Understanding Understanding

Everyone has their own way of understanding a math concept, whether it is concrete or abstract.  Students may be able to recognize a formula or explain solving a problem in their own words.  They may also only have partial understanding (Davis, 2006).  Understanding comes with time and it varies based on the type of mathematics being taught.  To enhance students' understanding of a concept, it is important to ask them to include symbols or models to show and interpret their work.  If they are able to explain their symbol, it is likely that they understand the concept (Davis, 2006).  Students who understand are also able to recognize how to apply a procedure in a new context.  In grade school, I struggled with knowing when to apply formulas to problems.  I could explain the process used in simple problems, but the problem came when there were multi-step problems which included new formulas.  I had to learn that some formulas only work when the numbers I was working with are in a specific order or form.  Last semester when I was in ETE 237, I worked with a student who had the same problem.  She knew how to apply the formula but if there were steps to take before applying the formula, she would misunderstand what the question was asking.  With the CT's help, I was able to guide the student in understanding when the formula works and characteristics a problem will have when a particular formula is ready to be used.

Students need to know how and why procedures work (Davis, 2006).  They also need to understand what the facts and generalizations of a formula are and why they are true.  Students need to be able to explain their thinking rather than just assuming the rules work because a long time ago someone decided that's how to do it.  They should work on making proofs of why formulas work.  By knowing the type of problem they are working with, students will understand when to apply a particular formula and when they need to make changes to a problem before they can use a formula (for example, needing to find a common denominator in fractions before they can add or subtract.) 

Two questions I have about understanding understanding mathematics are: 1. How do we know if students only have partial understanding when they are giving the appearance of knowing? 2. As future teachers, how can we challenge students more once they understand the basics of a concept?

Davis, E. J. (2006). A Model for Understanding Understanding in Mathematics. 100 Years of Mathematics Teacher,12(4), 190-197. Retrieved January 26, 2018.

From Wiggins, G. & McTighe (2005). Understanding by design. ASCD. Alexandria: Virginia