Elementary Math Frequently Asked Questions
“Research shows that children’s home environments and their family’s expectations, aspirations, and beliefs can greatly affect their attitudes toward mathematics and their level of achievement.” – Fan, 2001; Sheldon and Epstein, 2005
Q. Why do students have to show all of their work in mathematics? If the child can find the answer without showing one (or more) strategies employed, aren’t we simply adding to the confusion?
A. The fact is that while some students can do some steps in their head and get the correct answer, our goal in elementary mathematics is not just arriving at the correct answer for the current problem. Our goal is that students have the opportunity to learn strategies and practice the steps necessary to understand the mathematical concepts that underlie the problems. The more times the students employ the strategies and write the steps in simple problems, the easier it will become for them to repeat those strategies in novel situations. This repetition will help students to learn and remember the strategies, and after a while these mathematical practices will become automatic. Then, when students encounter more challenging mathematics in the upper grades, these foundational strategies and steps are automatic, allowing students to focus their energies on more the complex thinking required rather than having to worry about the more foundational mathematics.
Additionally, as students work through multi-step problems, their mathematical thinking becomes visible. This visible student thinking is beneficial to both the student and the teacher. For example, if students get an incorrect answer at the end of a lengthy problem after having done many of the steps in their heads, it is difficult for them to go back to the point of error. Instead, they have to do the entire problem over again. Not only does this slow the students down, but it is also likely to increase their frustration and reinforce the error. Without the work shown, the teacher is also unable to identify the exact error and assist the students most appropriately.
Q. Today’s math instruction looks very different from the study of mathematics that many of us had in school. What accounts for the difference?
A. Many of us learned mathematics through memorization. We learned to mimic a series of steps demonstrated by our teacher, and, if we carefully and obediently followed the model, we would be able to solve similar problems. That type of mathematics instruction works well for predictable problems, but it does not necessarily work well when students have to apply mathematical thinking to new and complex, real world problems. In those situations, students who lack the conceptual understanding of mathematics struggle. In all honesty, the work that we do in elementary school sets the foundation for algebra, which sets the foundation for trigonometry, calculus, physics, and beyond. The rote memorization of mathematics can only serve a small percentage of the population well; for the remainder, it results in a plateau in mathematical understanding and ability.
With time spent in the early grades on mastery of basic skills, fluency, and application of strategies, students can use their brain power in the upper grades to wrestle with the thinking necessary to approach the complex word problems (see sample problem). Instead of trying to remember the one correct series of steps necessary to solve a problem, students who know multiple strategies and steps can instead concentrate on the thinking required to solve the problem. Because the foundational skills have been internalized, time and effort can now be spent on understanding and applying new skills.
One thing that has not changed about mathematics is how one level builds on another. The sequence of topics in math has been carefully constructed based upon child development theory. The stronger the early foundation, the easier it is for students to build as the concepts become more difficult. The means to mathematical understanding is not through substitution or memorization, but through problem solving. While all the steps may seem like “extra work” now, the value of the approach is that the majority of students are well prepared to tackle increasingly difficult topics when they are first introduced to problem solving strategies in the early elementary grades.
Q. What can families do to support their children, especially if this style of mathematics is unfamiliar to them?
A. Many of our teachers have websites with videos which can support the material that was taught in class. Additionally, families have access to the ThinkCentral website, which includes electronic copies of the student book and instructional videos. If you are not certain how to access this website, please contact your child’s teacher.
We encourage families to support their children as they work through the steps of a problem, rather than focus their attention only on the right answer. Even when your child cannot complete a problem on a homework assignment, working through as many steps as possible benefits the child and reinforces long-term mathematical understanding. Moreover, the teacher learns a great deal as to what the children know and understand about the new learning. As a result, the teacher is more able to support the students because he or she knows exactly what the children are able to do independently and where the children need additional assistance.
As much as we are working on deep understanding of mathematical concepts, it is equally important that children are continually increasing their math fact fluency. A great way to help your child in math is to make sure that he or she practices math facts so that the basic arithmetic becomes automatic.
As always, please do not hesitate to contact your child’s teacher at any time.
Sample grade 4 word problem:
A group of 140 tourists are going on a tour of New York City. The tour guide rented 15 vans. Each van can hold 9 tourists. Did the tour guide rent enough vans? Explain your thinking.
The child who says “yes” may actually have analyzed the problem accurately and done a mathematical computation correctly (by dividing 140 by 9 and getting 15 remainder 5), but he or she is not able to apply the understanding of the mathematical concepts of like groupings and “remainders” to a real world situation. He or she may even write the explanation as “the tour guide needed to rent 15 remainder 5 vans.”
When a child shows all his work, the teacher can determine that the child had begun the problem correctly, thereby reinforcing the good thinking and clearing up the misunderstanding.
In order to arrive at the correct answer, the child needs not only to analyze the problem and complete the computation correctly, but must also visualize the groupings in order to interpret their solution and deal with the remainder accurately. He or she may write the explanation as “the tour guide needed to rent 16 vans as there would be 5 people left over after the first 15 vans were filled.”