As a former k-6 teacher of 30 years, I would like to share with teachers, parents, homeschoolers, tutors, and child caregivers what I know is the most successful way to teach math. I feel I must say something before any more time is wasted in "trying out" different approaches, when all that is really needed is common sense.
We are fortunate to have exposure to such a wide variety of opinions about the nature of math and how it can be learned. With easy access to all of this information, we can select what really works and use it. However, we must be aware that some promoted methodologies could be nothing more than short-term trends. Think back-remember "whole reading"? Well, phonics has been re-instated into reading programs, grammar and spelling are back from their hiatus, and now we have what's called "balanced language". I always say, "You don't know what you've got 'til it's gone." Let's not throw out the baby with the bath water in math too!
Most of today's math methodologies and ideas filter down into 2 major schools of thought- traditional and constructivist.
The constructivist way, also known as "reform", "progressive", or "discovery " math encourages practices such as learning by discovery, exploring numbers, much hands-on, ignoring established math algorithms, avoidance of the memorization of basic math facts, finding answers through reason/trial and error, child centered activities, and spiraling through the curriculum by quickly touching upon concepts without mastery.
The traditional way, also called "instructivist", encourages practices such as memorization of basic facts and terms, learning established procedural algorithms, plenty of practice and review, mastery of subject matter, sequential learning, and teacher directed activities.
Teaching in the extreme-favoring one school of thought over the other-is not helping our children.
Each philosophy or methodology has its merits and drawbacks. Neither is more important than the other. To tip the scales either way in favor of one methodology over the other is to offer students a one-sided curriculum with many inequities.
I applaud certain "constructivist" methods that develop number sense, curiosity, questioning, deep thinking, teamwork, and conceptual understanding. All math curriculums should include these methods, but in varying degrees of partial/total implementation, depending on the abilities and maturity of the students.
Here's an example. I had used a constructivist textbook and approach for 10 years and I found the following drawbacks:
By the time the children were in grades 4-6, they did not have a good recall of basic math facts. This hindered them tremendously in figuring out multi-step multiplication, division, fractions, and pre-algebra. Being that they were required to memorize only the 0,1,2,5, and 10x tables and derive the rest of the facts through nonretrieval strategies, they had to add extra steps to math problems that involved multiple steps already. This slowed them down, confused them, and they made many mistakes. The strategy approach is fine up until a certain point, but we must get practical. There comes a time when students MUST memorize their facts and utilize the most efficient and accurate algorithms to compute higher math.
Also, the children were too immature at these ages to work independently and responsibly in groups. Most groups allowed 1 or 2 individuals do all of the work, and the rest of the members thoughtlessly copied , not learning much of anything. Those that didn't understand or lacked the math knowledge needed to contribute did not participate. They felt uncomfortable and embarrassed due to their lack of skills, which now was very obvious in a small group setting of their peers.
When I began teaching in the late 1970's, I taught in a manner that was considered pretty traditional. However, I didn't use the traditional approach described today as rigid, reams of worksheet completion, drill and kill, boring rote, etc. My only recollection of that type of math fitting that description was as a student attending elementary school in the 50's and 60's. To differentiate between the forms of traditional math, I'm going to call this traditional math of the 50's and 60's ancient traditional math.
Having been taught in the manner of "ancient traditional" math as an elementary school student, I realized that I had poor number sense and lacked flexible thinking when challenged with higher math. I had such math anxiety!. I felt as though my brain would "freeze" when it came to more complex math. This made a big impact on the way I taught math. I made a promise to myself that my students would know the WHY and HOW of math and that I would make math fun.
Neither myself or my colleagues ever taught that "ancient traditional" way, even though we could be called "traditionalists". Our goal was to make math approachable, understandable, and do-able.
In the early 80's I was trained in developmental math. I fell in love with that approach and realized that I could help students understand math and develop basic skills/procedural expertise at the same time. So I started teaching in these steps:1. Concrete (3 dimensional, hands on), 2.Pictorial(2 dimensional) 3. Abstract (conceptual). There is nothing new about "number bonds" or some of the "reform" methods used today. We did the same but called it something else. Although I continued to emphasize the foundations, basic skill development, and established algorithms, I used learner friendly ways including active learning, games, manipulatives, projects, etc.
Besides the retainment of the memorization of basic facts and established algorithms, children should be taught to master the subject matter. Although the practice of spiraling is done in both spheres of traditionalist and constructivist teaching, it has a different application for each. Constructivists typically just "touch" upon concepts briefly, not expecting any mastery. Traditionalists spend more time on concepts, aiming for mastery. A lot of practice is provided, and it doesn't have to be available in just 1 or 2 forms. (like worksheets or typical flashcards) Creative teachers could teach the information in a myriad of ways, shattering the stereotype of "drill and kill" and "boring rote". This quest for mastery allows math to be gradually and sequentially learned. Children get the feeling of accomplishment and success at having learned something, rather than feeling frustrated and confused by a lack of mastery school year after school year.
We should, and could, offer elementary students the best of both worlds for a balanced math curriculum. We should use BOTH- the best traditional practices and the best constructivist practices if our students are to be successful in math.
©2013 Margo Gentile