A few words about unschooling math

[Ned & Luz]

A Few Words about Unschooling Math
 
FREEDOM TO CHOOSE
 
 Fingers & toes,  pattern blocks,  two by two,  4x4,  tape measure, #, scale, $, save, interest,  model,  profit (loss),  earn,  spend,  checkbook,  recipe,  batting average,  Captain May I?  third base,   thirty love,  fault, par,  birdie,  strike,  spare,  first down and ten to go,  penalty box,  map, scale of miles, compass,  Pokémon,  Candyland,  Monopoly,  Go,  Chess,  Sorry!  dominoes,  dice,  poker chips,  Bridge,  Crazy Eights,  charts, 
                Origami,  knit 1 purl 2,  weigh  +  pulley,  ratio,  chances, statistics,   average,  more or less,  even,  odd,  yards,  N scale,  area,  score,  speed limit, braking distance,  fourth dimension, sixth sense,  Indy 500,  build,  plan,  rate,   estimate.
                Predict,  revise,  depth,  angle,  trade,  straight,  spiral,  high tide,  low ball,  tempo,  %,  quarter note,  half pound,  forecast,  budget,  half price,  plus tax,   longitude,  light years,  escape velocity,  precession of the equinoxes (oh Best Beloved)  ?,  range,  set, 
                 Stitch,  sort,  size,  plot,  dozen,  $,  gain,  lose,  exact,  income,  allowance,  loan,  knots,  beads,  gear ratio,  minutes,  degrees,  fathoms,  grid,    meters,   Anno,  The Number Devil,  half pipe,  quarter turn,  full bore,   turning radius,  stacking,  nesting,  measure up,  @,  scale down,  abacus,  debit,  infinity,  first class,  equal share,  short shrift,  waxing,  waning,  rhythm,  balance,  cycle,   value,  graph,  perigee,  frequency,       
                Pennies,  double helix,  £, time zone,  millennium,  program,   binary,  generation,  epoch,  era,  nano second,  code,  puzzle,  fiscal year,  progression,  midpoint,  watts,  lumens,  ?,   horsepower,  ohms,  Great Circle Route,  52 Pickup,  ‘55 Chevy,  Hundredth Monkey,  altitude,  Lego,  Tangrams,   Fibonacci series,  height,  width,  length,  volume,  output,  Eureka!  displacement,  schedule,  time limit,  add up,  count down,  four score,  last full measure,  census,  Are we there yet?  a bushel and a peck,  postage,  efficient operation,  elegant solution,  gigabytes,  google, 
                 Powers of Ten,  increase,  <  >,  decrease,  supply  &  demand,  links,  contour lines,  Great Divide,  Bingo!  count down,  stock market,  daily log,  rent,  discretionary income,  arc,  geometric proportions,    geologic time,  navigation, 16 mm,    Stonehenge,  ¢,  grams/ounces, f stop,  low bid, dot-to-dot, orienteering, etc.

[JMommer]

THAT was quite a list...THANKS! (From a new homeschooling family who has yet to pick a curriculum beyond "real life.")

JMommer

[Moti]

The search engines probably like this post, which is the reason it exists.

[Ned & Luz]

The search engines probably like this post, which is the reason it exists.

Actually, I wrote this before I knew about search engines. It was a response to people who said they were "unschooling except for a couple of pages of math every day." Folks find it hard to believe you can learn math by living life and I wanted to give just a few ideas.
Luz

[Moti]

Then I apologize for my post. It is a common method for attracting search engines (smile) and I misunderstood your post.

Yes, you can learn Math from life but like much of unguided learning it can be chaotic and bring about solutions/algorithms/methods that apply in narrow situations but are used incorrectly in other situations as a result. It has to do with how we learn, and how we make deductions.

That said, for non-mathematicians, if Math has no relevancy to life that the student also knows and understand, it becomes a meaningless exercise, as the case is often when Math is taught in school.

[babaika]

Then I apologize for my post. It is a common method for attracting search engines (smile) and I misunderstood your post.

Yes, you can learn Math from life but like much of unguided learning it can be chaotic and bring about solutions/algorithms/methods that apply in narrow situations but are used incorrectly in other situations as a result. It has to do with how we learn, and how we make deductions.

That said, for non-mathematicians, if Math has no relevancy to life that the student also knows and understand, it becomes a meaningless exercise, as the case is often when Math is taught in school.

Allow me to make some comments:

1. Usually one CAN learn arithmetics and basic calculation from life (unless he or she, of course is Carl Gauss In 99% of the cases, however, it will not be "math". Enough to mention that dear to everyone issues of annuities or compound interest are related to the fundamental issues of numerical sequences or properties of a power respectively (to name a few concepts) and understanding of those requires a little bit more than just hands on experience

2. Math IS relevant pretty much to everything in life--another matter how one wants to describe this life around him (her) self and that is where the real problem lies--MATH IS not easy (I would agree) and unlike literature (which we may or may not like) requires a little bit different mental organization and approach. But will we make a conclusion about American literature without reading Hemingway and Steinbeck, or about German one without Heine or Schiller??? Not if we are thoughtfull, i would guess

[Moti]

Allow me to make some comments:

1. Usually one CAN learn arithmetics and basic calculation from life (unless he or she, of course is Carl Gauss In 99% of the cases, however, it will not be "math". Enough to mention that dear to everyone issues of annuities or compound interest are related to the fundamental issues of numerical sequences or properties of a power respectively (to name a few concepts) and understanding of those requires a little bit more than just hands on experience

Of course that it won't be mathematics as we study in the university, i.e. the theory of math and not applied. As trained as such I should know But given the discussion about school (elementry - high) math I was using "math" as such and not as mathematics. And of course, to understand some basic finance requires a much deeper understanding of mathematics that is even taught in schools [option trading would require stochastic processes and such for example].

2. Math IS relevant pretty much to everything in life--another matter how one wants to describe this life around him (her) self and that is where the real problem lies--MATH IS not easy (I would agree) and unlike literature (which we may or may not like) requires a little bit different mental organization and approach. But will we make a conclusion about American literature without reading Hemingway and Steinbeck, or about German one without Heine or Schiller??? Not if we are thoughtfull, i would guess

Math is indeed pretty much relevent to everything in life. BUT, not all math studied in school is relevant to what people would encounter. Take all the trigonometry proofs required [based on trig identities]. Why on earth do 99% of people need to be able to do those?<br>
You could argue that it developes logic and organization skills, but there are much better ways of doing so. In fact, all those "proofs", which are nothing but a sequence of exchanges of one term with another, can be done without thinking at all and just by a basic search approach.

Let me explain "Easy" as I mean it. Math (school, not Mathematics) is Easy. It is so because it can be done without abstract thought and has no open questions. It only requires a good mastery of the knowledge base, a systematic approach, and no fear to try. I know because I taught hundreds of students all of school's math in 3.5 months (about two 4 hours class a week), starting with 1+1 [I swear that was the first thing I wrote on the board] and ending with calculus. Can I say that they understood Mathematics? No way. Did they get high grades? Yep. The average grade was above 90%! [national standardized test]. Of course, they did understand the basic ideas of "Math". But not the real abstract ideas behind it.

So, Math is easy. Mathematics is not

[babaika]

Of course that it won't be mathematics as we study in the university, i.e. the theory of math and not applied. As trained as such I should know But given the discussion about school (elementry - high) math I was using "math" as such and not as mathematics. And of course, to understand some basic finance requires a much deeper understanding of mathematics that is even taught in schools [option trading would require stochastic processes and such for example].

Math is indeed pretty much relevent to everything in life. BUT, not all math studied in school is relevant to what people would encounter. Take all the trigonometry proofs required [based on trig identities]. Why on earth do 99% of people need to be able to do those?

You could argue that it developes logic and organization skills, but there are much better ways of doing so. In fact, all those "proofs", which are nothing but a sequence of exchanges of one term with another, can be done without thinking at all and just by a basic search approach.

Let me explain "Easy" as I mean it. Math (school, not Mathematics) is Easy. It is so because it can be done without abstract thought and has no open questions. It only requires a good mastery of the knowledge base, a systematic approach, and no fear to try. I know because I taught hundreds of students all of school's math in 3.5 months (about two 4 hours class a week), starting with 1+1 [I swear that was the first thing I wrote on the board] and ending with calculus. Can I say that they understood Mathematics? No way. Did they get high grades? Yep. The average grade was above 90%! [national standardized test]. Of course, they did understand the basic ideas of "Math". But not the real abstract ideas behind it.

So, Math is easy. Mathematics is not

Yes and No at the same time There are ways and methods which DO propell (say equivalent high school) students towards understanding namely of the concept and the first answer here is FLOW of the material. That is systemic development of the say program. Of course not all students, of course not all--but considerably larger portion of them that it is usually assumed within the framework of the most, well, public and many private schools' programs. When and where will those students decide to use (or not to use) La Grange, or Fermat's Theorem on Extremas, or whatever else (which is alltogether a separtae issue if to view strong physico-mathematical environment), but what is absolutely clear--students should be AFFORDED all this and taught all this. Especially so when "abstract" math ideas are supported by their reincarnation say in Physics--best example of which are quadratics in algebra and accelerated motions in physics. Essentially, two inseparable subjects (topics). And how nice after that to see how easily students beging to apply differentiation (and integration) in those concepts--without any problem

[Moti]

that the FLOW of material is critical. I believe I said it somewhere in this forum as it is the principles of my teaching philosophy when I write the books (or teach); namely that with the correct flow [and some other required elements] math can be easy. BUT, as I said, I differentiate between proving (and understanding of course) La Grange's theorem and being able to understand its idea and apply it.

I agree with you that almost all (some exceptions but much, much fewer than people think) can understand Math. However, to understand the higher level, that of the full theory development and concepts, relatively few can. Exactly like most can learn to play the piano. Few can be Motzart

[babaika]

that the FLOW of material is critical. I believe I said it somewhere in this forum as it is the principles of my teaching philosophy when I write the books (or teach); namely that with the correct flow [and some other required elements] math can be easy. BUT, as I said, I differentiate between proving (and understanding of course) La Grange's theorem and being able to understand its idea and apply it.

I agree with you that almost all (some exceptions but much, much fewer than people think) can understand Math. However, to understand the higher level, that of the full theory development and concepts, relatively few can. Exactly like most can learn to play the piano. Few can be Motzart

Agree!!! A lot depends on individuality of a student, well, in the end on his or her ability and approach should be personalised to a degree possible in particular circumstances. But, I think, the meaning of academic minimum should exists--that is the volume and scope of the material which is an absolute MUST. I personally consider the proofs of most theorems of plane geometry to be totally within the reach of overwhelming majority of students of age category 13-14 years old. Derivation of formula of the roots of quadratic equation--totally within the reach. All in all--a lot of things are, indeed, EASY. Same goes to derivation of main trigomometric identites and their sequels. And the list goes on. Results, however, manifest themselves very fast--very often within the duration of some department or even topic. The most profound example being possibly the whole spectrum of derivations from simplest formula of N-th term of arithmetic sequence to the derivation of explicit formula of arithmetic sequence and their total connection to linear functions. All this is easy as it is easy to teach them multiplication and division of large numbers (mentally--without any long hand, let alone calculators) using distributive property. Initially slow (usually 10-15 minutes)--then.........they love it!!!

[Moti]

(smile) it seems we do agree on much. In fact, my view is that because fo the flawed approach and books math is studied at too slow a pace. Especaily in the "concpet/theory" sense. For example, when I talk to 4th/5th graders I introduce the "Achiles and the Turtle" paradox [it did get solved recently] by playing it as a game. Few are quick to figure it out and EVERYONE gets it. From there we introduce the idea of infinitely small (which is the basis for calculus) and the idea of infinity in general. Even learn to compare countable sets vs. real numbers [a string with pins in it do wonder to show it ]. And the chess riddle [large numbers, leading to the infinite idea] and progression of such. Many BIG ideas in an hour, and they LOVE it and they get it!

So, yes, I do agree even big parts of mathematics, at least concept wise, are Easy. That said, using those concept and being formal about them is much more difficult and out of reach for most. But using them, and figuring out the applied proofs [meaning: not the real general theorems] is within almost all's reach, if done in correct flow, and explanation.

[babaika]

(smile) it seems we do agree on much. In fact, my view is that because fo the flawed approach and books math is studied at too slow a pace. Especaily in the "concpet/theory" sense. For example, when I talk to 4th/5th graders I introduce the "Achiles and the Turtle" paradox [it did get solved recently] by playing it as a game. Few are quick to figure it out and EVERYONE gets it. From there we introduce the idea of infinitely small (which is the basis for calculus) and the idea of infinity in general. Even learn to compare countable sets vs. real numbers [a string with pins in it do wonder to show it ]. And the chess riddle [large numbers, leading to the infinite idea] and progression of such. Many BIG ideas in an hour, and they LOVE it and they get it!

So, yes, I do agree even big parts of mathematics, at least concept wise, are Easy. That said, using those concept and being formal about them is much more difficult and out of reach for most. But using them, and figuring out the applied proofs [meaning: not the real general theorems] is within almost all's reach, if done in correct flow, and explanation.

Well, turtle problem is a classic example of increment and a slope within system of linear functions But here is the point where important clarification should be made. If to take math in isolation and at this stage--fine. This, however, changes dramatically when the serious physics hits. Majority of the physics problems already starting with simplest issues of molecular physics be it densities or Pascal's Law, let alone things of such nature as pressures (enough to mention Hydrostatic Paradox) or even rudimentary issues of levers and momentums--it is impossible to deliver them without fundamental explanation OUT of the framework of classic math proofs. Majority of the established laws and formulas there ARE derived with explanation of the physical nature of the process--otherwise they become moribund static cliches, which are impossible to use for solution of any practical problem. Things get even more complicated once students are subjected to basic dynamics, which can not be explained by merely naming variables within some formula--simply will not do any good. At that stage they HAVE to already understand some fundamental......Mathematical concepts (be it trigonometry, even plane geometry and even basic meaning of integral, which is very simple) and have to mentally coalesce already some important things into some unified vision. http://www.aimssoft.com/p_m_03.htm

This is an illustration of what and how is required for students to understand the concept in order to solve simplest problems on this topic. And this one, as well as hundreds of others, is not a situation when say gaming methods will do any good. Good (correct) answer at this stage is just merely means to an end, which is HOW to do this--and that is what will matter a great deal down the road. Well, I should say--spending all this Saturday with my daughter on-line, preparing (reviewing) her for the math test for Business School in UW. Approaches are virtually the same--needless to say, the test included problems on (for obvious reasons) Marginal Revenues and Costs (and whatever else--hate economics and........of course on clean cut physics (balloon problems) and clear calculus. And this ALL was about concepts and conceptual thinking--but that what colleges and universities in reality want, otherwise they will not do their OWN placement tests, even having SAT (ACT) scores on their hands--and that is the name of the game and entirely new paradigm, which our children will (and some already are facing) face in colleges. Thus, the question for us should be--whom and how child sees him or herself in nearest future and the earlier we will start asking that kinds of questions--the better THEY (and us will feel in the long run.

[Moti]

or you would have seen I am [well, until the end of this June] a Professor in a Business School [in Penn State, Graduated from Wharton] so I KNOW what business school ask for. Alas, the majority of courses do not ask for deep understanding of concepts (and I taught at Wharton, at Tulane, and at my position now). In fact, the biggest problem I had in my courses was that I did make them really think and understand the concepts and they were not used to it. Even in college now the approach is "here's a formula, can you spit it back at us on the test?" At both Tulane and Penn State, which are highly ranked business schools [in fact, my dept at PSU was ranked #1 for undergrads...] I asked (because they needed to know it to use in a case in my course) them about NPV [Net Present Value] which is a BASIC concept in finance. So basic that if a student who finishes a business school doesn't know it, s/he should not be hired in my mind. Uniformly they didn't know it, could not really explain it, and didn't even know how to use it [though they probably knew how to press the NPV function in Excel and get the result, well, maybe not even that ].

The point of it is that college education is not what you think anymore. It deteriorated strongly [due to SRTEs that make sure that professors would want to make the students happy, not knowledgable, and students who are used to being spoon fed in school, and of course the horrible multiple choice questions all tests use now].

Anyway, I agree with your first paragraph - I was a physics major as well [I started a dual degree in Math and Phsyics but then decided to focus on math and move to Operations Research].

And don't tell you daughter what I said - as long as she believes she needs to learn deeply, she would

And how can you hate economics? It's mostly math...[My research included lots of economic modelling ]

[babaika]

I was a physics major as well [I started a dual degree in Math and Phsyics but then decided to focus on math and move to Operations Research].

And don't tell you daughter what I said - as long as she believes she needs to learn deeply, she would

And how can you hate economics? It's mostly math...[My research included lots of economic modelling ]

I Will not She is still torn between business and communications, albeit I liked a problems on antiderivatives--mostly thinking, a good one. Well, can not say much about business schools--not my pedigree--technological schools are different. And, well, how precise--a lot, indeed, depends on personality of a professor--very often this is a difference between knowledge and....the lack of it. As for Operational Research for me personally--still imprinted in memory as endless sequence of mnemoinc tables and does not still improve the mood As for the falling of a standards already in universities--difficult to say, my good friend Ph.D teaches math in UCLA, for example--well there are as many questions to students as there are for universities' academic environments themselves--but this is precisely the issue, which is being discussed here--MANY students, very many come into colleges being simply INadequate to even moderate intensity of a program, very often lacking basic things. As for physics??? I better not start here--this is a sore spot, especially for the age category 14+. I had to write a book on the introductory Physics myself in order to help them (mostly basic calculational issues on molecular structure and beginning of statics and electricity) and later to patch up introductoryy dynamics and electrodynamics with thermodynamics through Tipler's and, of course, Halliday-Resnick classic Physics--this one is a true masterpiece and I use it a lot, once they pushed beyond basic things. But again, I would rather refrain from assessments on this whole....disaster on this forum.

[babaika]

And how can you hate economics? It's mostly math...[My research included lots of economic modelling ]

Because my degree is in....naval engineering Touching smthng related to things which have Marginal in their titles--kills me

PS. Although--once in a while I love to crack my skull over some economics problems. But those are mostly on differentiation (extrema).