Constructivist vs. conceptual math

[Rivka]

This is possibly a dumb question, but is "constructivist" math the same thing as "conceptual" math?

 

I've been reading various rants on the Internet about how awful "constructivist" math is, but many of the descriptions - emphasis on manipulatives, students encouraged to find their own ways of solving problems, focus on building understanding rather than drilling facts and formulas, etc. - seem to apply to much-praised conceptual math curricula like Miquon.

 

Is there a difference? And if so, what is it?

[usetoschool]

I think it depends on how you use the terms. In public school circles constructivism is about figuring out for yourself all of the knowledge and theories that have been discovered over the centuries and sometimes leaves kids to wander into incorrect solutions. It is a discovery process instead of a teaching process (no imparting of knowledge from on high as it were). I have some serious problems with it personally. I think if you use it as a guided process and create the aha moments it is useful. A lot of constructivism also ventures into brain development areas of guys like Piaget - who some feel (and there have been some discussions with strong opinions on this board) didn't quite get it right. It is a theory about learning that is used in more than just math.

 

Conceptual math is more about how the math is taught - with more emphasis on the why than the how, more about theory and less about algorithms and can, and should in my opinion, include a lot of discovery, but guided by someone who knows more than the student and knows which way the mathematical road is headed.

[Guest]

My impression is that , done well, constructivism is the guiding of kids through activities and thinking processes to "invent" or discover elements of math. An example: a teacher draws a line on the board, writes 0 at one end and 1 at the other, points a the middle and says, "What's in there?" The teacher proceeds to guide these kids, say around age five or so, through specific questions or comments, to the "discovery" of fractions, ordering fractions, and adding fractions. (This would take place over a number of hours.) The teacher does not directly tell them about fractions, introduce terminology of fractions, or give them algorithms to solve problems presented in a textbook until they have gone through this discovery period.

 

Conceptual math to me means something similar, in that it emphasizes understanding and mathematical thinking, but is more structured in terms of being taught more directly by the teacher. A teacher might lead older students in folding pieces of paper specified numbers of times and ways in order to get them to understand the concept behind multiplying fractions by one another, for instance. The lesson might center around a problem or around making up their own problems; it might include comparing different ways students think of to go about solving a particular problem, or thinking about why one approach doesn't work while another seems to. The emphasis would be on understanding the reasoning behind a "rule" or algorithm before being asked to work problems using it.

 

At their best, each draws on the other and incorporates the other. Badly done, I can see each getting a bad name -- but then so would direct, explicit, regular old teaching if it were badly done. i think to be a really good constructivist OR conceptual math teacher, you need to be a good mathematician and also well-versed in how kids think.

[sagira]

My impression is that , done well, constructivism is the guiding of kids through activities and thinking processes to "invent" or discover elements of math. An example: a teacher draws a line on the board, writes 0 at one end and 1 at the other, points a the middle and says, "What's in there?" The teacher proceeds to guide these kids, say around age five or so, through specific questions or comments, to the "discovery" of fractions, ordering fractions, and adding fractions. (This would take place over a number of hours.) The teacher does not directly tell them about fractions, introduce terminology of fractions, or give them algorithms to solve problems presented in a textbook until they have gone through this discovery period.

 

Conceptual math to me means something similar, in that it emphasizes understanding and mathematical thinking, but is more structured in terms of being taught more directly by the teacher. A teacher might lead older students in folding pieces of paper specified numbers of times and ways in order to get them to understand the concept behind multiplying fractions by one another, for instance. The lesson might center around a problem or around making up their own problems; it might include comparing different ways students think of to go about solving a particular problem, or thinking about why one approach doesn't work while another seems to. The emphasis would be on understanding the reasoning behind a "rule" or algorithm before being asked to work problems using it.

 

At their best, each draws on the other and incorporates the other. Badly done, I can see each getting a bad name -- but then so would direct, explicit, regular old teaching if it were badly done. i think to be a really good constructivist OR conceptual math teacher, you need to be a good mathematician and also well-versed in how kids think.

 

Great post. Since math has been my pet subject lately, I have much to think about.

 

I do think that Miquon takes the constructivist approach, and hence I don't want to use it as my main program, but I certainly think it's a wonderful way to introduce new concepts, which can be further cemented by adding in conceptual math. A conceptual program such as Singapore or Math Mammoth would be able, I believe, to fill in gaps that could result from using solely the discovery program. It would also provide the extra practice a program such as Miquon lacks IMO.

[jenbrdsly]

I'm a former elementary school teacher who was trained in Constructivsm. It's the main approach I use when teaching my son Bruce math. http://teachingmybabytoread.blog.com/2011/03/15/subtraction/

 

The other posters have been right on about the heart of Constructivsm being to help kids create and understand their own strategies for solving problems, instead of just blinding teaching traditional algorithms. I'd go further and say that one of the key goals of Constructivism is for kids to develop a strong conceptual understanding.

 

Some programs, like Right Start, have a lot of Constructivist approachs in them, but then throw in some traditional algortihm zingers that I try to stay clear of. If you really commit to the Constructivist approach, it's amazing. My six year old could not tell you what borrowing or carrying is, but he can now add 4 digit numbers and subtract 3 digit numbers with regrouping in his head faster than I can.

[nansk]

I think "constructivism" is an approach to teaching via discovery and discussion. "Conceptual" is (or ought to be) the goal of any teaching approach.

 

If this is right, then, I think Miquon and all the modern maths in the US (such as Everyday Maths) use a constructivist approach. However - and this is an important qualifier - I think Miquon achieves a conceptual understanding of maths, whereas, more likely than not, the modern books (and teachers) fail, which is why you are seeing so many "rants" online.

[morosophe]

Wow, what an interesting thread! I had no notion of "constructivism" in math before I read this.

 

I have to say, I would never, never take the "constructivist" approach with my oldest son. He has not a creative bone in his body, and would resent being forced to do activities without a known goal and a road map for getting there. Meanwhile, he's loving his Math-U-See classes, as "boring" and traditional as I'm sure they'd be to other kids.

 

So, wouldn't part of the success or failure of constructivism (in math or elsewhere) depend on the student?

 

Also, isn't this the kind of thinking best left for the logic stage in the classical trivium?

[Murmer]

I think constructivism math is a component of conceptual math. Constructivism is the developing of math ideas based on discovering for ones self the process and answer. Where it falls short is if the child is only constructing and never eventually conceptuallizing the math. IE I can have my child construct the number 5 out of a number of objects but until she is able to transfer that knowledge that the 5 objects is the same as the number 5 that she says when counting she has merely constructed 5 and not conceptualized it.

That is where many of the issues arise with the "new math" of the last 10 years. Child are creating and constructing and discovering but they are never making the jump to conceptualizing. A good math teacher would use the constructivist techniques to guide and push students to make the jump to conceptualizing the math concepts they are discovering instead of just letting the kids "play" math. But many teachers did not go into teaching because they particularly like math or were good at it (In teacher school I was the only student out of 20 that enjoyed my math classes and too calculus, in my teaching career reading masters degrees beat out math masters degrees 20 to 1).

[Soror]

I'm a former elementary school teacher who was trained in Constructivsm. It's the main approach I use when teaching my son Bruce math. http://teachingmybabytoread.blog.com/2011/03/15/subtraction/

 

The other posters have been right on about the heart of Constructivsm being to help kids create and understand their own strategies for solving problems, instead of just blinding teaching traditional algorithms. I'd go further and say that one of the key goals of Constructivism is for kids to develop a strong conceptual understanding.

 

Some programs, like Right Start, have a lot of Constructivist approachs in them, but then throw in some traditional algortihm zingers that I try to stay clear of. If you really commit to the Constructivist approach, it's amazing. My six year old could not tell you what borrowing or carrying is, but he can now add 4 digit numbers and subtract 3 digit numbers with regrouping in his head faster than I can.

The approach you used to get him to do the subtraction is straight out of Right Start.

[veggiegal]

Constructivism, as the name implies, has children "constructing" their own knowledge through exploration, etc. rather than the 'filling their head with info' model of learning. But it children can construct either conceptual knowledge of math, or simple algorithms that 'work', but without a deep understanding of the 'whys' behind it.

[Crimson Wife]

Constructivism can be either good or bad as an educational approach depending on the specific teacher and the student. A good math program will have a balance of discovery activities and practice with pen & paper calculations using the traditional algorithm. I'm not familiar with Miquon but Right Start and Singapore both definitely fall into the that category.

[LittleIzumi]

A lot of the complaints/articles I've read about constructivist math in PS centers around the fact that it is so focused on discovery, no other math is allowed. I mean that once the kids figure it out and figure out the algorithm for themselves, they aren't allowed to do it the faster way--they have to keep using the manipulatives & the long discovery methods for everything, not algorithms. Also, often if the kids don't "discover" the answer for themselves, the program moves on regardless because the program will come back to it later, but then the kid is lost as they move on. In addition, there is a complaint that math facts are completely left out, there are no drills for retention, etc. From what I've read, anyway.

[jenbrdsly]

The approach you used to get him to do the subtraction is straight out of Right Start.

 

 

:) So funny! And yet, he figured it out himself. I was refering more to how Right Start has you learn the traditional algorithm of borrowing and carring by using the abacus. We skipped over that part. My son usually starts on the left when adding or subteracting, instead of the right.

[Ali in OR]

A flaw with constructivism as applied in most public schools is a bias against traditional algorithms. While "constructing" knowledge can be a very effective way to begin instruction of a topic, eventually you want kids to become accomplished at using an algorithm to be able to solve problems quickly and with automaticity. You don't want them to have to painstakingly re-invent multiplication to solve 27x63. I have wondered if the teachers who favor constructivism the most are typically early elementary teachers. As a high school math teacher, I wanted kids who had mastered algorithms and the ability to do simple calculations quickly. I needed their brain cells to be grappling with more abstract ideas about algebra or trigonometry. That can only happen if the lower level math skills happen on an automatic level. Algorithms are great for that.

[Momling]

My impression is that , done well, constructivism is the guiding of kids through activities and thinking processes to "invent" or discover elements of math. An example: a teacher draws a line on the board, writes 0 at one end and 1 at the other, points a the middle and says, "What's in there?" The teacher proceeds to guide these kids, say around age five or so, through specific questions or comments, to the "discovery" of fractions, ordering fractions, and adding fractions. (This would take place over a number of hours.) The teacher does not directly tell them about fractions, introduce terminology of fractions, or give them algorithms to solve problems presented in a textbook until they have gone through this discovery period.

 

Conceptual math to me means something similar, in that it emphasizes understanding and mathematical thinking, but is more structured in terms of being taught more directly by the teacher. A teacher might lead older students in folding pieces of paper specified numbers of times and ways in order to get them to understand the concept behind multiplying fractions by one another, for instance. The lesson might center around a problem or around making up their own problems; it might include comparing different ways students think of to go about solving a particular problem, or thinking about why one approach doesn't work while another seems to. The emphasis would be on understanding the reasoning behind a "rule" or algorithm before being asked to work problems using it.

 

At their best, each draws on the other and incorporates the other. Badly done, I can see each getting a bad name -- but then so would direct, explicit, regular old teaching if it were badly done. i think to be a really good constructivist OR conceptual math teacher, you need to be a good mathematician and also well-versed in how kids think.

:iagree:Great post!

[Carol in Cal.]

My DD's geometry book was constructivist this year. Oh my goodness. What a pain.

 

She would derive the algorithms or the 'conjectures' but if she got them wrong in her notes, there was nothing to fall back on. Thankfully she had a really good teacher, but still, we were looking up a few holes on the internet because the book had no summary of the material, anywhere. If you didn't get it all into your notes, you COULD NOT do the work. That means that later when she is studying geometry for the SAT test we will have to buy another book, probably. I resent this entirely.

[jenbrdsly]

Wow, what an interesting thread! I had no notion of "constructivism" in math before I read this.

 

I have to say, I would never, never take the "constructivist" approach with my oldest son. He has not a creative bone in his body, and would resent being forced to do activities without a known goal and a road map for getting there. Meanwhile, he's loving his Math-U-See classes, as "boring" and traditional as I'm sure they'd be to other kids.

 

So, wouldn't part of the success or failure of constructivism (in math or elsewhere) depend on the student?

 

Also, isn't this the kind of thinking best left for the logic stage in the classical trivium?

 

 

You are having a very common reaction to Constructivisim, when people first learn about it. It's very though provoking, (and sometimes causes visceral reactions).

 

Before you decide 100% that Constructivism is not for your son, take a look at this lesson I just did with my little guy, on square roots. My six year old could really see how some numbers were square, and some were not. It's hard to achieve that same "aha!"moment, by just learning algorithms. http://teachingmybabytoread.blog.com/2011/06/09/square-numbers-and-square-roots/

[jenbrdsly]

A flaw with constructivism as applied in most public schools is a bias against traditional algorithms. While "constructing" knowledge can be a very effective way to begin instruction of a topic, eventually you want kids to become accomplished at using an algorithm to be able to solve problems quickly and with automaticity. You don't want them to have to painstakingly re-invent multiplication to solve 27x63. I have wondered if the teachers who favor constructivism the most are typically early elementary teachers. As a high school math teacher, I wanted kids who had mastered algorithms and the ability to do simple calculations quickly. I needed their brain cells to be grappling with more abstract ideas about algebra or trigonometry. That can only happen if the lower level math skills happen on an automatic level. Algorithms are great for that.

 

 

I agree with what you are saying, but I'm guessing that as a high school teacher yourself, you also don't want kids who ONLY know how to do algorithms. For example, a kid who thinks that in the problem: 562-399, when you borrow from the 60, it's really just a six.

 

The challenge for elementary school teachers is to 1) develop hard core mathematical understanding, and then 2) phase kids over into quick and automatic strategies (algorithms or not).

[Spy Car]

I agree with what you are saying, but I'm guessing that as a high school teacher yourself, you also don't want kids who ONLY know how to do algorithms. For example, a kid who thinks that in the problem: 562-399, when you borrow from the 60, it's really just a six.

 

The challenge for elementary school teachers is to 1) develop hard core mathematical understanding, and then 2) phase kids over into quick and automatic strategies (algorithms or not).

 

:iagree:

 

Your example (562-399) is one of those time when using the standard algorithm of column subtraction (vs a mental math technique) would be grossly inefficient.

 

The whole conversation is giving me a headache, truth told. There are so many scrambled definitions of what "Constructivist" math may (or may not be) that unless one is willing to settle for defined terms (bloody unlikely) the whole conversation will do nothing more than have people talking past one another.

 

For example, in some ways MUS is one of the most "Constructivist" math programs there is. Every topic is worked out (constructed) using physical manipulatives.

 

Other early math programs use physical manipulatives and a discovery method to make math comprehensible to young children in a concrete fashion that is appropriate to their intellectual development. It helps them bridge the concrete>pictorial>abstract progression of intellectual development so is a perfectly appropriate learning/teaching method for the early grammar stage, but something that ought not be necessary (for most) in the logic stage.

 

Comparing a program like Miquon (which is aimed at young children) with Everyday Mathematics because two highly dissimilar (in every way) math programs have somehow both been labeled as being "Constructivist" completely blurs the distinctions between them and makes the labeling a hindrance to understanding rather than shedding light on a method.

 

Bill

[poetic license]

You are having a very common reaction to Constructivisim, when people first learn about it. It's very though provoking, (and sometimes causes visceral reactions).

 

Before you decide 100% that Constructivism is not for your son, take a look at this lesson I just did with my little guy, on square roots. My six year old could really see how some numbers were square, and some were not. It's hard to achieve that same "aha!"moment, by just learning algorithms. http://teachingmybabytoread.blog.com/2011/06/09/square-numbers-and-square-roots/

 

 

I'm going to echo Bill on the issue of definitions. What you are showing as constructivism in your blog really just seems to me to be more like the approach of concrete-pictorial-abstract of Singapore.

 

My understanding of true constructivist math would not have you, the teacher, showing your son the square numbers or the splitting of numbers into hundreds, tens, and ones like you did for the subtraction. I could be mistaken, but I thought true constructivist math would involve more initiative from the students to develop their own approach, rather than teacher leading them along in this way. The approach you are describing, in other words, seems much closer to Right Start and Singapore as opposed to Everyday Math and the other so-called "fuzzy math" programs.

[jenbrdsly]

I'm going to echo Bill on the issue of definitions. What you are showing as constructivism in your blog really just seems to me to be more like the approach of concrete-pictorial-abstract of Singapore.

 

My understanding of true constructivist math would not have you, the teacher, showing your son the square numbers or the splitting of numbers into hundreds, tens, and ones like you did for the subtraction. I could be mistaken, but I thought true constructivist math would involve more initiative from the students to develop their own approach, rather than teacher leading them along in this way. The approach you are describing, in other words, seems much closer to Right Start and Singapore as opposed to Everyday Math and the other so-called "fuzzy math" programs.

 

I went through two years of Constructivist teacher training in addition to my credentialing work, when I was a teacher in California. You are right that Constructivism is not about imposing a teacher's thought process on a child.

In the square number example, this was a review lesson of something my son had already created for himself a year ago in his discovery of multiplication. I would not try to guide him into learning about square numbers, unless he had already had a whole bunch of experience working with arrays. I did write out the number labels myself however, because my 6 year old still does a lot of number reversals.

The subtraction example is bit different, because my son is a kindergartener in ps, who at the time was finishing the district's 2nd grade math program afterschool in preparation for GATE next year. The district's math book, Houghton Mifflin's Math Expressions, was teaching the traditional algorithm method, which we just skipped.

If it was a normal classroom situation with me as the teacher, I would have had student after student approach the same subtraction problem in his or her own way, with all of the other students learning from the thinking and reasoning of their peers. I would encourage all strategies and thinking, but give special attention to the most effective. In my classroom, we called this "whiteboard time" and did it in a small group setting. This was an activity my entire school did at every grade level.

In the homeschool environment however, there aren't any peers for my son to learn from. So I myself introduced several different ways to think about subtraction; money, the abacus, pictures, base ten blocks. My son developed his own strategies from there.

 

Could there have been a "purer" Constructivist way to do it? Sure. But now, five months or so later, my six year old can subtract 3 digit numbers in his head. So I'm still a big believer in Constructivism, but not too caught up on where we fall on the Constructivist purity spectrum.

[jenbrdsly]

I went through two years of Constructivist teacher training in addition to my credentialing work, when I was a teacher in California. You are right that Constructivism is not about imposing a teacher's thought process on a child.

 

In the square number example, this was a review lesson of something my son had already created for himself a year ago in his discovery of multiplication. I would not try to guide him into learning about square numbers, unless he had already had a whole bunch of experience working with arrays. I did write out the number labels myself however, because my 6 year old still does a lot of number reversals.

 

The subtraction example is bit different, because my son is a kindergartener in ps, who at the time was finishing the district's 2nd grade math program afterschool in preparation for GATE next year. The district's math book, Houghton Mifflin's Math Expressions, was teaching the traditional algorithm method, which we just skipped.

 

If it was a normal classroom situation with me as the teacher, I would have had student after student approach the same subtraction problem in his or her own way, with all of the other students learning from the thinking and reasoning of their peers. I would encourage all strategies and thinking, but give special attention to the most effective. In my classroom, we called this "whiteboard time" and did it in a small group setting. This was an activity my entire school did at every grade level.

 

In the homeschool environment however, there aren't any peers for my son to learn from. So I myself introduced several different ways to think about subtraction; money, the abacus, pictures, base ten blocks. My son developed his own strategies from there.

 

Could there have been a "purer" Constructivist way to do it? Sure. But now, five months or so later, my six year old can subtract 3 digit numbers in his head. So I'm still a big believer in Constructivism, but not too caught up on where we fall on the Constructivist purity spectrum.

 

Sorry, I didn't mean to sound like a snot there! I just wanted to explain my history with Constructivism and why I think it works so well.

[Spy Car]

Sorry, I didn't mean to sound like a snot there! I just wanted to explain my history with Constructivism and why I think it works so well.

 

You didn't to me. I have had every similar experiences with my (barely still) 6 year old. Many of the methods we used, including such things as discovery methods with addition and subtraction and using arrays to explore multiplication could, I suppose, be termed "Constructivist" (not that I think the term is particularly useful for being so vague).

 

I am very enthusiastic about appropriate use of these techniques as part of math education. I think it has made a world of difference in my child's math experience.

 

Does it mean I would hope children would be trying to discover multiplication and division themselves in a school setting in 5th Grade? Probably not.

 

But one of the upper level math programs that is widely considered among the most intellectually challenging options is the Art of Problem Solving, and it is my understanding (not having used it) is that AoPS includes "discovery" methods.

 

So I think it is hard (and potentially destructive to the aim of excellent math education) to generalize about math programs being "Constructivist" and assuming one knows what that really means, and whether the means and method are "sound" or not.

 

I know there are outstanding methods for teaching that include elements that might be called "Constructivist" and I image there are idiotic means that could fall in the same category. Which is why (to repeat myself) it is a term that has very limited usefulness in the real world.

 

Bill

[poetic license]

Sorry, I didn't mean to sound like a snot there! I just wanted to explain my history with Constructivism and why I think it works so well.

 

You sounded fine... it's neat to read your experiences. It's just such a nebulous term to pin down. From what I understand about learning and development, the conclusion that a child derives herself is most likely to be retained as compared to an understanding that is explained by someone else. If the teacher can, through such techniques, help a child to derive algorithms on their own, they are much more likely to remember them. (assuming they are not some of the crazy long winded algorithms in the fuzzy math programs).

 

Like you mentioned earlier, "quick and automatic strategies." I wish I had such a math education!

[jenbrdsly]

This thread is probably long dead now, but I finally wrote an extensive bit on my blog about what teaching math from a Constructivist perspective means to me: http://teachingmybabytoread.blog.com/math/

[Spy Car]

This thread is probably long dead now, but I finally wrote an extensive bit on my blog about what teaching math from a Constructivist perspective means to me: http://teachingmybabytoread.blog.com/math/

 

I enjoyed the blog post. We have some commonalities in approach and some differences. I don't understand the rationale of teaching "subtraction" as a topic after teaching addition, multiplication and division. Subtraction is the inverse of addition and (to my mind) ought to be taught in conjunction with addition.

 

The experience of number-bonds or play with C Rods shows the sum of parts makes a whole and a whole less a part yeilds a difference. This understanding seems fundamental, and developmentally sound.

 

As to mental math strategies there is something to be said for allowing a child to develop some of her (or his) own, but I don't think that precludes the value of introducing mental math strategies by direct instruction.

 

I have focused on our differences, but (what can I say? :D)

 

Bill

[jenbrdsly]

I enjoyed the blog post. We have some commonalities in approach and some differences. I don't understand the rationale of teaching "subtraction" as a topic after teaching addition, multiplication and division. Subtraction is the inverse of addition and (to my mind) ought to be taught in conjunction with addition.

 

The experience of number-bonds or play with C Rods shows the sum of parts makes a whole and a whole less a part yeilds a difference. This understanding seems fundamental, and developmentally sound.

 

As to mental math strategies there is something to be said for allowing a child to develop some of her (or his) own, but I don't think that precludes the value of introducing mental math strategies by direct instruction.

 

I have focused on our differences, but (what can I say? :D)

 

Bill

 

:) In a classroom setting, you can "rig it" so that you call on children you

Know will provide quick and efficient strategies. Then the other kids, who are not quite grasping the concept, will learn from the ones who do. Of course you also have to call on those children too and here their developing ideas.

In the afterschooling environment, I've struggled a bit since I just have one student! That's why I've had to go ahead and introduce some strategies myself.

[Spy Car]

:) In a classroom setting, you can "rig it" so that you call on children you

Know will provide quick and efficient strategies. Then the other kids, who are not quite grasping the concept, will learn from the ones who do. Of course you also have to call on those children too and here their developing ideas.

In the afterschooling environment, I've struggled a bit since I just have one student! That's why I've had to go ahead and introduce some strategies myself.

 

But how would having a child pick up a strategy from a classmate qualify as letting a child develop her (or his) own strategy? Not that I'm opposed to children teaching other children or shared learning experiences (I'm not) but it seems qulitatively different than trying to reason out strategies for oneself.

 

And I still don't understand the rationale for the subtraction sequence.

 

We are a tough crowd :D

 

Bill

[jenbrdsly]

But how would having a child pick up a strategy from a classmate qualify as letting a child develop her (or his) own strategy? Not that I'm opposed to children teaching other children or shared learning experiences (I'm not) but it seems qulitatively different than trying to reason out strategies for oneself.

 

And I still don't understand the rationale for the subtraction sequence.

 

We are a tough crowd :D

 

Bill

 

Hmmm... I was teaching at a school that focused on children initiated learning, so that might have been part of it.

The addition, multiplication, division, subtraction sequence was based on the idea that subtraction was the most difficult to learn. I don't remember if there was research cited to show this or not. But the upshot was that we introduced multiplication a lot earlier. I noticed in the section of WTM that describes Singapore that it introduces multiplication earlier than most programs too.

[poetic license]

I promise I'm not trying to be difficult, but am sincerely wondering: if it is child initiated, then why is there a "recommended sequence" of addition, then mult, div, and subtraction. Why not just let the child discover on their own in their own order? In my son's case, he ended up doing addition, then subtraction and multiplication simultaneously, and so far only very little division.

[jenbrdsly]

I promise I'm not trying to be difficult, but am sincerely wondering: if it is child initiated, then why is there a "recommended sequence" of addition, then mult, div, and subtraction. Why not just let the child discover on their own in their own order? In my son's case, he ended up doing addition, then subtraction and multiplication simultaneously, and so far only very little division.

 

In a home environment, this would the best way to go for sure! In a classroom environment, the teacher has to have math units that she plans out ahead of time etc.

[Spy Car]

Hmmm... I was teaching at a school that focused on children initiated learning, so that might have been part of it.

 

The addition, multiplication, division, subtraction sequence was based on the idea that subtraction was the most difficult to learn. I don't remember if there was research cited to show this or not. But the upshot was that we introduced multiplication a lot earlier. I noticed in the section of WTM that describes Singapore that it introduces multiplication earlier than most programs too.

 

I find myself in a strange position because I highly value cultivating reasoning skills in the individual child (including giving them the tools and opportunities to make math "discoveries" on their own, and also value cooperative group learning through problem-solving activities. These are very important to me.

 

That said, I also believe there are times for Direct Instruction. While I would criticize programs that have ONLY Direct Instruction as being intellectually limiting, I also have a problem with "Constuctivism" when Direct Instruction is eliminated for (what seem to me) to be ideological/educational-theory concerns that are out of sync with the practical realities of learning.

 

And I'm unmoved by the idea that subtraction should be taught last (of the four basic operations) because it is considered to be "hard." This is nonsensical to me.

 

Have you looked at Miquon?

 

I'm enjoying the conversation and am sure we would have much common-ground. Sorry that I'm focusing on the differences :001_smile:

 

Bill

Edited July 8, 2011 by Spy Car

[EKS]

But how would having a child pick up a strategy from a classmate qualify as letting a child develop her (or his) own strategy? Not that I'm opposed to children teaching other children or shared learning experiences (I'm not) but it seems qulitatively different than trying to reason out strategies for oneself.

 

 

Exactly! This is why teachers like mixed ability grouping--because the higher ability children can teach the others; unfortunately, the only people this approach seems to benefit are the teachers. However, if you want to all children to benefit from the constructivist approach, you need to have children at the same level grouped together.

[Spy Car]

Exactly! This is why teachers like mixed ability grouping--because the higher ability children can teach the others; unfortunately, the only people this approach seems to benefit are the teachers. However, if you want to all children to benefit from the constructivist approach, you need to have children at the same level grouped together.

 

I actually don't have a problem with children learning from higher ability children (some of the time). I think it can potentially be good for both parties. So I guess I disagree with the notion that only the teacher benefits. If a child can teach another I think they benefit from the teaching process in several ways, including getting them to systematize their knowledge so that it can be transmuted. And the child who is learning (if that is indeed the case) is also benefiting.

 

I would hate to take any one of these 3 modes (Direct Instruction, individual "discovery" or cooperative group learning) completely out of the mix, as I think all have their place. I'm very much a pragmatist, and am most interested in a practical mix that works.

 

Bill

[boscopup]

And I'm unmoved by the idea that subtraction should be taught last (of the four basic operations) because it is considered to be "hard." This is nonsensical to me.

 

:iagree: I didn't really have to teach subtraction. By teaching addition, my kids have figured out subtraction as the inverse. Plus our math programs use missing addend problems, which are essentially subtraction. I don't understand how subtraction could possibly be harder than multiplication and division? :confused:

[jenbrdsly]

:iagree: I didn't really have to teach subtraction. By teaching addition, my kids have figured out subtraction as the inverse. Plus our math programs use missing addend problems, which are essentially subtraction. I don't understand how subtraction could possibly be harder than multiplication and division? :confused:

 

I think the problem with strictly following any educational ideology too closely, is that it sometimes causes people to abandon common sense. Look at the Whole Language vs. Phonics issue. If you only taught Whole Language, that would be a problem. If you only taught Phonics, that would be a problem too. Teachers need to exercise common sense and be flexible in their implantation of pedagogy.