Memory, Mathemtics, Dot Plates, and Basic Facts


In order for basic facts to be known, in a fashion that can be accessed within a short period of time, there must be two types of memory involved. Short term and long term. Many instructional strategies don't get information from short term memory into long term memory. To move information into long term memory a couple of things need to happen. There has to be something in long term memory to which the information can connect. We think of memorization as being - committed to memory, learn by heart, putting something into an isolated box. It doesn't work that way. Consider a simplified three step process. Consciousness - an awareness of something for the learner to notice and think about. This is done in the short term memory which is limited by mental power. From here it may be possible the information is put into long term by organizing it with other information. The more connections during this organization the easier it will be to retrieve, be remembered, and retained. This is why I often tell students they need to learn more to remember less.

Research is now suggesting that long term memory has to be reconstructed from time to time. Our brain has to rebuild memories or lose them. So these associations are important. Too often we forget these necessary conditions or act as if it isn't important when we choose an activity without thought about all the previous experiences students have had. Learning in school isn't equal even when students are involved in exactly the same experiences. 

Story warning...  Skip paragraph if don't want an old man's story. One of my sons was in third grade where the curriculum called for all students to memorize all the basic facts of multiplication. At a parent teacher conference the teacher proceeded to tell us how proud she was that our son was the only one who had learned the seven - multiplication facts and obviously before anyone in the class. Being the proud parents we certainly didn't want to burst her bubble and I would certainly want to give my son credit that he was able to transfer his outside recreational experiences into the classroom. The truth of the matter being that he loved football and has played it since before preschool. When he played everyone who scored a touchdown got seven points. Extra points were automatic. So he knew all the touchdown multiples well before third grade. He also learned that multiplication can be related through repeated addition, so that he could use skip counting, which he did in an efficient fashion that anyone that didn't know, how he arrived at the answer, would have thought that he had committed them to memory. Which he did, first for the lower ones, later for more familiar ones, and eventually for all. Because of the importance placed on basic facts and operations, there wasn't much depth. Five touchdowns was 35 points and so forth. He didn't have a very deep structure for these basic facts or what multiplication was other than repeated addition. However, in this class that wasn't what was most important.

There are many different ways that people can commit something to long term memory. All of them require either a very memorable experience or an extreme amount of repetition or some combination between the two. How does the type of experience students have provide those needs? Do we select the experiences to be powerful enough (collapse of the World Trade Buildings) or repetitive enough to assist different people to remember. However, repetition alone doesn't work for some kinds of memory. More on that at the end.

There are a variety of different ways to remember and some of them work better for some of us and others work better for others. Too often teachers don't employ enough variety of methods to make memorization of basic facts accessible to all students. I am not targeting a teacher at a specific grade where the facts are specified to be mastered. I believe there are numerous activities that could and should be done early that position students better or not to relate their previous experiences to basic facts where they could discover that they already know them, before they are required to know them.

Children can add and subtract based on their understanding of number value with addition and subtraction being done by composing and decomposing numbers and for multiplication as another way of thinking about skip counting, arrays, combinations, and other patterns that they could have had in their pervious experiences. I have seen students in schools where they didn't have to spend class time drilling and learning basic facts, they had learned them in the many powerful experiences of mathematizing.

So my recommendation for anyone that is trying to assist students in learning the basic facts is to provide many experiences that can be as memorable as possible related to basic facts. 

Two example: one for addition and one for multiplication.

Dot plates or flash dots -
My dot plates are disposable plastic plates with each having 0-5 or 0-10 dots on them (video - thanks Brook). You hold them so students can't see the front with the dots and turn them over quickly to flash a look and then turn them back around. Or a computerized version of 1-5 at this link:

There is a 1-10 version also.

Both of these links are in the number value directory.

So how does this help? There are several things that make this activity very powerful. First, there is research that humans can know numbers in two ways. The first way is we recognize the values - 0, 1, 2, and maybe 3 just by looking at them. Not counting.

The research also suggests that some animals can also do this. So when we flash the plates people see them as different combinations of 1, 2, and 3 and will compose them to make larger values as necessary. This composition is the second way we recognize numbers - we create a number system.

How can we be sure?
Two ways. First, it is amazing that when they stick people into a machine and do a brain scan (fmri) they see scans of people who are using two completely different parts of the brain to recognize the smaller numbers and the larger numbers. The smaller numbers are being recognized in an area closer to the brain stem and larger numbers in an area more toward the frontal lobe. Interesting... The second way was discovered by people who study the historical development of language. They found that when you look back in time to see when words were to a particular language, that the number words used in the earliest times had words for only the values - 1, 2, sometimes 3 and then every other number value was referred to by something that meant - many. It wasn't until their culture invented a number system that they developed numbers for larger numbers. Interesting very interesting. 

There are other times that we associate a value without using math. Like recognizing a pattern through memorizing patterns like on a die or triangle where we first think triangle then 3. 

Back to the dot plates. When students get good at recognizing the number values, you can tell them you will flash a plate and before they tell you the answer, you will tell them if you want them to tell you the number of dots on the plate or one more. Later you can do one more and one less, and even later two more or two less, and maybe later five more five less or ten... Got the picture I presume.

So the dot plates assist students learning number value and addition facts. 

The other thing I think is very powerful is that when students look at the plate, they stop and consider different ways of arriving at the answer, and none of them is counting. Because when you flash it, they don't have enough time to count. This is important for two reasons. Getting kindergarten students to stop and say I can think of different ways to solve a problem - this is an incredible landmark to achieve. I can't emphasize this enough. One of the most important things to learn is to stop and survey the problem and say I can solve this a number of ways which way should I try first?

A second important thing is the counting. The worst thing we can do is turn students into habitual counters. If we do it will be the only way they want to solve problems. They will be expert counters and lousy problem solvers. Once children learn to count they should be encouraged to solve problems any other way but counting. When I see someone that knows how to count trying to solve a problem counting, I usually put my hand over what they are counting and say. I am sure you can think of another way. Maybe count by twos or group by five's, or ....

The second example for multiplication is something that I do see fairly regularly in classrooms. A wall in the classroom where there are charts for things that come in multiples (0, 1, 2, 3, 4, ...)  patterns that could be used to represent multiplication facts.

Enough for now, just need the example I promised earlier about how memory doesn't happen sometimes even with massive repetition. The example: Can you identify which of the pennies in the drawing is an accurate representation? Probably not. Most people can't and the older we are the more repetitions of seeing one we have had. Probably more than we have with 8+7 or 8*7. Remembering something isn't easy it takes effort and structure.

pennies image
Excerpt from - Why Don't Students Like School? By Daniel T. Willingham.

Comments? Questions?


Dr. Robert Sweetland's Notes ©