When a student sees the problem "150 divided by 30," he may use the method of placing the "30" first, then the division symbol with the "150" tucked neatly under the symbol. Or, the student may drop the zeroes in "30" and "150" and divide "15" by "3" to arrive at the correct answer. A study conducted by Janeen Lamb and George Booker of Griffith University concludes that students learn better when a concrete example of division is given rather than the problem just written out.
Take that example problem and change it into something that the students can visualize or see. In other words, with 30 students in the class, you have 150 candies to distribute among them all equally. Help them visualize 30 piles of candies into which one candy out of the total is placed into until all 150 candies are gone. Once the students have a grasp on the concrete processes of 30 sets of five candies each, they can use multiplication to check their answers.
Teaching the students the place values of the numbers is something that ideally is done in the elementary grade levels. For example, the number 2,346 has a "2" in the thousands' place, a "3" in the hundreds', a "4" in the tens' and a "6" in the one's place. In a problem of "2346" divided by "2," the students must understand the place values. If the whole number "2346" throws them off, break up the problem into three parts. Divide 2,000 by 2, 300 by 2, and 46 by 2. Add all the answers of the three parts and that is the correct answer. The written out answer would be "1000 + 150 + 24 = 1,174."
Subtraction is sometimes called the "opposite of addition." Technically that is true, but there are steps in subtraction that should be conceptually understood that are not present in addition. Again, as in division, a concrete example is helpful to students, who often have a hard time with subtraction. In a class of any number of students, have all students of one gender stand. Have these students walk to the front of the room. The subtraction problem is now "We have 30 students. If all nine girls leave the room, how many students will we have in class?" This is a concrete, visual method of teaching subtraction.
An understanding of the place values in subtraction is equally as important as it is in division. The problem "1235 -- 899 = " has a nine in the one's place being subtracted from a five in the one's place. Students must understand the concept of what is called "borrowing." The procedure of crossing through the "3" and making it a "2" then moving a "1" next to the "5" turns the "5" into a "15". The one's place borrowed the value of ten from the tens' place. This reduces the "30" to a "20" and enables students to subtract the "9" from the "15". This necessitates borrowing from the "2" in the hundreds' place and placing a "1" next to the "2" in the tens' place. The second 9 is subtracted then from "12" and not from "2."