As an adult learner in a teacher preparatory program, I was taught to teach math in three steps. The first step is to teach math concepts concretely. Students should use manipulatives and hands-on materials to solve math problems. Examples of concrete materials include base ten-blocks, fraction pieces, counting bears, chips, and connecting cubes just to name a few. After students are able to represent and understand how to solve math problems concretely, then they can move on to the second step, representing math in pictorial form. At this stage, students are required to transfer their concrete learning into representing math problems as pictures. Students are now required to represent and solve math problem by using only pictures. Pictorial representation can include pictures of base ten blocks, fraction circles, and graphs just to name a few. Once students have mastered the concept of solving math problems concretely and pictorially, the final step is to solve problems abstractly. Solving problems in the abstract includes number math equations (4x4, 13-4, 2+2, 54/2, etc.) and word problems. At this stage students should be able to apply concrete materials and pictures to solve equations and word problem. They are required to make the connection of solving a math problem concretely, transferring their thinking into a picture, to finally solving a word problem or number equation.
At times, my students have difficulty with representing math problems in pictures, once they've mastered representing equations concretely. I discovered this issue when teaching fractions. Students were eager to learn how pieces of various manipulatives were added together to make a whole. My class was able to show different fractions with fraction pieces, manipulate the fraction pieces to compare and order them from least to greatest or greatest to least, and to add or subtract fractions. The challenge came in to students applying this knowledge and solving fraction problems pictorially. Most of them had a hard time drawing fractions and comparing fractions. This made it even more difficult to introduce fractions in the abstract form of word problems and fractions in number form, such as 1/2, 3/4, 2/5. I plan to help students successfully move from the concrete stage into the pictorial stage. Once students have the capability to draw pictures and solve problems, then the abstract stage will be much easier to master and comprehend.
There are three common fraction models that are typically discussed in elementary and middle school textbooks: linear model, area model, and discrete model. Even though several other fraction models are taught, the three models mentioned are more widely found in textbooks. Teachers and students often represent fractions with concrete objects as well as static drawings. Moreover, there are many ways to represent fractions, and it is up to the individual to find which model strategy and hands-on materials work best to understand and manipulate fractions.
Some students do not realize that drawing a picture can help with counting, adding, subtracting, multiplying, and dividing. Pictures can also help with drawing shapes and figures in geometry and with fractions. In this unit I illustrate how pictures and figures can make solving math problems and word problems easier. Students normally draw tally marks, sticks, and circles. They need to understand that these are not the only pictures they can draw. For an example, students can solve the math equation of 4+4= by drawing four apples and then another four apples to get the answer. Students can draw numerous objects to represent numbers to solve math problems. Here I focus on using pictures to assist students to understand fractions.