Editor's Note: This article originally appeared in the January 2012 issue of The Old Schoolhouse® Magazine, the trade magazine for homeschool families. Read the magazine free at www.TOSMagazine.com or read it on the go and download the free apps at www.TOSApps.com to read the magazine on your mobile devices.

Headlines abound detailing the dismal performance of American students when they make the leap from elementary to upper-level math. In the areas of creative problem-solving, fluency of ideas, and mental agility, our students are falling short. Why?

Could it be we are expecting teens to give up an important part of the learning process? In early years, we use pattern blocks and toothpicks to give our students pictures of new math concepts. Yet, when students graduate to Algebra, Geometry, or Calculus courses, the tangible tools of math (manipulatives, games, and hands-on activities) yield the floor to more “mature” learning techniques. 

Unfortunately, what gets lost in that transition is a matter of brain function. More complicated math operations call more of the brain into action. Even though the sequential processing needed to perform a calculus problem may come from the left hemisphere, the right hemisphere is needed to access the big picture. Removing the tangible tools for seeing that big picture inhibits the student from tackling the problem with both sides of the brain.

A Question of Outcomes

Exceptional educators know that by attacking a problem or concept from the concrete to the abstract to the theoretical, students are able to interact with the material in a 360-degree fashion. My husband’s high school Physics teacher was a genius at taking the abstract algebraic and calculus concepts used in physics and making them meaningful to the teenage mind. He used architecture, footballs, and model rockets to generate interest and make them think. He understood that math games don’t become obsolete when students reach a certain age. Instead, they simply morph from beans and teddy bear counters into activities with substantially more “wow” factor. 

That Physics professor’s example forms a compelling case for the use of tangible education tools in the upper grades. It also raises the question of ultimate outcomes. What are we really trying to accomplish through math education? The discipline of math rests on a foundation of analytical abilities. Three of these—problem-solving competence, reasoning ability, and flexible thinking in application—are skills that cultivate a quick and agile brain by utilizing both hemispheres. Let’s take a closer look at how tangible math provides that type of 360-degree comprehension. 

Creative Problem-Solving

Mathematical concepts occur in relationship to one another. They build on each other, parlay off one another, and because math is a step-dependent discipline, each step requires a correct answer to move the problem forward. Relationships like these are best discovered and analyzed with the help of symbolic representation. This is where tangible tools shine. As students are given a pictorial peg to hang a concept on, they can work through each step and see relationships they may otherwise overlook. In this way, manipulative tools accelerate understanding and let the mind process relationships, leading to creative solutions. 

Fluency of Ideas

Real success in upper-level math courses rests on the student’s ability to think mathematically rather than plugging numbers into formulas. Evidence of mastery involves explaining why a solution is valid. Hands-on activities provide an impetus for students to take a concept, internalize it, and bring it to a logical conclusion. After solving a problem in this manner, students have the confidence and understanding to defend their position.