Diving Deep into Early Elementary Math

You will notice as you read this blog that I do a lot of research.  One of the things about being a virtual pilgrim is that your mind is always running off, following trails, and jumping into rabbit holes.  I like to journey to the heart of things and I don’t feel satisfied until I have explored a subject thoroughly.  Usually, by the time I feel ready to write about something, I have thought about it a lot.  And then thought about it some more.  Sometimes I am tired of thinking about it.  😛  So, if my posts are long and overwhelming, that is why.  And much of the time they overwhelm me too, but writing them out helps me feel a little lighter.

An additional math resources I use at home with my first grader is Singapore Math.  I started using Singapore with her when we were homeschooling during her preschool year.  That year we went through Singapore’s Earlybird math program.  After we finished that, we used Singapore’s Power Math Kindergarten workbook (I’m not sure if this is available anymore).  Then in kindergarten we took our time and went through Singapore 1A, waiting to move on until she had her number facts for 1-20 down solidly.  Now, halfways through first grade, we are nearly 2/3 of the way through Singapore 1B and I hope to finish it up in May and work on Singapore 2A over the summer.

At school, she uses Saxon Math.  Saxon is a good program but is very, very different than Singapore.  I would say that Singapore’s strength is in teaching math concepts, while Saxon’s strength is in teaching procedures and its built-in review of those procedures.  Put another way, Singapore teaches the why and Saxon teaches the how. Let me explain.

The first half of the Singapore Earlybird curriculum doesn’t look much different than your standard kindergarten math program.  Kids learn how to count, write their numbers, put them in order, name their shapes, sort, and so on.  But in the second half of the year, a major concept is introduced that Singapore relies on heavily throughout the remainder their elementary program: number bonds.

If you don’t know what number bonds are, here is the general idea.  Every number is made of multiple parts.  For example, the number 5 can be broken down into the following parts: 0 and 5, 1 and 4, and 2 and 3, and then written in opposite order: 3 and 2, 4 and 1, and 5 and 0.  Here is how Singapore illustrates this concept (please forgive my uninspiring diagrams):

An example of a number bond

In this example, the 5 is the whole number, and the 2 and the 3 are the parts.  From here, the student can be taught that the two parts added together equals the whole.  They can also see that if you subtract one of the parts from the whole number, what is left is the other part.  The numbers are all related to each other.  Singapore then encourages the kids to solve problems that reinforce the relationship between these numbers (number families):

2+3=5, 3+2=5, 5-2=3, 5-3=2

So, if a student forgets that 5-3=2, all he needs to do is visualize the number bond.  When 5 is the whole number and 3 is one of the parts, the other part is 2.  No counting backward.  No procedure to memorize.  Just an understanding of the relationship between numbers.  This is also demonstrated through heavy use of manipulatives, especially linking cubes.  You can use two colors of linking cubes, one color representing one part (for example, 2 ones) and one color representing the other part (3 ones) and show that when you link them together they make five ones.  Then you can show that 4 cubes of one color and 1 cube of another color also make 5.  And so on.

Manipulatives illustrating number bonds for 3+2=5 and 4+1=5

The concept of number bonds becomes extra important when the students start to learn number facts from 0-20 in first grade.  Singapore urges the teacher not to move on until the student has mastered their number bonds from 0-10 and can add and subtract these numbers without effort.  Why? What’s the big deal?

The big deal is this: if students understand their number bonds from 0 to 10, they will never have to memorize a single addition fact.  No timed drills.  No tricks. In order to accomplish this, Singapore focuses heavily on the concept of “making a 10”.

Making a 10 is simply an extension of the number bond concept.  Students are given a number, say 8, and asked what number they would need in order to make a 10 with the 8.  If they think about their number bonds, they will know that when the whole number is 10, and one of the parts is 8, then the other part must be 2.  This concept is used to help students with their 11-20 math facts.  Here is an example:

Let’s say the math problem the student is trying to solve is 8+7.  Most procedural math programs would just have students drill, drill, drill until they have memorized that the answer is 15.  Not Singapore.  Singapore wants the student to understand why the answer is 15, not just memorize it (which will come later, after the concept is understood and the steps become automatic).  Here is Singapore’s approach:

8+7=?  How do we solve this?  First we need to make a 10.  We need to take some ones from the 7 to make a 10 with the 8.  How many ones do we need to make a 10 with the 8?  We know that if 10 is the whole number and 8 is a part then the other part is 2.  So we need 2 more ones to make a 10.  We need to take those 2 ones from the 7.  If we take 2 ones from the 7, how many ones are left?  We know that if 7 is the whole number and 2 is one part then the other part is 5.  So if we take 2 away from the 7 then there are 5 ones left.  When we take 2 ones from the 7 and add them to the 8, the 8 becomes 10 and there are 5 ones left over.  One 10 and 5 ones is fifteen.  8+7=15.

The same concept also applies to subtraction problems.  Here is an example:

15-8=?  How do we solve this?   First let’s look at the number 15.  How many tens are in 15? One ten.   How many extra ones are in 15? 5 extra ones.  Can we subtract 8 ones from 5 ones? No, there are not enough ones.  How many ones are in a ten? 10 ones.  Can we subtract 8 ones from 10 ones?  Yes!  We know that if 10 is the whole number and 8 is one part, the other part is 2.  So if we subtract 8 ones from 10 ones there are 2 ones leftover.  So we have 2 ones left from our ten.  We also have the five extra ones.  So if we add the five ones to the 2 ones, what do we have?  We know that when 5 is one part and 2 is the other part, the whole number is 7.  So we have 7 ones.  15-8=7.

Here are some examples of how you could write these problems out so your student can practice this concept (the student would fill in the numbers in red):

Using the "make a 10" concept to show that 8+7=15 (click to enlarge)

Using the "Make a 10" concept to show that 15-8=7 (click to enlarge)

And here is how you would have your student show you with manipulatives:

Adding 8 and 7: Link together 8 cubes of one color and 7 cubes of another color

Adding 8 and 7: Take 2 from the 7 to make a ten with the 8.

Adding 8 and 7: You now have a ten and 5 ones. You can also see the original 8 and the original 7 and how they have been rearranged to make 10 and 5.

Subtracting 8 from 15: Start with 10 cubes of one color and 5 of another color to show the ten and the extra ones.

After concluding that there are not enough extra ones, subtract the 8 from the 10. This leaves 2.

Get rid of the 8 and connect the two green cubes to the 5 pink cubes. The student can see that the answer is made up of the five extra ones (pink) and the two green cubes that were leftover after the 8 was subtracted from the ten. Together, these make 7.

The concepts are introduced first with manipulatives, and they are used until the student can do each of the steps on their own without much prompting.  Then the manipulatives are taken away, although visual aids may still be used (drawing out the number bonds, etc.).  Once the student can go through all of the steps on paper (like the examples I wrote out above) then they are encouraged to do the entire process mentally, just writing down the answer.

Here are some example problems that reinforce number bonds for 0-10 (the student fills in the blanks):

And here are some example problems that reinforce breaking a number down into tens and ones using number bonds:

This may seem like a very long, complicated process.  Indeed, when you first introduce the concept it is quite time consuming to do a problem.  But with some repetition the student learns how to complete the steps and come up with the answer.  In the process, the student will most likely memorize the answers to some of these math facts.  That’s fine, because if they forget the answer, they now have the tool to find the solution, without counting on their fingers.  This same concept can be used for much larger numbers as well, and makes the transition from problems like 8+7 to 18+17 an easy one because the underlying concept is basically the same (just the process of adding 8+7 and then you add the two tens).  In fact, we recently started adding double-digit numbers and my six year old picked it up intuitively during the first lesson, which I know was due to the strong foundation that using the Singapore method gave her. While teaching math facts with the Singapore method takes up more teaching time in the beginning, it pays off in the long run in terms of retention and ease of applying the concept to more difficult problems.

Now, Singapore is not a perfect program.  Its weakness is that it presents math concepts in separate units and there is not much formal review of previous concepts throughout the year.  So if you are only using the textbook and workbook, you won’t be getting much review.  But the home instructor’s guide (which is a must buy in my opinion) offers mental math review sheets and various activities you can use to keep up your student’s retention.  There are also extra workbooks.  I use the Intensive Practice workbooks and the Challenging Word Problems books, these provide extra practice and review and give the student a chance to put the concepts he or she has learned to use in different ways.  If you use the tests, they are cumulative so concepts are reviewed there as well.

Saxon, while much more procedural than Singapore in the way it presents topics, also includes a lot of built in-review.  This is one of the benefits of this program.  It also includes more topics than Singapore, things like reading a thermometer, using geoboards, extensive calendar work, etc.  Singapore doesn’t cover as many topics as Saxon, but those that it does present are done so very thoroughly.

If you are interested in reading more on the differences between the U.S. standard approach to math and the Asian (Singapore) approach, I highly recommend Liping Ma’s book: Knowing and Teaching Elementary Math.  Ma was a teacher in China and came to the U.S. as part of her college program.  When she sat in on elementary classes while she was gathering research for her dissertation, she was very surprised at the lack of understanding of math concepts by both the students and the teachers.  In China, math is taught by a dedicated math teacher even in grade 1.  This teacher has specialized in math and has a deep understanding of the underlying concepts.  The way Ma describes it, the students are given a profound understanding of math concepts at each level.  Here in the U.S. we don’t emphasize the need for a highly skilled math teacher when kids are still learning things like 2+3=5. Math isn’t taught by a dedicated teacher with a specialized degree until around the pre-algebra level.  By then, many children have huge gaps in their conceptual knowledge of math.  After reading Ma’s book, I strongly believe that most students that struggle with math in upper levels do so because they do not deeply understand math at its most basic levels.  There is a point in math where being able to follow a procedure won’t cut it anymore if the underlying concept is not understood and students have never developed a strong number sense.  Focusing on this in the early years gives students an advantage when they get to that point.

I will share more about conceptual math as we cover new concepts!

2 CommentsLeave a comment »
  • February 17, 2011 Reply
    Christy said:

    Very interesting! How high does Singapore math go? Do they teach multiplication and division the same way? This is very interesting to me because I think in “number bonds,” although I never had a name for this thought process until now. I group numbers together, using 10’s as my anchor. I have little “tricks” and “short cuts” I use to add numbers together and after reading what you wrote about number bonds, it all makes sense to me. I don’t think I was taught number bonds, although I could be wrong. I do remember taking timed drills in elementary school but if I were to take one today, it would probably take me a long time. When I add, I don’t think math facts, I always use my grouping method. I know this may sound strange, but it makes perfect sense to me 🙂 When I subtract, I will use your example of 15-8, my first thought is always, what plus 8 is 15 and that is how I solve it. Anyways, thanks for your eye opening post. I look forward to more!

    • February 17, 2011 Reply
      Virtual Pilgrim said:

      Singapore goes all the way through high school math. We just started multiplication, and for now at the first grade level it’s basically showing that 2+2+2+2 is the same as saying 4 groups of 2 or 4X2. We haven’t started learning multiplication tables yet and we will do division next but I think it’s just the same kind of intro, like: there are 10 apples and five children, if we hand out the apples so everyone gets an equal amount, how many apples will each child have? I’m not sure what technique they use when they really start getting into those concepts, I think it goes much deeper in second grade. I will let you know when we get there!

      I think of numbers in terms of 10 also, and I also do the same thing you do when subtracting, I had never even thought of doing it the way Singapore teaches it, so what you say makes perfect sense to me too. 🙂 I wonder if some people just develop their own number sense whether they were taught the concepts or not. Or maybe our elementary math educations were better than we thought they were, haha. 😉

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