Student: “Do I need to show my work?”
I can still remember sitting in middle school algebra and hearing our much-loved teacher urge students over and over to show their work. In my mind, it was to prevent cheating and the copying of answers from the back of the book. While this recurring mantra occurs ad nauseum in math courses everywhere, it was not until my work as a classical educator teaching upper school math that I really came to appreciate the depth of this request.
What does it mean to justify your answer in math?
To justify an answer in mathematics is to show, step by step, why it is correct: the operations performed, the properties or laws that permit each one, and the reasoning that connects them. A justified solution lets another person follow the path from what was given to what was found and check every move along the way.
The pathway to ordered thinking paves the way to appreciate justice as we seek to understand God’s order. The study of mathematics leads students to grow in understanding the art of number, but it teaches students so much more. Rich mathematics requires students to show their work, teaches that truth is something to be pursued with vigor, and trains students to preserve equality. While justice is the destination of rightly ordered logic, mathematics is the art of ordered relationships between quantities.
Does showing your work have a purpose other than preventing cheating?
Math and formal logic are in many ways like two sides of the same coin. Especially in the upper school, each mathematical solution path is a logical argument leading toward resolution. This integrated perspective shows that the steps taken must present a well-constructed persuasive argument. Showing your work is evidence of right-thinking using sound construction based on established laws. This trains our students in the discipline of communicating right thinking with consideration of truth that will transcend beyond the study of mathematics.
Showing your work as a persuasive argument
Does showing your work make me believe you understand the problem?
In mathematics, the correction solution is only half the journey. You can’t arrive at a destination without having traveled somewhere. Communicating that thinking to others is a type of rhetorical presentation which can persuade me to believe you or not. The discipline of mathematics gives mentors the unique opportunity to reinforce that truth is not relative. There are wrong answers. There are poor arguments. A well-constructed logical argument leads to a sound conclusion. In algebra problems, we are often pursuing what is unknown and working to understand what is known. Likewise, in a court of law, the keeper of justice is working to discern truth with rightly ordered thinking.
Preserving equality in every step
Does showing your work demonstrate you understand the importance of preserving equality?
While the equal sign is the most written mathematical symbol, it is often the least appreciated. For nearly 500 years, adults and children have been using those to even lines to represent balance between two propositions. A just leader understands that balance and equality are to be held close. Laws are in place for its preservation. Mathematical laws help guide students in understanding how actions must preserve equality in an equation and the arrival at a solution must not have compromised that relationship. Math teaches the right use and application of the laws of number and when students show their work, they are through repetition reinforcing this pursuit of order through the upholding of the laws of number.
Recording the journey toward a sound conclusion
Our students are walking into a world that not only doesn’t know ultimate truth but to a fault denies that objective truth exists. What better way to pattern the mind to appreciate walking out a question in an orderly way to a sound conclusion that in an arena that can step away from more heated issues of moral, ethical, or purely spiritual natures? The orderly world of number is God’s gift and beautiful vantage point into his just nature. Let’s record the journey.
For ways to build this habit of giving reasons from the very first years of school, see Socratic Teaching from Kindergarten to High School.
