**The first 4th grade decimals assignments: **

**I. From the Everday Math Student Math Journal **[click to enlarge]:

**II. From the Singapore Math**[click to enlarge]:

*Primary Mathematics Workbook***III. Extra Credit:**

What are Singapore Math students missing out on by not starting out with Base-10 blocks and iconic symbols?

## 15 comments:

Actually, if you go and read the teachers manual, Singapore Math students are supposed to start with manipulatives such as base 10 blocks.

Right, but does one find this in the teacher's manual for

fourthgrade?There are still students in fourth that benefit from concrete to pictorial. That page is for them.

The situation in school is just like invented spelling. Good for some students,but the ones that are past that stage should be using the dictionary. Here, they should be using pictorial or abstract, if they are past concrete. But if we are running full inclusion, we can't do that because ncga is in effect.

The difference is that SM quickly moves on from artificial squares, dots and lines--which don't actually visualize fractions at all--and soon turns to actual divided shapes. EM's is just a counting exercise, SM is a fraction exercise.

I like manipulatives for any age. My objection to the EM pages is the insistence on using the square, lines and cubes. The effort that goes into understanding this as a writing system is effort that could be better spent in using real mathematical symbols. And what do you do if you need to represent one one-thousandth or one ten-thousandth?

It's typical of these fuzzy curricula that they teach approaches that only work for a small subset of real-world problems, instead of teaching universal algorithms that always work.

Yes, Katharine, you do. Why would I have said it if you didn't?

Though I admit to skimming past fractions as the kid was already doing them in school on the same level. (Not using Everyday Math, using HSP.)

Look, I'm not going to deny that EM has some serious problems, but nobody is helped by claiming things about Singapore Math that just aren't true. "They're not using manipulatives at this stage!" is probably true for many homeschooled students, but given that it IS in the teachers manual I doubt it is true for students in classes that use it.

A more crucial difference between the two pages might be that EM pushes the explanation of the manipulatives into the workbook, taking up valuable space and causing them, apparently, to put down fewer practice problems. This, in turn, means that students in SM progress further before the end of the unit.

I just used Base 10 blocks with 4th grade math. (MEP4b, not Singapore or EM.)

My daughter was confused and worried at the first introduction of adding decimals, so I broke out the blocks to help her visualize how the numbers match up. Just like in the EM picture, flats were ones, rods were tenths, and unit cubes were hundredths.

Modeling three addition problems that way got her perfectly clear on the necessity of lining up her numbers so that tenths are being added to tenths, hundredths to hundredths, and so on. I'm not going to sneer at using manipulatives at 4th grade level.

(Drawing base 10 blocks as an ongoing thing, though? Goofy.)

The thing I find the oddest about the EM page is that it needs to introduce a new symbolic language (symbols for lines, blocks, and squares, not in clear scale) in order to explain the symbolic language (numbers) the students ought to be learning. This isn't pictorial like the SM, it's symbolic. It's more confusing than SM, which is consistent about moving from concrete, through pictorial, to symbolic. EM lines up the three things (symbol, fraction, decimal) as three distinct symbolic languages for the same thing, complicating the decoding that SM works to simplify.

I believe it is true that the kids have to understand it in both a concrete and a symbolic way, but adding another type of symbols just makes it more complicated, not less.

My son learned decimals and fractions quite easily at 8 using SM and talking to me. It helps a heck of a lot to already have the habit of lining things up for multi-digit addition, subtraction, multiplication, etc. so you have a good understanding of place value.

IIRC, SM also uses a lot of currency, which is a good concrete system. He was surprised to hear that some kids find it confusing. I believe it is not inherently confusing but is usually presented in a confusing way. EM does that.

I can send you a scan, but here's section from the Primary Mathematics 4A Teacher's Guide U.S. Edition, p. 4:

"Use concrete manipulatives to explain place-value concepts for decimal numbers, including base-ten blocks, fraction squares (see next page for some you can copy for your students to color in or which you can copy onto transparencies) and number disks. Number disks will be particularly useful in the next unit where they can be used on place-value charts to illustrate addition, subtraction, multiplication and division."

They then have "fraction squares" on the next page, which look exactly like 2-d base-ten blocks. I usually just use the number disks.

Cassandra Turner, isn't that a guide commissioned originally by Son Light and not something produced by or for the Singapore market?

I have the Parker Baldridge text on teaching SM and they specifically caution against using base ten blocks past a very early stage. Absolutely not in fourth grade.

I don't know what actual Singapore training says about this.

Contacted Singaporemath.com to see what is in the teacher's manual for 4A, regarding manipulaties. Here is their response:

"the Teacher's Guide does mention base-ten blocks as possibility of a more physical representation of fraction squares, such as those pictured on p. 14 of the Standards edition textbook. The hundred flat looks exactly like this model. It is only mentioned as a possibility in the introduction as an alternative to the paper models that need to be copied and cut out. The lessons themselves do not use base-10 blocks but rather use what Parker and Baldridge call an area model, p. 10 and p. 14 of the Standards edition textbook, and include blackline masters of them. The lessons in the guide also include place-value discs, as on p. 15 of the textbook as well as on other pages. Use of the base-10 flat would not in any way be similar to the use in the Everday Math example at the blog, which is more symbolic, as one blogger there did mention. If a teacher chooses to use the base-10 blocks rather than the paper models, he or she would presumably use it in the same way as the textbook uses what Parker and Baldridge call the area model."

Stealth Jew: I have been working with schools implementing and using Singapore curricula for 6-7 years. Some people call it training, I consider it Singapore Math learning. After the first unit in second grade, I recommend teachers use number disks instead of base-10 blocks. I agree with Barry, that the use of them in the Singapore materials is different than the use in Everyday Math.

The biggest problem with the base-10 blocks as I see them is that it's pretty hard to show either the multiplication or long division algorithms. Once you divide problems like 2347 by 3 with number disks a couple of times, the algorithm is pretty easy to master.

Side note: I had a 7th grade teacher in Canada who told me he never really understood subtraction until we did the algorithm with number disks. I think we were doing something like 4000 - 237.

Interestingly enough... This got me thinking about the Singapore materials. I have a full set of Shaping Maths and My Pals Are Here (MPAH) for third grade. I pulled open the unit on addition and subtraction within 10,000 and found that in the Shaping Maths, they use number disks for the textbook, but in MPAH they used base-10 block representations.

MPAH is the curricular basis for the Americanized Singapore math series, Math in Focus.

Always learning...

Cassandra, that's interesting. I was referring to the training given to teachers in Singapore -- whether they would use base ten blocks at a fourth grade level. Not training for using Singapore math given to people in North America. Katharine's original question was posed regarding students in Singapore.

Stealth Jew,

Since they are used in the Singapore MPAH Teacher's Guide I purchased in Singapore, I'd say that it's possible that Singaporean teachers are taught how to use base-10 blocks. In my teacher prep program, we were not. Barry, did you learn what a place value manipulative was in your teacher training program?

I was in the secondary math teaching program (i.e., middle school/high school), so we didn't get into that. My math methods courses really didn't get detailed into any kind of teaching method other than in very general ways that embraced constructivist philosophies, differentiated instruction and other faddish disasters.

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