Affirming comprehension and preempting misconception in children's mathematics learning
While trying to engage my children at random in a coursera.org assignment, I asked my six-year-old to answer the question on the right, seeing we already covered basic exponents up to multiplication rules. She erroneously chose the 3rd option at first before changing her choice to the correct answer when I corrected her. I knew she understood the correct process, because she emphatically underscored her choice when her brother was attempting to answer my request to answer that same question.
So, what is a possible reason for her first choice? I think there can be several reasons for the inadvertent error. (1) Overall student is still growing mathematically; (1b) the student is yet to grasp the concept that some processes are not linear. That is, while 4x - x = 3x, x^4 - x does not equal x^(4 - 1). (1c) The student has not yet thought about the fact that a binomial is at best the product of two factors including another binomial; that is, excluding zero coefficients, we the product of two monomials doesn't generally yield a binomial. That is, order notwithstanding, x^3 = x(x^2); while x^4 - x = x(x^3 - 1). This is also an application of the distributive property, which we have also discussed before; it is the angle I plan to take next time I recast this problem (my children are on their play break and homeschool is over for the day).
This conversation is one of the reasons why I believe students should be taught mathematics in an organic way, in a manner conducive to their abilities, whether this translates to algebra at age six or arithmetic at age ten. Balkanizing mathematics into strict grade bands is not always the way to go. Yes, there are arithmetic requirements that should preempt algebra, but arithmetic alone is not sufficient to establish a mathematical foundation. Pre-introducing some fundamental concepts as soon as students can bear them may also be a useful goal. For there are enough misconceptions in mathematics to keep us busy in thought from infancy to adulthood.