The Department of Education has released a report that reflects a dip in a student’s understanding (Comprehension) of mathematics. [National Center for Education Statistics (2009). The Nation's Report Card: Mathematics 2009 (NCES 2010–451). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington, D.C.]
However, we are not here to examine why there is dip in math’s skill levels, but to understand what exactly understanding mathematics means.
A CNN article on the same topic, Report card shows U.S. students need to improve math skills reveals:
According to the Department of Education, some of the skills required to achieve a basic level of understanding of mathematics at the fourth-grade level include performing simple computations with whole numbers; showing understanding of fractions and decimals; and solving simple real-world problems.
To achieve a basic level of understanding at the eighth-grade level, students must be able work with whole numbers, decimals, fractions and percentages; be able to solve word problems and use diagrams, charts and graphs; and be able to solve simple algebra and geometry problems.
Similarly at a higher level, Understanding Mathematics a study guide by Peter Alfeld says, you understand a piece of mathematics if you can do all of the following:
Explain mathematical concepts and facts in terms of simpler concepts and facts.
Easily make logical connections between different facts and concepts.
Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.
Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)
Question to reader: Where exactly do you stand with reference to understanding mathematics?