My students come to me for math tutoring because they continue to flounder with the “new math” curriculum. For a complete description of what is being taught and how it feels for students, see Part I of this series. Part I – The New Math: Why We Have It
If expert mathematicians have redesigned the curriculum, why aren’t the results better?
I believe it’s because the experts aren’t taking into account the developmental stages of most students, and because they really aren’t aware of the problems most classroom teachers are faced with.
The new math teaching methods are mainly designed to create:
1.) the ability to work in cross-disciplinary teams;
2.) understanding (now viewed as even more important than being able to compute); and
3.) innovative and divergent math thinkers–the three characteristics increasingly required of white-collar jobs in industry today.
Yet the new math curriculum is failing to achieve these goals. Let’s take a look at WHY, by seeing how these things actually play out in most classrooms.
How These Three Goals Actually Work Out in Classrooms:
1. Creating an ability to work in cross-disciplinary teams. The idea is clearly that “putting students in groups to solve problems” will create this ability. However, there are TWO IMPORTANT REASONS why this is not happening in most classrooms. The first reason is BULLYING, and the second reason is STUDENT ATTITUDE and LACK OF MATURITY.
Middle-school, when most students are first put into math-solution groups, is the age of the MOST EXTREME BULLYING (although bullying starts in Kindergarten). Students are usually left to sort themselves into groups, and usually, in-crowd friends choose each other, while the remaining students are randomly forced into groups with students who regularly bully them. This same situation continues in many high-school classes, and is sometimes worst of all in the smallest schools where there is only one math class per grade.
It takes an extremely effective teacher who can give groups precise tasks, direction, and rewards based on individual effort to get a group to make effective progress. Generally what happens is one of several things. The students don’t understand what they are doing at all and therefore have no idea (or motivation) even to try. They end up wasting time and talking about non-math-related matters. Or, at best, one or two students do understand and do the work, while the others loaf and do nothing, but coast on the group grade (if there is one), having not done the work, and not understanding the work that was done by the others. Or, those who are friends in the group use the hour as a social time, while the unwanted group members spend the time staring at their papers, feeling excluded, and just wasting the whole hour.
Requirements for effective group work are: 1.) being in a group with others you like or respect, and others who like or respect you; 2.) Having enough background in the subject, that when given A SPECIFIC TASK, all the individuals in the group can work on it; 3.) Being able to effectively subdivide tasks; and 4.) Having individual accountability for one’s contributions to the group. Most teachers do not have either sufficient time or experience to be effective in all these ways and rely on immature students who are not willing/able to these things themselves (as an adult work group would be able to do).
2. Creating understanding of WHY methods work, rather than merely learning computational solutions. This is an admirable goal, but it is not being correctly implemented at the proper ages, in the proper stages, or in the proper ways.
Mental maturity, and ability to deal with abstract concepts arrives at different times for different students. Abstract thinking arrives for a very few students in the lower elementary grades, for a few more students in the upper elementary grades, for about half of students by middle school, and for at best two-thirds of students by high school and early adulthood. For some people, it never arrives at all. Having taught a great variety of math topics over the years, some students grasp one topic at a young age, but don’t grasp another until many years later, if at all. Since every student has a unique profile of what they grasp or don’t grasp, this is the origin of the “spiral curriculum,” where each year, many topics are introduced, and each year, the math texts cut slightly deeper into each topic (assuming the school is still using math texts).
Let us take telling time as an example. A few students are able to grasp telling time well in kindergarten, while others, no matter HOW much time is spent in the classroom in grades two and three, just cannot grasp it until fifth grade. Then suddenly, something “clicks.” Their brain has arrived at the right level of mental maturity.
Unfortunately, today’s curriculum introduces so many topics that few are actually mastered. Thus, many students move up through the grades NEITHER understanding, NOR being proficient in calculating. Most students need and WANT to become proficient at calculating and getting the right answer in the elementary grades. This builds their confidence. They also want to know in what situations they might use those skills (which gives learners motivation, and is often an area neglected by teachers). Those who do not become proficient at calculating lose confidence in themselves and are certainly even LESS likely to be open to any discussions of “understanding.”
A current controversial topic in the math field is whether students need a certain amount of proficiency before they can understand “why” things work. After two decades of experience teaching math at the elementary and middle-school levels, I come down hard on the side that it IS necessary. Young elementary students can appreciate that a correct answer can be found through several different methods, but it is a waste of precious class time AT THAT AGE to spend a lot of time on WHY (an abstract concept which despite the weeks spent on it does not actually increase their understanding) instead of on developing proficiency and thereby building students’ confidence and excitement about learning more.
It was not the intent of the math experts, I am sure, in revising math curriculum, to have students wind up being neither able to understand, NOR be able to calculate! Their intent was to WIDEN the curriculum to INCLUDE more understanding. But with only four-to-five hours a week (at best) of classroom time to teach math per week, at least half of the available time is being taken up with “understanding” (which is not being understood by the majority of students), and not enough time for most students to become proficient at calculating. Those who do become proficient are generally having additional support from parents and tutors. Furthermore, homework has been greatly reduced from a decade ago (approximately cut in half) which means that more students than ever before are not mastering basic procedures. When students get into middle school and one-third of them still cannot determine the answer to 3 x 8 without consulting their calculators, it is highly unlikely they will gain any “higher understanding.”
3. Creating innovative and divergent math thinkers. Criticisms of the past were that students were memorizing times tables and learning to calculate, but not understanding what those calculations meant; students were unable to take even a simple story problem and know which calculations to perform.
After two decades in the classroom, I can easily see this problem did not stem from memorizing or calculating. This problem stemmed from teachers throughout school not teaching children how to TRANSLATE between English words, and math language. In most cases, elementary teachers are not math majors. In fact, most became elementary teachers because they are math-phobic! They teach the calculations, and generally skip all the story problems (as did I when I first began to teach math). Yes, it is partly a time problem, but the REAL problem is that most teachers are afraid they will not be able to explain to students how to do story problems, because they never learned themselves! Speaking as someone who did not learn this skill myself until I was an adult, I see that this is the number one area that students need the MOST help with. I find myself wondering if students in India, China, and Japan are getting this sort of help from a young age, while students in the West are not?
Rather than wasting precious elementary time on esoteric math subjects, and making “arrays” for WEEKS in order to “understand” multiplication, students would be much better served learning to calculate, and having DAILY GUIDED PRACTICE on particular types of story problems, both in order to recognize types of problems, and to be able to readily understand how to translate the English language into MATH language.
What the math “experts” who design curriculum are not realizing is that showing students all the different possible ways to solve every type of math problem does NOT create the “divergent” innovative thinkers they are looking for.
As for math majors, sometimes (not always), those who were brilliant in math are unable to explain it clearly to those who are having trouble, because the teachers never experienced those same troubles themselves. Sometimes (not always) teachers who were not good math students are able to master math, and are far better at figuring out where and why students are “stuck.” Lucky children with difficulties have those teachers! The very first requirement for becoming a divergent thinker is self-confidence in one’s own abilities. This comes from being sure that one knows at least ONE way to get the right answer every time, even if one knows that other ways do exist. The main thing is to MASTER at least one method.
Beyond competence, creating divergent thinkers is more of a personality-trait question. This question has more to do with motivation and stimulating interest, and comes from the sort of child who always asks, “Why?” Most children don’t ask why, and most don’t care about why. To create more innovative, divergent thinkers, every teacher in every classroom, in every subject, needs to challenge ideas and get students excited about learning. And yes, teachers need to be “entertaining,” too! Innovative thinkers aren’t usually innovative in just one area (such as math). Most innovative thinkers draw their ideas from multiple sources and synthesis of ideas from multiple disciplines. Students need help becoming competent, and beyond that, to be inspired enough to pursue their own interests in a self-directed way. Curriculum which forces students to calculate by many different methods fatigues many students and actually de-motivates them from further self-directed learning.
It is difficult for a new or average teacher to overcome these difficulties. Hopefully with time and experience, Western society will adjust to the new math curriculum, but I am afraid it will be later, rather than sooner.