Talk:Basic Algebra/Working with Numbers/Distributive Property

Notes for Educators
It is obvious to most educators in the classroom that students must have a good number sense to comprehend mathematics in a useful way. A critical part to have number sense is understanding multiplication of real numbers and variables that stand in the place of real numbers.

Students also need as much practice as possible with counting principles. Explaining multiplication and the distributive property as above helps to solidify some counting principles knowledge in the minds of the students.

In order to teach the distributive property an educator might be interested in how students first perceive knowledge of this kind. The better we understand how the brain obtains knowledge the more responsibly we can guide it.

Piaget model of cognitive development sets up level of understanding that the students minds passes through.

According to this chart, the distributive property would sit in sensory-motor or perhaps the pre-operational stages. Piaget's work has been largely criticized, but few doubt that it is a good starting place to think about how the brain acquires mathematical understanding.

Annette Karmiloff-Smith was students of Piaget and many believe that she brings his ideas forward. She believes that human brains are born with some preset modules that have the innate ability to learn and as you have experiences you create more independent modules. Eventually these modules start working together to create a deeper understanding and more applicable knowledge. The person moves from implicit to a more explicit knowledge which helps to create verbal knowledge.

Education, and specifically mathematics education plays a role during the process of moving from the instictually implicit stages to the more verbal explicit understanding. A student acquires procedural methods then learning they theory behind the procedure. This runs parallel to mathematics education. If you accept this model of how the mind comes to understand a concept, it would be critical to teach the students the procedural methods and mechanics of how the distributive property must be carried out. It would then be just as important to show them why this works out the way it does, or at least provide them with the educational opportunities to explore why it works out.

This exploration should take three stages. First the students needs to master the mechanics of the distributive property. In math ed terms, this might be considered drill and kill. The next step would be asking the students to reflect on why they think the distributive property has such a behavior. This could be related to encouraging metacognition with your students. Have them reflect not only on the procedure of the distributive property but also on why they think that. Hopefully the third and final step would be a the last two steps coming together in the students' minds as a solid understand of the distributive property.

Since this knowledge would probably first be link in the students mind as a procedure only helpful in a math classroom, it might also be beneficial to encourage the students to stretch this concept across domains. After all, one of the main purposes of a public mathematics education is to encourage logicality among the populous.

One of the most common errors for students to make is to just multiply the first number in the parentheses by the number outside. For example

$$2(x+1)=2x+1$$

This could initially be remedied by explaining the distributive property as taking 2 groups of (x+1) and adding them, like multiplication means to do.

This might lead to another misunderstand though. It might be confusing to think about things like $$.5(x+1)$$ or $$\frac{2}{3}(x+4)$$ because it is hard to think about .5 groups of (x+1) or $$\frac{2}{3}$$ of $$(x+4)$$. When a student first learns about multiplication they are told that it like grouping things together to simplify the addition of the same number multiple times. Once they have mastered this concept multiplication is extended to all rational number. Now multiplication is better thought of as a scaling process. You are taking one number and scaling it by a factor of another. This same mental leap is needed to think about distributing a rational number because the distributive property is still just multiplication.

An effective method to explain multiplication as a scale factor is to have two number lines, one right above the other. If you are multiplying by $$\frac{1}{2}$$ then the scale factor is $$\frac{1}{2}$$ and you can draw guide lines from the top number line to the bottom number line that scale every number down by one half. So a line will be draw from 2 on the top number line to 1 on the bottom number line. Another line will be drawn from 3 on the top number line to 1.5 on the bottom number line and so on. Of course this method is easier to use if you have an interactive applet or program of some kind that allows you to update the scale factor immediately. Without this instant gratification the students may find this explanation too cumbersome to follow.