Goods of the Mind

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Is This Too Much Math?

10 Feb 2017

Sometimes I see discussions on parent forums that discuss the need for teaching math to young students, especially problem solving. Some parents say that it is perfectly fine if a student does not learn much more than just how to read and write until they are in their teens. At that age, anything would be rapidly absorbed as if it had happened over 6 or 7 years of study. They argue that anything that takes effort to learn during elementary schooling will be learned faster and easier later on. But then, why do we offer early math education and why do we advise parents to enroll in it? Is early math education, specifically problem solving, a crucial building block of general education, or not?

Sometimes, we may not be aware of the importance of developing effective work habits from the earliest age. The levers that enable mass education do not promote the ability to focus, to learn independently, and, most importantly, to create original ideas. These side-effects, once solidified into habits, are difficult to undo. Moreover, since they happen at a level of thinking and execution that is largely invisible to the parent or to the tutor, the mystery of why some students ‘max out’ and start underpeforming later on, usually in high-school, remains unsolved for many observers.

Let us discuss a few examples of how students may adapt to learning environments in ways that are not leading to progress (a more detailed analysis can be found in my book Parents’ Guide to Competitive Mathematics):

Worksheets with problems in increasing order of difficulty create the expectation that students progress in a linear manner. Behaviorally, such worksheets provide a perfect excuse for the student to retract in a zone of apparent comfort wherefrom progress is as difficult to elicit as it is to oust a bear from its warm cave in the dead of winter. The student anticipates that after a certain point the remaining questions will require some effort and, upon reaching this point, stops to ask for help without attempting to solve further. The fact that the student doesn’t even read the more difficult problems, to see what they are about, let alone attempt them, is a work habit that prevents many students from making progress.

To me, it is clear that worksheets with problems in increasing order of difficulty are counterproductive. If people had always done only what they knew how to do already, there would have been no scientific or technological progress! The difficulty of a worksheet has to vary in a less expected manner. Without the anticipation of the level of difficulty, the student will at least read the problem statement, or maybe even attempt to solve it - and the attempt is very often much more important than the ability to solve. The ability to solve can be trained, by learning more theory, but the lack of drive to try things out is extremely difficult to turn around. We want to keep students curious, driven, and in a constant state of discovery. For this, we need variable difficulty within the range of questions posed.

Another format that should be used only sparingly is the multiple choice answer. Obtaining the correct answer to a problem by plugging in all the possible answer choices, or using elimination and guessing, does not constitute an actual solution to a problem. When students rely on strategies that do not require them to learn anything, they are not actually becoming more skilled at anything. Our methodology is based on our belief that learning is a process that leads to skills that are applicable in order to survive, create, and accomplish, not only a good GPA, or a good college admission, or acing standardized tests, but original, valuable work. Although we we prefer to look beyond the short term, learning how to solve problems in a proof-based manner pays off even in the short term.

The multiple choice answer format is easy to grade, easy to standardize, and easy to feed into the big data systems that most institutions use to make decisions. In making so many things simple, other desirable goals have been lost. The student is not incentivized to work on paper or to produce drawings and explanations such as a real-life job situations would require. This type of questionnaire does not require the student to create a model for the solution at hand - a model that can be corrected, improved, generalized. Only the correct answer counts, not the quality of the method used to obtain it. However, the quality of the method is key: the student should work to optimize solutions, seeking more clarity as well as a shorter execution time. All these important workplace skills are left out of the multiple choice questionnaire - instead, the student trains to trump the purpose of the exam by honing guessing skills and, as we see all too often in the field, resorts to outright random answering.

Here at Goods of the Mind we have built experience teaching thousands of gifted students and we notice that work habits play a great role in a student’s ability to make progress. It is mainly for this reason that we strongly advise parents to provide problem solving education earlier, rather than later in their student’s life! If the student learns early on to summarize data on paper, to draw simple but meaningful diagrams, to compare and notice differences, to draw conclusions from their work, they will build a strong framework on which progress will be made throughout their lives. These skills are much more difficult to learn later, when un-learning bad work habits is extra work that both they and their instructors must muddle through.

Our early problem solving classes focus on:

Try us out! We have lots of experience teaching young students.