monitoring progress, probably to adopt them after the initial mistakes are made and resolved). Two of the many improvements that the CCCM will bring are differentiated instruction and meaningful assessments that go well beyond the current one-size/technique-fits-all presentation methods and multiple-choice abomination testing regimes. Adapting presentation material to capitalize on each student's learning strengths is much more feasible via computing than what a given teacher can provide in a classroom situation, and the reach extends to wherever the student happens to be when studying. Dynamic adaptive assessment will require substantial implementation, but once it's up and running, it will be an enormously powerful tool in helping identify students' strengths and weaknesses, where they need personalized help from a teacher, and probing students' knowledge from a wide variety of perspectives that is simply impossible in the traditional education system.

Then there's the entire landscape of mathematical computing-based tools that enable students to explore a much wider array of concepts and techniques than could ever be covered in a traditional classroom situation. Mathematica alone provides the means for students to perform calculations and computations, even complex ones, so rapidly that they can try them out on numerous problems to see what happens when input values are changed in terms of coefficients, exponents, ordinate/independent value ranges, numerical signs, various operator combinations and permutations, data set populations, etc. Yes, we still want students to start out plotting points by hand, but once they've proven that ability there's no reason why they should not only be able, but must be encouraged to explore what happens as they change equations, functions, initial conditions, etc. Even miniscule changes in physical constants can have extreme effects on the behavior of every component in the universe as we know it. Then there are the nearly uncountable features in computing tools such as Mathematica, SAS, SPSS, etc., that students may wonder about and investigate on their own, if not suggested by their teacher. This is what learning is really all about - not the presentation of static representations of isolated facts, but instilling a spirit of adventurous curiosity about what lies behind the next corner/turn/bend as they expand their knowledge horizons.

I guess my enthusiasm might be showing, so I apologize to anyone who thinks that paper and pencils are going to be enough to properly educate young people to take over operating (let alone improve and expand) the food, transportation, energy, manufacturing, information, research, etc., infrastructures we've managed to build during the past century, or so, after we geezers and codgers exit Stage Left, rapidly, even. We passed the point where we can feed the still-growing population of the planet without computing in the 1980s, the other domains listed above by the 1990s, and the education infrastructure is long overdue for a real overhaul to integrate computing properly. The Pi is a very small, but vitally important and necessary, however, not sufficient, step in achieving the latter.

EDIT: I forgot to mention a couple of more very useful things presented at the CCCM workshop. They encourage teachers to present a Problem of the Month to not only each class, but their parents, other teachers, and the entire school. A Problem of the Month is meant to require the part-time efforts of multiple people upwards of a month to solve. Although the linked pages are password-protected for use by districts paying for access (check with SVMIMAC or your district to see if yours is a member to get the password, or get yours to join), the page which lists examples that can stimulate thought and imagination is at:

http://www.svmimac.org/problemsofthemon ... month.html

This stems from a book by George Polya, (1887 - 1985), the Father of Problem Solving: “How to Solve It”, 1945, in which he said, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems." Polya also said that a problem that can be solved in a few hours isn't a problem at all, it's merely an exercise used to warm up to solving real problems.

You can get a sense of a small part of what the CCCM workshops cover in this PowerPoint presentation:

http://www.svmimac.org/images/2011CI.Day1.ppt‎

Re: Teaching kids real math with computers

Posted: Mon Feb 24, 2014 1:14 am
Clearly Jim Manley is MUCH more tuned in to this than myself, but from a theoretical standpoint, I don't see the applicability of computers to math(s) anymore than to philosophy, or I suppose literature. Excepting of course you can look up novels and philosophers much more easily with a computer, and similarly you can display applied mathematical results much more easily, but I don't see how computers help with training math concepts more particularly than any other discipline. Leibniz, Newton, Peano, Frege, Russell, Whitehead et. al. seemed to get by without computers. What am I missing?

Re: Teaching kids real math with computers

Posted: Mon Feb 24, 2014 2:24 am
Douglas6 wrote:Clearly Jim Manley is MUCH more tuned in to this than myself, but from a theoretical standpoint, I don't see the applicability of computers to math(s) anymore than to philosophy, or I suppose literature. Excepting of course you can look up novels and philosophers much more easily with a computer, and similarly you can display applied mathematical results much more easily, but I don't see how computers help with training math concepts more particularly than any other discipline. Leibniz, Newton, Peano, Frege, Russell, Whitehead et. al. seemed to get by without computers. What am I missing?
Hi Douglas6 - the issue isn't whether a given person can learn math(s) with or without computing (notice that I don't use the hardware term "computer(s)" - there's a LOT more to computing than just the hardware or even the software). It's how effectively as many people as possible can learn math(s) (or any of the other disciplines you listed, for that matter). I think we can agree that Leibniz, Newton, Peano, Frege, Russell, Whitehead, et al, were probably "smarter" than the average bear, and definitely better-prepared to be educated than the typical student with whom I interact on a daily basis (bears make it through circus training all the time, but I challenge you to get any teenager to do any of those tricks bears can do with aplomb! ). This is a point that is missed by almost everyone both within and outside the education community, so don't feel bad if you missed it as an innocent bystander - WE ARE NOT TRYING TO EDUCATE JUST THE LIKES OF LEIBNIZ, NEWTON, PEANO, FREGE, RUSSELL, WHITEHEAD, ET AL.

Sorry for yelling, but it's damned hard to get that point across to almost anyone, especially the more advanced someone is in the bureaucracy and particularly within academia. Those folks were already interested in the subject matter, partly because they were often trying to solve a problem - how to gain an advantage in billiards in the case of Leibniz, you probably didn't know, or why the planets traced the orbits they did, in Newton's case (yes, there were theoretically interesting aspects, too, but Necessity IS a Mother!). We should also be mindful that the pace of advances in STEM areas has become a bit more intense than in the times of Leibniz and friends - they had to wait decades to centuries for the Next Big Thing to be discovered or created. However, we witness them now on a nearly daily basis, or could if only we weren't so distracted by the Sports Illustrated Swimsuit Edition, Cool Ranch Doritos Tacos, and yammering spokes-geckos. A big reason for that increased pace is because we've managed to educate a whole lot more people to much higher levels in ever-widening swaths of real estate, much more efficiently on a resource basis, if not an average-outcome basis (increased knowledge and abilities).

However, the modern-day educator has to try to compete with the likes of Biebers, Kardashians, Honey Boo-Boos, Snoop Dogs/Lions, YouBoob Kitties ... well you probably get my drift. Home situations are not the Leave It to Beaver kind (or whatever is considered 1960s family "normal") these days - there is more often than not no father figure or even any mature male influence in most kids homes, and that's a known recipe for disaster when it comes to creating low self-esteemed, depressed underachievers, and even criminals of the future, much less hoped-for model citizens with advanced degrees - there are way too many distractions, including lack of good nutrition in the worst cases (80% of the kids in my school district qualify for the national free school lunch program). I'm not saying we need to all become better rappers, music video producers, nasty spoiled housewives, misbehaving celebrities, etc., than those constantly in the headlines, but we need every other edge, trick, advantage, and surprise that we can conjure up, and computing can help with that.

If you can make demonstrating a mathematical principle interesting, why wouldn't you do that instead of just a plain-vanilla presentation that's been done the same way since Pythagoras was roaming the streets of Athens? For the life of me, even at my advanced age (I'm older than Sputnik), I can't believe how some educators can drone on doing every lesson exactly the same way as they have for decades before, and then wondering why Jane or Johnny can't calculate, let alone compute. Even multi-millionaire professional sports athletes often have to change up their pitch/swing/delivery/pass-offense/stroke/bowl/etc., in order to remain relevant on the field/court/pitch/alley for very long, and I would find the boredom of decades of such repetition enough to want to end it all. I often imagine what the Great Minds of the past could do with the tools we now have - they certainly wouldn't turn their noses up at them for every single thing they did for work or entertainment (well, except for maybe Socrates, but he was always such a crank!). Ask Stephen Hawking what he thinks of computing technology the next time you get a chance, OK?

Of course, it doesn't help when a student doesn't have a driving need to learn anything, especially something that requires mental effort as education does, because Mommy and/or Daddy and/or The State will always provide. It's one of the side-effects of giving away everything - eventually everyone expects everyone else to fulfill their needs even when they have the capacity themselves, and that's the definition of a societal collapse in the making. In the worst case, we may have to wait for an existential threat to make itself obvious as to why kids need an education. In the meantime, we'd better make the best use of the tools at hand to get them interested, at a minimum, before they find themselves yet-another art historian (oops! ) flipping burgers or mopping floors for a career and we find ourselves unable to defend ourselves via ever-more-complex technologies, keep our economies going through viable infrastructure, and even keeping ourselves adequately fed. For those who may say, "Well, if they can't pull themselves up by their own bootstraps, screw 'em.", we never know where the next Hawking, Einstein, Newton, Leibniz, et al, is going to come from, and it could very easily be some homeless kid who in no way was responsible for his plight. We don't have the luxury of only educating those who are lucky to wind up in nice private schools due to the "genetics lottery" as Bill Gates puts it. Most kids really are curious, thoughtful, and imaginative at a young enough age, until the educational system and societal "norms" beat that out of them (HR weenies and drone managers hate curious, thoughtful, imaginative people, partly out of jealousy and mostly out of fear of being usurped and tossed aside, usually for good reason).

I hope I've shed some light on where the problems lie that are unique to our circumstances, and what computing can do to help, but not necessarily by any means.

Re: Teaching kids real math with computers

Posted: Mon Feb 24, 2014 4:39 am
Wow Thanks, Jim, I don't think there's a sentence there I disagree with, and I'm glad to hear it all, and hope others will listen. What struck me about this thread was the association of math(s) with computing, and computing with math(s), which I think is prevalent but misguided, and I think you eloquently cleared this up.
I totally get the challenge to motivate the average student (I was one) and if computing can do it (it probably can) let's use it, creatively, whether in mathematics, or philosophy, or literature or history or the arts.

Re: Teaching kids real math with computers

Posted: Mon Feb 24, 2014 11:45 am
Douglas6 wrote:I totally get the challenge to motivate the average student (I was one) and if computing can do it (it probably can) let's use it, creatively, whether in mathematics, or philosophy, or literature or history or the arts.
There has been a movement in some schools to integrate the arts with STEM, called STEAM, but I say that it really should read SCHTEAM, for Science, Computing, History, Technology, Engineering, Arts, and Mathematics. I split History out from the Arts because I'm volunteer senior docent at the Computer History Museum in Silicon Valley and as a student I wasn't a big fan of history courses, but I've learned my lesson in order to avoid repeating the mistakes of the past (as in, "Those who ignore the lessons of history are doomed to repeat them."). The reason I didn't like history courses was precisely due to the way they were presented, as linear progressions of events with dates on timelines that were divorced from everything else happening in parallel. Exploring the vertical dimension of horizontally-oriented timelines is key to tying together related events and this is much easier to do via computing than traditional methods.

Some might just scoff and call SCHTEAM something old-fashioned named SCHOOL, but my experience is that unless you break complex things down into their component parts in obvious ways, most people won't get the message. Plus, SCHTEAM has the coincidental good fortune to be an amalgam of the words SCHOOL and TEAM, and that's very important as some non-STEM educators are jealous of the attention that STEM has gotten. Attention means money, and when it's spent on flashy equipment that's ostensibly meant for a limited purpose initially in certain departments, those in other departments can get their panties in a twist, especially if they're technophobes or are violently in favor of spending money on things that smell of printing presses (or preferably, quill pens and ink), binderies, and embossing of leather. The good news is that the newer teachers entering the system are comfortable with technology and understand its benefits and how to integrate it into their classroom to varying degrees. The bad news is that most are soon disgusted with the grandfathered tenured teachers who never had to earn a permanent credential that now takes 5 ~ 7 years of postgraduate study and classroom teaching after earning a BA/BS, using self-created exemplar materials subject to approval by the grandfathered tenured teachers who never had to produce such material to earn tenure. I just ran across such a disillusioned English teacher who left teaching to work in Home Depot stocking shelves and helping customers find items rather than deal with the bureaucracy, incompetence, and lack of support in schools.