For math to be enjoyable and satisfying to a child, the material needs to be accessible and challenging. Math that is out of a child’s reach is frustrating and distressing. Math that is too easy is tedious and stultifying. Every child in the class needs to be given math that is at her real level, and it should never be assumed that a child’s real level corresponds to her grade level. Often it doesn’t. In many classrooms, most of the children are working on the wrong level of material most of the time, struggling with problems they lack the foundation to solve or stagnating in concepts they have already mastered.
Many educators think it is best to define each grade level of math as being composed of a set of concepts and hold everyone in the room to this definition of appropriate math for the year. The following year, the child is advanced on to the next grade level in math even if he has failed to grasp much or most of the preceding grade’s material. If a child learns 40% of the material in fourth grade, he is allowed a fresh start in fifth grade. He begins the same fifth-grade material as a child who had learned 95% of the fourth-grade material and both children are expected to move along at the same pace.
Because so much of mathematics is sequential, this is disastrous. There are certain critical concepts and procedures which are necessary for advancing in math. If a child can only add by counting on his fingers, learning multiplication will be terribly cumbersome. If a child has not mastered fractions, large parts of algebra will be hopelessly confusing. If a child can’t divide numbers, polynomial long division will make no sense at all.
Advancing children to the next level before they have mastered the necessary foundational concepts almost guarantees that the longer a struggling child stays in school, the worse he will do in math. It also means that higher grades will have many more failing children in them.
Public school math classes typically contain an enormous range of levels of math proficiency. If we treat an entire class of children as if they are all at the same level and try to get them to learn a set of pre-determined topics, we fail to meet the needs of most of them. When children are asked to solve problems and take tests on material they lack the background to understand, the result is usually frustration, confusion, embarrassment, and, quite often, poor behavior and an apparent inability to focus. Some children genuinely have ADHD; others have a situationally-triggered inability to focus. They are being asked, over and over, to attend to tasks they lack the skills to succeed at and they respond by humming, banging, poking others with pencils, or doing anything else they can think of to relieve the very unpleasant sensations of inadequacy, shame, and boredom. Given the right level of math and the right support and feedback, all this humming and banging and poking will, for many children, come to an abrupt halt.
Denying a child more advanced math when she has mastered her grade’s topics is equally damaging and can result in behavior problems as well. When children are obliged to work below or above their real level, they learn very little, are prone to distract others, and can develop a lasting dislike for math.
We need to let go of the outdated goal (which was somewhat understandable, though still damaging, in the days before the internet) of getting every child in the room to work on a set of pre-determined topics.
This goal should be replaced by a higher and much more practical goal: to engage every child in the class at the highest level possible for that child for the entire length of every math period.
The focus should not be on uniformity, but rather on maximizing every child’s growth and pleasure in math.
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Suppose, in a fourth-grade class, Andy is still confused about carrying and borrowing, while Tanisha is ready for trigonometry. Each of these children’s needs is pressing and urgent. Neither child should be focusing primarily on fourth-grade math.
If Andy, over the course of the year, can master all the concepts he did not learn in first, second, and third grades (or even half of these), he will be much better off than if he is pressed to struggle with fourth-grade topics, some of which he might be able to memorize without understanding and others of which he will be so confused by that he won’t be able to get any traction with them at all. If the teacher takes the attitude that nothing can go too wrong with Tanisha, the highly advanced student, because she is so far ahead, and at most Tanisha should be offered a few enrichment problems which may take her a little deeper into the same math the rest of the class is doing, Tanisha will not thrive either. Just like the struggling Andy, she needs math at her level.
If the teacher notices that a child in the class has advanced to a level beyond the teacher and the teacher can no longer offer meaningful help when the child gets stuck, this should not be a signal that the child needs to hold back on learning. Instead, the child should be offered more and more challenging math, and — if this is possible — the teacher should invite a volunteer parent, university student, or member of the community who has a deep love and grasp of math to come into the classroom during math period and offer help and inspiration to the most advanced math students.
Many parents, even very busy ones, would relish the chance to make a real difference in a classroom and play a significant role in the learning process and would actually prefer this to making cupcakes or a financial donation. There are also many people in the community, including retired scientists or engineers or mathematicians, who like being connected to young people and would very much enjoy being of great service for an hour a day and making a huge difference to some children. As well as offering inspiration and instruction to students whose math level is beyond that of the classroom teacher, these volunteers could also provide individual help to any struggling child who needs it.
The most efficient way I have found to provide every child in a classroom with the appropriate level of math is by using Khan Academy. Khan Academy’s free website allows students to zero in on their level and work through problem sets which cover math topics from kindergarten through calculus. This makes it possible to tailor the level of math to the needs of the students in a way that would otherwise take hours of preparation each day on the part of the teacher. While the students are working on Khan Academy, the teacher can move around the room, guiding and helping.
~ This post is an updated chapter from my book Math Class Redesigned: how to teach children to love math.
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