Using Mental Images

 Is there a better way to help students internalize number combinations and differences than counting-on or counting back?  Do efforts to have students memorize math facts (addition and subtraction) remain successful over time?  Even in middle school, math students can still be observed to be saying, 9, 10, 11, 12, 13, 14.  Number construction has not formulated.  The memorization method has grown stale.  They continue to use counting to find sums and differences.

Students that have been introduced to thinking in collections have better thinking strategies and flexibility in solving math facts of addition and subtraction.  They can quickly visualize situations and construct in their own minds.

Counting is a central math activity for a child when entering school.  They need to see the patters of of sequence of numbers.  As these counting patterns strengthen, we can begin as early as first grade to introduce collections of dot sets.  This is an abstract concept of number when he or she has mental imagery associated with numerals.  This concept has to be practiced with the students daily with dot cards, tens frames, dominoes or your own made up collection.  It should be a consistent tool that you use.  The tens frame is probably the best by first grade.  The "Quick Draw" process at the overhead for two seconds with dot cards or ten frames will increasingly encourage and challenge students interest to see collections instead of counting the the dots.

If students are allowed to use the visuals to make construct visual addition and subtraction solutions it is superior to memorizing facts.  This takes the drill and timed test out student learning.  Wirtz (1980) observed that children enjoyed 'pushing the dots around in their heads'.

The teacher is demonstrating addition of 8 + 7 before and after using dot patterns.

Using collections, young students are encouraged to create an abstract model or concept of number without  resorting to the procedure the find the number each time by counting.  Encouraging counting each time can  actually interfere with the construction of number.  That is, because it is a rote operation and may be devoid of any mathematical meaning.  We cannot say that counting in itself is not meaningful but that alone may be a substitute for sense making.  (Wheatley, 1999)

This teacher wants the student to look at the dots and move five above in the tens frame  (top) to  the bottom  tens frame.  When complete, there will be one filled tens frame (10) and a tens frame with one 2 dots.The student should be able to visualize the sum of 12 without counting on,  memorizing or finger counting.

To conclude:  Students and teachers will and should always count in the beginning grades in order to establish numbers and their recognition and patterns in the base ten number system.  The point to be made is that by the time students are beginning to combine and deconstruct numbers, dot patterns can be introduced.
Tasks that encourage students to think in collections provide rich opportunities  to create mathematical relationships and become powerful problem solvers.  Using collections rather that counting is a more effective way to encourage students to construct mathematical units.

I have personally used tens frames and dot cards with my young and primary students.  We have used them to construct problem solving examples.  This is exciting for them and although we will have to convert to the numerical algorithms after we learn the abstract way, they already have a visual concept to rely on.  These activities only foster self-reliance, confidence and a sense of mastery.  It enables the student to create with their hands and mind, a solution without relying on counting each time.  It actually helps the students remember their facts. Have blank tens frames and dots that can be purchased (multiple colors) to allow the students to make their own.  Have them create their own frames individually one to ten.  Give them a baggy for storage in their math area.  The possibilities are endless.  They can partner and problem solve.  You can also do the same thing with blank tens frames and bingo counters.  This way you can constantly change the values when you problem solve individually.  My favorite manipulitive is the pictured below.  It is tens frame that has magnetic dots to interact with the teacher, Wow!

These are called magnetic answer boards.