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Math: Comprehension vs Calculation

Calculation vs ComprehensionMath is more than just memorization. Many of us have had the experience where we have practiced counting by “5s” to 100 with our children until we think they’ve got it. What happens when they get to 100? Many times, it’s 101…102…103. Wait – what happened to counting by “5s”? They may have memorized all of the fives from zero to one hundred, but did they really understand the concept? Were they able to apply it to a new situation?

I was recently in a classroom where a student, who had memorized her basic math facts and could recall them quickly, was asked to measure an object and write the answer on a worksheet. The object was 21 inches long. She used two rulers and measured 12in on one and 9in on the other. The answer that she wrote on her worksheet was “129”. I then asked her what “12+9” is and she promptly answered “21”. However, she could not connect this answer with the answer on her worksheet. She could add, but she didn’t understand when to apply this skill in the real world.

Understanding the basics of addition and subtraction can help lay the foundation for true understanding of more complex concepts. Addition and subtraction are so concrete – “If I have 4 apples and Johnny gives me 5 more, how many apples do I have?” – that children can really hold onto them beginning at a young age. Our schools have put so much emphasis on memorization and regurgitation of math facts, but students cannot seem to transfer their understanding of one concept to an unfamiliar situation. Students with faster recall abilities are put into higher level math programs (regardless of their ability to comprehend), and other very capable students may be left unchallenged because they cannot memorize math facts quickly. One of my sons belongs to that latter group. In the beginning of the second grade, he struggled with a basic “What is 7+6”, but after a short introduction to base 10 blocks, he was quickly able to add “232+1319”.  He could also just miraculously come up with the answer to complex word problems, even though when you dumbed them down to their basic equations, he would stutter and second guess himself. Why? I think it’s because understood the process. He just didn’t have the facts memorized that would help him to compute the answer. In the local school system, he would have been kept back in his math group until he mastered the memorization portion, even though he had amazing problem solving skills and was eager to be challenged.

At home, I let my kids use manipulatives, calculators, the Internet and anything else that will help them to solve a complex math problem. We don’t focus on the right answers (though those are nice), we focus on the steps that need to be taken to reach those answers. Our society encourages rote memorization in an age where memorization is just not necessary. It encourages hand calculations when we, as adults, just don’t calculate by hand anymore. Wolfram Alpha is a computation engine that is readily available on my phone, ipad or computer. It is lots of fun to play with – you should try it out. I’m not saying that my children should be counting on their fingers for eternity. What I’m saying is that at my house, we’ve taken the focus off of memorization and put it on comprehension. As they practice their math skills, they rely on computational help less and less. I am fully confident that my children will be able to perform mental arithmetic with more confidence and ability than most teenagers and adults, because they won’t get that sinking feeling of inadequacy that many people get when they are faced with a math problem. Math can be exciting, challenging, and fun. Or, it can be worksheet based, boring and anxiety inducing. Whether we’re looking at this a parents, home schoolers or teachers, we all have the opportunity to intervene and make math a positive experience for our students.

Check out this TED talk by Conrad Wolfram about his organization’s push for computer based math.

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