# Transitioning Instruction from Concrete to Abstract Math Problems

Instructor: Hilary Agnello
This lesson briefly discusses the difference between concrete and abstract math and then describes how to transition students from concrete to abstract problems.

## Concrete vs. Abstract Mathematics

For many students, grasping concrete mathematics is the easiest part of learning math. Some students suddenly face obstacles and frustration in their math education when content moves from concrete to abstract.

Consider Amy, a very eager and disciplined student who understands everything the teacher presents and completes her work without much assistance. Suddenly, content seems just out of reach, and Amy finds that no matter how hard she tries, she can't seem to connect to the new math. This is most likely when content has become too abstract for Amy. Without help that allows Amy to understand the material — without taking away from her own critical and analytical thinking development — Amy will experience frustration and unfortunately begin to be turned off by math.

Teachers who can properly facilitate this transition can make huge differences in their students' math confidence and long-term academic success; however, this is no easy task! Concrete mathematics is present in all grade levels contingent on the content being delivered. Concrete mathematics can be as high-level as college level mathematics, and abstract mathematics can be as low-level as kindergarten. The qualifier is in the student.

Concrete mathematics is mathematics that has a tangible aspect that allows the student to learn the content through direct demonstration or example. Abstract mathematics requires the student to tie previous mathematical knowledge to bridge the gap between past and new knowledge that has not been demonstrated to the student.

A middle-of-the-spectrum example is to consider a seventh-grade math class that is learning how to multiply and divide fractions. The concrete aspect of this math is to demonstrate how to multiply fractions by direct instruction and possibly a tangible example of using fraction pieces. The abstract to this would be for students to learn to divide fractions and determine the rule to multiply by the reciprocal, through presenting the question, providing the manipulatives, and leaving the students to determine the rule on their own.

• Concrete: 1/2 * 1/3 = 1/6 (This can be directly modeled by the teacher for a concrete math concept.)
• Abstract: 1/2 ÷ 1/3 = 1/2 * 3/1 = 3/2 (This is abstract because the student cannot perform division of fractions the same way as multiplication, and the concept of tying dividing fractions to multiplying by a reciprocal is most likely not in the student's mathematical knowledge base.)

### Why Bridge the Gap?

When students bridge the gap between concrete and abstract, the level of understanding increases, and there is more depth in comprehension. Students develop the critical thinking skills necessary to determine mathematical processes they have not yet practiced and also can learn to recall and connect previous math concepts to future math concepts.

### How to Bridge the Gap Between Concrete and Abstract

To bridge the gap between concrete and abstract mathematics, a teacher must be aware of the prior content students have been exposed to and the future math content that students will be learning. Let's consider the previous example of multiplying fractions to dividing fractions. The prior knowledge in this example would be adding and subtracting fractions and the future math around this concept would be applying order of operations to fractions.

Once the prior knowledge is established, the teacher's best tool to bridge the gap between concrete and abstract mathematics is to scaffold, or provide slightly more challenging material and prompting questions that guide a student toward the desired learning objective. Scaffolding also includes providing the prior knowledge that students can apply to new concepts.

For this specific example, the teacher could present leading examples and questions that could help students make the connection between multiplication and division of fractions, such as ''What is the difference between multiplying by 1/2 and dividing by 2?'' This provides students an opportunity to use prior knowledge to determine the connection between multiplication of fractions to division of fractions.

One of the most challenging abstract mathematical concepts for students to comprehend initially is how a variable stands for a number in an equation or expression. Because a variable represents a number that is not yet known and can change per the equation or expression, this concept presents a concrete to abstract jump that all students in a classroom will not likely be cognitively prepared to make at the same time.

• Concrete: 2 + 5 = 7
• Abstract: 2 + x = 7 (This is abstract for many students because the variable x in this case stands for 5, yet the variable x will stand for a different number when presented in a different equation. The changing nature of variables is a very challenging concept for many students to comprehend initially.)

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