Math is learned in three stages: concrete, representational and abstract. Although concrete materials are associated with early childhood math concepts, they can be used all the way through college mathematics to reinforce the meaning and contexts for the math skill.

When learning math, young children gain number sense and quantity concepts using concrete materials. Concrete items are countable, stackable, groupable and movable After using concrete objects, students move to representational presentations. This is where we see the workbooks with pictures of these familiar items, and the student circles or draws to indicate the ways that they can add, subtract, multiply and divide them. Tally marks, using counters that represent tens and hundreds, and drawing groups on the white board are representational for math lessons. Finally, the student moves into the symbolic, or abstract view where numerals are used to represent quantity. This symbolic stage requires abstract thought and the ability to create a mental context of countable objects that are represented with number symbols. When a student can count by 5 by rote memory, this does not indicate that they are understanding that each number that they say means that five distinct items have been added to their quantity. It may mean that they have a good ability for rote auditory memory.

To help a student who is struggling with a new math concept, take them back to the concrete and representational stages to build meaning and context. For example, if a student is struggling with dividing fractions, you can draw a rectangle on the white board and show how dividing it by 1/2 would be half of the shape. This would be the representational stage. To take it all the way to the concrete stage, you may take a cup of colored index cards and create a rectangle on the table using the cards as the fractional parts of a whole. You can then mark on the cards or remove them to demonstrate the result of multiplying 1/2. I like to emphasize that the word "times" shows the student how many times the group is to be counted. 3/4 times 1/2 means that 3/4 will be counted one half of one time.

Take the time to move your student back to the concrete and representational presentations of the math problems and they will develop conceptual understanding of the math strategies. Keep those popsicle sticks around for a few more years and they will come in handy!