The experience of early childhood education suggests, and brain research confirms, that they way you *think* effects the way your brain develops. The process of myelinisation in a child’s early years (i.e., the process whereby the brain grows a fatty sheath around developing nerve fibres to better transmit its neural pulses) almost literally cements in the child’s way of thinking, and the “filing system” his brain is developing as he grows and explores his environment. This development of the brain doesn’t cease at five or six, it continues right up until we’re twenty-four or so.

It’s a two-way process. To paraphrase Winston Churchill, We build our brains, thereafter they build us.

California neuroscientist James Fallon studies the brains of psychopathic killers; people with badly built brains—who almost literally have got something missing. But what he also discovered was something “about libertarians that—just might—keep them peaceful.”

And oddly, the part that might keep them peaceful is precisely the part that neuropsychologist Steven Hughes and Montessori researcher Angeline Lillard identify as the very part that Montessori education builds up (and that nannying and “helicopter parenting” discourages): the executive function.

How ‘bout that.

Interesting. Sorry if I've missed you saying before, but can you recommend a good layman's level book on neuroscience. I've read a bit of Domasio but found it dubious from an epistemological/psychological perspective.

ReplyDeleteHow about feeding the learner (i.e., a child who is still in a primary school level) knowledge that they were regarded as too early by education experts? Such as teaching them differential & integral calculus or physics equation of motions and kinematics (translational & rotational classical mechanics)? Will such advanced learning effort develop their brains or not? If not then why not?

ReplyDeleteI say this because this is exactly what I am doing to 9 year old kids. They're currently learning integral & differential calculus. They can solve simple double integration (function of 2 or more variables: in multi-variable calculus). Multi-variable calculus is a topic that is only taught at final year level University calculus courses, but 9 year olds can already solve (by hand) some introductory problems in multi-variable calculus such as square-bounded double integral problems.

Examples in certain calculus topics (yep, there are more to calculus than just single or double integral/s) are shown to them first, and then followed by problems to be solved by hands (to make sure they understand the calculation rules/methods first). I let them mark their own answers by using Maple, which is one of the best commercial CAS software (computer algebra system) that's available today. The good thing about Maple software is that it shows all the middle steps of solving the problems up to the answer.

That means that one can type in any mathematical problem in algebra or calculus and then Maple solves it step-by-step up to the answer. The user can see which step that he got it wrong, because in algebra or calculus, once you got an error in any of the middle steps, all steps afterward are wrong (including the answer). So, I let Maple do the tutoring (by showing them which middle steps/procedures that they got it wrong) in their problem solving. This is good because, they don't have to wait for me to give them problems to solve. They can make up their own. Attempt to solve those and check them in Maple.

I have seen that they can create their own problems randomly (i.e., random equations that just pop out from their heads) in which they first solve by hand, then they can confirm what they're doing on the whiteboard is correct or wrong when they type the same problem to Maple to solve. If the answer for a specific problem on the whiteboard is not the same as that shown by Maple (after inputting the problem into it for checking), then they can click the student mode in Maple where Maple will show all the middle steps including the answer. They can quickly see where the mistake was made (in which middle step).

I told them that Great Physicist as Richard Feynman started learning higher level math at about 9 or 10 years old, not because of his own volition to do so, but it was introduced to him. I bet that he wouldn't have voluntarily done learning calculus at that age of 9/10 without him being introduced to it. His parents were both university professors, so one can inferred that his family wanted him to achieve higher level learning at a younger age which would give him better opportunities when he reached tertiary education level. By the time Feynman was 15 he had already mastered differential & integral calculus (according to wikipedia).

So, I believe that children's learning can be enhanced by introducing them topics/concepts that are too advanced for their age & level of understanding (perhaps according to so called education experts). This will help them develop their brains.

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ReplyDeleteI call my teaching method for 9 year old kids,

Irag Surge Invasion Strategy(ISIS). I didn't set out at the beginning of this year to call it that way, but I had been taken by surprise myself of how fast they can absorb/grasp math concepts as weeks passes when I introduce them to new topics every week. I have surged ahead and taught them University level calculus problems such as double integration and partial derivatives, without them knowing geometry and other topics. Well, I have only touched on geometry once (i.e., finding angles of triangles, classifying triangles into isosceles, scalene, etc...) and I intend to go back for more thorough tutorial on geometry because it is important.See, the US forces surged ahead to get to Baghdad in their 2003 invasion thus mostly ignoring other cities, because they thought that once they capture the difficult heavily defended main city and Baghdad fell, the rest is history. Other cities would surrender if Baghdad fell. Well, to be precise, that's exactly what happened. What I have noted is that when simpler concepts are being introduced (e.g., finding roots of quadratic equations), it took me less effort in terms of explanation, because I think that their brains got used to be bombarded with more complex uni-variable & multi-variable calculus equations and when they see something more simpler as quadratics, they don't get intimidated at all but grasp it fairly fast.

I believe that once the kids are introduced to something complex (at young age), then anything simpler that those concepts will be much easier for them to understand or grasp. I definitely know that when I am going to introduce geometry, it would be a walkover; there is no doubt in my mind about that, since I’ve seen how fast they grasp problem in solving quadratic equation roots this week which is a new topic that I have just introduced.

Someone from a NZ group called the Brainwave Trust www.brainwave.org.nz, gave a talk at our son's Montessori preschool on this. They were founded to educate people on this very topic.

ReplyDeleteBasically if a child is not exposed to the right experiences and/or is abused from 0-3, then at best they will never reach their potential, and at worst end up psychotics.

For the first few years the human brain has a near infinite number of potential connections; too many to take into later childhood. What evolves is that only the connections which are used will survive and strengthen (in a process called mylenisation); the rest will die out from misuse.

From what I understood it is possible to reestablish those connections in later years from conscious learning and training, but it's a lot more difficult.

Peaceful libertarians?? Can you take us through your passionate support for the invasion of Iraq and Piekoff's enthusiasm for nuking Iran and bombing the "ground zero mosque" again? Yeah you guys love peace and non-violence. How 'bout that.

ReplyDeleteJudge Holden

@Judge Holden: You're confusing pacifism with peace-loving.

ReplyDeletePacifism in the face of aggression is not a recipe for peace; it's an open invitation to aggression.

That's a point that's been well exercised here---even if you've yet to grasp it. Pacifism kills.

@Jeff: My consultants on this recommend these two:

* 'The 21st Century Brain,' by Steven Rose; and

* 'What's Going On in There?: How the Brain and Mind Develop in the First Five Years of Life' by Lise Elliot;

along with all the reading recommendations at Steven Hughes' site, www.goodatdoingthings.com

@FF: AS we talked about over the weekend, it's all very well teaching nine-year-olds calculus, and there's no doubt that they can do it--what might be in doubt is whether they can understand what they're doing down to first principles.

That's the thing about the hierarchy of knowledge--the most neglected issue in education. There are many things a student needs to have understood and integrated long before he can understand

whyhe's doing calculus, and what his manipulations mean.If he has, then great. If he hasn't, then he's just a well-trained monkey. ;^)

PS: I can't point you to specifically what subjects and concepts a student needs to have learned, understood and integrated long before he gets to calculus (but I'll bet among them would be complex algebra, binomial and trinomial equations, functions, change & rate of change, infinity, areas, graphs & slopes, parabolas and hyperbolas, mathematical series etc.) but you'll get a feeling for what I mean by seeing the hierarchical series David Harriman has sketched out for teaching science, in his Periodic Table of the Sciences--to be read from left to right, and from bottom to top.

@Falafulu Fisi: You should really ask to have demonstrated to you the Montessori binomial cube and the trinomial cube.

ReplyDeleteYou'd be amazed what they allow a five-year-old to understand (when they're first presented) and, later on, what they allow an eight-year-old to do.

PC said...

ReplyDeletewhat might be in doubt is whether they can understand what they're doing down to first principlesYep, derivative first principle (DFP) has been introduced. Well, solving the symbolic definition of DFP is something they can do, i.e., manipulation of DFP formulas to get the answer. The understanding of the whys & the hows is something that they haven't fully understood yet, but they will grow into it as their knowledge grows (simply by doing lots of exercises). Why? I haven't taught them function limits, which one needs to understand before understanding DFP. This is the reason I called my method

Iraq Invasion Surge, because I concentrate on trying to embed knowledge about rules in certain math topics (as function differentiation & integration), without getting bogged down too much on learning other parts/knowledge-nuggets that they need to know first (as what you yourself calledknowledge integration).The question to ask is if my teaching methods of trying to integrate knowledge in a "top-down" manner compared to a "bottom-up" manner, which is what you're arguing about. This means that I teach them to know higher level math concepts first even before they understand the lower-level stuff that they must have integrated to form those higher-level concepts/knowledge. Is there a deficient in my method/s or worry about

knowledge integrationshould be done in a "bottom-up" manner. My answer is: No. None at all. I believe (anecdotal) that the surge learning method (try to get to higher-level knowledge/concepts as fast as you can but that depends on your students' learning abilities + better teaching methods) which is a top-down manner, is the best.See, the binomial cube manual activity that you quoted from the Montessori website above, is no different from what I am doing. Here is a description from the link:

PURPOSE:The child is at the stage of the absorbent mind. She child is not asked to understand the formula, but is using the cube in a mathematical way. The child will build up a predisposition to enjoy and understand mathematics later.Now, can you see that it is exactly what I am doing? Get them to familiarize with mathematical Rules first, then introduce first principles later (or perhaps they can be taught at the same time or one after the other). The only difference in the Montessori is that I do surge teaching (spearheading in introducing higher level stuff first before coming to first principles later which is vital that they must understand - as you said

knowledge integration).[Continue...]

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ReplyDeleteBut if you insist PC that first principles must be understood by the students first (which is something that I by-pass because I race to get to Baghdad before the troops lost the drive to fight or before the students reach saturation level in their brains to absorb any new math concepts or knowledge given/introduce to them), then tell me where is the first principles of binomial expansion in Montessori teaching?

I have introduced the first principles in binomial expansion, but that came later after they understood polynomial differentiation & integrations. In bottom up manner (i.e., the way they're being taught at public school), students need to know binomial first before learning calculus. The reason I introduced binomial expansion recently is that I needed to introduce them quadratic & cubic equations. Again, quadratic was introduced to them after my students already know how to solve function differentiations & integrations.

Here is how I did it:

Expand : (2x + 3)(-5x + 2)

Split into:

2x(-5x + 2) + 3(-5x + 2)

-10x^2 + 4x - 15x + 6

-10x^2 - 9x + 6

It doesn't matter if the binomial expression was : (2x + 3)^2 or (-5x + 2)^2 or something else, the identity method of additive distribution (first principle) is applied. Throw them any binomial or cubic they will apply the additive distribution principle to it. I haven't told them that the method is called addition distributive law, because I don't want to bombard them with too many words, rather they should concentrate on symbolic manipulations of mathematical symbols. They'll know about the distributive law at some stage, no doubt, but that is less important as this stage, Baghdad is the crown and not Tikrit/Mosul (getting them to understand the rules of binomial expansion is more important than knowing what's the first principle used in solving it call).

Giving manual based activity to demonstrate mathematical concept/s (exactly as the

ReplyDeleteBinomial cubethat you quoted) is how I started at the beginning of the year. Integers is a topic that is not taught untilintermediate or highschool year level which 9 year olds won’t be introduce to yet. At the beginning of the year I had a laminated number-line sheet running from the interval -100 (left) to +100 (right). So, here is how I did it:Eg #1) Calculate: 9 - 14

Steps:

------

a) Position the highlighter (ie, marker) on the first number (the starting point) along the laminated number-line sheet.

b) If the sign of the second number is positive, then move the marker to the right by that number of units (specified by the second number) on the number-line. Where one stops is the answer.

c) If the sign of the second number is negative, then move the marker to the left by that number of units (specified by the second number) on the number-line. Where one stops is the answer.

So, to solve the problem in Eg #1 ; 9 - 14 , then the following steps are applied (this is a manual activity)

=> position the marker at 9 as starting point

=> move the marker to the left (since it is a minus/negative) from the starting point (ie, 9) by 14 units

=> the 2 actions above come to a stop at -5

Now the interval in the number-line (from -100 to +100) is limited where generalization is impossible, because the answer to calculation of large numbers such as:

631 - 874

can't be done as a manual activity since the number-line's interval is restricted (from -100 to +100).

Since, the concept of -ve and +ve has been introduced (well, that's first principle, there is no number concept that's more fundamental than -ve and +ve integers), I then show how to generalize addition & subtraction of integers, ie, any integers (it doesn't matter how large or small).

Here are the rules for adding/subtracting integers when they’re aligned in a column:

Rule #1) Always put the larger number (regardless if it is –ve or +ve) at the top of the column (which is row 1) and the smaller number underneath (ie, at row 2).

Rule #2) If the bigger number is +ve, then proceed to add/subtract in the normal way (regardless if the smaller number underneath at row 2 is -ve or +ve. Well everyone knows how to ad or subtract 2 numbers.

Rule #3) If the larger number at the top of the column is -ve, then change the sign of both numbers to their opposite (+ve changed into –ve and –ve being changed into +ve) , which makes the top number become +ve. Next is to put a –ve sign at the bottom space where the answer is to be placed after the calculation. Proceed to Rule #2) to find the answer.

Now I’ll give some examples,

Eg #2 : 867 – 472

Obviously Rule #1) and Rule #2 are applied.

Eg #3) : 472 – 867

Apply Rule #3 first, then find the answer using Rule #2

-867

+472

-------

?

+867

-472

-------

- ? (use Rule #2)

Eg #2 : -867 – 472

Apply Rule #3 first, then find the answer using Rule #2

-867

-472

-------

?

+867

+472

-------

- ? (use Rule #2)

Now, they don’t need a number-line sheet now to do addition/subtraction manual activity.