Although answers are often set in stone in the math classroom, methods of deriving said answers can vary tremendously. It is a common misconception that the best way to be a mathematician is to follow a uniform roadmap of steps. In fact, giving students some flexibility in their approaches can make them grittier problem solvers. Once multiples strategies have been shared, students can compare and critique using concrete examples. It is critical that we allow students the opportunity to use The Standard for Mathematical Practice #3 and to debate in the math classroom.
Our first resource involves the image at the top of this post. Students are often taught a set of steps when it comes to solving. In this set of steps, distributing comes first. Is this always the ‘best’ first step? In the problem above, the student followed the traditional steps of solving and distributed first. Wouldn’t it be more efficient to begin by multiplying both sides by the reciprocal? Multiplying both sides by 4/3 would eliminate all messy fractions right off the bat and leave the student with a simple one step solving problem. Different problems call for different approaches and efficiency does not always follow a recipe.
If you haven’t already spent some time on the Open Middle website, I highly recommend that you do! Thank you to Barbara Rappaport for reminding me just how powerful this site can be. The notion is that every problem posted has an open middle. That is, the beginning and end of the math problem are closed/fixed, but the middle is extremely open for interpretation. The website includes hundreds of open middle problems organized by domain and ranging from Kindergarten to High School.
Finally, how do we assess argument in the math classroom? I have created a rubric that specifically examines the Standard for Mathematical Practice #3: construct viable arguments and critique the reasoning of others. Feel free to download and adapt the rubric from dropbox.