Thoughts from NCTM session on Singapore Math + Technology

Last week I presented with Lauri Susi at the National Council of Teachers of Mathematics (NCTM) 2012 Annual Conference in Philadelphia.  Here was our session description:

470 – Technology + Singapore Strategies = Number Sense
Lead Speaker: Cassandra Turner
Co-Speaker: Lauri Susi

Visual reasoning is a powerful tool for making sense of mathematics. Learn successful visual strategies and instructional methods from Singapore that allow students to develop a deeper understanding of number concepts using hands-on manipulatives and software. Walk away with strategies for guiding students’ learning that you can use tomorrow.

We displayed the above image of lions on the screen while discussing the Concrete-Pictorial-Abstract progression of understanding. A teacher raised her hand and said something along the lines of:

I don’t like that picture. There are male lions and female lions, they aren’t the same. I can’t add them together and students get confused in upper grades when they think that these can be added.

Which is such a great comment. Why? Because this illustrates one of those interesting points that isn’t always in a student textbook and as the teacher you have to be aware of it : labels matter. Yet it isn’t so obvious at a kindergarten level.

2 male lions and 3 female lions make 5 lions altogether.

Well, they’re all lions and we’re looking at a part-whole understanding of addition. Here’s another image from the kindergarten book:

2 boys and 3 girls make 5 children.

2 daisies and 2 tulips make 4 flowers.

So how does this apply to later, more advanced concepts? Consider:

  • 2 ones and 3 ones make 5 ones.
  • 2 tens and 3 tens make 5 tens.
  • 2 tens and 3 ones make 23 ones.
  • 2 dimes and 3 pennies make 5 coins and they also make 23 cents.
  • 2/5 and 3/5 make 5/5
  • 2/5 and 3/4 make…hmmm, we need some common terminology here.

Thanks kindergarten and first grade teachers for laying this foundation!

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Making math masters: A brief overview of Singapore Math®

My students love math class. In fact, many will tell you math is their favorite subject. Why? They’ll tell you it’s because Singapore math is fun. I’d say it’s because once they understand how math works, they become confident in their abilities. So what exactly is Singapore math?

Wait, math from Singapore? Isn’t that some little island in Asia?

Primary Mathematics is based on a program of study introduced by the Ministry of Education in Singapore in 1981, a time when Singapore’s students were middling in math. Fifteen years after the adoption of its new Primary Mathematics Syllabus, Singapore students led the world in global Math achievement tests (Singapore topped international rankings first in 1995, and again in 1999, 2003 and 2007).

The Singapore math success story—from mediocre to world-class in a generation—is no secret. The curriculum provides students with a solid foundation in mathematics by focusing on visual understanding, connections, number sense, mastery, and word problems.

Concepts in Singapore Math® are taught in a concrete – pictorial – abstract sequence

Hands-on manipulatives or real life objects are used to demonstrate the concept, then students use and create pictorial representations. This interim visual step is typically missing from many curricula used in the U.S. It provides a transition from the words to an abstract algorithm. The goal is always to use the concrete and visual components  to get to a standard algorithm.

To gain number sense, students are taught to make connections between topics. While first graders will still work on “fact families”, Singapore math also uses a pictorial representation called a “Number Bond” to help students see the connections between addition and subtraction.

Fact Families:                  Number Bonds:

Understanding numbers and operations is critical to mathematics

Singapore materials focus on place value to provide a deep knowledge of numbers. As students work with and manipulate numbers, they work towards fluency by learning and using mental math strategies.

For example:

“If I know that 7 and 3 make 10, I could solve the problem of 47 + 8 by breaking the number 8 apart into 3 and 5. Adding the 3 to 47 gives me 50, then I can easily add on 5.”

These mental math skills show flexible thinking and provide a “check” students use when the algorithm is learned. I was in a first-grade classroom last week where the teacher was talking about addition and subtraction strategies with her students. They were working with numbers like 9 + 5 and the teacher had asked the students how they got their answers:

“I counted on from 9”
“I took 5 apart to 1 and 4 and made a ten first”
“I used automaticity!”

To get to mastery, students work on focused concepts and skills. U.S. curricula are typically criticized for being “A mile wide and an inch deep”. Topics continually spiral and “It’s ok if kids don’t have their multiplication facts memorized this year, we’ll reteach them again next year.”

And next year and next year…

Not so with schools using Singapore Math®. In first grade, students will learn multiplication of twos and threes within 40. In second grade, they’ll master multiplication and division by 2,3,4,5 and 10. Each year builds on the prior foundation and extends student understanding. By the end of third-grade students will have mastered all of their multiplication tables as well as multiplying and dividing by a single digit. Yep, they will even become proficient with the dreaded “long division algorithm”.

Understanding problem solving

Another component of mastery is the ability to take what you already know and apply it in a new context. Remember being tortured in school with story problems? The heart of the Singapore curriculum is an emphasis on problem-solving — and that means word problems. They are incorporated throughout the materials to provide context to each topic as it’s taught. The key to solving these begins with a bar model or pictorial representation of the word problem. For instance:

1)  Ginny has 40 cherry and grape gumballs in all. She has 24 cherry gumballs. How many grape gumballs does she have?

gumballs (1)

 

2)  Ginny has 24 gumballs. She has 3 times as many gumballs as Paul does. How many gumballs does Paul have?

paul_and_ginny

3) 2/5 of the students in a class are boys and the rest are girls. There are 35 students in the class. How many boys are in the class?

students

4)  The ratio of the number of boys to girls in a class is 2:3. After 6 boys join the class, the ratio becomes 5:6. How many boys were in the class at first?

Students 2

 

This is a sixth-grade problem from a unit on changing ratios. Can you see the answer? Note that the number of girls doesn’t change.

1 unit = 6 boys

4 units = 24 boys

Mastering Math Makes Math Fun!

Singapore Math® is a great foundation for elementary math success. Working with teachers in their classrooms, I see the impact the materials have on students every day. Singapore math can help make every child in every classroom a competent and confident mathematics student.

 

Answers to Word Problems

1) Ginny has 16 grape gumballs

2) Paul has 8 gumballs

3) There are 14 boys in the class

4) There were 24 boys in the class at first

 

Bar Models generated from ThinkingBlocks.com

 

 

 

 

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Twelve Days of Singapore Math

10 Base-Ten Blocks

On the first day of Christmas, my teacher gave to me…
a mental math addition strategy.

On the second day of Christmas, my teacher gave to me…
2 ways to skip count, and a mental math addition strategy.

On the third day of Christmas, my teacher gave to me…
3 subtraction questions, two ways to skip count, and a mental math addition strategy.

On the fourth day of Christmas, my teacher gave to me…
4 division worksheets, 3 subtraction questions, two ways to skip count, and a mental math addition strategy.

On the fifth day of Christmas, my teacher gave to me, 5 number bonds…
4 division worksheets, 3 subtraction questions, two ways to skip count, and a mental math addition strategy.

6th day — 6 part-whole bar models
7th day — 7 hundreds chart games
8th day — 8 connecting cubes
9th day — 9 two-step word problems
10th day — 10 base-ten blocks
11th day — 11 fraction circles
12th day — 12 place value disks

And Best Wishes for an Outstanding New Year!

 

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Second Grade Math Wall

I visited a wonderful second grade classroom last week!

Number bond, words and a place value chart for the number 152.

In the classroom, students were working on adding two-digit numbers. Some examples follow:

Adding 34 + 5:

Adding 24 + 32 on a place value chart
(from p. 29 of the Primary Mathematics Standard Edition 2A Textbook):

Decomposing addends to tens and ones to add 24 + 32:

A little harder to see…this student didn’t need to decompose the 24 and 32 into tens and ones, she just grouped the ones and the tens:

Next up, subtraction problems. Maybe with less circling/grouping of tens and ones.

 

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Implementation challenges: It’s not necessarily the curriculum

Sometimes the strategies used in the Singapore Math materials look different. Number bonds, bar models, place value charts, arrays and area models can be unfamiliar to parents. Most schools adopting Primary Mathematics host Parent Nights to walk parents through the new materials, the program and to share why the school decided to switch from their old curriculum.

I’ve done dozens of these parent nights and common concerns run through the questions. After a presentation and Q & A parent session this week, a parent approached me about her son’s home enjoyment that afternoon.

“How am I expected to help my fourth grader?” she asked, “This Singapore Math is so different.”

She opened her son’s 4A workbook and showed me the problems she couldn’t help her son with:

Unfortunately for this parent, there’s nothing “Singaporean” about order of operations.

She couldn’t help her son because she didn’t remember elementary school math. In fact, she had never heard the term “Order of Operations”. I tried, “Please Excuse My Dear Aunt Sally,” something many adults were taught back in the day… Nope, that drew a blank as well.

After I explained what the order of operations is, I offered the parent two pieces of advice.

1. Include a note with her son’s home enjoyment, something along the lines of “We struggled with these problems.” The teacher needs to know this. Home enjoyment is practice for your student and provides feedback for their teacher. The teacher needs to know if the lesson taught that day at school could be completed individually by the student that night at home. If not, there was a breakdown somewhere.

This parent note tells the teacher a couple of things:

  • The student was able to complete the first two parts of the assignment that included problems with two operations, but couldn’t work the final exercise where the problems had three operations.
  • Something was lost between the teacher’s lesson and this student. Was he in the bathroom? Did the teacher’s lesson and guided practice not include three operations? Are there attention issues? Did the student just not “get it” and not ask questions? Many things could have created this disconnect. Was it a single student, or are there more that struggled?
  • This parent may not be able to help her son with 4th-grade math.

2. Pick up a copy of a book on elementary mathematics. One of my favorites is Arithmetic For Parents: A Book for Grownups about Children’s Mathematics.  In the foreword, the author includes insights from his time in an elementary classroom:

Elementary mathematics isn’t simple at all. It has depth and beauty.

The book is written for parents that want to be an active participant in their child’s studies, as well as the “reader who wishes to return to his childhood mathematics, from a different angle.”

Good math teaching is just good math teaching. While there are some differences in the strategies used in the Singapore books, teaching math so students understand the concepts as well as master the algorithm or rule is the goal.

Do you help your child with their home enjoyment? What happens when you are fuzzy on the concepts your child is learning?

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