Math: Why memorizing procedures isn't enough

[domenice111]

sounds good to me, but when a test rolls around, I'm generally not given time to think. The problems on the examinations generally take about as long to complete, as the time allotted.

The goal of a student is to do as well on a tests as possible, therefore they often gear their methodology to best suit the test scenario. This means memorizing patteners, rather than comprehending problems.

The problem isn't the students'.

[jamesblackburnlynch]

@domenice111 I haven't taken your tests, but I do know my own and those I took over the years. It's much easier (that includes much faster) to do well when the student really understands rather than memorizes. Why? Because if all you do is memorize, you begin a problem having no idea what to do. You read the problem (maybe...) and then guess what method to apply. If you misremember some part of the procedure, then you have do redo the problem. None of this happens with understanding.

[jamesblackburnlynch]

@domenice111 By the way, the rule professors (I am talking college here) is that an hour test should take 10-15 minutes for them to do, all written out. Now that's not a reasonable amount of time for anyone who has not WRITTEN the test. When I go to do my own test, I already know what each question says and how the answer is going to go. I don't have to spend any time thinking "what am I supposed to do." But purely mechanically, it shouldn't take the full time.If it does, the test is too long

[huijingwu]

I agree with you 100% as that is my problem. I look to apply concepts but don't see the whether the question/problem makes sense. Then I get stuck as to where to begin. There has to be common sense when looking at problems. I was completely ignoring the common sense part and looking for the answer in the back of the book like you said. Thanks for this vid!

[studying]

Mastery of a skills necessarily includes understanding - if not, you haven't really mastered it.

Oh, and any fifth grader can do this in that time (fourth graders for the higher sections). Forcing students to abstract and 'understand' why on each step is so slow and inefficient - a better approach is more real-life problems and repeated practice.

Skills should come first, understanding can develop naturally later - same thing with languages.

[jamesblackburnlynch]

It is not that case that mastery of skills includes understanding. Many people can use the quadratic formula but have no idea when to apply it. Say you have the equation x^4+3x^2-5=0. Can you use the quadratic equation? How about x^2-2x+6=12?

And I don't agree with your conclusion about what comes first. Why? Because I have endless student evidence that it doesn't work. Understanding doesn't develop later for those that just memorize. Hatred of math does. Common sense is unrelated.

[studying]

You have way too narrow a view of 'mastery of skills' - no one would even use the quadratic formula first to factor those equations (and they can't be applied bec. ...).

I said master, not memorize. Yes, you memorize, but not necause someone told you, but because of necessity, of doing it again and again, of finding a better way of doing it, and why. Then round out your knowledge of the skills. As you progress to other topics, you learn to apply past lessons and it all connects.

[jamesblackburnlynch]

But the quadratic formula can be applied in both cases. The first just by making the substitiution y=x^2 (now it is all in y^2) and the second by subtracting 12 from both sides.

If you're going to define "mastery of skills" as understanding them and being able to do them efficiently, then I agree with you. But, if you do so, you will be confused in converstions with math educators and parents, because that is not the generally accepted definitio of "mastery of skills."

[studying]

Math hatred is cultural. And 'understanding first' is problematic in a couple of ways:

1. mathematical sense is painfully hard to acquire, and common sense is not the way to get it.

2. highly redundant/slow

3. Mathematics is highly circular and self-defining (2.5 is not 3, there is no mathematician that wouldn't accept this reasoning)

4. Often, you'd get incomplete skills (like not knowing how to factor your equations because the quad formula doesn't work).

5. slow problem-solving skills

[jamesblackburnlynch]

1. I don't agree at all. People make math seem hard, but it's not. It's true, there are hard parts, but we aren't talking about them. For example...logs...people make such big deals about them. They aren't hard to understand. No harder than division really. But they sure are if all you do is try to memorize a bunch of rules.

2. How is "understanding" redundant? It only is if you think math is memorizing skills. Then you might think getting at understanding is redundant.

[jamesblackburnlynch]

3. True.

4. I'm not following your example. If the quadratic formula doesn't work, then you can't factor. Do you mean that? I can't understand your comment here.

5. That can be true. As I've said repeatedly, understanding first, but then mastery of skills (with the usual definition). Without that eventual mastery, you will have "slow problem-solving skills."

[studying]

Crap. This is what the 'mastery of skills' approach is.

(2+3)/2 = 5/2 = 2.5, not 3. Easy, two to three seconds.

(x+4)/x = I don't know, because I don't know what x is.

Then the teacher'll ask: what is x then?

So cross multiply, move x to other side, get 4 = 3x; 4/3 = x. So the equation will only be true if x is 4/3. Easy, five seconds.

Did I think about it? - not really. Could I explain why I did it - of course (subtraction cancels addition, not division; or else cross multiply).

[jamesblackburnlynch]

Yes, the students could do what you did too. They have mastered these skills. But the point is they don't really understand them. And my proof, as offered in this video, is that when asked to not use the skills, just use common sense, they couldn't do it. They HAD to use the skills. They didn't realize that they didn't need them. This is caused by not understanding what the skills are for.

[jamesblackburnlynch]

The point of the lesson was to help them use common sense to figure out which rules are true, and which aren't. In 2^3*2^4 do you mulitply the exponents or add them? The point is that you don't need to memorize the answer. You can see that 2^3*2^4 is just 2 multiplied 7 times. So you add them. There are too many rules to memorize. Common sense (or simple examples, really) will help differentiate. But students just try to apply skills without thought. We are trying to help with that.

[studying]

Sorry I forgot to check up on my comments. That said, if they couldn't understand the relationship between addition and subtraction, they haven't really mastered it (or they forgot). The 'common sense', as you describe it, comes as you repeatedly do the tasks and understand the governing principle behind it all. Plus mathematics and common sense don't mix well, and is really misleading for kids who want to continue to higher maths.

[jamesblackburnlynch]

Okay, here is where you and I disagree deeply. Math an common sense mix together perfectly. If they don't, then something is wrong with the math. Sometimes, the math gets so sophisticated that the common sense gets hidden (and the math can go beyond it), but all math is based on common sense, because "common sense" is what the world actually is. And math tries to duplicate that.

Common sense doesn't just come from repeatedly doing something, it comes from observing the world.

[xghoulxx]

I think that if students are not able to grasp that simple of a concept, then they have to work twice as hard in order to keep up in a higher math course. I say this because math is a course were you have to understand instead of like what you said "memorize" when I first saw your problem I instantly broke it down in my head to 2/2+3/2=3 since you have to treat the numbers as individual cases instead of being lazy and slashing out a 2 while completely ignoring signs, and "common sense."

[augiedogp]

You teach this in college? Wow... Americans are really behind...

[jamesblackburnlynch]

Behind whom? Many folks, that's for sure. But this problem is common around the world. There's a growing recognition that many students need these kind of developmental classes in most countries.

[macnolds]

james, good point. however...you're forgetting one crucial thing: why should one delve so deeply into mathematics? students usually have 4-6 classes to worry about. each teacher believes his/her subject should get the most time. the student's goal is to get good grades, so he/she can get the required level of education necessary to be able to pursue a specific career. that's it. if one is already an A student, why sacrifice more time without any kind of tangible gain?

[jamesblackburnlynch]

I agree that these students (who were going to do anyway) needn't take this class. And, clearly, they could survive (and do well) without really being able to answer some very simple questions.

My point is: if the people who do well can't do it, then what about the people who are doing poorly? How far from understanding are they. What do they think they are doing when they are "learning" the material? How can we change whatever that is to a simple, commonsense way of learning math.

[freetothinkify]

a lot of children are afraid to think, or they want to follow the leader exactly. YouTube "chimpanzee vs human child learning."

[jamesblackburnlynch]

I know what you are talking about it. I saw a special on that research on Discovery a few weeks ago. Fascinating. I wonder what the kids would have done with that clear box (if you saw the special you know what I'm talking about) when they were a bit older. And what age were the chimps? Very interesting research regardless.

[bg2250]

soooo you're saying that by not teaching them procedures, we will be able to teach them to think. That makes no sense at all. OF COURSE they need to think. No one disagrees with that.

[jamesblackburnlynch]

Absolutely not. The title of my video is "Why memorizing procedures isn't enough" not "Why we shouldn't teach procedures." Procedures are important. Memorization is important. But neither are worth a thing without understanding.

The essential point that I keep making (and keep seeing over and over again in my work) is that many students think math is nothing but memorization of procedures. They think none of it has any meaning whatsoever.

[jamesblackburnlynch]

They never think about whether something is true or false using common sense. Because they think math and common sense have nothing to do with each other.

That is what I'm talking about. People shouldn't need to memorize all that much in math. Most of it simply makes sense. And can be recovered through simple examples. (Like is sqrt(a+b)=sqrt(a)+sqrt(b)? Students use that a great deal and never think to actually just check with a couple of numbers if it's true or not.)

[bg2250]

So why are you critiquing McDermott? Your comments are irrelevant to hers.

[bg2250]

Or did you thinkshe was advocating for meaningless procedures? She would be quite an idiot to do that, and i did not see that in her presentation at all. Quite the contrary, she is advocating for meaning, by making the meaning easy to see--with a simple procedure. Your comments are totally irrelevant to McDermott's.

[jamesblackburnlynch]

I'm not sure what your understanding of "totally irrelevant" is, so I can't really follow your comments here.

She is advocating focusing on the practicing of efficient procedures as the most important part of math education. The Saxon books she recommends are designed with this intent.

Again, the "standard algorithm" is NOT the simplest procedure. It is very efficient. But it's efficiency does hide some of the way it works.

[jamesblackburnlynch]

First of all, this video isn't a critique of hers. This is a separate video. I'm still thinking about all this stuff. I always do. MJ McDermott made that video long ago. I responded because I wanted people to hear some of the reasons those books were created.

She's actually not a particularly important part of this debate. I appreciate her bringing it up in a professional way that has interested others.

[wv05vq]

When you asked them whether it was true or false perhaps you could have restated the question, quite often when there is a lack of understanding you need to do this.

P.S. Math, and in particular exists outside our physical world so going to real world examples is not necessary, and quite often not easy if at all possible to come up with.

[jamesblackburnlynch]

I think your "P.S." might be missing a word or two. I don't really follow it. At this level of mathematics, I can't think of many things that don't have easy applications to the real world.

I would have liked to restate the question. In terms of money. I wish I had.

[wv05vq]

'and in particular your example (2+3)/2.

As for the first part I meant restating how you ask it... you asked "Is it true or not". I may have said, "Is it true or not, what I mean is that if i replace the numbers with other numbers will this cancelling always work, or does it even work here".

[jamesblackburnlynch]

I don't know. Your "rephrasing" seems quite a bit more complicated than the original question. I am giving them a very particular, simple example. You are asking them to think about what would happen if they change numbers to some "other" numbers. That is asking for quite a bit more than I did. And they weren't really able to handle just this specific case.

[honeybbqgrundle]

It's so hard to change the "math=memorization" mentality once they hit high school. My AP calc students choose to (try to) memorize the derivatives of tan,sec,csc, etc instead of just using the quotient or chain rules.

Sure there are a few people who can be helped, but realistically, it's too late by that time. I often think I could do more good teaching elementary schoolers, but I can only teach math so that won't happen.

[jamesblackburnlynch]

I agree that it's hard. Even the students who want to understand more are so in the habit of learning by memorization that they can't easily do it. But that doesn't mean it can't be done. The best way to help is to make it worth it. If we put problems on tests where memorization just doesn't work, then they will learn to shift.

They won't really believe you until the first test, but then they will. It's still hard to change, but now they have to.

[honeybbqgrundle]

That's a good point. I suspect it's a little easier in college though. If too many students fail my class, and they would if I made it more challenging, I may be out of a job.

It doesn't help that my school ignores my repeated requests for a pre-algebra class and shoves these kids who sometimes can't do -5-3 into algebra 1. If kids, or maybe moreso their parents, would just understand that it's not necessary for them to take AP calc in high school things might be better.

[honeybbqgrundle]

A lot of the memorizing mentality comes from kids who are in 3 or 4 AP classes and think they're going to Harvard or something. If they'd just stop for a second and think then they'd realize they don't have to do 50 volume of revolution problems because they're all pretty much the same.

But sometimes you get to have a lot of fun with them. One day I just derived the formulas for the triangular numbers, the sum of the first n squares/cubes, etc in a few different ways. They seemed to enjoy that.

[honeybbqgrundle]

I also showed them Archimedes' balancing method for discovering the volume of a sphere and a few of them seemed interested. Usually when they ask why I did a step or how someone would think of something I can explain.

But with that argument, when they asked how he ever thought of it, all I could say was I have no idea, that's just Archimedes for you.

[jamesblackburnlynch]

The bad news is that, if they want to go to college, it really is in their interests to take AP classes. I was just listening to a presentation by our head of admissions discussing what we are doing to recruit the best students. And one thing they do is look at the AP or college-prep courses students take. The GPA just isn't a reliable measure anymore.

I think the -5-3 thing is something many people don't understand. Negative numbers seem like more nonsense to lots of people.

[honeybbqgrundle]

Just wondering, did you say bad news from my perspective, or do you agree? I really wish the whole AP program would be done away with or completely overhauled. I have AP students who can't graph a parabola or solve |5x-6|<10. They shouldn't be in advanced math classes, but they are because they think they need to be to get into a good school.

I wish they would just learn the elementary material better before moving on. It would save time in the long run.

[jamesblackburnlynch]

Not being a high school teacher (nor are my kids in HS yet), I don't really have much of an impression of what you've said about the AP students. In my time, few people took those classes. I don't really know what it's like today. So, I meant it from your perspective.

But I do know those students. We have kids who got A's in Calculus who can't waive our pre-algebra class. They are so mad that we insist they take it, but they really do need it.

[bg2250]

That's like saying if youhaven't "memorized" the sound that m makes, then you'll be able to read, because the sound of m is big ol' creative deal in reading. Like multiplying is a big ol creative deal in math. Memorize that part, so you have some mental space for the truly creative part. Why waste brain cells on trivia you should have memorized by grade 3?

[jamesblackburnlynch]

Because there is an important lesson children need to learn about math.

That math makes sense. That it can be understood. That if it doesn't make sense to you, you just need to figure it out. It will.

This lesson must be learned early enough that it cannot be undone. 

If the first experiences in math are simply memorization of "efficient algorithms" that's what they will expect. And as it gets harder, students will understand less and less.

This is what is happening.

[sleeper2345]

What should students memorize? What do you mean by "memorize"? I don't think anyone is saying that certain information should not be memorized, but the question is how it comes to be memorized.

[Zahlman]

What they should memorize is what they already understand. This is the approach I grew up with (in a Montessori school), and in university (even in a rather challenging engineering program) I found that I almost never had to *consciously* memorize anything, because by the time it was necessary, I had already unconsciously done so - through my understanding.

[bakerstreet221]

I struggle with ideas about what we should teach the general K-12 population. All mathematicians are in a position to influence K-12 math ed so it matters a lot what our opinions are. At home with my own kids we do puzzles (like probably all mathematicians) to teach the building blocks of mathematics: logic, symbolic thinking, induc./deduc. reasoning, geometric intuition, those kind of things. But I'm not so sure a general K-12 math ed based on that is wise....

[bakerstreet221]

I think math instruction should separate computation (at the K-5 level is arithmetic) and problem-solving+intuition. Intuition is the wrong word, but I don't know what to call that feeling when you know something is wrong -- that subconscious way an experienced mathematician looks for certain patterns. Anyhow, computation (or algorithm application) != problem-solving. Do both? Can't because K-12 math hrs/day are limited. School boards have to choose. I don't know the answer...

[bakerstreet221]

Following up. IMHO one of the big problems with K-12 math education is that mathematicians don't teach K-12 math. So kids don't learn from those that who like math and who have a deeper understanding -- i.e. it is not just a bunch of algorithms. One of my most influential HS teachers was a physicist who left industry to teach math, but the rest of my math teachers seemed to be English majors ;-| Then again some of my worst teachers EVER were my math professors in college.

[jamesblackburnlynch]

Of course, I agree with virtually everything you've said. We come from the same background and probably have lived similar lives in many ways. (God knows some of my worst teachers were math researchers.)

Knowing the students who go into elementary education, and there feelings about math and how brittle their understanding of it is...it's hard to imagine them being able to handle doing much math "reform." And, they probably couldn't care less about it, lots of times.

[bakerstreet221]

I think math intuition is very hard to learn. For some minority, it comes easily but most not. You give the example of students trying to see if (X+3)/X is correct and getting lost in the algorithm -- rather than just sticking in X=2. I encounter the same problem all the time with my post-doctoral fellows (stats). They are very skilled at math -- better than me -- but routinely they get trapped 'missing the forest for the trees'.

[jamesblackburnlynch]

Is that really an example of "math intuition?" I think the point there is that the students do know (intuitively) that (2+3)/2 isn't 3. But since they don't think math should make sense, they don't bother trying to see if it does.

I do think intuition is very hard to learn. At a deep level. But I believe if we can just convince the students that math will make sense, if they think about it, most of the time they will have sufficient intution to do the level we are talking about.

[bakerstreet221]

You are right the (X+3)/X is not an example of lake of 'math intuition'. But I'm not sure it is an example that 'they don't think math should make sense' either. I see the same problem in my post-docs and myself -- and we are applied mathematicians. It looks more to me like a problem of the mind categorizing a problem and then getting stuck in that category. In the (X+3)/2 case, the mind sees an algebra or algorithm problem -- and it is harder to figure out the answer.

[bakerstreet221]

For example, last week I knocked myself out on a problem because I saw a multivariate Markov process with variable transition probabilities (that happens to be the sort of thing I work with a lot). I ran around in circles for a week trying to come up with unbiased fit to my dataset. Finally, I stepped back and realized I could recast the problem as a cohort model and had the analysis wrapped up in an hour. That I think is a version of the (X+2)/X near-sightedness problem.

[jamesblackburnlynch]

I agree completely. I think we are just using different terms to describe the same thing. To me, the idea that "math doesn't make sense" is WHY they get caught up in the mind categorization that this is simply an algorithm. Because there is no other option. When faced with a math problem, you are supposed to go find the right formula or process. No thought is needed. The formula does all that for you. Your job is just to memorize the formulas. And guess the right one to use.

[giovabao]

your students are not very....bright

[ehal3386]

Oops...I feel, not feet, like switching colleges...ehem

[ehal3386]

Wow, so far away from CA! Anyway, I checked the wikipedia entry for your college and I thought it sounded really cool. I feet like switching colleges already..haha;-)

[ehal3386]

I like you , James. You make me think^-^! I was brought up to learn mathematics through rote learning. Now I don't think I used my brain that much when I was young. Where do you teach, btw? I am now a college student in UC Davis, California.

[jamesblackburnlynch]

I teach at Berea College in Kentucky.

[snakeinthegrass20]

Ha! this is great. I'm re-doing my maths education before I start my science degree in Sep. I'm learning number theory and then going to move on to Saxon algebra 1/2.

[jamesblackburnlynch]

What do you mean by "number theory?" To me, that is a high level abstract mathematics subject. It would be strange to move on to Saxon after that. Are you referring to arithmetic?

My impression of Saxon is not good.  It is the Saxon methond, I believe, that memorization is the key to success in math. That's exactly what I'm arguing against in my videos.

To be fair, I haven't actually seen the Saxon materials. This is just my impression from what has been described.

[rostromojado]

The Saxon method is that "practices makes perfect." Students do not grasp a concept on the day it is first encountered. Mathematics is learned by *doing* and that is what the Saxon method emphasizes. Once the students have a firm foundation in the fundamentals of mathematics, learning the theoretical aspect of mathematics should be easy.

[jamesblackburnlynch]

I wish it were so. If it were, we wouldn't have the problems we do today. But, from my experience, what happens with most students who are taught by the "practice method," is they internalize the notion that mathematics is just something to practice. It doesn't actually make any sense (in a "common sense" way), so there is no point in every wondering if it is "true" or not. The example in my video above is typical.

[rarrzero]

The Saxon method definitely does divorce math from common sense.. I, too, fell victim to the "(x+4)/x = 4" fallacy for quite a while after I was introduced to algebra due to a procedural elementary education.

There are a lot of parents who worry that reform math somehow harms the very bright students; I don't see how this can possibly be so, because it extends the way those bright students learn math to everyone, and makes explicit the mental shortcuts commonly used by numerate people.

[jamesblackburnlynch]

Well, if you understand mathematics to be a series of formulas that simply need memorizing (which many people seem to think math actually is), then "understanding" the formulas is just a waste of time. Just stop being lazy and memorize the stupid things!

We just believe that math actually makes sense and this memorization system is actually very inefficient.

[archAntebullem]

Im a junior in Algebra 2 . Do you have any study tips ? Reason asking, I study a minimum of 1 hour a day , but still only mange to accomplish Ds on my tests.I can never picture myself doing well on said tests and i always get paralyzed mentally when im quizzed. I can't say that it was teachers fault because i primarily learn from the book assigned, also I usually do fine on the book's problems at home(i check my answers from the back). Would you say that test anxiety is a likely factor?

[jamesblackburnlynch]

Studying one hour a day should be more than enough to do better than a D. What methods are you using to study?

Do you ask questions in class the moment you are confused? Math is the kind of subject that you should know if you know it or not. If there is anything that doesn't make sense, then you don't know it. That's when to ask. Don't wait until you have so many questions that you don't even know what your question is.

[jamesblackburnlynch]

Test anxiety certainly sounds like a big problem for you as well. But the only way to really get over that is to be absolutely sure you know the math. To do that, you have to be very honest with yourself and recognize: 1) what you really understand, and 2) what you don't. Don't overlook #1. That's just as important. You can build on those things and use them to deal with #2.

Do you get paralyzed mentally when a friend quizzes you or just a teacher?

[archAntebullem]

I mostly stress in class. Looking back at old tests, I can tell I was over focused on things. I suppose I put strain on my tests because math is something I really enjoy and its a field i want to look promising in . While in class I want to look the best with the highest scores but at the same time I also fear that I wont be able to do that . So in other words I want be good at it but I fear that I won't be able to. Does that make sense?

[archAntebullem]

With the issue broken down it can be said that I cant do as well of math in class than outside because i set Leibnz standards for myself that i know i probably cant achieve. What Im really looking for is the promise that i will understand through my efforts. I guess i just lack humility in the subject , which is something that I have been applying and I feel alot more comfortable in class these past two weeks.

[jamesblackburnlynch]

Glad to hear it.

[jamesblackburnlynch]

It does make sense. But if you get D's on tests, do you really have the highest scores? I think I might be missing some of your point.

But certainly the reason why math anxiety is so high is that people connect it with their own sense of self/intelligence. This is really true of any kind of test when it matters.

Still, the goal is to know the math so well that, even nervous, you can still do it well.

[WindCorsia]

Happy to have teachers like you! Still, especially in math, sorry, you can't teach people to think. It's an internal light bulb that they themselves have to initiate. Also, you're too specific on what young people aren't doing, but that's true only when compared to your own position (what you can do).

Your experience accounts for a lot of the wisdom that's still missing in them. Give them time, some of them will eventually see the light.

[jamesblackburnlynch]

It may be that you can't "teach" people to think, but you certainly can encourage. You can even make it so they have to think or not succeed. This is as opposed to encouraging them to never think. And allowing them to "succeed" without ever thinking.

[Idontlikebanjo]

Its all about logic and working things out in a logical fashion , the younger generation are losing this type of skill due to the computerisatiion of everthing, ( the computer says no mentality) they are not letting their mind do the questioning, its also about lazy parenting and teaching, although not one particular factor on its own is the cause, its a multitude of things. I also think we live in a society were math is becoming less and less important in everyday life.

[jamesblackburnlynch]

How is math less important in every day life? Specifically, what did people need to do, mathematicall, 10, 20, 30 years ago that they don't need to do today?

As for "logic and working things out," you do recognize that these are exactly the skills required to interact with computers, right? "Kids today" can be very savvy with these things. But only in a context that they recognize.

[Idontlikebanjo]

math is becoming less important as we now can fire numbers into a computer and it will do the crunching,( evolution i suppose). Lots of education is now done via a pc, and this pc can do all the tasks that used to be performed by pencil, eraser and the grey matter, and this promoted questions in the individuals mind of, how & why.

The importance of maths is lessening due to the fact we can utilise other avenues to find the solutions eg pc`s, internet etc. enjoyin your guitar lessons ;)

[jamesblackburnlynch]

Always happy to do the guitar lessons.

But how is what you are saying a new development? Calculators were able to do most of what you are saying for 40 years. Computers have been doing it well for nearly 30.

[Idontlikebanjo]

i know its no new thing, but the methods and speed at which processing the data has changed significantly thus making access and use more widespread.

[jamesblackburnlynch]

While I agree that what you are saying is true, I don't think it really has had an impact on how math is used on a daily basis for most people. 30 years ago people weren't using the quadratic equation in their lives and they still aren't today. They were using statistics (whether they knew it or not) and still are.

To me, that's not the issue we're seeing that's causing the phenomenon in my video. that was there 40 years ago.

[crawhip2]

it's the SAME people who cause road accidents at busy intersections, they just don't THINK!

[ernieelsrulez]

This is an interesting way of quantifying a problem some people experience. I am in a high level high school math class and some students cannot see past the "procedures" to the actual applications. In my opinion it is crucial to master the theory/concept/applications before you can start attacking a problem. Not only does it make the problem easier, but it allows you to know what your answer means and vindicates all the effort put in.

[ernieelsrulez]

by the way I enjoy your guitar work and appreciate your lessons

[jamesblackburnlynch]

Thanks.

[jamesblackburnlynch]

If only I could convince students that it's easier this way. But I guess it's not easier if you have to learn a completely different way of approaching all math. It'd be worth it in the end, but the original cost isn't that small.

[ernieelsrulez]

yes, it was a realization that I had to make on my own. I had gone through math (my best subject) easily until I hit AP calc 1. The pace was so fast that you had to get the concepts befor struggling throught the difficult problems. I think the realization helped because my grade went from a 78 in the first quarter to a 96 in the 4th quarter. It is almost impossible to teach this unfortunately...I think each person must realize it for themselves.

[jamesblackburnlynch]

Well, hopefully, we can put students in a situation where they can come to that conclusion on their own more easily. And earlier than calculus would be nice too.

[crawhip2]

You'll find the strangest people in this world , I'll tell ya...these are probably the same people who KNOW how to socialize but when it comes to 'tech'...it's 'sorry i think i'm inthe wrong class'...i can write a book about these people and they think that the 'quiet' people are nerds! hahaha totally untrue....(It's probably just a phase that they're going through...or they weren't hit over the head by their parents, spare the rod spoil teh child hmmmph....)

[wii041]

Very good point. I'm still in high school, in an upper level math, but I (like the students you mentioned) sometimes find myself not thinking first and trying to apply a procedure instead. I completely agree that it is fundamental that we are taught the thought process first (to understand a problem) then the procedure later (to solve the problem). Anyways great video! 5/5

[jamesblackburnlynch]

It's interesting that students say this themselves. The students in the class I'm talking about in the video really want to do more "thinking" as well. They are dissatisfied with just memorizing math, doing okay in classes, but feeling like they don't really understand what they are doing.

And, still, they can't just choose to change. Old habits cannot simply be changed. Particularly when there's so much history to relearn.

Thanks for watching!

[crawhip2]

I used to have students like that in my senior hight school classes, they knew how to dress, and act more or less like grown-ups but they were not trained how to think 'logically ' at an early age. it takes them years to get their drivers license even because of their lack of thinking skills.

P.S I'm not Jerry Seinfeld -I'm not trying to be funny either.....