Thoughts on Conceptual vs. Traditional math, please?

[dessertbloom]

I'm trying to learn more about the pros and cons of each. Having grown up with traditional math, I'm so much more comfortable with it. How do you see these two approaches in terms of classical education and the ages and stages of children's development? I have a couple STEM loving kids and want to give them a great start. My oldest did MM 1A at home, then went to private school where she had already learned all of her first grade math before the get-go. I'm trying to decide how to work with her over the summer. My other STEM loving kiddo is just 4, so we have time with him, but he's teaching himself and wants to do school so I'm looking at MEP Reception. I'd love to hear thoughts on this topic!  THanks

[LMD]

We spent a lot of time on conceptual maths, logic, puzzles etc. Dd did some Singapore and miquon and then we started with Beast academy. The beast release schedule was slow so we filled in the time with bits and pieces - Singapore cwp/ip, problem solving books, life of fred etc. I was very relaxed about a grade book a year.

 

Last year, early 5th, she took part in a mathematics competition, in the 5/6th grade section. First ever maths test and I didn't prep at all. She hadn't quite finished all the BA4 books at that point. She got a distinction, but more interesting was the breakdown against the average - she scored twice the national average in problem solving skills.

 

Traditional maths is fine as long as they understand the underlying concepts. Procedure alone won't help with a differently presented problem.

 

I'm sure much wiser people will come and reply soon!

[RoundAbout]

Personally I don't think traditional, procedural math is enough for a 21st century education. Obviously basic fluency in arithmetic is necessary for everyone, but the thinking and problem solving involved in programs like Singapore or Beast Academy are going to better prepare a student for higher level STEM coursework, and are just a lot more interesting to boot. Those programs may not be a good fit for every student, but I would definitely lean towards as much conceptual work as the student could handle. The big thing is that it doesn't have to be an "either/or" proposition.  I am a big fan of teaching conceptually as much as possible but then practicing math facts and faster calculations through games and puzzles (and even the occasional worksheet if the student needs work in one area). 

[SparklyUnicorn]

Well, I'm not entirely sure what is meant with conceptual vs. traditional.  Guessing what I think you mean, something like Singapore really has both.  I don't know that I'd want something that only does one or the other.

 

 

[8filltheheart]

I don't think they are mutually exclusive. Most math programs do teach concepts. Posters on here will say emphatically that Horizons math does not teach concepts. It makes me wonder what they are teaching bc I have used it for over 20 yrs with 8 kids, 6 all the way through their 6th grade book. (It used to stop there, so so do I bc I have formed my own math progression sequence.). The concepts are most definitely taught and my kids go on to be incredibly strong math students in high school and college.

 

Does it teach like Singapore? No. But I am not a huge SM fan.

 

More than 1 math approach can lead to solid mathematical understanding.

[HomeAgain]

It is handy to know the traditional algorithms.  Most math curricula will teach that part well (most.  Everyday Math is not intuitive and cumbersome).

 

I want my kids to see past the algorithms and know when to use them, how to get past something when they get stuck, and to approach math like a puzzle to solve.  MEP is very good for that - it teaches traditional methods, it makes sure kids understand conceptually what they're doing, and it throws puzzles at them to get them to think through using the skills they have.  I could spend all day teaching the 'how-to steps', but unless a kid really understands it the work becomes pointless.  It leaves them with a shaky footing in arithmetic before getting to algebra because they have to rely solely on memorized information.  It doesn't mean you have to use a lot of manipulatives, though they certainly help.  Our go-to items are c-rods and MUS materials, but often it's just learning how to think about the problem differently.

[Ellie]

I'm trying to learn more about the pros and cons of each. Having grown up with traditional math, I'm so much more comfortable with it. How do you see these two approaches in terms of classical education and the ages and stages of children's development? I have a couple STEM loving kids and want to give them a great start. My oldest did MM 1A at home, then went to private school where she had already learned all of her first grade math before the get-go. I'm trying to decide how to work with her over the summer. My other STEM loving kiddo is just 4, so we have time with him, but he's teaching himself and wants to do school so I'm looking at MEP Reception. I'd love to hear thoughts on this topic!  THanks

 

I don't even understand what that means. :confused1:

[MeganW]

I was 100% convinced that my kids would going to have a thorough conceptual understanding of math that was so strong they wouldn't even NEED to memorize algorithms.  That was sort of like knowing everything about parenting before you have kids.  :)  

 

One of my kids has everything I hoped for.  He can logic through any math problem.

 

Two of my kids understand most of what they have learned conceptually.  They rely on the algorithms, but can tell you why if you probe.

 

The fourth, well, I'm just working toward her being able to solve common problems and get a correct answer.  I keep trying for conceptual, but to be honest, it isn't happening with this kid, at least not at this point.  Maybe once she can solve most arithmetic procedurally, the concepts will begin to make more sense.

[HTRMom]

I used Kumon as an elementary child, which is completely process - learn to follow the formulas as quickly as possible. Then I learned more conceptual problem-solving thinking in middle and high school math. I ended up a chemical engineer.

 

Isn't the grammar stage supposed to focus on the grammar of math, mastering the mechanics and memorizing facts, and the logic of math comes in the logic stage? That's how I've been expecting to do it. (Oldest still in preschool.) I guess I would have said that there's nothing wrong with more conceptual problem-solving work in elementary years, as long as it doesn't detract from memorizing math facts and getting really good at basic mechanics of math. You don't want to be taking calculus but getting out your calculator for 12 x 15.

 

Am I way off base? Do kids not really need to get good at arithmetic in the era of calculators?

 

 

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[RoundAbout]

I used Kumon as an elementary child, which is completely process - learn to follow the formulas as quickly as possible. Then I learned more conceptual problem-solving thinking in middle and high school math. I ended up a chemical engineer.

 

Isn't the grammar stage supposed to focus on the grammar of math, mastering the mechanics and memorizing facts, and the logic of math comes in the logic stage? That's how I've been expecting to do it. (Oldest still in preschool.) I guess I would have said that there's nothing wrong with more conceptual problem-solving work in elementary years, as long as it doesn't detract from memorizing math facts and getting really good at basic mechanics of math. You don't want to be taking calculus but getting out your calculator for 12 x 15.

 

Am I way off base? Do kids not really need to get good at arithmetic in the era of calculators?

 

 

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I think the better analogy is that teaching conceptual understanding is akin to teaching reading via phonics instead of just memorizing whole words. I don't think most people are talking about super abstract thinking at this age, more like using manipulatives to understand how to make 10's, break apart numbers and reconfigure them, and things like that with C-rods or other tools of choice. 

 

I teach a couple of Kindergartners once a week and yesterday one of my students added 37 and 34 by counting each and every bead past 37 on the AL Abacus. Once I showed him that we could just start by counting the full rows of 10s it was like a lightbulb went off and he got to the answer much more quickly. That's the kind of thing I think of when people say "conceptual understanding" at the elementary level - number sense instead of just memorizing facts and rote algorithms. 

[HTRMom]

Oh! That makes sense. Thank you. I was thinking that people were suggesting algebra in second grade, or tons of abstract word problems or something.

 

 

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[MistyMountain]

There definitely is a difference in approaches. A conceptual program really teaches the why of what is happening and has a bigger emphasis on really understanding what is happening when you are doing a problem so you can visualize it by either being strong on mental math tricks, place value, subitizing and visualizing a problem by seeing what the numbers are doing. It might show things concretely first with manipulatives especially in ways that are not just counting them up. A procedural math teaches algorithms for solving problems and spends less time on the why and they have more memorizing with facts.They both teach concepts but with emphasis on different aspects when teaching. Things like Signapore, Miquon, Beast Academy MEP, Right Start would be more conceptual while Rod and Staff, Saxon are more procedural.

[LMD]

I do, definitely, value learning the facts cold, times tables especially. 6x7=42

 

But conceptual adds the understanding. So they know that 6x7=(6x6)+6, or (6x10)-18 etc. They understand the relationship, they see how the numbers fit together and are comfortable playing and manipulating them.

This is more to do with discussion with the teacher, rather than specific books or curriculum necessarily.

 

Another story -

I was in the car with my six year old. He asked how many weeks in a year, I answered with 52. I then continued the discussion by saying that half of the year was 26 weeks, 26+26=52. My son then continued to ask, 'so, 27+25 must also be 52! And 28+24, and 29+23...'

 

After a few of these, he got quiet, and then suddenly piped up with, 'hey mama, that means that 37+15 also equals 52!' He understood the concepts and was able to mentally manipulate the numbers, it wasn't rote learning - he's only 6, he hasn't learned the algorithm for double digit addition with carrying, he wasn't following a procedure.

Edited March 15, 2017 by LMD

[dessertbloom]

Thanks everyone, this was helpful.  If anyone is interested, this was the most useful thread I found when first looking into the topic:

 

 http://forums.welltrainedmind.com/topic/527884-help-me-understand-conceptual-vs-traditional-math/ 

 

I guess I'm just so indecisive when it comes to choosing math! I have R&S grades 1 and 2, and all of Math Mammoth, and Miquon, and of course MEP is free online...  I keep looking at Singapore.  I guess I have to wrap my brain around the more conceptual Asian style teaching of math before I'm comfortable teaching it. I keep seeing in different posts that MM teaches itself, that it is written to the student. Am I missing something? I read the Introduction for each chapter, but within the Worktext I don't really see how it's teaching the student.  I only see one or two examples, and then lots of problems to solve. I saw that she has added videos online, that could help I guess.  Am I going to be more confused by Singapore switching books and workbooks every 10 minutes, or would it help that it has a Teacher's Manual? Is MEP more or less the same thing, in that it's scripted (and free?)   Or should I just stick with old fashioned, tried and true R&S?  Ahh...  sorry for the stream of consciousness!  

 

[LMD]

Desertbloom, any of those will be fine. The best way to decide is to go with your gut and jump in and try. What will make or break any of those is the teacher, if you are engaging and explaining, having discussions and using manipulatives then you'll be fine. Truly.

[ScoutTN]

MM is written to the student, but I teach it to my 3rd grader, for sure! I taught my older child all the way through Right Start, so I am now comfortable explaining the concepts. I can help him by explaining a different way if something is confusing and I can easily see where he needs more practice. Before I taught Right Start, I could not have done this. (Humanities mom with a stupendously weak math education here.) 

 

My Dd transitioned into MM from 5 years of Right Start and it is mostly self-teaching for her. I add explanations or find a video (MM or Khan academy) or give a hint when she is stuck. I make sure she doesn't get bogged down and that she reviews well. But MM does the primary teaching. This student is diligent, responsible and enjoys math, though she is an average math student. It might not be so independent with a different type of student. 

 

 

[Farrar]

I think the thread you linked is a good one. But generally there are lots of ways to break down math programs and that's just one lens for looking at them. But I second what 8 said about how most programs are both to some extent. It's a continuum. If you look at some of the vintage math texts, they're very "conceptual." I think it's a Ray's? Or, no, I think it's a Strayer-Upton that I saw that has a page that teaches left to right subtraction basically the way Right Start teaches it.

 

Thanks everyone, this was helpful.  If anyone is interested, this was the most useful thread I found when first looking into the topic:

 

 http://forums.welltrainedmind.com/topic/527884-help-me-understand-conceptual-vs-traditional-math/ 

 

I guess I'm just so indecisive when it comes to choosing math! I have R&S grades 1 and 2, and all of Math Mammoth, and Miquon, and of course MEP is free online...  I keep looking at Singapore.  I guess I have to wrap my brain around the more conceptual Asian style teaching of math before I'm comfortable teaching it. I keep seeing in different posts that MM teaches itself, that it is written to the student. Am I missing something? I read the Introduction for each chapter, but within the Worktext I don't really see how it's teaching the student.  I only see one or two examples, and then lots of problems to solve. I saw that she has added videos online, that could help I guess.  Am I going to be more confused by Singapore switching books and workbooks every 10 minutes, or would it help that it has a Teacher's Manual? Is MEP more or less the same thing, in that it's scripted (and free?)   Or should I just stick with old fashioned, tried and true R&S?  Ahh...  sorry for the stream of consciousness!  

 

As for programs... pick what's going to work for you and your kids. You may love the idea of a program, but if you can't figure out how to teach it or if your kids simply can't seem to get it, then it doesn't matter if it's "the best." To your specific questions, I would say MEP and Singapore are pretty different in some key ways. MEP is great. But the different book options in Singapore make it useful for different learners who need more practice or more challenge. Part of what teaches in a well written program is the problems themselves - you'll find that's especially true in a program like Miquon, that's discovery based. The problems lead the student to "discovering" things about how the numbers work in different situations. But even in a program like MM, that's where the teaching is. A lot of problems.

 

I agree with the advice to just pick something and dive in. The worst thing that happens is that you switch programs, which is really not the endo f the world.

[Spy Car]

Oh! That makes sense. Thank you. I was thinking that people were suggesting algebra in second grade, or tons of abstract word problems or something.

 

 

 

 

On the other hand, one can have something as simple as two Cuisenaire Rods where one (the longer of the two) represents the "whole" and the other represents one "part."

 

Early C Rod work/play involves having children find the value of the missing "part." The missing "part" can become the "missing addend." 

 

In an equation, the missing addend can be shown as 5 + [  ]  = 9.

 

It can also be shown as 5 + x = 9, and suddenly a 5-year-old is going algebra.

 

Bill (who recommends the Liping Ma book to anyone who is unclear about conceptual approaches vs analog-only ones in early math education)