SingaporeMathTopicMatrix < Kitchen

28 Nov 2005 - 23:57

Singapore Math topic matrix

from the AIR report (pdf file)

I'm going to try to rustle up the equivalent chart from TRAILBLAZERS so we have a direct comparison.

This seems to say that in Singapore children are using all 4 algorithms in the first grade.

the TRAILBLAZERS track

Christopher told me yesterday that he learned how to subtract with regrouping in 2nd grade. 'It was hard,' he said. He learned the multiplication algorithm for the first time at the end of 2nd grade.

From what I can see, the TRAILBLAZERS track could be as much as a year slower than Irvington's old track, though it's hard to tell:

This unit extends students’ work with place value to four-digit numbers and helps them build an understanding of our number system, the base-ten place-value system. The activities in this unit lay the conceptual groundwork for performing multidigit addition and subtraction. Two-digit addition is reviewed. Three- and four-digit addition and subtraction algorithms are developed in Unit 6.

This is from Unit 3, Grade 4 (there are 20 units altogether).

Here's a passage from the Teacher Implementation Guide (pdf file):

In kindergarten and grade 1, students using MATH TRAILBLAZERS practice their counting skills. They learn to count past 100 by 1s, 2s, 5s, and 10s. They count forward and backward from any given number. They group objects for counting. Students use counting to solve addition and subtraction problems. They learn to write numbers up to and beyond 100. The 100 chart is introduced and used for a variety of purposes, including solving problems and studying patterns. Students partition, or break apart, numbers in several ways (25 = 20 + 5, 25 = 10 + 10 + 5, and so on). These activities help children become familiar with the structure of the number system. Beginning in kindergarten, a ten frame is frequently used as a visual organizer.

The first passage is intended for parents.

The second passage comes from the Teacher Implementation Guide.

subtraction algorithm mastered by the end of 4th grade

It's probably worth skimming through pages 5 through 7 in the Teacher Implementation Guide. This passage lays out the TRAILBLAZERS content, sequence, and timing for teaching the subtraction algorithm. Once again, it's difficult to nail down the exact 'scopt and sequence' TRAILBLAZERS follows:

Later in grade 2, systematic work begins on paper-and-pencil methods for subtracting two digit numbers. Students are asked to solve two-digit subtraction problems using their own methods and to record their solutions on paper. The class examines and discusses the various procedures that students devise. At this time, if no student introduces a standard subtraction algorithm, then the teacher does so, explaining that it is a subtraction method that many people use. The standard method is examined and discussed, just as the invented methods were. Students who do not have an effective method of their own are urged to adopt the standard method.
Problems that require borrowing are included from the beginning. Though this differs markedly from traditional approaches, we view it as important in developing a sound conception of subtraction algorithms. Giving children only multidigit problems that do not involve borrowing encourages the development of a rote and faulty algorithm that may not carry over into problems that require borrowing.
By the beginning of grade 3, students have a strong conceptual understanding of subtraction and significant experience devising procedures to solve subtraction problems with numbers up to 1000. They also have some experience with standard and invented paper-and-pencil algorithms for solving two-digit subtraction problems. In grade 3, this prior knowledge is extended in a systematic examination of paper-and-pencil methods for multidigit subtraction. This work begins with a series of multidigit subtraction problems that students solve in various ways. Many of these problems are set in a whimsical context, the TIMS Candy Company, a business that uses base-ten pieces to keep track of its production and sales. Other problems are based on student-collected data, such as a reading survey.
As in grade 2, the class discusses and compares the several methods students use to solve these problems. Again, any method that yields correct results is acceptable, but now a greater emphasis is given to methods that are efficient and compact. This work leads to a close examination of one particular subtraction algorithm. (See Figure 3.) Students solve several problems with base ten pieces and with this standard algorithm, making connections between actions with the manipulatives and steps in the algorithm. After a thorough analysis of the algorithm, including a comparison of the standard algorithm and other methods, students are given opportunities to practice the algorithm. Practice in paper-and-pencil methods for multidigit subtraction is distributed throughout grades 3 and 4.

TRAILBLAZERS delays mastery of the subtraction algorithm until the end of 4th grade.

This is certainly consistent with the constructivist belief that premature teaching of the algorithms closes off conceptual understanding.

TRAILBLAZERS whole number operations scope and sequence

What they've done here is to use the idea of a math curriculum based in problem-solving to justify not teaching the algorithms.

All of the problems being solved in these first years are of the type:

"How do I add, subtract, multiply, and divide without knowing any algorithms?"

This is not remotely the case in the Singapore series.

Like TRAILBLAZERS, PRIMARY MATHEMATICS is a problem-based curriculum.

But in PRIMARY MATHEMATICS children use the standard algorithms to solve problems. That's why, in Singapore, children can begin using bar models to solve simple algebra problems in the 3rd grade. The bar models help them perceive which algorithms to use in what sequence.

If you don't know the algorithms, a bar model's not going to do you much good.

Introduction to Math TRAILBLAZERS
TRAILBLAZERS (TIMS) Teacher Implementation Guide: Math Facts
TIMS Teacher Implementation Guide Laboratory Method

key words: scope and sequence Singapore Math Primary Mathematics Trailblazers

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This seems to say that in Singapore children are using all 4 algorithms in the first grade.

It's probably more accurate to say that they're using all 4 operations in first grade. For addition and subtraction, they're definitely using algorithms, but for multiplication and division, they're limited to a subset of the single-digit facts in first grade.

In third grade, they would do things like 123 * 4, but it isn't until fourth grade that they do 123 * 45.

One thing I found striking about the Singapore timeline was how they started things earlier but spread them out. I'm pretty sure that when I was in 3rd grade we did all of whole number multiplication in one year: starting with both factors single digits and then eventually progressing to both factors being multi-digit. But it takes the Singapore curriculum four years to cover that material, building it up in a systematic way.

-- RudbeckiaHirta - 29 Nov 2005

right

they're using the 'operations'; they know they have to add, subtract, multiply, and divide

what they don't know is how to do it

they are expected to invent ways to do it, which, typically means counting

I found a math journal Christopher kept in the 2nd or 3rd grade where he explained how he got various answers

in every case, he counted

This is my point: the 'problem-solving' nature of TRAILBLAZERS is all centered in the one core problem: I need to add, subtract, multiply, or divide and I don't know any algorithms.

-- CatherineJohnson - 29 Nov 2005

The Singapore kids never invent algorithms.

They use the algorithm they've learned to solve word problems.

-- CatherineJohnson - 29 Nov 2005

otoh, I seem to be working with some kids who were involved in the piloting of TRAILBLAZERS, which means they've had the curriculum for a large chunk of their school years. These are highly talented kids, but when we do a word problem they often have no idea which operation to use.

You can't learn math conceptually; you have to do it.

Unless you've solved problems using the operations—the operations and/or the algorithms—you haven't learned how, and you don't have conceptual knowledge.

This is something I've learned personally from reading math-free books on math, the kind of book written to explain math to the non-mathematician.

I'm not especially attracted to these books anymore, because I find that I don't know much more about math than I did before I read the book.....

I assume it's like reading a book on how to write.

Reading a book about writing isn't remotely the same thing as writing.

-- CatherineJohnson - 29 Nov 2005

What I found most noteworthy about the Singapore curriculum is that while the addition and substraction algorithms are taught fast and early in Singapore, multiplication and division are verrrrryyyy sprreaaaad ouuuuut.

-- RudbeckiaHirta - 29 Nov 2005

This is my point: the 'problem-solving' nature of TRAILBLAZERS is all centered in the one core problem: *I need to add, subtract, multiply, or divide and I don't know any algorithms.*

Wow, it's no wonder all the kids in those Trailblazer playlets are complaining that math is hard.

And it will never get any easier for them, I guess.

-- CarolynJohnston - 29 Nov 2005

Multiplication and Division are first introduced in the second half of first grade (1B) in Singapore Math. It's more like an introduction to grouping at this stage and decomposing at this early level. Still, it is not a spiral in any sense of the word. Topics aren't just repeated ad infinitum (ad nausem). Each topic gets built on and expanded in each grade and is constantly practiced in the interim.

-- KDeRosa - 29 Nov 2005

Ken—I think the AIR report calls PRIMARY MATHEMATICS a spiral curriculum&mdash

hmm

let me check

-- CatherineJohnson - 29 Nov 2005

sheesh

looks like it's going to take forever to run the search

here's my question

primary mathematics does revisit topics each year, but does essentially no reteaching of topics (apart from review)

oh—this is interesting

apparently Singapore calls the curriculum a spiral:

The topic matrix shows the logical development of mathematical content; new topics build on prior mathematical content as students progress from grade to grade. Singapore calls this process of building and deepening content over successive grades a “spiral approach” (Ministry of Education, Singapore, 2001a).

I need some way to define the difference quickly and succinctly when talking to parents, teachers, and administrators.

-- CatherineJohnson - 29 Nov 2005

Wow, it's no wonder all the kids in those Trailblazer playlets are complaining that math is hard.

no kidding!

if you really stop and think about having to literally reinvent the wheel......inventing a wheel is pretty much starting at the top in terms of hardness

-- CatherineJohnson - 29 Nov 2005

constantly practiced in the interim

I haven't seen this, but I may not be looking closely enough. (For instance, I wouldn't say that the section on nets has kids practicing percent or decimals.....though I could be wrong)

I'll have to check my books again.

otoh, I'm not 'complaining'; they know what they're doing.

They seem to know how long they can set a topic aside while learning a different topic.

This is what happens when you FIELD-TEST A CURRICULUM

-- CatherineJohnson - 29 Nov 2005

I think the Singapore "spiral" is somewhere between the DI "spiral" and the constructivist "spiral," but closer to the DI "spiral."

For example, at the end of 1A, you briefly leave subtraction and do some geometry, but by lesson 1 in 1B you are right back to subtraction, this time with bigger numbers.

-- KDeRosa - 29 Nov 2005

Yeah, definitely.

Skimming a bit more in the AIR report (I've actually read a good half of it very closely) they coupled the word 'spiral' with the word 'mastery.'

The Singapore curriculum does, indeed, spiral back to previously taught topics, such as fractions.

But it assumes mastery of the previous material on the topic, and teaches new material.

-- CatherineJohnson - 30 Nov 2005

For example, at the end of 1A, you briefly leave subtraction and do some geometry, but by lesson 1 in 1B you are right back to subtraction, this time with bigger numbers.

Oh, that's interesting.

So you have a pause for some distributed practice.

How is this structured into the textbook?

I've been using textbooks for 3A and up, and haven't noticed anything like this. It may be there; I haven't been following these texts closely.

-- CatherineJohnson - 30 Nov 2005

Next Monday I'm going to be meeting a Russian immigrant dad who is homeschooling his kids with Singapore Math.

The head of the After-school program was impressed when he told her that's what he was using.

-- CatherineJohnson - 30 Nov 2005

WebLogForm
Title:	Singapore Math topic matrix
TopicType:	WebLog
SubjectArea:	CompareAndContrastPosts, SingaporeMath, TrailBlazers
LogDate:	200511281857

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Singapore Math topic matrix

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