## i-Excel: Heuristic and Model Approach

From a discussion on the Well-Trained Mind Forums is a question about some the supplemental material available for the Primary Mathematics curriculum:

On the (SingaporeMath.com) site there are some supplemental books I don’t recognize, but I can’t tell what they’re for–if they’re new-new or replacing-CWP-new

Math Works?
i-Excel?
Math Express?
Brain Maths?

I wrote here about the MathExpress: Speed Maths Strategies. i-Excel Heuristic and Model Approach (Update 2/2013 – Series going out of print and will be replaced with: Process Skills in Problem Solving)
Author: Li Fanglan
Levels 1-6 available in the U.S. from Singaporemath.com

The i-Excel: Heuristic and Model Approach is a completely different type of workbook than the MathExpress series.  i-Excel books focus on problem solving. Can you deduce the meaning of “heuristics” based on the following description from the introduction?

At primary levels, Model Approach has been proven to be the most versatile and effective method to help pupils solve many difficult problems.Heuristic Approach, on the other hand, helps them handle the higher level problem solving by unconventional means.

##### Heuristic Approach

The first part of each book includes eight different Heuristic Approaches and then focuses on different Model Approaches based on topics taught at each level. Heuristic Approaches in Level 2 include the following:

1. Guess and Check
2. Act it Out
3. Draw a Diagram I
4. Make a List
5. Look for a Pattern I
6. Draw a Diagram II
7. Draw a Diagram III
8. Look for a Pattern II

Upper level Heuristic Approaches include: Simplify the Problem, Work Backwards, Make a Supposition, Solve Part of the Problem, Use Before-After Concept and Restate the Problem in Another Way. Each Heuristic Approach has a worked example, four or five practice problems and one more challenging problem to solve. (Some levels put these challenging problems into a separate unit.)
Here’s the challenge problem from Level 2 – Draw a Diagram II:

A pizza was cut into halves.
Jolene took one half and ate 2/3 of it.
a) What fraction of the pizza did she eat?
b) What fraction of the pizza was left?

Completed guided examples are included in the Answer Key at the end of this section. Make a note, however,  not all questions have worked solutions.

##### Model Approach

The second portion of the books works with the Model Approach as it applies to mathematical topics.

• Levels 1-3 focus on applying the Model Approach with the four operations. They also have yellow, pink and blue rectangular stickers in the back of the book for students to use with the model drawing problems. These are important for students who are learning or struggling with drawing proportional bar models.
• Level 4 has two parts consisting of 28 units on the Model Approach: Whole Numbers and Fractions. There is also a Part IV: Non-Routine Problems – Challenge Yourself 4.
• Level 5 has sections on Whole Numbers, Fractions, Ratio, Decimals, and Percentage. Part VII includes 3 assessments.
• Level 6 parts include ratio and Proportion, Percentage and Speed. The final section is entitled “Examination Practice”. Some of these problems are double starred for extra challenge.

Here’s an example of a ** problem from that unit:

Grace had a total of 120 red and blue pens in the ratio of 3:5. After she gave away an equal number of each type of pens, the number of red and blue pens left was in the ratio 3:8. How many pens did she give away altogether?

Good news if you’re scratching your head right now. All problems in Level 6 have detailed solutions worked in the answer key.

##### In the classroom

Most who have used Primary Mathematics would agree; the focus is on the bar model as the main problem solving strategy. The i-Excel series brings explicit instruction in additional problem solving strategies into the classroom. This is one of my favorite supplemental books to use with students. The challenge and variation makes it a favorite of students as well. I have incorporated the heuristics into a self-directed activity and have used the challenge problems for a “Problem of the Week”.

The Level 3 book includes a unit entitled “Act it Out” that became a great independent activity/group center. The example given is to use 10 coins to form the figure below. Moving only one coin at a time, what is the least number of moves to turn the shape upside down? After working this example as a whole group activity, I could now have chips (coins) and new problems available in the classroom: as part of centers, as a substitute lesson plan, or for students to use as a quiet desk activity.

Have you used the i-Excel or Brain Maths series? Share your experiences in the comments below. I’d love to know how the books have worked in a classroom or in your home.

## MathExpress: Speed Maths Strategies

From a discussion on the Well-Trained Mind Forums comes a question about some the supplemental material available for the Primary Mathematics curriculum:

On the (SingaporeMath.com) site there are some supplemental books I don’t recognize, but I can’t tell what they’re for–if they’re new-new or replacing-CWP-new

Math Works?
i-Excel?
Math Express?
Brain Maths?

First, these aren’t replacing CWP (Challenging Word Problems series), but they are fun books that are great supplements for homeschooling families or classrooms. MathExpress: Speed Maths Strategies
Author: Li Fanglan
Levels 1-6 available in the U.S. from Singaporemath.com

I recommend MathExpress if you are interested in becoming faster and more fluent with mental mathematics. Book levels 1 – 3 focus on basic mental math strategies with the four operations. Beyond that level, if you’re using them with students, some of these approaches can start making math look like a series of algorithms to memorize or tricks. It’s important that students understand the reason why these “short-cuts” (as they are referred to in the books), make mathematical sense.

An example from Level 1- Express Strategy 13:

Can you get the answer in 10 seconds?
26 + 49 = ?
58 + 37  = ?

There is a visual & written solution for each problem on the next two pages. Here’s the first written solution:

1 and 49 make 50.
Rewrite 26 as 25 + 1.

An example from Level 2 – Express Strategy 9:

Can you get the answer in 10 seconds?
342 – 190 = ?
237 – 172 = ?

Again,  there is a visual and written solution page for each problem. Here’s the first:

190 is 10 less than 200. Subtract 200 from 342 before adding 10.

After an explanation of the strategy, there is a page of practice, a page with two word problems and a Speed & Accuracy Test.

There are six volumes in the series:

Levels 1 and 2 include addition and subtraction.
Level 3 adds in multiplication and division.
Level 4 includes all four operations and decimals.
Level 5 has fractions and decimals.
Level 6 has strategies to check answers, percentages & advanced problem solving. Here are two 10-second problems from Level 6 and the first solution:

125 x 25 x 32
1/2 x 50 x 28 x 11

4 and 8 are factors of 32.
24 x 4 = 100 and 125 x 8 = 1000.
Multiply 25 by 4 and 125 x 8 before multiplying the two products.

Before working the solution to the second problem, the book provides this word problem:

A rectangular tank measuring 50 cm by 28 cm by 11 cm is half filled with water. Find the volume of water in the tank.

I highly recommend the upper level books for adults looking to improve their mental math abilities. I should also mention that the books have two diagnostic assessments at the back of the book, along with an Answer Key and Detailed Solutions. (You’ll find the solution to the 6th grade level problem above on 69.)

##### In the classroom

My students have so much fun working on these strategies. In a classroom, I’ve used these books with students AFTER concepts have been mastered to help students become faster with their mental computation. As an example, in a 3rd grade classroom, I would focus on the strategies in the Level 2 book for the first half of the year, then, depending on the students’ understanding of multiplication and division, I’d introduce some of the strategies from Level 3. These can conclude in wonderful mathematical conversations. Here’s an Express Strategy from Level 3 that should lead to an interesting discussion:

Can you get the answer in 10 seconds?
26 x 5 = ?
148 x 5 = ?

And the strategy:

2 fives = 10
Multiply 26 by 10 instead, then half the product.
26 x 5 = 26 x 10 ÷ 2

= 260 ÷ 2
= 130

I’ve used the i-Excel and Brain Maths series. I’ll post reviews on those also. If you’ve used these materials, please share how in the comments below. I’d love to know how they worked in a classroom or in your home.

## Comparing Singapore Math Materials: Workbooks

In Part 1 and 2,  I shared examples from the Teacher’s Guides and Textbooks from four sets of materials used in Singapore and the United States. The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Following are the practice pages from each workbook that correspond with the lesson on addition within 10,000 that introduces regrouping in the hundreds. As before, each thumbnail links to a full-sized file. Once again, there are minimal differences between the U.S. and Standards editions of Primary Mathematics.  Problem #1 changes pictures from towels hanging on a clothesline to boats. Problem #2 has one small change. The equation for  letter B changes from  4107 + 5 to 4105 + 5. Finally, on problem #4, “Weihua” becomes “Will” U.S. Edition Workbook 3A:   Standards Edition Workbook 3A:   The My Pals Are Here Workbook is perforated and 3 hole punched. Perforated pages would be a great change to make to the Primary Mathematics workbooks! Neither My Pals Are Here nor Shaping Maths have any word problems tied to this practice lesson, in fact, there are very few word problems in the books at all. My Pals Are Here Workbook 3A Part 1:  Shaping Maths Activity Book 3A part 1:  Were you expecting less practice in the materials from Singapore?

## Comparing Singapore Math Materials: Textbooks

In Part 1, I shared some examples from the Teacher’s Guide from four sets of materials used in Singapore and the United States.

The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Following are the pages from each textbook unit on addition within 10,000  that introduces regrouping in the hundreds. As before, each thumbnail links to a full-sized file.

There are minimal differences between the U.S. and Standards editions. The Standards edition is in color and there are two additional prompts asking students to estimate their answer first, then check for reasonableness. The first example shows regrouping in the hundreds. Problems 1-5 ask students to recall addition with regrouping the ones or tens or hundreds.

U.S. Edition Textbook 3A:   Standards Edition Textbook 3A:

My Pals Are Here includes two pages of instruction, another page with directions to a game and a final page exploring regrouping in the hundreds. Problem 5b on page 29 is the only problem that demonstrates  regrouping in both the ones and hundreds places,  although students are only asked to find the missing ones value in one addend.

Note the example #2 on page 29 that spells out the concept in words  (5 hundreds + 8 hundreds). This is a great reminder of how teachers can model this concept in a classroom and is included in the Teacher’s Guide for Primary Math both U.S. and Standards editions ( 7 ones + 5 ones = 1 ten 2 ones).

My Pals Are Here Textbook 3A:

The Shaping Maths lesson is two pages of slightly more abstract description than My Pals Are Here. Place value disks are used instead of images of base-10 blocks.

Shaping Maths Coursebook 3A:

## Comparing Singapore Math Materials: Teacher’s Guides

### First in a three-part series

I thought it might be interesting to provide examples of how a lesson is presented in four different sets of Singapore Math materials. Part 1 compares the materials where a lesson begins – the Teacher’s Guide. The rest of the series will include textbooks and workbooks.

All four sets of materials are listed below. The two editions of Primary Mathematics are currently in use throughout the United States. My Pals Are Here and Shaping Maths are currently in use in Singapore.

This overview doesn’t include the Math in Focus series by Marshall Cavendish and exclusive United States distributor Great Source (A division of Houghton Mifflin Harcourt)  which will be available soon in the United States.  Representatives at the NCTM Conference in Washington, D.C. stated that the Math in Focus content is based on the  Singaporean edition of  My Pals Are Here, with U.S. money and measurement the main additions. A listing of key topics can be found on the Great Source site.

The materials are all from the third grade level:

1. Primary Mathematics U.S. Edition (2003)  from SingaporeMath.com
2. Primary Mathematics Standards Edition (2008)  from SingaporeMath.com
3. My Pals Are Here Maths (2007) obtained in Singapore from Marshall Cavendish
4. Shaping Maths (2007) obtained in Singapore from Marshall Cavendish

Most people will display the cover of a book. You get the title, authors and little else. The back of the book,  however, contains more interesting information. (Exception? The Primary Mathematics-U.S. Edition) These are from the 3A Teacher Guides and provide a brief overview of each series. For your reading ease, each thumbnail links to a full-sized file.    For comparison, I will be using the first lesson in the 3A materials from the unit on addition within 10,000  that introduces regrouping in the hundreds. All materials use the term “renaming” except My Pals Are Here, which uses “regrouping”.

Below are the names, pages and the stated objectives for the lesson from the corresponding Teacher’s Guide along with one page as an example: Primary Mathematics-U.S.: Adding Ones, Tens, Hundreds and Thousands (p. 22) Primary Mathematics-Standards: Adding Ones, Tens, Hundreds and Thousands (p. 74)

• Review of addition of numbers up to 3 digits.
• Adding thousands with another number up to 4 digits with renaming once. My Pals Are Here: Addition With Regrouping in Hundreds (p. 40)

• Add two 4-digit numbers with regrouping in hundreds using concrete representation.
• Show regrouping of hundreds to thousands and hundreds.
• Carry out vertical column addition by adding the hundreds first then the thousands with regrouping in the hundreds place
• Add without place value charts. • To add 4-digit numbers with renaming once.
• To add 4 digit numbers with renaming more than once.

Both My Pals Are Here and Shaping Maths have a larger sized Teacher’s Guide (A4 size). This allows the publisher to include each page from the textbook, surrounded by:

My Pals Are Here: instructional objectives, instructional procedures, key concepts, materials, additional activities, individual work, heuristic for problem solving and thinking skills

Shaping Maths:  objectives, lesson, materials, classroom organisation, vocabulary, general learning difficulties, IT, notes and textbook practice

The Teacher’s Guide for Primary Mathematics – Standards provides much more guidance.  The layout of the material is more familiar to most teachers and looks a bit more like a typical American teacher manual.  Although the Standards Edition provides California Standards, it would be easy to correlate these to another state’s standards. As a design, I like the spiral binding that allows the book to lay flat.

As we will see in the rest of the series, you should not choose a set of materials based on the Teacher’s Guide alone.