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Conceptual Learning

Why not just teach steps?

Students take information from the world using all their senses. Why should learning algebra be much different?

Let's look at how athletes learn to play sports. Imagine taking a child to watch a professional baseball player for 50 minutes a day and then be expected to win a game. It seems ridiculous, because, in order to play the sport, the player must feel the bat and gloves in their hands. They must run the bases to determine how long it will take. They need to imagine hitting the ball into the outfield. No matter how much you watch the pros, until you get your whole body involved, there is no way to really excel at the sport.

This is not much different than a student watching a teacher for 50 minutes and then be expected to do their homework perfectly. Truly understanding algebra involves more than just memorizing steps. In order to best reach this level of understanding, our teaching methods involve the use of moving and placing algebra tiles in order for students to feel and see what is really happening in a problem and therefore be able to predict a reasonable outcome.

Doesnít this make the student dependent on the tiles?

Students move away from using tiles at their own pace. Typically, students follow the following learning pattern:

Concrete > Representation > Abstract



The concrete stage involves actually moving and touching the tiles by the students. The students use the visual aid in order to form number sense and develop special understanding of the different types of numbers and unknowns. If compared to elementary arithmetic, students use cubes, beans, stuffed animals or chips to add or subtract.

The representation stage is the intermediate point. Students quickly tire of moving the tiles and start to draw the different forms instead. When compared to arithmetic, this is the equivalent of students counting on figures or creating tick marks when adding. The students find this method quicker than taking out the manipulatives.

The final stage is the abstract stage. In my opinion, it is best to allow the students to determine when they are ready for the abstract stage themselves. They will move away from the pictures and use the traditional symbolic representations of variables, unknowns and numbers. Again, if compared to arithmetic, students use the Arabic numbers (1,2,3,4,5,? to symbolically represent a quantity instead of showing tick marks on a page.

Of course, every student learns in their own way. Some students skip the representation stage and go directly to the abstract. Others might never move away from the representation stage. What is important is to keep the sense of alternate methods of solving problems so students donít feel trapped into having to rely on only their memory to solve a problem. Students can always revert back to the concrete stage or representation stage in order to solve a more demanding question. These alternatives foster problem-solving techniques and lessons for everyday life. Basically, use what you know to get through unknown territory.

What if my student doesnít want to learn this way?

No problem. Although we encourage students to give conceptual learning a try, Aplusalgebra.com teaches skills using various methods. Each skill can be presented in a variety of ways. Encourage your student to view the various methods so they can pick the lessons that are best for them.