### How We Solve Arithmetic Problems - How We Solve Societal Problems

Multiplication and division involving either fractions or decimals that are less than one persist as major challenges in fifth grade classrooms. Punching these decimals into a calculator, provided that the correct buttons are pushed, can of course lead to the right answer. Knowing how to do these arithmetic operations by hand likewise can be successful, but the question of whether a child really understands the concept of multiplication and division involving fractions remains to be addressed. Without such understanding, a careless handling of a calculator or an error in computation by hand can easily happen and at the end, a child is completely unaware that his or her answer does not make sense. A recent study involving seventh graders from a public school near Pittsburgh shows that students continue to fail in this area.

The work, scheduled to be published in the Journal of Educational Psychology, looks at how well a student answers an arithmetic problem involving decimals. The following are sample questions.

The results are quite revealing. Students do well with addition and subtraction problems presumably with the notion that adding leads to a bigger number while subtraction always reduces the number. The following table summarizes the findings: Above copied from Conceptual Knowledge of Decimal Arithmetic. Lortie-Forgues, Hugues; Siegler, Robert S. Journal of Educational Psychology, Aug 15 , 2016, No Pagination Specified. http://dx.doi.org/10.1037/edu0000148
With multiplication and division, rules derived using numbers greater than one, such as multiplication leads to a larger number while division makes a number smaller, do not apply to decimals less than one. 5 multiplied by 0.29 is less than 5 and 5 divided by 0.29 is greater than 5. The study also examines the explanations behind the judgment made by the student and the following table shows that indeed, learning requires a great bit of unlearning. Above copied from Conceptual Knowledge of Decimal Arithmetic. Lortie-Forgues, Hugues; Siegler, Robert S. Journal of Educational Psychology, Aug 15 , 2016, No Pagination Specified. http://dx.doi.org/10.1037/edu0000148

"Operation and operand" in the above table means that the student recognizes that not only the operation (either multiplication or division) but also the operand (whether these are less than or greater than one) decides if the answer will be larger or smaller. "Unconditional operation" refers to a student invoking the general rule that multiplication makes things bigger while division makes things smaller regardless of the operands. Surprisingly, those who try to estimate (Computational estimation) gets the wrong answer more than half the time.

These arithmetic problems mimic in so many ways how we, as adults, often struggle with problems society faces. We have strongly held preconceived notions that lead us to wrong conclusions. We are obstinate. We therefore hardly learn from our mistakes. The results shown here are from seventh grade students, not fifth grade. Surely, these children have seen multiplications and divisions involving fractions or decimals that are less than one. It therefore seems onerous to accept and digest something that goes beyond what we have known for so long. One thing the authors write that is worth repeating here is this: "Confidence is often a good thing, but misplaced confidence is not."