Theoretical math question RE: conceptual vs. traditional

[birchbark]

Well, I have gone from "conceptual math" back to traditional! And when I say traditional, I mean pre-60's math. This also excludes spiral curricula, such as Saxon. It is not just because it is simpler to learn and teach, but, I believe, more effective for non-mathy students.

 

My firstborn spent grades K-5 in conceptual math programs, mainly Miquon and Singapore. We were both underwhelmed.  I switched him to a little-known curriculum with a different approach last year, and the difference has been remarkable.

 

I agree that the goal is for students to think conceptually, the question is how to get there. I'll post a few articles here that have impacted my thinking, and that also explain things far better than I could!

 

Mathematics Education: Being Outwitted By Stupidity

 

The Myth About Traditional Math Education

 

It's Not Just Writing: Math Needs a Revolution Too

 

Is It True That Some People Just Can't Do Math? (This is a Willingham article. His book is a must-read.)

 

What's Missing From Math Standards?

 

The Decline Of Math

 

I guess that was more than a few.

 

Strayer-Upton and Ray's are two popular elementary curricula among homeschoolers that fit this bill. Another curriculum that I am interested in (but have not seen in person) is Mastering Mathematics. For grades six through Algebra 2, there is Systematic Mathematics.

 

 

[MomintheMountains]

"The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.†- See more at: http://www.educationnews.org/education-policy-and-politics/barry-garelick-math-education-being-outwitted-by-stupidity/#sthash.VZnDoaHw.dpuf"

 

Yes! Thank you for sharing these links. This is what I have been trying to say (not very well) in this thread. This is exactly how I have always learned math. I learn the algorithm first and then I can understand the concept. My children seem to learn like me, too.

[Korrale]

"The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.†- See more at: http://www.educationnews.org/education-policy-and-politics/barry-garelick-math-education-being-outwitted-by-stupidity/#sthash.VZnDoaHw.dpuf"

 

Yes! Thank you for sharing these links. This is what I have been trying to say (not very well) in this thread. This is exactly how I have always learned math. I learn the algorithm first and then I can understand the concept. My children seem to learn like me, too.

This is how I learnt also.

[Ellie]

"The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.†- See more at: http://www.educationnews.org/education-policy-and-politics/barry-garelick-math-education-being-outwitted-by-stupidity/#sthash.VZnDoaHw.dpuf"

 

Yes! Thank you for sharing these links. This is what I have been trying to say (not very well) in this thread. This is exactly how I have always learned math. I learn the algorithm first and then I can understand the concept. My children seem to learn like me, too.

 

:iagree: :iagree: :iagree:

[SorrelZG]

The old pre-"traditional" texts I prefer very clearly teach the concepts involved in beginning arithmetic (making obvious the connections between addition, subtraction, mult., div., and fractions), AND provide varied drill for the purpose of mastery.

[smileyme]

I like my Asian Math too.. I grew up in India, and we had pretty competitive exams, even in the first year. I consistently scored a full 100 percent most times. Of course my gran made me do the 50 or so problems a day... But I believe this also helped me handle the competitive exams later for the grad and post grad entries. I am seeing my first grade daughter here get me a set of pages, all done correctly... but her assement is but 10 problems. How can they determine her whole mathematics concepts by just 10 problems beats me. And when I test her, she is doing perfectly even the second grade problems, and some third grade ones, quite fine. But, she started math late, and  her teacher is using the same pace with her. She comes home saying she is bored by the math her teacher does for her, and some of her friends are doing the higher math, and she would like to do that. I am trying to understand what types of assessments works here. Maybe the new system has its own advantages, but I need to understand it better.

An answer to the original question:  Yes!  I know a 13 year old who has only done Saxon, and is now part of a very competitive university program for gifted middle and high school math students. 

 

Having said that, I have always loved math.  I have a degree in physics.  But, like Crimson Wife, I learned more teaching Right Start B and Singapore 1 and 2 than in 16 years of previous education... 

 

So I am a huge fan of "Asian math".  I add drill to Singapore because I  think we can have our cake and eat it, too.  :-) 

 

If you choose an Asian math, just do your homework.  Read the teacher's manuals.  Watch the videos at EducationUnboxed.  Understand before you teach.  Otherwise it's just the blind leading the blind, so to speak! 

 

I have only been HSing a few years.  But it has become clear to me that the program you teach best is more important than the program that matches your student's learning style.  If you can't teach Asian math no matter how hard you want to or try to, then don't do it.  Teach the program where you can confidently answer your child's questions and give additional information if necessary. 

 

And now, having said THAT, I don't think Asian math is that hard to grasp for most of us.  It's not like learning a new language.  Reading the teacher's manuals is more like hitting yourself in the forehead constantly and saying, "Why didn't anybody teach me this way?  Why am I just now spotting this connection?!"

 

[wapiti]

"The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.†- See more at: http://www.educationnews.org/education-policy-and-politics/barry-garelick-math-education-being-outwitted-by-stupidity/#sthash.VZnDoaHw.dpuf"

 

Yes! Thank you for sharing these links. This is what I have been trying to say (not very well) in this thread. This is exactly how I have always learned math. I learn the algorithm first and then I can understand the concept. My children seem to learn like me, too.

 

FWIW, the quoted article is criticizing Fuzzy Math programs (eta, referred to in the article as "reform math"; examples would include Everyday Math, Connected Math, TERC, etc.), and rightly so.  Fuzzy math programs are at the opposite end of a continuum from traditional math and tend to lack sufficient instruction and practice with algorithms.  In contrast to fuzzy math, programs such as Singapore and MM take a balanced approach to teaching concepts and algorithms.

 

Whether to teach the algorithm before or after the concept is more of a pedagogical issue that I can't really comment on, except to say that it may be an individual thing.  My one kid who especially just wants to know how to do something will not remember the procedure later and really doesn't want to know why at all - he needs to learn the concept up front in the lesson and also think more deeply about it with problem solving or it does not stick (BTDT).

 

I just skimmed, but the article touches on an important point, that many students need explicit instruction, which may be lacking particularly in a fuzzy math program or which may vary among math programs more generally.  That's a separate issue from concepts vs algorithms, although I can imagine that some programs may teach one more explicitly than the other.  How much explication is sufficient will depend not only on the student but also on the teacher and how much oral instruction is given along with the text.

 

I do think that the Math Wars have unfortunately focused on a false dichotomy between concepts and algorithms when there are existing programs that teach both, together, as they should.

[Crimson Wife]

I like my Asian Math too.. I grew up in India, and we had pretty competitive exams, even in the first year. I consistently scored a full 100 percent most times. Of course my gran made me do the 50 or so problems a day... But I believe this also helped me handle the competitive exams later for the grad and post grad entries. I am seeing my first grade daughter here get me a set of pages, all done correctly... but her assement is but 10 problems. How can they determine her whole mathematics concepts by just 10 problems beats me. And when I test her, she is doing perfectly even the second grade problems, and some third grade ones, quite fine. But, she started math late, and  her teacher is using the same pace with her. She comes home saying she is bored by the math her teacher does for her, and some of her friends are doing the higher math, and she would like to do that. I am trying to understand what types of assessments works here. Maybe the new system has its own advantages, but I need to understand it better.

If she is bored with math and has clearly mastered the particular topic, why on earth would you insist on her doing MORE of the same types of problems? That's the surest way IMHO to make a bright kid absolutely LOATHE the subject.

 

I hated, hated, hated math growing up because I was assigned pages upon pages of problems that were far too easy for me. I would've liked math so much better and gotten much more out of the time spent on it had I been given 10 really challenging problems (like the ones in Singapore's "Intensive Practice", Beast Academy/Art of Problem Solving, MEP, Royal Fireworks Press "Problemoids", etc.)

 

Less can sometimes be so much more when it comes to math...

[kiana]

FWIW, the quoted article is criticizing Fuzzy Math programs (e.g. Everyday Math, Connected Math, TERC, etc.), and rightly so.  Fuzzy math programs are at the opposite end of a continuum from traditional math and tend to lack sufficient instruction and practice with algorithms.  In contrast to fuzzy math, programs such as Singapore and MM take a balanced approach to teaching concepts and algorithms.

 

Whether to teach the algorithm before or after the concept is more of a pedagogical issue that I can't really comment on, except to say that it may be an individual thing.  My one kid who especially just wants to know how to do something will not remember the procedure later and really doesn't want to know why at all - he needs to learn the concept up front in the lesson and also think more deeply about it with problem solving or it does not stick (BTDT).

 

I just skimmed, but the article touches on an important point, that many students need explicit instruction, which may be lacking particularly in a fuzzy math program or which may vary among math programs more generally.  That's a separate issue from concepts vs algorithms, although I can imagine that some programs may teach one more explicitly than the other.  How much explication is sufficient will depend not only on the student but also on the teacher and how much oral instruction is given along with the text.

 

I do think that the Math Wars have unfortunately focused on a false dichotomy between concepts and algorithms when there are existing programs that teach both, together, as they should.

 

Very much so.

 

Some students learn better by learning the concepts first, and some learn better by consistent practice with the algorithm. In a program designed for students in public schools, *both* should be provided.

 

Constructivist math may work very well if the teacher actually understands the math that he or she is teaching. But when a student is aided by a teacher who only understands the math algorithmically if at all, and peers who also do not understand it at all, it is the blind leading the blind and nothing is learned.

 

[Corraleno]

FWIW, the quoted article is criticizing Fuzzy Math programs (e.g. Everyday Math, Connected Math, TERC, etc.), and rightly so.  Fuzzy math programs are at the opposite end of a continuum from traditional math and tend to lack sufficient instruction and practice with algorithms.  In contrast to fuzzy math, programs such as Singapore and MM take a balanced approach to teaching concepts and algorithms.

 

:iagree:

 

Also, I think some people assume that if a math program mentions the concept behind the algorithm, it is "teaching concepts." There's a big difference between a program that teaches the algorithm, then mentions the "why" behind it in a few sentences, then provides problem sets that really only reinforce the algorithm, versus a program that leads students, step-by-step, through the concepts behind the algorithm, followed by lots of varied problems (including challenging word problems) that test and reinforce conceptual understanding as much as application of the algorithm.

 

Kids who use the first type of program tend to think they're good at math, but find complex word problems too difficult and confusing. That's because they don't have the conceptual understanding of what the algorithm does; they're just able to apply it automatically when they see a problem that is already set up for that algorithm. 

[Dicentra]

So what do we do about math in the elementary years?  Having just finished teaching an instructional methods course in elementary math at the local college, I feel like throwing up my hands in despair.  Very few elementary teachers are math majors (or science majors, for that matter) and do not seem to have the understanding of math needed to teach conceptually.  When they attempt to teach conceptually, it's as kiana mentioned - the blind leading the blind.  The best remedy would be to get math specialists into the elementary system but that doesn't seem to be happening.  So is the lesser of two evils to simply let the non-mathy teachers teach algorithmically?  I feel like at least then the students would have SOMETHING coming into high school.

 

This ties in with Wishbone Dawn's thread on developing our own teaching skills, as well.  If a parent isn't mathy, maybe a curriculum or program that's heavier on algorithms and less so on conceptual understanding (not completely lacking conceptual explanations but they definitely take a back seat) is a better choice.  Not the ideal best choice but the best of the options available.

 

The same can be said regarding other subjects as well.  Writing is NOT my strong suit and I need a program to help me teach writing that's more formulaic and that holds my hand.  It's probably not the ideal writing program and, if I'm honest, my dd won't get the best writing instruction she could get if I had the option of outsourcing.  She's getting the best of the options available to us, though (I hope), and I'm continuing to self-educate so that I can be the best writing teacher I can for her.  Ideally, I think the key is to recognize our weaknesses as educators (whether that's home educators or public/private school educators) and to ameliorate them as best we can through continuing self-education and seeking advice from others who are experts in our weak areas.  The worst thing we can do is to bury our heads in the sand and ignore our weaknesses or become defensive and come up with excuses as to why we don't need to face our weaknesses.

 

Just some thoughts.

[wapiti]

So what do we do about math in the elementary years?  Having just finished teaching an instructional methods course in elementary math at the local college, I feel like throwing up my hands in despair.  Very few elementary teachers are math majors (or science majors, for that matter) and do not seem to have the understanding of math needed to teach conceptually.  When they attempt to teach conceptually, it's as kiana mentioned - the blind leading the blind.  The best remedy would be to get math specialists into the elementary system but that doesn't seem to be happening.  

 

One option might be for more concept teaching to be included in the text, written directly to the student, without key information being contained in a TM that might not even be read, much less understood and then orally communicated to the student as it should be.  I worry that far too much is contained in TMs that falls thru the cracks in the classroom, and/or kids aren't really paying attention.  At least if the instruction were in the student's text, it would be available for parents as well when it's time to do homework.

 

eta, I like Wendy's idea of math specialists as a way out until a new generation of elementary teachers who do have sufficient understanding can join the ranks.

 

So is the lesser of two evils to simply let the non-mathy teachers teach algorithmically?  I feel like at least then the students would have SOMETHING coming into high school.

 

This ties in with Wishbone Dawn's thread on developing our own teaching skills, as well.  If a parent isn't mathy, maybe a curriculum or program that's heavier on algorithms and less so on conceptual understanding (not completely lacking conceptual explanations but they definitely take a back seat) is a better choice.  Not the ideal best choice but the best of the options available.

 

 

I vote that the *exact opposite* would be the better choice, that the parent/teacher lacking a sufficient math understanding would point toward using a program with more thorough concept instruction.  However, it may be that the instruction needs to be more direct, and/or more explicit in that case, such that the parent learns along with the student.

 

The worst thing we can do is to bury our heads in the sand and ignore our weaknesses or become defensive and come up with excuses as to why we don't need to face our weaknesses.

 

Agree!

[birchbark]

FWIW, the quoted article is criticizing Fuzzy Math programs (eta, referred to in the article as "reform math"; examples would include Everyday Math, Connected Math, TERC, etc.), and rightly so.  Fuzzy math programs are at the opposite end of a continuum from traditional math and tend to lack sufficient instruction and practice with algorithms.  In contrast to fuzzy math, programs such as Singapore and MM take a balanced approach to teaching concepts and algorithms. . .

 

I do think that the Math Wars have unfortunately focused on a false dichotomy between concepts and algorithms when there are existing programs that teach both, together, as they should.

 

While Singapore is a huge step up from the fuzzy math these articles refer to, it still has more "concept" and less "practice" than what many children need. It also is not as incremental and systematic as it might be. This may be fine for mathy students (and mathy teachers!), and provide that extra challenge that they need. I think Singapore is actually recommended in one of the articles I linked.

 

But despite Singapore's popularity, this forum abounds with complaints about the program. I believe the author of Math Mammoth created her curriculum to improve on these weaknesses of Singapore, and make it more homeschool-friendly, since Singapore was created for math teachers in classrooms. MM is one that I often recommend.

[Dicentra]

One option might be for more concept teaching to be included in the text, written directly to the student, without key information being contained in a TM that might not even be read, much less understood and then orally communicated to the student as it should be.

 

 

I vote that the *exact opposite* would be the better choice, that the parent/teacher lacking a sufficient math understanding would point toward using a program with more thorough concept instruction.  However, it may be that the instruction needs to be more direct, and/or more explicit in that case, such that the parent learns along with the student.

 

 

I hear you, wapiti - it would be the better choice.  But is it the most realistic choice?  This is what I struggle with when considering these issues.  Ideally, a balanced (algorithmic & conceptual) math program is best for students no matter where they're learning - home or at a b&m school.  I think, though, the natural human inclination IS to either ignore our weaknesses or become defensive and excuse them away.  It may be that homeschooling parents and/or public school educators feel they don't have the time, inclination, and/or energy to self-educate and learn the conceptual math.  Some educators (home or public) have had such a poor math experience themselves they refuse to even consider attempting to re-learn math conceptually.  So, realistically, what do we do?  This is what frustrates me (not you, wapiti - the problem! :)).  That's why I thought that maybe the lesser of the evils (algorithmic programs) would be the better choice in the sense that it's the more realistic choice and at least you'd end up with students (homeschooled or public schooled) that had SOME kind of math skills going into high school.  I've had students in my high school teaching days who went through elementary programs where the completely conceptual math programs were pushed and "algorithm" was a dirty word.  Unfortunately, they had elementary math teachers who didn't understand the math they were teaching and were essentially forbidden to teach algorithms.  It was a NIGHTMARE when those kids reached high school.  They had no conceptual understanding AND they had no algorithms to fall back on.  If I'm supposed to be teaching them stoichiometry in Grade 11 chem but they don't even understand what a fraction is or how to work with one, what can I do at that point?  I can't take a week out of a 20 week, semestered chem course to teach fractions, decimals, ratios, and percents - I'd have to drop some of the chemistry which then puts them behind for Grade 12 chem.

 

Sigh.  I don't know.  There's what should happen (educators self-educating to overcome weak areas) and what realistically will happen (some might, many will not).  What do we do?

[Crimson Wife]

So what do we do about math in the elementary years?  Having just finished teaching an instructional methods course in elementary math at the local college, I feel like throwing up my hands in despair.  Very few elementary teachers are math majors (or science majors, for that matter) and do not seem to have the understanding of math needed to teach conceptually.  When they attempt to teach conceptually, it's as kiana mentioned - the blind leading the blind.  The best remedy would be to get math specialists into the elementary system but that doesn't seem to be happening.  So is the lesser of two evils to simply let the non-mathy teachers teach algorithmically?  I feel like at least then the students would have SOMETHING coming into high school.

I think the best solution would be to use a textbook that has the conceptual teaching already in it. Something like Right Start or Math Mammoth. I didn't know what the heck I was doing when I started homeschooling, but I learned along side my oldest. RS and MM both teach the traditional algorithms as well as the underlying concepts.

[Corraleno]

So what do we do about math in the elementary years?  Having just finished teaching an instructional methods course in elementary math at the local college, I feel like throwing up my hands in despair.  Very few elementary teachers are math majors (or science majors, for that matter) and do not seem to have the understanding of math needed to teach conceptually.  When they attempt to teach conceptually, it's as kiana mentioned - the blind leading the blind.  The best remedy would be to get math specialists into the elementary system but that doesn't seem to be happening.  So is the lesser of two evils to simply let the non-mathy teachers teach algorithmically?  I feel like at least then the students would have SOMETHING coming into high school.

 

I think this is a really good question. Obviously the ideal solution would be specialized math teachers starting in the early grades, and I do think that's what schools should be working towards. However, since that seems extremely unlikely, or at least very far off, I'd like to see schools try the "flipped classroom" idea, where kids learn math via video instruction from a really good teacher (like Ed Burger), and the teacher just helps with homework. And since the teacher would need to watch the videos, too, it would serve as teacher training as well as student instruction. (ETA: and I would ensure that the curriculum was more along the lines of MM or Singapore, not Everyday Math!)

 

That sort of system would have problems of its own, but at least it might be an improvement over the current situation. I know it would have been a vast improvement over what my son was learning in PS; listening to his teacher try to explain fuzzy math concepts that she clearly did not grasp at all, was truly painful and one of the main reasons I started investigating homeschooling.

[wapiti]

. 

 

[Dicentra]

I think the best solution would be to use a textbook that has the conceptual teaching already in it. Something like Right Start or Math Mammoth. I didn't know what the heck I was doing when I started homeschooling, but I learned along side my oldest. RS and MM both teach the traditional algorithms as well as the underlying concepts.

 

Ideally, yes.  (See my above post.)  But will this work out realistically?

 

I'm sorry.  I must sound so pessimistic.  I just learned over years of teaching high school (and now college) that what seems to be the best option "on paper" doesn't turn out to be the best option realistically. :(

[Dicentra]

I think this is a really good question. Obviously the ideal solution would be specialized math teachers starting in the early grades, and I do think that's what schools should be working towards. However, since that seems extremely unlikely, or at least very far off, I'd like to see schools try the "flipped classroom" idea, where kids learn math via video instruction from a really good teacher (like Ed Burger), and the teacher just helps with homework. And since the teacher would need to watch the videos, too, it would serve as teacher training as well as student instruction. (ETA: and I would ensure that the curriculum was more along the lines of MM or Singapore, not Everyday Math!)

 

That sort of system would have problems of its own, but at least it might be an improvement over the current situation. I know it would have been a vast improvement over what my son was learning in PS; listening to his teacher try to explain fuzzy math concepts that she clearly did not grasp at all, was truly painful and one of the main reasons I started investigating homeschooling.

 

I like the video instruction idea. :)  I know from a purely academic standpoint, having separate teachers for separate subjects (each being a specialist in that subject) is the best option but I've also seen research that shows that that's not a great idea at the elementary level.  Some young children (not all) have a difficult time having multiple teachers at the early elementary level - lack of consistency, etc.  I'd like to see more schools try the video instruction idea, though - the classroom teacher is still there with them so that helps address the consistency problem.

 

 

The trouble with that is that it's a vicious cycle.  I don't know what the answer is except that I don't think learning the concepts needs to be unrealistic.  However, I do think that the advocacy of fuzzy math has added a very unfortunate dimension.  The development of Everyday Math, for example, was unseemly, with the mathematicians leaving the project upon frustration with the professional educators.  I don't really understand why school purchasers get "sold a bill of goods" so often.

 

 

I agree, wapiti - it is a vicious cycle.  I'm having a hard time seeing a realistic way out, though.  I don't mean to say there isn't a realistic way out, just that I can't see one. :)

 

And I do agree - I don't think that the actual learning of conceptual math is unrealistic.  I think that virtually everyone is capable of learning math conceptually.  What I'm questioning (depressing pessimist that I am) is whether we can convince educators (home or b&m) to actually put the effort, struggle, and time in to go ahead and do it.  You can lead a horse to water...

[kiana]

I think this is a really good question. Obviously the ideal solution would be specialized math teachers starting in the early grades, and I do think that's what schools should be working towards. However, since that seems extremely unlikely, or at least very far off, I'd like to see schools try the "flipped classroom" idea, where kids learn math via video instruction from a really good teacher (like Ed Burger), and the teacher just helps with homework. And since the teacher would need to watch the videos, too, it would serve as teacher training as well as student instruction. (ETA: and I would ensure that the curriculum was more along the lines of MM or Singapore, not Everyday Math!)

 

That sort of system would have problems of its own, but at least it might be an improvement over the current situation. I know it would have been a vast improvement over what my son was learning in PS; listening to his teacher try to explain fuzzy math concepts that she clearly did not grasp at all, was truly painful and one of the main reasons I started investigating homeschooling.

Or honestly, put them all in TT and let the teacher facilitate by making sure they stay on-task. That would let them pace independently as well so that you don't have the slow kids frustrated by the quick ones and vice versa. Move towards group classes once you get to more qualified teachers in (hopefully) middle school and definitely high school.

 

I shouldn't be so negative. And I really would much rather have small-group instruction with qualified teachers, preferably math specialists, starting in kindergarten. But I just don't see it happening any time soon on a wide scale, although I have some really outstanding elementary education majors in my precalculus class this semester.

 

The other really sad thing is that it only takes one terrible teacher to undo years of good math instruction.

[wapiti]

I like the video instruction idea. :)  I know from a purely academic standpoint, having separate teachers for separate subjects (each being a specialist in that subject) is the best option but I've also seen research that shows that that's not a great idea at the elementary level.  Some young children (not all) have a difficult time having multiple teachers at the early elementary level - lack of consistency, etc.  I'd like to see more schools try the video instruction idea, though - the classroom teacher is still there with them so that helps address the consistency problem.

 

I think this is where Khan is onto something, though last I looked (a long, long time ago), the videos and the program more generally were leaning much more toward procedural.  If the videos were, on the other hand, more well-done from a conceptual standpoint, then yes, I think this could be very useful.

 

On the horse/water/drink aspect, that's not only true of teachers but many many students as well.  The old saying that there's no royal road to learning is so true - the students need to intellectually engage with the lesson.  (Even my kids don't typically engage with the lesson, LOL, when they're watching well-done aops videos, and then afterward they ask how to do the problem :glare:.  So, I don't make them watch the videos anymore but sit with them one-on-one and we do the lesson problems together, socratically.)

[MomintheMountains]

I'm sorry, but there is something wrong when elementary math is thought to be so hard that nobody can adequately teach it.  Math is not hard, at least not the kind of math that most people need for their every day lives.  Maybe people would stop being math phobic if we could wrap our minds around that basic idea.  

[Corraleno]

I'm sorry, but there is something wrong when elementary math is thought to be so hard that nobody can adequately teach it.  Math is not hard, at least not the kind of math that most people need for their every day lives.  Maybe people would stop being math phobic if we could wrap our minds around that basic idea.  

 

Then why teach math at all beyond 8th grade? That's all the math that "most people need for their everyday lives." And why not apply that same logic to science? Or history and literature? Do we really want to limit education to just what the average person "needs" to get by in their daily lives?

 

The fact that so many American teachers can't adequately teach math isn't because it's "too hard" — it's because they were never taught properly. I don't think the solution to that is to just teach computation and give up on true conceptual understanding. IMHO, conceptual math is no more difficult to learn, or to teach, than purely procedural math — assuming it's taught right, by a teacher who understands it.

 

I think the reason so many people are math phobic is because math seems like a completely abstract language full of random rules that make no sense. The cure for that, IMHO, is to teach kids how to "read" math fluently, instead of asking them to just memorize a few tourist phrases. Of course they also need to know the facts & algorithms, but without a true understanding of what they're doing, kids just end up floundering in higher level math. 

[konglish]

 

Also what does it mean to say one's education served them just fine? 

 

Not strictly related to math, but I never find the "I had _____  in my childhood and I turned out fine" to be a very compelling argument.  Usually because I'm thinking "you turned out sort-of fine...and we don't know how you might have turned out with the alternative."  

 

I'm enjoying this thread!  

[Corraleno]

I keep struggling with this point in this discussion.  Is it really true that there are a large number of elementary teachers who don't understand elementary level math?  I have a hard time wrapping my brain around that.  I can understand the breakdown once moving into algebra, but I can't figure out what would be difficult to understand conceptually about elementary math.  I'm hoping that doesn't come out sounding rude.  Can anyone offer an example of a concept that a teacher might not understand well enough to teach it effectively?

 

Liping Ma's book, Knowing and Teaching Elementary Mathematics, is full of examples:

 

Teachers who think the zeros in multi-digit multiplication are just "place holders" and not "real zeros"

Teachers who think 4 divided by 1/2 = 2

Teachers who can't remember the difference between perimeter and area

Teachers who don't understand the commutative and distributive properties

Teachers who have no idea why "invert & multiply" works

etc.....

 

It's an eye-opening (and really depressing!) book.

[kiana]

I keep struggling with this point in this discussion.  Is it really true that there are a large number of elementary teachers who don't understand elementary level math?  I have a hard time wrapping my brain around that.  I can understand the breakdown once moving into algebra, but I can't figure out what would be difficult to understand conceptually about elementary math.  I'm hoping that doesn't come out sounding rude.  Can anyone offer an example of a concept that a teacher might not understand well enough to teach it effectively?

 

Liping Ma's book "knowing and teaching elementary mathematics" has multiple examples of flawed explanations from teachers. The four main areas are subtraction with regrouping, multidigit multiplication, division by fractions, and exploring relationships between perimeter and area. The fourth part was really seeing how they explored things they hadn't thought of before. The book also explains why these ideas are mathematically problematic. Here is an example: A teacher is teaching subtraction with regrouping, and explains "But you don't have enough ones to subtract, so you go to his neighbor here who has plenty". On the surface, this is a reasonable explanation. But it reinforces two incorrect ideas that students consistently have:

 

1) If you want something, you just put it there. This is also seen in fractions, where students frequently will simply add a denominator without doing anything to the numerator, because they do not understand that when establishing a common denominator they are really multiplying by 1. It is also seen in equations, where a student will be told to manipulate an expression, and will simply start squaring it, multiplying it by something, et cetera.

 

2) These are two different numbers. In reality, they are different parts of the same number, and this must be repeatedly emphasized.

 

ETA: Haha Jackie and I were cross-posting, but this is a really good book.

[ravinlunachick]

I keep struggling with this point in this discussion. Is it really true that there are a large number of elementary teachers who don't understand elementary level math? I have a hard time wrapping my brain around that. I can understand the breakdown once moving into algebra, but I can't figure out what would be difficult to understand conceptually about elementary math. I'm hoping that doesn't come out sounding rude. Can anyone offer an example of a concept that a teacher might not understand well enough to teach it effectively?

I posted not long ago about my dd arguing with her (public virtual school) teacher after the teacher insisted that any number divided by zero equals zero. To make matters worse, it was during a discussion of using inverse operations to check your work! So, yes, I believe such a situation is absolutely possible.

[Dicentra]

I keep struggling with this point in this discussion.  Is it really true that there are a large number of elementary teachers who don't understand elementary level math?  I have a hard time wrapping my brain around that.  I can understand the breakdown once moving into algebra, but I can't figure out what would be difficult to understand conceptually about elementary math.  I'm hoping that doesn't come out sounding rude.  Can anyone offer an example of a concept that a teacher might not understand well enough to teach it effectively?

 

I guess it all depends on how "understands math" is defined. KWIM? :)  I think there are teachers who are well versed enough in the "invert and multiply" algorithm for dividing by a fraction to teach the algorithm fairly effectively to students.  But do they understand why they are inverting and multiplying?  Maybe we should try a little experiment. :D  Everyone who thinks they can, post an explanation that would be understandable by, say, kids in Grade 4 or 5 on why we invert and multiply when we divide by a fraction.  No Googling, no copying from a text.  (And math people - hold on for a bit until you post. ;))

 

(My point is that many of us may think "Oh, elementary school math - that's easy to understand!" but what we're thinking is "easy to understand" is actually how to use the algorithms, not the theoretical explanation behind the concepts.  In the Number Theory course I took in university, we had to prove very basic tenants of math in a very complicated way (at least, that's how I remember it :P - I joke that it took me a page and a half to prove that 2+2=4)).

[a27mom]

I guess it all depends on how "understands math" is defined. KWIM? :) I think there are teachers who are well versed enough in the "invert and multiply" algorithm for dividing by a fraction to teach the algorithm fairly effectively to students. But do they understand why they are inverting and multiplying? Maybe we should try a little experiment. :D Everyone who thinks they can, post an explanation that would be understandable by, say, kids in Grade 4 or 5 on why we invert and multiply when we divide by a fraction. No Googling, no copying from a text. (And math people - hold on for a bit until you post. ;))

 

(My point is that many of us may think "Oh, elementary school math - that's easy to understand!" but what we're thinking is "easy to understand" is actually how to use the algorithms, not the theoretical explanation behind the concepts. In the Number Theory course I took in university, we had to prove very basic tenants of math in a very complicated way (at least, that's how I remember it :P - I joke that it took me a page and a half to prove that 2+2=4)).

I actually tried to do this last month, when helping my friends 6th grader with his math homework. He used Everyday Math, but obviously had no idea "why" he was doing what he was doing (or supposed to be doing). (He was working on multiplying and dividing mixed numbers, and had been confused about fractions for almost 2 years)

[Crimson Wife]

I guess it all depends on how "understands math" is defined. KWIM? :)  I think there are teachers who are well versed enough in the "invert and multiply" algorithm for dividing by a fraction to teach the algorithm fairly effectively to students.  But do they understand why they are inverting and multiplying?  Maybe we should try a little experiment. :D  Everyone who thinks they can, post an explanation that would be understandable by, say, kids in Grade 4 or 5 on why we invert and multiply when we divide by a fraction.  No Googling, no copying from a text.  (And math people - hold on for a bit until you post. ;))

I use Hershey bars as my example for this, LOL!

[Dicentra]

Anybody willing to type in and share his/her explanation? :)

[wapiti]

Anybody willing to type in and share his/her explanation? :)

a recent thread http://forums.welltrainedmind.com/topic/505867-dividing-fractions-why/

[Dicentra]

a recent thread http://forums.welltrainedmind.com/topic/505867-dividing-fractions-why/

 

Thanks, wapiti!  Didn't know about that thread. :)

[rainbird2]

I'm reading Liping Ma's book right now.  Definitely an eye opener.  (I'm kind of peeved about how I was taught math in school, but I won't go there right now.) 

 

Are there any recommendations for a follow up to Ma's book?

[smileyme]

Being Pro-math all my life, I think math is about concepts, but the concepts get more clear, as you solve more... Hence some genuine quality practice. So not necessarily text books, but definitely conceptual practice books helps. All kids probably have it hidden some corner, and practice gets that out... Of course it depends on interest too.. If kid likes art more than math .. well encourage it... art helps math, math helps art.. whatever works.

[strange_girl]

Thanks for all the replies!

 

DD is using MM, which is pretty conceptual, so we'll probably stick with that. She has also been liking working through Horizons Math 2 on the side, so between the two she'll be okay. :)

[smileyme]

Remember visiting this thread years back. DD is going to middle school and problem solving workbooks helped her speed up much better than concepts overloading, she was in GT problem solving and enjoyed that. I believe for kids to ramp up problem solving works best. Singapore math and beast academy.