Help me understand conceptual vs. traditional math

[Erin]

The ones from a century ago were still quite excellently balanced on this point, especially in early arithmetic. They started going downhill shortly thereafter, though ;)

 

And yet it's "modern math" that sent men to the moon, invented the iPad and created surgical procedures from a mere beam of light (laser).  

[vaquitita]

This is the exact concept I am working on with my 3 year old in math right now: if you have 5 counters and then split them into a pile of 2 and a pile of 3, you still have 5 counters. And then if you move one counter from your 2 pile and add it to your three pile, you now have a 1 pile and a four pile and you still have 5 counters. I am confident that by the time he is ready for Singapore 1A that he will have no trouble seeing that 9+6 = 10+5.

 

This, learning all the combinations that make up a number, is in Ray's arithmetic and strayer Upton which I would consider traditional. And it is in the HIG to spend time learning all the combinations that make 6, 7, 8, 9, 10. But I do think it would be very easy to move quickly through this part, not noticing that you should be spending time on this. And many do not even use the HIG in the early years. I read somewhere that in Singapore they start at 7. Which makes a difference too, if you are doing this with a 5yo vs. a 7yo.

 

I am right now using Singapore PM 1 with my 7yo. Alongside it we are using Franklin primary arithmetic, which gives more practice. We are spending lots of time 'drilling' these math facts, just not with a traditional worksheet. We hit it orally with Franklin, we hit it with games (go to the dump from RS), we hit it with blocks and pictures and worksheets with Singapore. Since Singapore is 'advanced' I'm just making myself not worry about finishing each year at grade level. I was considering study time math, traditional, but the large number of problems on each page would not work well for my pencil phobic oldest DS. Singapore seems to work for him, the color, the few problems per page. He gets concepts, hates worksheets. After considering lots of ways to utilize the extra Singapore books to get that much needed practice, I finally went with strayer Upton practical arithmetic. It works well orally, and it includes lots of drill. I actually like that the scope and sequence is totally different from Singapore. That way if he isn't quite ready for something in Singapore math, it will be hit again later in SU. I cover it, he understands it, even if it isn't totally mastered, and we move on, knowing it will come up again in strayer Upton. I think it's totally possible to get enough practice with Singapore, if you use multiple books, which it is intended for you to do. I just find strayer Upton easier to juggle. I hope I'm getting the best of both this way. 😃

 

Basically I think you need both. With a 'conceptual' program you need to be sure to do plenty of practice. With a 'traditional' program you need to be sure and teach with concrete objects.

 

Eta: I do wish there were more good reviews of traditional math programs here. When I was searching because we needed to switch from MEP I just found over and over Singapore, miquon, and right start. Saxon was mentioned repeatedly, but usually something along the lines of we used this our first year of HS and it was torture. R&S was mentioned too, but since it's religious my charter school won't purchase it, so I didn't even look at it. I would love to hear what programs the pro-traditional math people are using?

[Meadowlark]

This, learning all the combinations that make up a number, is in Ray's arithmetic and strayer Upton which I would consider traditional. And it is in the HIG to spend time learning all the combinations that make 6, 7, 8, 9, 10. But I do think it would be very easy to move quickly through this part, not noticing that you should be spending time on this. And many do not even use the HIG in the early years. I read somewhere that in Singapore they start at 7. Which makes a difference too, if you are doing this with a 5yo vs. a 7yo.

 

I am right now using Singapore PM 1 with my 7yo. Alongside it we are using Franklin primary arithmetic, which gives more practice. We are spending lots of time 'drilling' these math facts, just not with a traditional worksheet. We hit it orally with Franklin, we hit it with games (go to the dump from RS), we hit it with blocks and pictures and worksheets with Singapore. Since Singapore is 'advanced' I'm just making myself not worry about finishing each year at grade level. I was considering study time math, traditional, but the large number of problems on each page would not work well for my pencil phobic oldest DS. Singapore seems to work for him, the color, the few problems per page. He gets concepts, hates worksheets. After considering lots of ways to utilize the extra Singapore books to get that much needed practice, I finally went with strayer Upton practical arithmetic. It works well orally, and it includes lots of drill. I actually like that the scope and sequence is totally different from Singapore. That way if he isn't quite ready for something in Singapore math, it will be hit again later in SU. I cover it, he understands it, even if it isn't totally mastered, and we move on, knowing it will come up again in strayer Upton. I think it's totally possible to get enough practice with Singapore, if you use multiple books, which it is intended for you to do. I just find strayer Upton easier to juggle.

 

Basically I think you need both. With a 'conceptual' program you need to be sure to do plenty of practice. With a 'traditional' program you need to be sure and teach with concrete objects.

Yep, the first paragraph sums up my experience with Singapore. I have the HIG, but never knew which games to play, what activities to add, or when to even do them. I found it confusing. I can see it's just not my teaching style. For people who can easily implement it, I think there's a much higher chance of success.

 

We're changing to CLE on Monday. I have to admit, I was a little sad putting Singapore away though. I keep thinking maybe I haven't given it a fair shake, but I need to try something a little more laid out for me-something to tell me what facts to drill every day, etc. This conversation has instilled in me a need to teach math both ways...so I need to balance CLE out if I find it needs it, and I am prepared to do that thanks to this thread and all of your opinions.

 

It was also a breath of fresh air to hear from a few of you that Singapore and other Asian math programs are not necessarily superior. I think I was feeling like a failure because I couldn't make the "best" program work for me. I felt like I was failing my kids. Thanks for making me see that a traditional program can also give my kids a superior math education.

[mohini]

Can you give me an example? And can you tell he how non-conceptual (or Asian) programs approach a similar thing? I didn't grow up in Asia, but the math program we used as kids wasn't that different. Examples can help me understand what you mean.

 

Well, I gave a couple of examples before but here's another.

 

The way we look at numbers in "traditional" vs "conceptual"

 

Traditional teaches 45 is 4 tens and 5 ones - when you want to manipulate the number (+/- etc..) you will start with the algorithm. Yes, you should comprehend how the algorithm works but you will work through the algorithm largely using facts that you have stored (though of course there are some strategies for remembering and extending those facts)

 

Conceptual says the same (of course) but adds an endless array of possibilities for breaking down the number depending on what you are going to do with it. 

If you are adding 45 + 55 then you should know that 45 is a 40 and a 5 and 55 is a 50 and a 5  - so 40+50=90 and 5+5=10 Therefore  45+55=100

 

If you are subtracting 36 from 45 (45-36) you might need to make 45 into 50 by adding 5 and if you do that you might need to add 5 to 36 too. So you would have 45+5= 50, 36+5=41  so 50-41= 9 (you may even go a step further and say 50-40= 10 and 10-1=9) Therefore 45-36=9

 

You might be adding 45+7 in which case you have to say "7 is a 2 and a 5." 45+5=50 and 50+2 = 52 therefore 45+7=52

 

There are 2 aspects of this process that I would categorize as "Logic Stage thinking." The first is the "therefore" statement. This is a statement of causation - causation is a form of analysis that is traditionally relegated to logic stage thinking. It is a logical syllogism. If 7 is a 5 and a 2 and 5+5= 10 then 7+5 is the same as 10+2 therefore 5+7= 12.

 

The second is the process of choosing which strategy will be appropriate for a given problem. That requires a logical analysis of the problem. "It will be easier (more logical) if I....." That analysis constitutes a form of reasoning that is again based in understanding the underlying "cause" of the problem. And - after analyzing the problem, the student must turn to the number and devise a reasonable strategy to manipulate the number around the problem.

 

Mind you - all of these strategies are super cool and very appropriate to teach to a kid who is ready to make that leap - but it is a leap. Grammar stage thinking would focus on fact acquisition, observation, repetition and comprehension (which I differentiate here from "understanding.") to build a body of knowledge that could later be applied to those types of strategies.

 

Anyway - like everybody keeps on reminding everybody else, most programs teach both. The emphasis may be more towards one or the other, but the basic ideas are included in any solid program.

 

I really only started the whole hullaballoo to say that there is nothing wrong with introducing the conceptual aspect later on and in fact, there may be reasons to wait. If you read the thread about "when did your kid start reading" there are all kinds of different answers. Some kids read at 3 and some didn't touch a book until 9 but now are breezing through Victorian novels.

 

Sometimes introducing a concept at the right time makes all the difference.

[Roadrunner]

Well, I gave a couple of examples before but here's another.

 

The way we look at numbers in "traditional" vs "conceptual"

 

Traditional teaches 45 is 4 tens and 5 ones - when you want to manipulate the number (+/- etc..) you will start with the algorithm. Yes, you should comprehend how the algorithm works but you will work through the algorithm largely using facts that you have stored (though of course there are some strategies for remembering and extending those facts)

 

Conceptual says the same (of course) but adds an endless array of possibilities for breaking down the number depending on what you are going to do with it.

If you are adding 45 + 55 then you should know that 45 is a 40 and a 5 and 55 is a 50 and a 5 - so 40+50=90 and 5+5=10 Therefore 45+55=100

 

If you are subtracting 36 from 45 (45-36) you might need to make 45 into 50 by adding 5 and if you do that you might need to add 5 to 36 too. So you would have 45+5= 50, 36+5=41 so 50-41= 9 (you may even go a step further and say 50-40= 10 and 10-1=9) Therefore 45-36=9

 

You might be adding 45+7 in which case you have to say "7 is a 2 and a 5." 45+5=50 and 50+2 = 52 therefore 45+7=52

 

There are 2 aspects of this process that I would categorize as "Logic Stage thinking." The first is the "therefore" statement. This is a statement of causation - causation is a form of analysis that is traditionally relegated to logic stage thinking. It is a logical syllogism. If 7 is a 5 and a 2 and 5+5= 10 then 7+5 is the same as 10+2 therefore 5+7= 12.

 

The second is the process of choosing which strategy will be appropriate for a given problem. That requires a logical analysis of the problem. "It will be easier (more logical) if I....." That analysis constitutes a form of reasoning that is again based in understanding the underlying "cause" of the problem. And - after analyzing the problem, the student must turn to the number and devise a reasonable strategy to manipulate the number around the problem.

 

Mind you - all of these strategies are super cool and very appropriate to teach to a kid who is ready to make that leap - but it is a leap. Grammar stage thinking would focus on fact acquisition, observation, repetition and comprehension (which I differentiate here from "understanding.") to build a body of knowledge that could later be applied to those types of strategies.

 

Anyway - like everybody keeps on reminding everybody else, most programs teach both. The emphasis may be more towards one or the other, but the basic ideas are included in any solid program.

 

I really only started the whole hullaballoo to say that there is nothing wrong with introducing the conceptual aspect later on and in fact, there may be reasons to wait. If you read the thread about "when did your kid start reading" there are all kinds of different answers. Some kids read at 3 and some didn't touch a book until 9 but now are breezing through Victorian novels.

 

Sometimes introducing a concept at the right time makes all the difference.

O.K. This is how math is taught in general. I guess I don't understand how else one would teach addition. I disagree greatly that this is a logic stage thinking.

[nannyaunt]

I am sitting here giggling to myself.  Who knew that all those years I spent doing 7x8 = ... is 6x8 = 48 + another 8 is 48 + 2 = 50 + 6 = 56, I wasn't being mathstupid, I was increasing my understanding of math concepts.

 

[wendyroo]

If you are subtracting 36 from 45 (45-36) you might need to make 45 into 50 by adding 5 and if you do that you might need to add 5 to 36 too. So you would have 45+5= 50, 36+5=41  so 50-41= 9 (you may even go a step further and say 50-40= 10 and 10-1=9) Therefore 45-36=9

 

You might be adding 45+7 in which case you have to say "7 is a 2 and a 5." 45+5=50 and 50+2 = 52 therefore 45+7=52

 

There are 2 aspects of this process that I would categorize as "Logic Stage thinking." The first is the "therefore" statement. This is a statement of causation - causation is a form of analysis that is traditionally relegated to logic stage thinking. It is a logical syllogism. If 7 is a 5 and a 2 and 5+5= 10 then 7+5 is the same as 10+2 therefore 5+7= 12.

 

The second is the process of choosing which strategy will be appropriate for a given problem. That requires a logical analysis of the problem. "It will be easier (more logical) if I....." That analysis constitutes a form of reasoning that is again based in understanding the underlying "cause" of the problem. And - after analyzing the problem, the student must turn to the number and devise a reasonable strategy to manipulate the number around the problem.

 

A grammar stage student, by definition, should be immersed in the grammar and phonics of the language...

 

Little Johnny comes to the sentence, "The banshee, the marshal and the dishonored chef are using the special eggshells to enshroud the midship which will surely be the talk of the nation."

 

Little Johnny knows the Logic of English rule:

"SH spells /sh/ at the beginning of a base word and at the end of the syllable. SH never spells /sh/ at the beginning of any syllable after the first one, except for the ending -ship."

 

Okay, first word, "the", check.  

 

Next word, banshee, Johnny flips through his mental Rolodex of approximately 20 (!!) syllabication rules that may or may not apply in this case and decides it probably (because none of the rules are absolute) breaks into ban and shee.  Perfect, /sh/ at the beginning of a syllable after the first, therefore I pronounce it...?  Not a clue, except clearly we know that in this case it is pronounced /sh/.

 

Marshal?  Mar-shal?  No, Little Johnny, say that one /sh/ too.

 

Dishonored?  Remember, Little Johnny, in that one the "dis" is a prefix and therefore we split the syllables between the s and h.

 

Oh, but chef has French origins, therefore that "ch" is not pronounced /ch/, but rather /sh/.

 

Special and nation don't have "sh" in them at all, but remember that phonics rule number 562 (yes, I'm just making that up) covers "ci" and "ti" sometimes, in some words saying /sh/.  

 

Eggshell?  Well, don't think of those as syllables, but rather whole words in a compound word, therefore the whole "no /sh/ at the beginning of a syllable after the first" doesn't apply.  And enshroud and midship are really prefixes attached to a base words, therefore, don't consider shroud and ship to be the second syllables, but rather the first and therefore they should also be pronounced with a /sh/.

 

Then you just have to remember that sure is an exception, and therefore it gets pronounced with a /sh/ sound just because.

 

Isn't this a funny story, Little Johnny?

 

--------

 

At least the beauty of the math strategies is that they can all lead you to the correct answer.

 

23 - 16

 

Little Johnny says he wants to use the strategy of adding 7 to each number.  Okay.

 

30 - uh oh, what is 16 + 7?

decompose the 7, move 4 to the 16, so 23.

 

30 - 23

 

Did that help, Little Johnny?  Not really?  Okay, so try something different.

 

In math, there is a right answer, but no one right way to get there.

 

Wendy

[Ad astra]

Basically I think you need both. With a 'conceptual' program you need to be sure to do plenty of practice. With a 'traditional' program you need to be sure and teach with concrete objects.

 

 

I agree. I grew up in East Asia and both were equally emphasized in math classes. We learned concepts in a similar way to the "Singapore" but also did lots of facts worksheets for tests. When solving problems, various methods were encouraged as long as you can get the right answer. We didn't need to show the whole process of calculation like Common Core Math requires.

 

I haven't been on this forum for long but honestly felt there existed some kind of "myth" about SM or conceptual math. IMO it shouldn't be the one way or another. I used SM1 and didn't find any superiority in it to other programs. Sure, it's a solid program and IP and CWP are more challenging but that doesn't mean this is one fit for all mathy students. And it is true the reviews built in SM alone would be no way enough for average students to excel on tests in my home country.

 

As students, we tried to avoid getting fixated with one certain method or presentation and used several different math workbooks that include various methods, math fact drills and challenging problems. I believe it's not so different in other East Asian countries whose students also far surpass American peers on international math tests.

[Free]

I get that, but I also think that it shouldn't be the way 9+6 is taught to kids, if that makes sense.  In other words, I think the concept is important, and to teach the concept it's easier to show it with smaller numbers at first, but I don't think that's the way they should be teaching kids to learn or be "comfortable" with single digit addition in the early elementary years.  So I think what you're talking about -- breaking numbers down to make problems easier -- is a different lesson than teaching a 1st grader that 9+6=15.

 

I disagree that small children should be taught 9+6 as a discrete "fact" to be memorized without any understanding of what it represents or how the same numbers can be represented in different ways. As a child, I was never taught addition or subtraction facts at all. Every time I had to add or subtract, I had to count over with my fingers. The memorization of the fact came with the repetition of the concept.

 

It is quite possible that memorization feels like the intuitively right way to teach math to you, because that is how you were taught. Whereas making children manipulate the numbers over and over again till it becomes automatic is how I feel is the right way to teach, because that is the way I was taught.

 

[EmseB]

I disagree that small children should be taught 9+6 as a discrete "fact" to be memorized without any understanding of what it represents or how the same numbers can be represented in different ways. As a child, I was never taught addition or subtraction facts at all. Every time I had to add or subtract, I had to count over with my fingers. The memorization of the fact came with the repetition of the concept.

 

It is quite possible that memorization feels like the intuitively right way to teach math to you, because that is how you were taught. Whereas making children manipulate the numbers over and over again till it becomes automatic is how I feel is the right way to teach, because that is the way I was taught.

 

 

Sorry my post wasn't clear, I don't think it should be memorization without understanding.  I have yet to find a program or person that uses such.  I was speaking specifically about the idea that if a student isn't "comfortable" with 9+6 then they should be taught that it is 9+1+6-1, so 10+5, so 15.  Before I went down that road, if my student was uncomfortable with the idea that 9+6=15, then I would probably show them, either with a drawing or objects that 9 things + 6 things is 15 things.  That concept of what addition actually is, if my child was uncomfortable with it, is what I would teach and cement before going on to manipulating the numbers (with further addition/subtraction!) to make the problems "easier".  And I'm talking about single digit addition here.

[Free]

That is like saying that if kids do copy work they will be resistant to learning to formulate compositions later. It's not true - but the concepts must be introduced at the right stage of course. They will not have to re-learn anything but will be enlightened to the complexity and number relationships with which they are already familiar.

 

Nobody here is saying that we should be teaching PURELY math facts, it is a matter of where the emphasis is placed. And FWIW, we actually do teach kids to memorize phonics rules precisely so that they can decode different words later. We don't start out by emphasizing all the different exceptions to the rules, we build up a set of rules for them to use so that, as they become proficient in reading, they can analyze them to decode and later to spell. Then, in the logic stage, we teach them WHY there are so many different phonics rules (different linguistic roots, colloquialisms, strange etymologies etc....) but we don't teach them WHY the phonics rules are the way they are from the beginning. That would be far too much analysis for a grammar stage student.

 

Addition and Subtraction are grammar stage skills. In fact some kids pick these up even in preschool. With all due respect, I think it is a little absurd to suggest that understanding conceptual arithmetic requires a child to be in logic stage.

 

Also your analogy with transition from copywork to original composition is not accurate. Copywork is a way to train the brain to absorb the structure and patterns in language so that it becomes easier to recreate similar language in one's own compositions.

 

Phonics too is a way to get children to see patterns in spelling and pronunciation so that those patterns can then be applied to other unfamiliar words.

 

Similarly, conceptual mathematics (breaking and joining numbers in various ways) is a way to train the brain to see numbers as flexible and observe patterns and connections between different numbers.

 

If anything, using math facts for teaching arithmetic is more analogous to using whole word for teaching reading.

[birchbark]

I'm another fan of traditional math. 

 

After putting my older DS through several years of Miquon and Singapore, as well as reading a lot about math instruction, I think mohini is right on the money. There have been many "traditional vs. conceptual" math threads on these boards over the years, but she has done an exceptional job articulating the case for traditional math.

 

I posted some articles on the subject in an older post. (And I do realize that the conceptual programs mentioned in some of the articles are the more extreme form used in PS, but Singapore leans enough that way that it produces many of the same frustrations.)

 

A pp mentioned the lack of traditional program reviews. I agree. Especially surprising considering this is a classical education forum. I'll just throw out this link to a blogger's excellent review of Strayer Upton. She spends a lot of time explaining how the curriculum develops concepts and understanding. 

[Roadrunner]

I am going to repeat this again. Singapore is as traditional as it can get. Having grown up with Russian math, I don't see anything in SM that is in any way deviating from how math is taught in traditional classrooms in countries outside of Asia.

 

Classical education strives to excellence, unless we define it as an education in Classical Greece. I would expect math resources that build strong problem solving skills to be more in line with Classical spirit than material not digging deep enough.

[Kathie in VA]

Wow this is as hairy as a discussion of mastery vs spiral or vertical vs horizontal phonics. Lol.

 

I switched to SM not because it was "conceptual". I hadn't heard of that concept. I did know we wanted a mastery based prgm because it works for me and it works for my kids. So I considered a few if them, prob both traditional n conceptual. SM caught my radar because of the step by step lessons in word problems (my weak area).

 

Most programs seemed to require memorizing the basic facts outside the typical lesson and seat work. It would be great to see one have a daily sheet in with the daily seat work to focus on the facts but that would be crazy for the students who don't need it. I gathered that keeping it separate allowed in individual attention to this issue.

 

As to the breaking expressions into parts, this I liked. My kids seemed to understand it well enough, and even got faster then me quickly. Although the other way the someone mentioned was neat also...taking the time to notice all the different ways to add up to a number like 15. I didn't learn that way either. We just did the facts and then did harder problems and moved on from there. Both seem like great approaches. Neither seem like a logic stage level to me. It seems very similar to the level it takes to think through reading or spelling after learning all the phonograms and rules. They did the grammar stage thing and memorized the phonograms but they are also capable of manipulating them to read or spell. Now this just doesn't work for some kids at all. They get all frustrated with all these options and rules and don't know which to use when. This doesn't mean that this is a logic stage level lesson; instead it just means that a different approach may be preferred. It also doesn't mean that one phonics approach is better then others or even more appropriate for a classical approach. Same is true for mastery vs spiral math.

 

Oh one more point that I don't think was mentioned. Breaking the facts down into parts is just the beginning. It is done to learn how to do such a thing. Once they get that idea it is put to work on larger problems. Pretty quickly they can consider 179+26=205 as a very easy problem that can be done in their head. This becomes a confidence builder. So yea it may seem needless to learn this process for 9+5 but it makes sense if you know it is just a starting place (the basics) to spring off to more complex calculations.

[MeaganS]

As long as nobody is having their kids drill 7 + 9 = 16 without understanding it, all is good. If anybody is blindly memorizing facts than they should reconsider. I have seen enough posts on this board that sound like blind memorization. I want to caution against that approach (lots of people read this board). No reason to take things personally.

I kind of am with my oldest.  It isn't something that I would recommend for everyone (and I probably won't be doing it this way for my younger children), but it is how she learns.  Now, granted, she is autistic, but she has been able to learn everything on grade level so far and most people can't tell she's autistic when they meet her.  Most of her struggles are with language.  I think the way she learns just shows an extreme version of a type of child, though.  With her, I did Singapore Essentials A&B, Miquon Orange, Kitchen Table Math on and off, and half of Math Mammoth 1 before I realized she really does just need a more traditional math approach.  I really like conceptual math and feel very comfortable teaching it, but every example I gave of how to manipulate numbers just went right over her head.  She could do it at the time, but she never internalized any of it.  Once I taught her that 4+4=8 though, she started seeing that everywhere.  We could manipulate a problem and she would see that it ended up just being the fact and all of a sudden it was like a light bulb went off.  4+4 is always 8, no matter how you look at it.  For her, she really did have to learn the facts before she could understand the concept.  We are using Saxon with incredible success with her right now.  I think it helps that she also has a much higher attention span for spiral programs with many types of problems on a page than for mastery with pages of the same type.

 

I'm pretty sure I didn't say he memorized them, I think in the portion you quoted I used the same word that you are using (mastered).  I think the nitpick is important, because kids who master math facts in traditional programs don't just memorize them either.  No, they don't regroup them every time they do a computation (single-digit), but they know that 9 things + 6 things = 15 things and it isn't just a fact without meaning.

I think what we're doing with the math facts is partially this, though.  I try to make sure that dd understands the concrete meaning of the fact, but honestly, at this point, that is just secondary.  Saxon definitely teaches the way JodiSue is explaining here, and it is considered to be a more traditional program.

[MeaganS]

I've been enjoying this discussion a lot and honestly I tend to agree that most of the math programs mentioned on this forum will get you where you want to go and are quality programs (CLE, MM, Singapore, Saxon, etc.)  I also heartily agree that traditional vs. conceptual, spiral vs. mastery matter, but only inasmuch as they fit the student and not because they are inherently better. I think I do have a more conceptual-math is better bias, though, and I think it is because it seems like conceptual math just does more with math.  Maybe this is because I would have loved to have learned math conceptually, although I can't honestly remember how I learned math in PS, just that I was bored most of the time because it was too slow.

 

I have found the discussion of logic stage = conceptual math very interesting and I can't decide if I agree or disagree.  I guess that means that this is a good debate with excellent points on each side. Thank you guys!

[kiana]

Sorry my post wasn't clear, I don't think it should be memorization without understanding.  I have yet to find a program or person that uses such.  I was speaking specifically about the idea that if a student isn't "comfortable" with 9+6 then they should be taught that it is 9+1+6-1, so 10+5, so 15.  Before I went down that road, if my student was uncomfortable with the idea that 9+6=15, then I would probably show them, either with a drawing or objects that 9 things + 6 things is 15 things.  That concept of what addition actually is, if my child was uncomfortable with it, is what I would teach and cement before going on to manipulating the numbers (with further addition/subtraction!) to make the problems "easier".  And I'm talking about single digit addition here.

 

With this clarification to your previous statement, I agree 100%. Clearly there was some misunderstanding on my part as I did not understand that you were talking about a child who did not understand yet that 9 + 6 meant 9 things and 6 more things or 6 things and 9 more things. I would absolutely not start trying to teach mental addition strategies to a child who did not yet understand what addition meant.