When they are first learning a concept (at the beginning of the year), you do all or most of the work in the concrete stage. As they progress, you add the pictorial and abstract stages, doing less with the concrete stage but not eliminating it all together, even after they show they fully understand. This type of learning is not a quick "how many can you finish in 2 minutes" type of learning. I tried to never do more than 10 problems per day, sometimes only doing one to three problems. The children will spend a lot of time on whatever stage they are most comfortable with. In the concrete stage, all they want to do is play with the manipulatives (whatever you are using to demonstrate the concept). When they start to understand the concept better, they will get very detailed in their drawings for the pictorial stage. They want to put whiskers on the cats or make the "x" representing the dogs have a face to match that animal in the concrete stage. That's okay. It means they are progressing in understanding the concept. When they start to feel comfortable in the abstract stage, you still want them to demonstrate the other stages, but you will want to start using story problems rather than just the raw numbers and symbols. If a child understands the basic concept, go on to harder numbers but don't change the overall concept you are working on for the year. Also, don't forget to include learning about money at each stage. That will have to be after they've worked with the concrete items and are starting to feel comfortable with the pictorial or abstract stage. Measurements can also be added, but be careful not to go too fast. If you're teaching the counting concept that year, think about using other objects besides rulers, etc., when you measure. "That couch is 10 of Daddy's shoes long!" Don't introduce fractions of a measurement before you have talked about fractions, for example. It will be very frustrating and your child will probably regress.
As a teacher, I couldn't believe how many 4th and 5th graders did not fully understand counting! Teaching it this way ensures that they know what they are doing it and can use these basic math concepts when they get to harder problems. Let me give you a couple of examples at different ages.
(1) When my daughter first stayed home, she could count but she could not match the numbers with objects (one-to-one correspondence). For the "concrete" step, I gave her a small box and had her fill it with stuffed animals. She lined them up in the living room (this took an excruciatingly long amount of time because she had to explain to each one why he was chosen before the others, but that was okay because she needed to do that). We then took some store-bought number flash cards and put one number next to an animal, in order. Then we went down the line, counting them as we went. I would do things like trade the places for the animals and we would count again to see if the total changed. That was math for the day (it took about 30 - 45 min. to do it). We did this several days using My Little Ponies, Barbies, Noah's Ark animals, etc. so she could get the concept. She would never have gotten tired of playing with her toys, so I had to decide when to change the activity - which was about 2 weeks. (I was bored, but I wanted her to really understand so I had to find a way to be excited about it). At the time, she had a severe fear of writing (thanks to public school), so we had to skip the pictorial and abstract, but if we didn't skip it, the pictorial would have been to draw a circle or an "x" for each animal and write the total number next to it. The abstract step would have been to write tally marks (teaching her to bundle them together) and put "=18", or whatever the total was.
(2) Addition was done with the same toys. Concrete: we got 3 plush dogs and 2 plush cats and put them side by side. We counted how many dogs we had, then how many cats and then talked about how many animals there were total. (I picked out the toys we used this time). Since my daughter was still afraid of writing, I did the drawing of the pictures, but I was just very basic. I used circles for dogs and x's for cats. Then I wrote the number 3 under the dogs, the number 2 under the cats. Then I circled them and wrote 5 outside the circle. It would have been better if she had drawn it, but with her fears, I had to take what she was willing to give, so she leaned against my arm and we talked through everything I drew. On the same page, I wrote 3+2=5. We talked about the problem, I had her point to the right animals, the drawings, and the abstract numbers so that I knew she understood the concept. I think 5 was the maximum we did. As she learned how to add, I let her pick the toys we would use. For subtraction, we did the opposite. For multiplication and division, I will use what I did when I taught school (my daughter isn't there yet). We would take objects and group them.
(3) Multiplication is simply repeated addition (3+3+3=9 or 3x3=9). I taught them that the "x" symbol meant "groups of," so 3x3 should be read "three groups of three." Then we used these objects (we used much smaller items in the classroom than stuffed animals because we're dealing with large numbers) to create the problems. 7x8 meant 7 groups of cheerios with 8 in each group. Then we practiced counting by multiples to find out how many of the items we actually had. Pictorial stage was shown by putting an "x" inside a circle for each group, separating the groups and writing the numbers outside each circle, then drawing a large circle around the entire group and putting the total number near that circle. Abstract would be to write a multiplication problem. After they understand this concept very, very well, then you can teach the entirely abstract concept of larger multiplication. Look into Lattice multiplication when you get to this stage. I taught both the traditional method and Lattice Multiplication. Every problem had to be done twice. We only did 10 problems per day, but they knew what they were doing!
(4) Division is taught just the opposite of multiplication. Division is simply repeated subtraction (9-3-3-3=0). You start with the total amount and divide the manipulatives into smaller, equal groups with remainders. Draw the pictures by drawing the total amount and then circling groups out of the whole. In the abstract stage, though, you will want them to write every division problem 3 different ways.
54÷6=9
9
54√‾₆‾ (sorry, this one didn't come out on Word well)
54/6=9 (should look like a fraction)
Go on to "long" division only after they fully understand the basic concept. I did not teach "short" division to every child because it should only be taught if they fully know what they are doing with long division.
(5) This method can be continued with the higher math skills, as well. In Singapore, they require all three stages (concrete-pictorial-abstract) to be used through the top grade in High School. One vital concept in algebra is to make sure both sides of the equation are equal. Use a balance scale to get this concept across. The pictorial stage is very important, because it's hard to find different weights to show these concepts. You can use different lego blocks sitting on a scale drawn on paper to show the concept
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