Four Thinking Styles of Perception/Process

[nyiregyhazi]

They had memorized and regurgitated math facts.  We'd go back to the concept level as it is supposed to be taught in gr. 1 & 2.  When they got that, then BOTH their algebra, and their basic arithmetic improved.  HOW we do things has a significant impact on how well we do.  Facts such as 2 + 7 = 9 need to be known.  But if you know:
2 + 7 = 9, therefore 7 + 2 = 9, therefore 9 - 2 = 7, therefore 9 -7 = 2, algabraically, meaning seeing the patterns, maybe picturing it as things being swished around - then you can make leaps.  It also becomes interesting, and interest is a great booster of learning.

I'm not convinced that the above supports the initial point, where you say:

"Most of the older students had problems with things like algebra because they did not have a true grasp of what + - x / actually was."

How can a child not understand what adding and subtraction is? If they cannot understand the concept of addition and subtraction to the extent that all of the logical follow ups to 2 + 7 = 9 can be derived, they have probably not even understood the baseline concept of numbers at all-  before even going into sums. Anyone who can fully understand the concept of numbers up to 10 should automatically have grasped all that any two numbers added together make the same result, regardless of which goes first. If they have not, I'd be amazed to think that they could have memorised all the basic sums, without noticing such a glaring fact. How can a child memorise all those sums, without grasping the meaning of the small numbers? It strikes me that either they're going to both, or that they're not going to be able to memorise the sums at all.

The only way to have memorised every calculation involving numbers below ten yet not to grasp what they mean is to have failed to grasp what numbers up to twenty (mostly up to ten or less) mean in isolation. I'm quite prepared to believe that the kids had problems, but your interpretation that they had memorised all these sums yet failed to grasp the most basic recognition of what numbers actually refer to just doesn't ring true. What you seem to be talking about is actually gaps in their understanding of LOGIC- not failure to understand what addition and subtraction mean or inability to translate the symbols to something worldly etc. It's dependence on worldly issues that leaves people unable to do algebra. You have to let go attempts to visualise everything as a quantity and use logic instead. That's about as far from real-world visualisations as anything could be. When I look at your equations I don't visualise a single quantity. Even when processing small numbers like 7+2 my brain uses a totally abstract process. Only when I had to hand somebody 9 of a given item would literal visualisation of a quantity come into the picture. Every other process is 100% abstract. If a kid has yet to realise how those equations follow from the first, grasp of logic is what is missing. It's nothing to do with understanding things in the real world.

The study seems to reinforce precisely these issues. It says that intelligent kids already know the results. They don't do any counting. Logically, it's very likely that the kids who can't do algebra would be those who are lost in counting and in trying to visualise quantities. Repeating needless counting would stop them developing the simpler logic that algebra functions around. I'm not questioning whether you managed to help those kids, but it's far more logically probable that they were TOO caught up in exactly what those symbols mean- instead of trusting a logical internalisation that has no need to visualise quantities direct.

[keypeg]

That is exactly what I am proposing, and this comes both from my work and training in teaching fundamental concepts at the primary level, and then helping students at the intermediate level whose problems seemed to stem from the primary level.  What can happen is that the young kids get fast forwarded into memorizing "number facts" through flash cards etc.  It's possible to do oodles of arithmetic homework and get right answers through the memorized number facts, yet never truly grasp what it all means.  That explains the problems with early algebra, and also why these kids with problems can recover so quickly once this hole is patched.

[nyiregyhazi]

That is exactly what I am proposing, and this comes both from my work and training in teaching fundamental concepts at the primary level, and then helping students at the intermediate level whose problems seemed to stem from the primary level.  What can happen is that the young kids get fast forwarded into memorizing "number facts" through flash cards etc.  It's possible to do oodles of arithmetic homework and get right answers through the memorized number facts, yet never truly grasp what it all means.  That explains the problems with early algebra, and also why these kids with problems can recover so quickly once this hole is patched.

I'm not convinced. I think you perhaps missed the crux of my point. If a kid cannot understand 5+4=9 and make his own translation  of what that means in practise, he has not yet understood the concept of the numbers 4 5 and 9 well enough. This is the most foundational ability of all. I don't believe kids memorise their sums without having first understood the meaning of simple numbers. I'm quite sure there's something else under the surface here. The association to reality theory doesn't ring true here at all. It could only make sense if the meaning of numbers aren't understood in the first place.

Also, the translation to alegbra stems from logic- NOT what numbers mean in the real world. I don't believe this is truly the hole at all. These kids couldn't have handed you 4, 5 or 9 sweets if requested? They almost certainly understood what was referred to perfectly fine. Even if the approach you used helped them, the explanation of why that happened seems improbable.

[nyiregyhazi]

Also, when I first learned algebra, the teacher merely told us that both sides of the equation must balance- so you must always perform the same action to both sides. That's all I needed to know. I never once needed to see what was going on in a less abstract context. The teacher could have replaced x with a digit to prove that it works in particular examples- but that doesn't either educate or change the logical procedures required to solve equations. It just reminds you that it does work.

In that respect, it's purely a trust issue, when it comes to bringing it to reality. If a kid can't do it, either they're not good enough at stepping away from that which they can visualise and learning to use logic instead of that, or they're too stubborn to TRUST that logic works. The reason I never found algebra hard is because I was never visualising real numbers in the first place, but instead using logical procedures. I was happy not to give a damn what x represented until logic derived it for me. I honestly think some people are just too stubborn to trust that logic will reveal it, without them needing to try to picture everything in their head. I never visualised quantities throughout any of the maths I did, algebraic or numerical. I trusted that I COULD translate any written digits to a quantity if I wanted to and then happily did no such thing. That side was learned by most kids before they got to school. Who ever lost the ability to count out up to twenty of an object, by not practising? It's literally just the ability to count like this. The quicker you depart from repeating a timewasting distraction that was already mastered by most five year olds, the better. The study supports that completely directly.

[lostinidlewonder]

wth are you guys on about? What has this got to do with this thread?

[keypeg]

wth are you guys on about? What has this got to do with this thread?

Faulty Damper had written about an article that he believed supported his conjectures about math. learning, and that's what I responded to.  Since the topic of this thread is "thinking styles", we had branched off to educational matters some time before.

It actually does have a lot to do with music or piano study, and my teacher and I have discussed math. often in relationship to music.  It took me a while to understand what he meant.  Music has patterns, and so does math.  There are mathematical patterns in music.  If you understand the patterns in music, then it is easier to play and understand music.  I was doing this when self-taught without being aware of it, because without theory and names, there is nothing to hold on to.  It looks like I'm not good at verbalizing it either.

For sure, however, how you perceive these things can affect your fluidity and ease.  For example, when students painfully memorize "Every Good Boy..." and hunt for notes, or fill out workbooks without making any real connections, this gives something different than when connections are made and real understanding is built.  A lot of times this is not purely intellectual.

Actually the insight of what my teacher meant came to me one day when a cashier took the large handful of loose change that I had, rapidly spread it over the counter in groupings without counting (if you have 4 quarter you instantly see $1.00 without actually thinking 25 X 4 = 100), had the total in an instant and then said "I love doing this.  I'm a mathematician."  All of the rote memorization that I had done as a child with flashcards flew out of the window, and in an instant I had a new way of perceiving math, and also understood what my teacher had really meant when he said that music was "mathematical".

[faulty_damper]

Most of the older students had problems with things like algebra because they did not have a true grasp of what + - x / actually was.  They had memorized and regurgitated math facts.  We'd go back to the concept level as it is supposed to be taught in gr. 1 & 2.  When they got that, then BOTH their algebra, and their basic arithmetic improved.  HOW we do things has a significant impact on how well we do.  Facts such as 2 + 7 = 9 need to be known.  But if you know:
2 + 7 = 9, therefore 7 + 2 = 9, therefore 9 - 2 = 7, therefore 9 -7 = 2, algabraically, meaning seeing the patterns, maybe picturing it as things being swished around - then you can make leaps.  It also becomes interesting, and interest is a great booster of learning.

It's not important to actually be able to count e.g. 5+2=7.  (Those numbers are meaningless anyway.)  What is important is what the numbers represent which are abstract ideas.  However, in order to get to abstraction, one must first know the concrete.  This is why knowing math facts are vital.  Then, once it is known, it is then important to manipulate them to know that 2+5=7, 7-5=2, 7-2=5, etc.  This is the development of identifying patterns, which is abstraction.



Here's how abstract thought comes about:
1. Concrete - 2. Manipulation - 3. Abstraction

It always follows this linear progression.  The inability for abstract thought is a direct result of a failing at one or both of the preceding.

It's important to follow this linearly because a lot of people will argue which is more important, e.g. knowing math facts vs. seeing number patterns, when in reality, they are discussing different areas of thinking.  This is just like "Whole Word vs. Phonics" debate in the subject of reading.



Tying this in with piano using the linear model:

(1)Learning how to depress keys - (2)experimentation - (3)improvising

The reason why most classically trained pianists can't improvise is because they didn't (2)experiment.  They only learned how to (1)depress keys in particular order that recreated Chopin's Nocturne but that's it.  

In contrast, when Chopin (1)learned how to play the piano, he (2)experimented so much so that he could (3)improvise for hours.



But back to mathematics:
The reason why students struggle at the (3)higher levels of mathematics is a direct result of failing to (2)manipulate or (1)learning math facts.

[nyiregyhazi]

For sure, however, how you perceive these things can affect your fluidity and ease.  For example, when students painfully memorize "Every Good Boy..." and hunt for notes, or fill out workbooks without making any real connections, this gives something different than when connections are made and real understanding is built.

While this is all true, I think it warrants a clarification- seeing as we've been talking about the issue of memorised facts. The problem here is not having memorised something. Rather, the problem is having memorised something  indirect that leaves a necessary CALCULATION to be performed via counting, before you can arrive at the piece of information required. If you know each note directly at a glance, you have genuinely good memory. If you only know how to count up and derive it using the mnemonic  you are doing the equivalent of the student who does 2+3 by forever counting 2,3,4,5  rather than by memorising direct results. I used this same counting analogy in my blog  post on the fundaments of musical reading.

Superfically, your example might appear to imply a warning against rote-learning- but if you consider beneath the surface it's actually an example of the perils of inadequate memorisation. This style of memorising leaves the brain with too much calculating to do rather than a direct memory to retrieve- just as is detailed in the article that was linked.

Actually the insight of what my teacher meant came to me one day when a cashier took the large handful of loose change that I had, rapidly spread it over the counter in groupings without counting (if you have 4 quarter you instantly see $1.00 without actually thinking 25 X 4 = 100), had the total in an instant and then said "I love doing this.  I'm a mathematician."  All of the rote memorization that I had done as a child with flashcards flew out of the window, and in an instant I had a new way of perceiving math, and also understood what my teacher had really meant when he said that music was "mathematical".

I think to refer to "logic" would be more accurate than maths itself, when relating to music.  Of course, logic is used in maths, but it clarifies what can otherwise seem abstract. That said, your example here is HUGELY founded upon memorisation of such logic. All of these things are built on having memorised facts about arithmetic. Then you use logic and then you go on to memorise the results of that logic. How do you know that four quarters is a dollar? Did you calculate that on the fly?

There are a few ways you could have done it. You might know that x/x=1. A quarter is 1/4 therefore your need 4 of them. You might visualise what a quarter looks like on a pie chart and count how many are needed. You might even be aware of your 25 times table and use that to get there. All of these things are founded upon having learned things by rote, applying them and then memorising the process until you get to the point where you can just do it, thanks to the memory of prior calculations. Most likely, you've simply memorised the fact that four quarters is a dollar via experience and didn't even think about the mathematical reason why. A person who doesn't even comprehend maths could do so- even if they only understand things via terminology rather than  by mathematical concepts.

Although you seem to be trying to frame your experience in a way that might contradict what that study says about fact retrieval over derivations from scratch, it actually supports it entirely.

[nyiregyhazi]

Also, it strikes me that these memory retrieval issues are at the heart of good sight reading of rhythm.
 
Let's take a very basic rhythm of a minim and two crotchets. Do I consider that the first lasts for two and the next two each last for one? Not at all. I have memorised that the first note lands on 1 and that the next land on three and four. The note lengths sort themselves out by my memory of that. Obviously, I have to be capable of making calculations as to what beats/subdivisions every note lands on and that requires consideration of the note length. You have to be able to view it from either side. However, once I know certain patterns I think not of the note lengths but of where they land in relation to beats. Inexperienced players have a tendency to fixate on the textbook length of each note- but don't necessarily tend to realise that they need to make calculations from these figures and then concern themself with learning where events fall in relation to consideration of EVERY beat of the bar.

Again, it's about using logical processes to decode things but REMEMBERING the results of calculations, rather than having to perform the same calculations over and over and over.  With time, fewer and fewer things have to be directly derived on the fly.