s an AoPS Academy campus director, a big part of my job is meeting with parents of prospective students. One of the most common complaints I hear is that their children never show any work. Parents are surprised when I push back, gently, on the underlying assumptions. In fact, showing work is sometimes a bad idea, and forcing students to show work unnecessarily can be harmful.
Doing all math work mentally is, of course, not a good habit. Adults understand this because they have been through the entire course of schooling, and they know what lies ahead for their children. They know that Algebra, Geometry, and other higher level classes will require students to lay out their thoughts on paper in order to have any hope of solving the problems.
But many students who come to us can get everything right in their school math classes without showing any work at all. Why should they? The directive to write down lots of intermediate steps often feels like something that is done for the teacher’s benefit, not for the student’s.
In fact, our intake assessments include guidance on how to do certain problems more quickly, and often without any writing! Mental math can be very positive; no one should be writing out the steps for “13 x 1,000,” after all. I often have to stop students, well trained by the prevailing regime, from writing down such easy steps. They are often delighted to have a teacher challenge them to do a little mental math.
Even something that looks a little tricky can be done mentally with the right conceptual understanding. One of my favorite problems asks students to compute 6×27+23×6. Now, using the usual algorithms taught in school, this is a fairly boring problem. However, we actually use this problem to introduce the Distributive Property, which helps students recognize that the problem is equivalent to 6 times 50, which is easy to compute mentally as 300.
Even for more complex problems, the pedagogical goal is usually to help students cut down on the number of necessary steps. A classic problem is to add all the whole numbers from 1 to 100. Of course, this can be done by brute force, but not easily. A much better approach is to pair up all the numbers from the outside in: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, etc. We can then turn what would be a very tedious addition problem into a quick multiplication problem: 101 x 50 = 5,050. Some students can do this problem entirely mentally!
When Do You Show Work?
My point in the above examples is not to deride the idea of showing work, but rather to put it in the proper perspective. In order to help students see the value in putting work on the page, they need problems they simply cannot do in any other way. Even then, they will often need specific guidance and structure as they take their first steps towards becoming well-organized problem-solvers. Helping students productively “show their work” is one of my major goals as an AoPS Academy instructor. So, I want to share some of what I have learned about how an adult can be most helpful to a young person who does not yet know how to write out their work.
Early in their first AoPS course, many students will start making mistakes, much more frequently than they’re accustomed to. This is not surprising, since they often have never been asked to do any problems that are truly challenging or that need to be broken down. In those instances, I do encourage the students to write out their work, because they are simply trying to keep too much in their heads at once. This is a much easier conversation to have than when they are getting everything right!
Getting a problem wrong can help spur students to see the need to approach a problem differently. Still, simply encouraging them to write more may not be enough. I have found I can help my students more by suggesting what they write, why that specific information needs to be written, and even where on the page to put it.
What: Usually some intermediate computation whose value will be used to find the actual answer. The thing they need to write could also be a general relationship between two quantities, which they can then use to solve the specific problem at hand.
Why: It’s often worthwhile to explain to students that they need to put this information onto the page so they don’t forget it before getting to the final answer. Depending on the student, it may be worth going into more depth, and pointing out that this is how almost all real mathematics problems are solved: by keeping track of smaller “sub-problems” until we arrive at an answer.
Where: This part is often not intuitive for adults, since it can seem very basic, but many students need it. If I don’t specify where on the sheet to write that middle step, it will often end up half a page away from the problem!
Once they are on the path towards the answer, I then ask them to finish on their own, and move on to a different student. After all, greater independence in problem-solving is one of the main goals of our program. Especially at first, though, many students need as much specificity as possible in how, exactly, they should write out their work.
There are several other situations that can make it easier for students themselves to see the benefits of clearly organized mathematical notation. Educators can structure their classes around opportunities for these situations to arise.
- When there is a need to communicate to others about their process. For example, when they are working in pairs on a problem, and their partner has a different answer. The only way to resolve such a disagreement is to go to the evidence, i.e., the mathematical work that led to the differing answers.
- When there is a geometrical component to a problem, and the diagram is not provided. In some problems, creating an accurate diagram or graph is the task! Connecting pictures with symbols is an important step in the learning process. This relationship is one of the many reasons it’s critical to fold geometry into the curriculum every year, rather than walling it off and then trying to do it all in a single year.
- Even if the topic is not related to something specifically geometric, making diagrams can be a great problem-solving strategy. Many problems in counting and probability, for instance, can be made clearer through a visual representation. These situations can be chances for the students to invent their own notation or diagram system, either on their own or with a partner.
Of course we want all of our students to develop the ability to write clear, organized solutions to serious math problems. For some students this skill comes naturally, but for others it can be an uphill climb. Everyone can learn how to show their work, with persistence, just like any other skill. However, it is important that we put students in situations where they are not simply directed to write things down to please an adult. They need meaningful problems that are complex enough that writing feels natural. They need to be with peers who are reliable mathematical partners. They need teachers who can help guide them through the early stages of this process, but then know to get out of the way. When those elements are in place, I have found, students will grow not only in their own mathematical understanding, but also as mathematical communicators.