F.T. is learning Singapore Math in his second grade class at Great Hearts Monte Vista. The school recently hosted a workshop for parents of 2nd, 3rd, and 4th graders so we could learn more about the curriculum and find ways to support our children’s learning at home. I took notes during the meeting, and promised my friends who couldn’t attend that I would type them up and share them. I hope these notes will also be helpful to families who are considering enrolling their children at Great Hearts in the future, and to families who may use Singapore Math for their homeschool curriculum. If this blog post has mistakes, please blame me, not the teachers. Likewise, this blog post does not do justice to the energy and enthusiasm of the Great Hearts teachers.

First I will share some of the basic principles and terminology. Then, I will give examples of the types of problems the students are working on. Finally, I will share the teachers’ tips about materials to use and ways to practice at home.

### Basic principles

- It’s more important for students to have a real sense of what numbers mean, than for them to memorize a process and crank it through—the dreaded “drill and kill.”
- The teaching approach starts with the concrete and moves to the pictorial and then to the abstract. A concrete example might involve hula hoops on the floor, and groups of students standing in the hoops. A pictorial example could be drawing circles on a piece of paper. An abstract example would use numbers and symbols (
*e.g.*, +, -, =). - Classroom instruction starts with a math game, and then moves to group work so students can build communication skills while sharing and justifying their answers. The way to show that you really understand a problem is to be able to explain it to a friend—without getting frustrated. Finally, students do individual work and homework.
- In real life, we have access to all kinds of conveniences (calculators, smart phones, etc.), so it’s not strictly necessary to learn how to do math in our heads. But, learning how to solve math problems teaches problem-solving skills that are useful later in school and life.

### Terminology

- Number sense: having the ability to use number in real-life situations, and understanding what numbers mean. Students with good number sense are able to read a story problem and understand it, and also to create their own story problems.
- Number bonds: a way of drawing math problems that illustrates the relationship of the parts to the whole, and differences between parts, using circles and lines. Here is an example of a number bond:

- Place value chart: a chart with columns for ones, tens, hundreds, thousands, and so on; reinforces the idea that the place of numbers is significant

- Bar models: a way of drawing math problems that uses bars of different lengths to represent different numbers, as a way of visually building up parts into a whole, or comparing a part to a whole.
- Making tens: starting with a number between 0 and 9, and adding a number to it to make 10.
- Mental math: solving problems in your head.
- Math facts: having quick mental access to basic information,
*e.g.*, 2 + 2 = 4 or 3 x 3 = 9.

### Classroom examples

During the workshop, the teachers gave examples of the types of problems the students are working on at each grade level.

In second grade, the students are practicing with number bonds, which consist of three circles: the circle on top represents the whole, and is connected by lines to the two lower circles, which represent the parts.

In this example, using a number bond to break 83 into 80 + 3 makes it easier to do the mental math of 3 + 2.

Number bonds can also make it easier to make tens—making the mental math easier. This example uses number bonds to make it easier to split off the 3 and add 7 to make a 10.

In third grade, the students start with a story, and draw pictures to explain it. For example, let’s say the students are working in a math group of four, and two new students join the class. You could draw stick figures or smiley faces, but wouldn’t it be easier to draw squares?

The bar model above shows the parts and the whole; it helps you find how many altogether—an addition problem. Then, let’s say that three students get the flu; you can use squares to show that, too.

The bar model above shows a comparison; it helps you find the difference—a subtraction problem.

Bar models can also be drawn without the lines between individual boxes; the relative length of the bars shows their value.

This bar model shows multiplication: 25 x 3 is really 25 + 25 + 25; you can see that the bottom bar is three times as long. Drawing math is a problem-solving strategy; if you can draw it, you can explain it.

The transition to fourth grade involves tackling larger numbers: second graders learn thousands, third graders learns millions, and fourth graders learns billions.

Taking a large number and rewriting it in expanded form makes it easier to do the mental math.

Dividing 90 by 2 in your head might be hard at first, but you know it is close to 100, so you can go down from there.

For multiplication, it’s effective to teach the stacked notation and also an expanded notation that uses number bonds. For multiplying 63 x 9, it may be easier to start with 63 x 10 and then go down from there.

Bar models can be used to teach order of operations. (Remember PEMDAS?)

Here’s the story. Sean has 15 baseball cards. Adam has twice as many baseball cards, and then he gets a present of 10 more. (Boo Adam!) We can mentally divide the 15s into 5s to find the size of a 10.

Here are the questions. How many baseball cards does Adam have altogether? As mentioned earlier, the bar model can be used to put the parts together into a whole. Next question: How many more baseball cards does Adam have than Sean? That question uses the bar model to make a comparison bar model.

These bar models can also be written out using numbers and symbols, such as 15 x 2 + 10. The bar models are a reminder to do the operations in the right order: multiplication, then addition.

For stacked math problems, the old terminology was “borrowed” and “carried.” In Singapore Math, the terminology is different. When adding, “ten 1s” is renamed as “one 10.” When subtracting, “one 10” is renamed as “ten 1s.” Here is an example:

If you are looking for more detail about what gets covered in each grade, and how the curriculum cycles back over those topics, go online to read the Singapore Math Singapore Math scope and sequence.

### Materials

- A handful of pinto beans on a divided place can be used to illustrate parts and whole.
- A handful of pinto beans and an egg carton can be used to practice division.

- A white board is useful for drawing place value charts because it’s easy to erase ten 1s and rename them as a ten, and so on.

- Math journals: third grade students keep a journal of what they are working on in class, as a way to show their work to their parents.
- Flash cards: make your own using index cards, or order online. (We have sets for addition, subtraction, and multiplication; these flash cards are the ones we use at home, but are not specifically endorsed or recommended by the school.)

### Practice at home

- Second grade: Practice making tens. We do this in the car on the way to school; little sister picks a number from 0 to 10, and then I ask F.T., “Four plus what equals ten?”
- Second grade: Play the card game war because it requires quickly comparing numbers.
- Fourth grade: Play the sequence game. Start with two numbers (between 1 and 9); add the last two numbers and write the ones place; keep going until the numbers start to repeat. At the parent workshop, we started with 5 and 9, and found these values:

As an experiment, you can try starting with different numbers, and see how long it takes for the numbers to repeat.

- Fourth grade: Use flash cards to practice multiplication math facts up to 12 x 12.
- Find ways to practice math in real life. On Sundays, the kids and I go shopping at the Quarry Farmers and Ranchers Market. It’s a great place to practice math because many of the prices are round numbers. Yesterday, G.N. picked out this $3 watermelon. I gave F.T. at ten dollar bill to pay for it, but first I asked him how much change he would get back. He thought about it for a minute, and then realized that 3 and 7 make 10.

I feel like F.T. is on the right track by learning Singapore Math at Great Hearts. The curriculum is designed so that everyone (not just gifted or AP students) will be taking Calculus II by 12th grade. F.T. is learning what the numbers really mean; when he’s ready to use the symbols and algorithms, he will understand problem solving methods, too.

If you would like to read a broader curriculum discussion, please see this earlier post (Español) about the Great Hearts Monte Vista open house in March 2014. This week, the school will hold a parents’ workshop on Spalding phonics, and I look forward to writing about that, too.

A caveat: If you are a Great Hearts parent from Arizona, please keep in mind that the Great Hearts Monte Vista curriculum may be a little different from yours, whether due to modifications by our local school leadership, or as part of the adaptation to the TEKS (for example, requiring math facts up to 12).

Does your school (or homeschool) use Singapore Math? If so, please leave a comment to share your study tips.

Super Great notes! I didn’t see anything that I would say is different than what I learned. Are you all doing regrouping from left to right for subtraction? For Math in Focus, which is what we’re using… it’s doing the “American way” or right to left. We also say “regrouping” for when the numbers are traded one direction or the other. Other than that, I think all this is right for Math in Focus, too. Really super love Singapore Math! 🙂

These are all great! I really do appreciate this and thank you for sharing.

Singapore Math focuses on teaching students the tools to fully comprehend mathematical computation by making use of visualization techniques, which includes number bonds and bar modeling. Thanks for sharing this terminology.