concepts- not for everyone

[regentrude]

Interesting example about the equal's sign!!! BAsed on what my Aunt tells me about the level of understanding of high school students she teaches, many, many of them what make the very error reported in that article.

 

I observe the same kind of stuff in my college classes. These are students who have gone through calculus and they still do those run-on expressions with equal signs where the sides are not equal.

This is something I am instilling in my 6th grader right now.

[Capt_Uhura]

Regentrude - my Aunt is also teaching pre-Alg to community college students. They do not have the fundamental knowledge of place value. It's very sad..... to be paying college tuition for these remedial classes.

[forty-two]

There is a great video out there where recent MIT engineering school graduates are asked if they can light a light bulb with a wire and a battery. Only one out of many (a hundred?) was able to succeed. This is the importance of concepts.

 

:iagree: This is me when it comes to science :glare:. All this theoretical info swirling around in my brain, but I'm entirely incapable of applying it to anything outside of the narrow parameters I learned in class. Because I *don't* get the concepts. Whether because I wasn't taught them, or I just was unable to extract them from the practice problems, or I just didn't do *enough* problems - I was a math/science student through college, and I can't even begin to use any of that to solve the simplest practical problem :glare:.

 

You can get along in real life without knowing anything about history or science or geography or literature, too — does that mean they're not worth studying? Many people seem to regard math as merely a skill rather than an academic subject like history or science. There is just as much depth and richness to the study of math as there is to the study of biology or classical literature or medieval history. Sure, as long as an adult can read a newspaper and balance a checkbook, they can "get along in real life," but I don't think anyone here on the WTM boards wants to settle for that.

 

Different parents will emphasize different subjects, and for some, teaching the basic math skills needed to "get by" will be as far as they want to go. But for those of us who see math as much more than just a skill, that's like saying "As long as my child can name most of the presidents and has memorized a dozen or so dates, that's all they need to know about American history. Half the population isn't capable of learning more than that anyway, so what's the big deal?"

 

Exactly. Classical ed is about studying the good, the true, and the beautiful, for its own sake and to strengthen the mind. I want to teach math in such a way as to show the truth and beauty of it.

 

I have absolutely no debt and understood compound interest. I like word problems. They are fun, real life problems. I have no trouble solving them.. I don't know. I mean I read stuff like :

 

<snip overly complicated mathy explanation that I wrote in another thread>

 

I'll be honest, I don't understand what she said in that at all. I mean I teach the ones, tens, hundreds etc. Is that what she means?? You have to add the same kind of numbers together... I just don't get it. You just add it. Why does it have to mean anything??? If I have an army of 230 and I have 570, then I will have 800. You just add....

The gist is that there are multiple levels of why. It's not just a choice b/w teaching by rote or explaining "the concepts". There are layers of concepts, depends how far you want to delve into the subject (or any subject). The place value vs field axiom example I gave was (trying :tongue_smilie:) to contrast two different layers of why. And the field axioms aren't the end-all, be-all, either - you can derive them with a different axiom system.

 

Teaching hundreds, tens, and ones would be teaching place value. And, really, if all you want to do is calculate, place value is all you need. Not everyone *cares* about why things are the way they are :tongue_smilie:, or not to the same depth, at least. I mean, I am perfectly happy with the standard field axioms - I'm not losing sleep wondering where those axioms came from, kwim :D - but other people clearly felt differently, and decided to figure it out. And that led to lots of interesting (so they say :lol:) discoveries.

 

Anyway, the point is that there is way more to "conceptual math" than just place value. And there is way more to it than the field axioms, too. But imo, teaching to the level of the field axioms gets to actual math, while only going to the level of place value doesn't. And that matters to me. But it obviously doesn't matter to lots of other people ;). If you don't care, and you don't *want* to care, then don't worry about it.

[Storm Bay]

:iagree: This is me when it comes to science :glare:. All this theoretical info swirling around in my brain, but I'm entirely incapable of applying it to anything outside of the narrow parameters I learned in class. Because I *don't* get the concepts. Whether because I wasn't taught them, or I just was unable to extract them from the practice problems, or I just didn't do *enough* problems - I was a math/science student through college, and I can't even begin to use any of that to solve the simplest practical problem :glare:.

 

.

I agree with much of this, but shortened it to save space. We need practical as well as conceptual! But there is no one formula that is best for everyone.

[bbrandonsmom]

Has this already been mentioned-in LM's book, she mentions that it's better to know more than one method. She has an example where in one class, the three groups came up with 3 different methods to solve an addition problem, and the teacher then they all talked about what way might be better (usually it's the standard way), but for some people another way is better. It's knowing the different ways to solve a problem.

 

That said, my aunt was a math professor-so in her generation learned math from 30-40 yrs ago. We were talking the other day about Singapore and various ways to solve a problem. She said straight out that wasn't the way she learned, but can see how useful it is. She learned some of the concepts in college I think.

[Poke Salad Annie]

This may have already been shared, but I recently read an article that argued that this misunderstanding of the equals sign is a major problem with American math education.

 

http://educationresearchreport.blogspot.com/2010/08/students-understanding-of-equal-sign.html

 

 

This is why I am teaching all of our problems from MEP or any other source I use on the dry-erase board to show that the equations we write must maintain the integrity of being *equal*, by use of the *=* sign to show the RHS and LHS. I teach early on that whatever is added, subtracted, etc. to the RHS of the equation, must be done to the LHS, or the whole equation becomes "wonky". I illustrate this by physically demonstrating how my arms would be one higher than the other, such as a balance scale would be out of balance. I demonstrate the equation in balance by standing with both arms out at my side, as a balance scale in balance would be.

 

When I write a problem on the dry-erase board, I set it up just as I learned in my algebra class in jr. high---that's where I learned the foundations for equations--and I feel I need to teach at that level to develop understanding from my child. We set the equation up, then work it sequentially, bringing down each step and writing it in, making sure to keep both sides equal (I stress that the *=* sign is the controller of the problem). I hope I'm making sense here, but I just wanted to explain how we use this in our work, and I can already see the benefits. I have used this since Year 1 in MEP, and there is no confusion about the *=* sign.

[LimitBreak]

Here's a link to a mathematician's opinion about math and advancement in an industrial society. One of his points is that mathematics will be outsourced in the future and that we should teach kids to be innovative in math in order to compete:

 

 

"For many years, we have grown accustomed to the fact that advancement in an industrial society required a workforce that has mathematical skills. But if you look more closely, those skills fall into two categories. The first category comprises people who can take a new problem, say in manufacturing, identify and describe key features of the problem mathematically, and use that mathematical description to analyze the problem in a precise fashion. The second category comprises people who, given a mathematical problem (i.e., a problem already formulated in mathematical terms), can find its mathematical solution.

Hitherto, our mathematics education process has focused primarily on producing people of the second variety. As it turned out, some of those people always turned out to be good at the first kind of activities as well, and as a nation we did very well. But in today's world, and the more so tomorrow's, with a growing supply of type 2 mathematical people in other countries - a supply that will soon outnumber our own by an order of magnitude - our only viable strategy is to focus on the first kind of ability, and hope we can hold our own in that category.

In other words, the only mathematical niche I can see for the US - and, fortunately for us, it is a crucial niche in today's world economy - is at the innovation end. Fortunately, innovation is an area where we still lead the world, in large part because our political system allows and rewards innovation, and also because it is very much a part of the American character"

 

 

 

Here's the link: http://www.maa.org/devlin/devlin_07_10.html

[Nan in Mass]

So that (the first) is what the infamous Everyday Math is trying to do. And probably succeeding, although at the price of the second.

-Nan

[Colleen in NS]

Hmm...ok, I find this personally convicting. I'm all about wanting my children to read classic literature, memorize poetry and scripture, be aquainted with beautiful art and music, study history...and I never, never, never looked at math in that same light. I was able to do math in school, but always lookead at it in the light of how do I get the answer. Just show me what to do. I could do it, but it didn't click with me. I didn't connect with it. Do you know what I mean?

 

But wow, if I look at it like this, that's an entirely different perspective. And I think, actually, that's something I could get excited about.

 

Thanks for that little epiphany.:001_smile:

 

I think I'm there with you! I keep revisiting the whole math thing in my mind every few months, esp. when these threads come up. I wonder if it's sort of like when you memorize history dates/lists/events, learn how to plot on a timeline, learn how to narrate or outline something in a history book, but then never *use* those skills to go on and *study history?* When does the change in math take place - is it around when you change from arithmetic to pre-algebra, or arithmetic/pre-algebra to algebra? I mean, sure, a kid has to understand place value, fractions, what a number means, etc.; but is arithmetic the time to acquire skills (both through conceptual understanding and memory drill and practice in problem-solving) and then start applying them to study algebra, etc. and does that mean the same thing - using previous skills to go into a deeper study of math? Or does algebra/geometry also mean skills to study deeper math? Each time I revisit this, a few more things become clear. I never knew until the past year or so, that math could be so interesting.

[Spy Car]

This is why I am teaching all of our problems from MEP or any other source I use on the dry-erase board to show that the equations we write must maintain the integrity of being *equal*, by use of the *=* sign to show the RHS and LHS. I teach early on that whatever is added, subtracted, etc. to the RHS of the equation, must be done to the LHS, or the whole equation becomes "wonky". I illustrate this by physically demonstrating how my arms would be one higher than the other, such as a balance scale would be out of balance. I demonstrate the equation in balance by standing with both arms out at my side, as a balance scale in balance would be.

 

When I write a problem on the dry-erase board, I set it up just as I learned in my algebra class in jr. high---that's where I learned the foundations for equations--and I feel I need to teach at that level to develop understanding from my child. We set the equation up, then work it sequentially, bringing down each step and writing it in, making sure to keep both sides equal (I stress that the *=* sign is the controller of the problem). I hope I'm making sense here, but I just wanted to explain how we use this in our work, and I can already see the benefits. I have used this since Year 1 in MEP, and there is no confusion about the *=* sign.

 

Thank you. You've given me (us) some very good ideas to "chew on". I will use this!

 

Bill

[Snickerdoodle]

:)

Edited October 13, 2010 by Snickerdoodle
Should not type when this tired...

[5LittleMonkeys]

The problem I have with this assessment is that that is a lot of assumptions as to what a teacher might or might not do.

 

But, to assume that teachers ignore or don't follow up on concepts that are actually being taught in textbooks b/c teachers don't value them and teach them correctly and thereby make more traditional programs "non-conceptual" is a disingenuous argument.

 

Not all teachers do this but many do. I sat in on ps classes with both of my older girls for many years and many of the teachers did indeed drop the concept after the initial lesson and concentrated on just getting the dc to memorize the formula for how to solve the problem. Why? I didn't ask at the time but I would think because it was easier and faster and got the results they were looking for so that the child had the skill needed to pass the test. The tests that the ps administered did not test for conceptual understanding they were mostly looking for an answer to fill in the blank after the =.

 

Luckily my older understood the conceptual part easily. My younger did not retain the conceptual part because it wasn't practiced nor was she capable of retaining\memorizing the procedure.

 

To make matters worse, when I brought the second oldest home I did not understand, until just recently, that she would do better with an deeper understanding of concepts. I assumed that the easiest way for her to get through math would be to just memorize the steps. So, I too was guilty of doing the same thing. When you have a dc that struggles with all aspects of math you get caught up in just trying to get through it with as few tears as possible. I am now planning on going back several grade levels and teaching her a more conceptual foundation in math hoping that will help her work out a problem when she is unable to instantly recall those facts and procedures.

This is why I am teaching all of our problems from MEP or any other source I use on the dry-erase board to show that the equations we write must maintain the integrity of being *equal*, by use of the *=* sign to show the RHS and LHS. I teach early on that whatever is added, subtracted, etc. to the RHS of the equation, must be done to the LHS, or the whole equation becomes "wonky". I illustrate this by physically demonstrating how my arms would be one higher than the other, such as a balance scale would be out of balance. I demonstrate the equation in balance by standing with both arms out at my side, as a balance scale in balance would be.

 

When I write a problem on the dry-erase board, I set it up just as I learned in my algebra class in jr. high---that's where I learned the foundations for equations--and I feel I need to teach at that level to develop understanding from my child. We set the equation up, then work it sequentially, bringing down each step and writing it in, making sure to keep both sides equal (I stress that the *=* sign is the controller of the problem). I hope I'm making sense here, but I just wanted to explain how we use this in our work, and I can already see the benefits. I have used this since Year 1 in MEP, and there is no confusion about the *=* sign.

:iagree:One of the first pages I looked at on the MEP website was one with a balance problem on it. I immediately knew that this was something I wanted my dc to understand. My dd7 now knows through rote memorization that 8 is the answer to 5+3 but she also knows that (5+3) is equivalent to (8) and also that (8x1) is equivalent to (5+3). When my dh looked at one of her pages from last week he immediately noted that much of it was beginning algebra concepts. He was impressed. This way of thinking about math has been exciting for us and I can't wait to bring my struggling child into this program.

 

One more thing and then I will stop rambling. Are concepts for everyone? IMO, yes, everyone should learn basic concepts however how deep you go depends on the interest and ability of the individual. Two of my dc seem to be able to intuitively understand concepts and they really see no division between understanding all the deeper whys behind the facts and procedures that they are learning. Its all rolled up into one package for them. I think a lot of people are like these two; they are learning conceptual math without really realizing it. However, for my other dd, I believe a deeper understanding of concepts is going to be her ticket to finally being able to feel comfortable with math even though it is going to be difficult for her. Right now math is something of an enigma. To her the only way to get both sides of the equation to balance is to memorize some magical formula that constantly eludes her. I'm hoping that conceptual math will finally demystify it for her.

[Colleen in NS]

:lol: Examples like these (well, basically all the "conceptual" examples) are why I am so perplexed at the snubbing of many traditional homeschool textbooks. That is exactly the way Horizons teaches every concept and is why I posted in the other thread that I believe a lot of the "conceptual" discussions are distorted. Asian programs are not the only ones teaching conceptual math.

 

The problem I have with this assessment is that that is a lot of assumptions as to what a teacher might or might not do.

 

to assume that teachers ignore or don't follow up on concepts that are actually being taught in textbooks b/c teachers don't value them and teach them correctly and thereby make more traditional programs "non-conceptual" is a disingenuous argument.

 

I am SO glad you wrote this stuff! I've been thinking about it for days, as I question my own teaching of math. I'm in my 5th year of using R&S (book 8 now) and have loved it. But I have "just" followed it, going by its reputation. I have seen conceptual teaching in it, and then reinforcement of concepts through doing review problems. I have understood things I never understood before. But, lately I've gotten a bit slack in my teaching times, in the interest of saving time - plus, my oldest "gets" math. Then I started getting worried, reading these threads. Today, I looked more closely at my books, and noted that some lessons in the grades 6-8 books contain "introduction" sections, which I figured out were, many times, designed to get kids thinking about previously learned concepts or to get them thinking about a concept coming in that lesson. And I haven't always paid close enough attention to that intro. section. So today I went through all in book 8, with ds, up to our current lesson, so I could find out exactly what his understanding is. I plan to have a little closer look at the intro. sections in book 7, and compare with book 8, to make sure I know what he understands. These sections do help the kids to look at problems in different ways. I never noticed it before. But I'd have to agree that these traditional texts, if followed closely (which I plan to do better with once again), will show concepts, and I can tweak my child's learning of those, as needed, as we come across them. Phew! Dd is in book 5, and I also plan to keep up with all the oral drill sections, even if she complains, because often I have found that there are items in it that are NOT covered in the lesson. I'm going to have to go back to my original trust that R&S publishers put those in there for a reason.

Edited October 14, 2010 by Colleen in NS