s/o On how YOU rate a math program as conceptual

[Snowfall]

I was reading the BJU vs. MM thread and saw 2 (I think it was 2 at the time I read it) people say BJU is very conceptual and 1 person say it isn't. But then I saw someone (I hope you don't mind if I paraphrase you) say that BJU was conceptual and she preferred teaching BJU because it's traditional math, not Asian style math. I don't know anything about what BJU is like, but from everything I've been able to read and learn about Asian math vs. traditional American math, the difference basically IS conceptual, in that traditional American math is very much about algorithms and procedures and Asian math is very much about conceptual understanding. So how could a program be traditional American and conceptual at the same time? I thought that was kind of an oxymoron. I understand that a program could focus on teaching the algorithms and conceptual understanding, but that's what Asian math curricula DO, so that can't be the difference. What am I missing?

 

I've read all the conceptual vs. traditional threads, all the Asian vs. American threads, all the "What is conceptual math?" threads, the Liping Ma book, multiple websites, etc. Now I'm just confused. lol

[daughterofsarah77]

Me too : ) I don't "get" it either. We are using CLE this last week after never settling in with MM or Singapore...and all I can say is dd is getting it. Is she memorizing her math facts? Yes. But that doesn't mean she ISN'T getting the idea behind it. As a matter of fact, CLE has been the only curriculum that has laid it out for me in a manner where I can explain it to her in a way she gets it.

We will still add in MM or Singapore for extra practice, but CLE has gone so smoothly, I can't deny it's impact in our homeschool.

I kept telling myself I COULDN'T do CLE because it's not "conceptual", but after reading Liping Ma's book, I realized that for me...I can try to add in explanations and extra activities to form a conceptual view. But, you need a solid foundation to build on.

[Veritaserum]

I have kids using MM and one using CLE. MM is more conceptual than CLE, but CLE doesn't solely teach via memorization and algorithms. I do sometimes give a more conceptual explanation for something taught in CLE, but overall I think it does a good job. MM does it better, but CLE is still an acceptable choice to me. :)

[ondreeuh]

I could give my son the most conceptual program in the world, but if it didn't "speak" my son's language, it wouldn't be effective. We use CLE and between the book's written explanations, it's illustrations that show each step, and my own teaching, he is understanding the concepts behind the math.

 

Singapore, which is touted as a wonderful conceptual program, does not include anywhere near the amount of review that I think it should. My son could understand the concepts the first go-round, but after leaving the topic for a while he was unable to recall the "whys" and just tried to follow the procedure. CLE may not spend as much time going over they "whys" but they stick, which leads to better overall understanding of the concepts for my son. Your mileage may vary, and all of that. :)

[TracyP]

I have thought about this a lot but I don't know if I have actually figured anything out. The one thing I have noticed is the teacher.

 

For example, a mom who is very good at math will look at a program and say it is conceptual. She will say when I got to (fill in the blank) I just pulled out the C rods and illustrated (fill in the blank) and my kids had excellent understanding of the concepts. I am the teacher who does not know how to do this unless I am told.

 

Next mom, knows so little about math that she thinks it is conceptual because......the advertising said so, someone's blog said so - THIS IS ME. It was after reading Liping Ma that I realized my kids need a superb math instructor who has a strong conceptual understanding.

 

To me this means that a "conceptual" program isn't the end all be all. A teacher with a strong conceptual understanding is far more important. If you (generic) are a strong math teacher you can use anything. I chose MM because I see Maria Miller as being that teacher for my kids.

[Snowfall]

For example, a mom who is very good at math will look at a program and say it is conceptual. She will say when I got to (fill in the blank) I just pulled out the C rods and illustrated (fill in the blank) and my kids had excellent understanding of the concepts. I am the teacher who does not know how to do this unless I am told.

 

Next mom, knows so little about math that she thinks it is conceptual because......the advertising said so, someone's blog said so - THIS IS ME. It was after reading Liping Ma that I realized my kids need a superb math instructor who has a strong conceptual understanding.

 

Let me make sure I understand what you're saying. :) Is it that the mom in the first example can make something conceptual, so to her it seems like it always was, because it came naturally to her to do that? That actually makes a lot of sense to me!

 

To me this means that a "conceptual" program isn't the end all be all. A teacher with a strong conceptual understanding is far more important. If you (generic) are a strong math teacher you can use anything. I chose MM because I see Maria Miller as being that teacher for my kids.

 

I agree with that - if you are good at these things, you can make anything conceptual, since it's all about how it's explained to the child. I was thinking of that when reading daughterofsarah's post. I don't think I could've done it a few years ago and I don't know if I could do it for anything past 1st grade math since that's as far as we've gotten. However, after having been at it for the last 2 years with MEP and Rightstart, I do feel confident that I could easily explain place value, base ten, addition with carrying/composing and subtraction with borrowing/decomposing in a conceptual manner, no matter what curriculum we used. Like I said, I'm only sure I could do it for K and 1st, but I could at least do it that far, just from reading and practice.

[shinyhappypeople]

I think the conceptual / non-conceptual debate is pointless. None of the many, many math programs I've used or researched strictly taught algorithms without going into the concept behind it. I don't see any of the "Just carry the one" talk that I was raised on. Every program I saw - even the "non-conceptual" ones - teach what carrying/regrouping is really doing.

 

I do think that some of the programs tend to spend way more time on the concept than on the algorithm, and others are more likely to give equal time or lean a little the other way.

 

Thank goodness for the variety. Heavily conceptual programs made my DD hate math.

[Crimson Wife]

I think the conceptual / non-conceptual debate is pointless. None of the many, many math programs I've used or researched strictly taught algorithms without going into the concept behind it.

 

Have you seen MCP? I looked through one of the late elementary books (4th or 5th grade, can't remember which) and it was exactly the kind of "here's the formula, now go do a bunch of problems" type stuff I remember from growing up. :thumbdown:

[RootAnn]

...The one thing I have noticed is the teacher. ... It was after reading Liping Ma that I realized my kids need a superb math instructor who has a strong conceptual understanding.

 

This is the main thing I got out of Liping Ma's book: The teacher makes the big difference.

 

So, if I know the "why" behind the "how" and the program I am using isn't getting the information through (or I want to explain it differently/better/ another way), I can use a different way to teach it. If all I know is what the program says or all I can show is "plug and chug," I will be like the less-than-stellar teacher examples in Ma's book.

 

My #2 child often doesn't get math concepts the first time. I end up showing / explaining the "why" three or four different ways and showing the "how" ten times in completely different formats. (This takes a number of weeks or months depending on the concept.)

 

To me, it doesn't matter if the program is conceptual or not. If it works for my kids & me (with whatever tweaking I have to do), I'm going to use it. And for us, :ohmy: it is (mostly) A Beka. :D

[TracyP]

Let me make sure I understand what you're saying. :) Is it that the mom in the first example can make something conceptual, so to her it seems like it always was, because it came naturally to her to do that? That actually makes a lot of sense to me!

 

 

Yes, exactly.

 

I think the conceptual / non-conceptual debate is pointless. None of the many, many math programs I've used or researched strictly taught algorithms without going into the concept behind it. I don't see any of the "Just carry the one" talk that I was raised on. Every program I saw - even the "non-conceptual" ones - teach what carrying/regrouping is really doing.

 

I do think that some of the programs tend to spend way more time on the concept than on the algorithm, and others are more likely to give equal time or lean a little the other way.

 

Thank goodness for the variety. Heavily conceptual programs made my DD hate math.

 

I think this is a big confusion in the debates too. Every program (that I know of) does show the concept. In a nutshell, some go on to drill the algorithm while a conceptual program actually keeps on going deeper and looking at it from different angles.

 

I agree, there is no right one size fits all answer in these discussions.

[mammaofbean]

I think the conceptual / non-conceptual debate is pointless. None of the many, many math programs I've used or researched strictly taught algorithms without going into the concept behind it. I don't see any of the "Just carry the one" talk that I was raised on. Every program I saw - even the "non-conceptual" ones - teach what carrying/regrouping is really doing.

 

I do think that some of the programs tend to spend way more time on the concept than on the algorithm, and others are more likely to give equal time or lean a little the other way.

 

Thank goodness for the variety. Heavily conceptual programs made my DD hate math.

 

i totally agree! and my dd gets the conceptual stuff pretty quick. she needs more time memorizing less time on concepts. possibly this will change when we hit geometry, but this is how it is going for arithmetic.

[SnowWhite]

I did not post on the MM/BJU thread because I don't have a way to compare (having never used MM), and there were people on both sides there already, so there seemed little for me to say. I can explain why BJU seems conceptually strong to me and CLE does not, though.

 

Here is one example (ds just tried CLE for one LU and thankfully let me steer him back to BJU):

 

For long multiplication, CLE merely presented the algorithm. The child was expected to move from multiplying a 4-digit number by a single digit to multiplying the same number by a 2-digit number, with NO physical representation or visual example of what was happening with the numbers.

 

The instructions stated "First, multiply 46 x 3. Do not use the 2 that is in the tens place. Multiply as you have already learned when using a single-digit number. (illustration of 46 x 23 vertically, with 138, the partial product, underneath) Now you will multiply by 2 tens, but first you must use a zero as a placeholder in the ones place. (illustration adding a zero under the 8 of 138) Write a zero under the 8 in 138. Now you are ready to multiply 46 x 2. (Do not use the 3 in the ones place.) Multiply and place your answer to the left of the zero in the ones place. (illustration of partial product 92, left of the previously written zero). Now add the two products. (Illustration of the completed problem.)"

 

By contrast, when BJU introduces the same topic in a couple of months, they will begin by multiplying multiples of ten, with physical and visual examples. The next day, they will show how to apply the Multiplication-Addition Principle, again with both physical and visual examples. The problems are first worked out horizontally, like this: 14 x 15, (10 +4) x 15, (10 x 15) + (4 x 15), 150 + 60, 210. An array is beside the problem, with green squares for 4x15 and purple squares for 10x15 so the student can visually see the 150 and 60 squares being multiplied and added. There are 3 more such problems for the student to solve using manipulatives or a visual array. On the third day, the topic is continued using the vertical model, still with an array to illustrate what is happening with each operation.

 

I was taught to multiply using the method CLE demonstrates. We didn't even bother with the zero. We simply "staggered the partial product to the left one space". It makes me shudder! "Use zero as a PLACEHOLDER". This is not even a correct statement. The zero is there because the student is multiplying by a multiple of TEN. The partial product ENDS in zero. I was so conceptually ignorant by the time I got to high school that algebra was a huge revelation! Things finally made sense. When I think of the hours I wasted acting as a human calculator, copying columns of numbers and multiplying and dividing huge numbers I feel like weeping for my lost youth.

 

Now I cannot speak for Math Mammoth, but having not heard about the physical manipulatives they recommend I would doubt they're strong on physical demonstration of concepts.

[Guest RecumbentHeart]

I did not post on the MM/BJU thread because I don't have a way to compare (having never used MM), and there were people on both sides there already, so there seemed little for me to say. I can explain why BJU seems conceptually strong to me and CLE does not, though.

 

Here is one example (ds just tried CLE for one LU and thankfully let me steer him back to BJU):

 

For long multiplication, CLE merely presented the algorithm. The child was expected to move from multiplying a 4-digit number by a single digit to multiplying the same number by a 2-digit number, with NO physical representation or visual example of what was happening with the numbers.

 

The instructions stated "First, multiply 46 x 3. Do not use the 2 that is in the tens place. Multiply as you have already learned when using a single-digit number. (illustration of 46 x 23 vertically, with 138, the partial product, underneath) Now you will multiply by 2 tens, but first you must use a zero as a placeholder in the ones place. (illustration adding a zero under the 8 of 138) Write a zero under the 8 in 138. Now you are ready to multiply 46 x 2. (Do not use the 3 in the ones place.) Multiply and place your answer to the left of the zero in the ones place. (illustration of partial product 92, left of the previously written zero). Now add the two products. (Illustration of the completed problem.)"

 

By contrast, when BJU introduces the same topic in a couple of months, they will begin by multiplying multiples of ten, with physical and visual examples. The next day, they will show how to apply the Multiplication-Addition Principle, again with both physical and visual examples. The problems are first worked out horizontally, like this: 14 x 15, (10 +4) x 15, (10 x 15) + (4 x 15), 150 + 60, 210. An array is beside the problem, with green squares for 4x15 and purple squares for 10x15 so the student can visually see the 150 and 60 squares being multiplied and added. There are 3 more such problems for the student to solve using manipulatives or a visual array. On the third day, the topic is continued using the vertical model, still with an array to illustrate what is happening with each operation.

 

I was taught to multiply using the method CLE demonstrates. We didn't even bother with the zero. We simply "staggered the partial product to the left one space". It makes me shudder! "Use zero as a PLACEHOLDER". This is not even a correct statement. The zero is there because the student is multiplying by a multiple of TEN. The partial product ENDS in zero. I was so conceptually ignorant by the time I got to high school that algebra was a huge revelation! Things finally made sense. When I think of the hours I wasted acting as a human calculator, copying columns of numbers and multiplying and dividing huge numbers I feel like weeping for my lost youth.

 

Now I cannot speak for Math Mammoth, but having not heard about the physical manipulatives they recommend I would doubt they're strong on physical demonstration of concepts.

 

I appreciate this detailed example. It took me months of reading through all the related posts/debates on the topic and all I could read of Liping Ma via Google's preview to figure out why I was so confused as to what the issue was. I finally figured out it was because I was actually taught conceptually and didn't realize there was another way to learn arithmetic. Even using phrases like "carry the one" weren't without understanding of what was going on, which is what confused me in the discussions. Your example of the difference helps me understand clearly where the problem lies and now that I have a better understanding of the issue, I do recall not being taught the "why" in high school math apart from being in the advanced math class (which was an elective subject on top of the compulsory math).

[mom31257]

Here is one example (ds just tried CLE for one LU and thankfully let me steer him back to BJU):

 

For long multiplication, CLE merely presented the algorithm. The child was expected to move from multiplying a 4-digit number by a single digit to multiplying the same number by a 2-digit number, with NO physical representation or visual example of what was happening with the numbers.

 

The instructions stated "First, multiply 46 x 3. Do not use the 2 that is in the tens place. Multiply as you have already learned when using a single-digit number. (illustration of 46 x 23 vertically, with 138, the partial product, underneath) Now you will multiply by 2 tens, but first you must use a zero as a placeholder in the ones place. (illustration adding a zero under the 8 of 138) Write a zero under the 8 in 138. Now you are ready to multiply 46 x 2. (Do not use the 3 in the ones place.) Multiply and place your answer to the left of the zero in the ones place. (illustration of partial product 92, left of the previously written zero). Now add the two products. (Illustration of the completed problem.)"

 

.

 

I was wondering because you said you tried one LU, did you have the teacher's manual or are those the instructions in the student's LU?

[Snowfall]

Now I cannot speak for Math Mammoth, but having not heard about the physical manipulatives they recommend I would doubt they're strong on physical demonstration of concepts.

 

The MM website recommends an abacus and links to a virtual abacus. She also says to use whatever manipulatives you like, I think. The worktexts have illustrations to show what's happening with the math.

[wapiti]

The MM website recommends an abacus and links to a virtual abacus. She also says to use whatever manipulatives you like, I think. The worktexts have illustrations to show what's happening with the math.

 

To expand on this, one can get a little bit of the flavor of the illustrations in the worktexts from the sample pages on the MM website. Here are a few random examples:

 

http://www.mathmammoth.com/preview/Multiplication_1_Array.pdf

 

http://www.mathmammoth.com/preview/Multiplication_2_Multiply_in_Parts.pdf

 

http://www.mathmammoth.com/preview/Multiplication_Division_3_Multiplication_Division.pdf

 

http://www.mathmammoth.com/preview/Division_2_Finding_Parts_with_Division.pdf

[ondreeuh]

For long multiplication, CLE merely presented the algorithm. The child was expected to move from multiplying a 4-digit number by a single digit to multiplying the same number by a 2-digit number, with NO physical representation or visual example of what was happening with the numbers.

 

The instructions stated "First, multiply 46 x 3. Do not use the 2 that is in the tens place. Multiply as you have already learned when using a single-digit number. (illustration of 46 x 23 vertically, with 138, the partial product, underneath) Now you will multiply by 2 tens, but first you must use a zero as a placeholder in the ones place. (illustration adding a zero under the 8 of 138) Write a zero under the 8 in 138. Now you are ready to multiply 46 x 2. (Do not use the 3 in the ones place.) Multiply and place your answer to the left of the zero in the ones place. (illustration of partial product 92, left of the previously written zero). Now add the two products. (Illustration of the completed problem.)"

 

Is this LU 403, lesson 6? If this is the only LU you used, I can see why you are confused.

 

In previous lessons, students practice multiplying by tens. 5 x 30 is multiplying 5 by 3 tens. So your answer is 15 tens, which is written 150 because there are 15 tens and zero ones. They do this over and over again, and see that you can multiply by the number of tens and place a 0 at the end of their answer. 3 x 20 = 60. So from there, it's not a big leap to show that when you multiply 46 by 2 tens, you place your zero and multiply 46 x 2.

 

It's really hard to get a good feel for how an incremental program teaches when you only see one part. CLE is not at all like MM (which is mastery based). It teaches step-by-step over time, so if you see one of the later steps you don't know how they got there and it can look like the program just jumped into it.

Edited January 27, 2011 by ondreeuh

[boscopup]

The MM website recommends an abacus and links to a virtual abacus. She also says to use whatever manipulatives you like, I think. The worktexts have illustrations to show what's happening with the math.

 

Right. It doesn't always explicitly say "Pull out this manipulative" (although it did for the abacus lesson in 1B), but the pictures are usually pretty explanatory anyway. If you need manipulatives, they're easy to add. I pulled out the base-10 blocks for my son this week when dealing with the early stages of mental math - making 10s for problems like 6 + 8 or 12 - 4. Usually he doesn't need manipulatives, but this new concept needed them for initial explanation. He worked a few problems using them, then did the rest without and was fine (and showed that he fully understood the concept).

 

I actually prefer this method for my son, since he doesn't usually need the manipulatives or want to use them, so I don't want a program that *requires* the manipulatives. I can certainly understand someone not being comfortable with MM not holding your hand on when to bring out manipulatives though. If your child really needs them and you don't know when to use them, MM probably isn't the best choice for you.

[TracyP]

Right. It doesn't always explicitly say "Pull out this manipulative" (although it did for the abacus lesson in 1B), but the pictures are usually pretty explanatory anyway. If you need manipulatives, they're easy to add. I pulled out the base-10 blocks for my son this week when dealing with the early stages of mental math - making 10s for problems like 6 + 8 or 12 - 4. Usually he doesn't need manipulatives, but this new concept needed them for initial explanation. He worked a few problems using them, then did the rest without and was fine (and showed that he fully understood the concept).

 

I actually prefer this method for my son, since he doesn't usually need the manipulatives or want to use them, so I don't want a program that *requires* the manipulatives. I can certainly understand someone not being comfortable with MM not holding your hand on when to bring out manipulatives though. If your child really needs them and you don't know when to use them, MM probably isn't the best choice for you.

 

:iagree: especially with the bolded. This is what makes MM so perfect for my dd.

 

I have used / am using CLE and I think it is wonderful. It is my #2 math if MM doesn't work for any child. However, I do not see it as being conceptual in the way MM, RS, SM, BJU(?I know nothing about) are. This doesn't make CLE bad, it just makes it an incredibly thorough, very good, traditional math program.

[SnowWhite]

I was wondering because you said you tried one LU, did you have the teacher's manual or are those the instructions in the student's LU?

 

This is fourth grade math... no, I did not have the manual. From the sample of the TM on the website, it looks like a copy of the student book plus a sentence or two further explaining the lesson- certainly nothing like the BJU manual I prize, with its physical manipulation activities.

[SnowWhite]

Is this LU 403, lesson 6? If this is the only LU you used, I can see why you are confused.

 

In previous lessons, students practice multiplying by tens. 5 x 30 is multiplying 5 by 3 tens. So your answer is 15 tens, which is written 150 because there are 15 tens and zero ones. They do this over and over again, and see that you can multiply by the number of tens and place a 0 at the end of their answer. 3 x 20 = 60. So from there, it's not a big leap to show that when you multiply 46 by 2 tens, you place your zero and multiply 46 x 2.

 

It's really hard to get a good feel for how an incremental program teaches when you only see one part. CLE is not at all like MM (which is mastery based). It teaches step-by-step over time, so if you see one of the later steps you don't know how they got there and it can look like the program just jumped into it.

 

Yes, that is the lesson. I can believe that they practice multiplying by tens, but the picky teacher in me wants them to use the correct terminology. (Placeholder. HMPH!)