Dot Configuration Activites

In my last post, I asserted that it is important for students to develop solid mental math skills in order to be proficient mathematicians. Mental math, sometimes called number sense, is an internal representation of numbers and number relationships. The basis for developing these skills is flexible thinking about numbers, where students can easily think of one number in a variety of ways. I demonstrated how this type of flexible thinking about numbers can help students solve problems using mental math strategies that are more efficient than using traditional algorithms. Van de Walle (Teaching Student Centered Mathematics) recognized four different types of number relationships that are important for number sense development.  (The following is quoted directly from Teaching Student Centered Mathematics)

Spatial relationships: Children can learn to recognize sets of objects in patterned arrangements and tell how many without counting. For most numbers, there are several common patterns. Patterns can also be made up of two or more easier patterns for smaller numbers.

One and two more, one and two less: The two-more-than and two-less than relationships involve more than just the ability to count on two or count back two. Children should know that 7, for example, is 1 more than 6 and also 2 less than 9.

Anchors or “benchmarks” of 5 and 10: Since 10 plays such a large role in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10.

Part-part-whole relationships: To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers. For example, 7 can be thought of as a set of 3 and a set of 4 or a set of 2 and a set of 5.

In this post I am going to propose activities that would be much more effective at developing flexible thinking about numbers, than the rote addition and subtraction drills that are currently available on the Khan Academy site.  One of the activities that Van de Walle recommended was dot configuration flash cards, to help students quickly recognize sets of objects arranged in familiar patterns, without counting. This not only helps develop spatial relationships, but also with part-part-whole relationships. The following are mock-ups I created for ipad apps using these dot configurations.  (The design of the following apps could be improved greatly and made to me much more game like and motivating, but the purpose of the mock-ups were to show the educational objective - not present the perfect design.)

In the above example, a student would be given a dot configuration and would try to identify the amount as quickly as possible.  They would practice this until they knew the dot configurations without counting. 

This example shows a dot configuration in a ten-frame.  Van de Walle used ten frames to develop the relationships between single digit numbers and ten.  A variety of different number relationships are obvious when looking at this ten frame.  It shows 6 dots, which is made up of 5 and 1 more.  It also shows that 6 is 4 away from ten.  It is very important for students to be able to quickly identify how far away a number is from the next benchmark number when doing mental math.

The above is another example of an app using dot configurations to build number sense.  The students would simply touch the number and move it over the the corresponding dot card.

Once the student has learned the dot configurations by heart, than we can extend the part-part-whole relationships by showing only part of dot configuration and asking the student to tell what is missing.  In the above example 4 dots are shown, and just by looking it is obvious that 1 is missing from the traditional 5 dot configuration.  This helps the student cement the fact that one way to make 5 is 4 and 1.

I have found an ipad app that does use dot configurations in a creative way to help students learn numbers (though not with ten frames).  The following screen shots are of the ipad app MathGirl Number Garden, which I bought for my 5 year old daughter.  We both love it, and I recommend both it and the Math Girl Additon House, to anyone with young children (though it is obviously geared towards girls).

 The above dot configuration of 9 makes it easy to see that 9 is made up of 3 threes, or even 6 and 3.  On this app, the student simply touches the correct number that matches the number of flowers. 

On the above dot configuration it is easy to see that 6 is made up of 2 threes.

On this dot configuration it shows 6 made up of 5 and 1.  Showing one number in a variety of ways increases the flexible thinking about numbers that is necessary for good mental math skills.

Again, here is another way to display six.  Displaying the same number in a variety of ways, helps to build flexible thinking about that number.  In this example, using the hands helps build relationships with the benchmarks 5 and 10.  It is easy to see that 6 is 1 more than 5, and easy to imagine that 4 more are needed to get to 10. 

In this post I hope I have successfully demonstrated the value of dot configuration activities, and how they can build flexible thinking about numbers.  I would love to see something like the above activities included in the Khan Academy's math scheme.  I think they would be much more effective at building the number sense required by today's mathematicians, than the rote addition and subtraction drills that are currently there.

In my next post, I plan to share some ideas of number line activities that would build the four different types of number relationships that I identified at the beginning of this post.