Pen/Pencil

“Wait, you do math with pen?” one of my students exclaimed yesterday in disbelief.

What started as a simple observation soon turned into a friendly debate across the entire class and then into a good conversation about thinking work versus presenting a solution in mathematics.

“But why??” the student inquired “I have always been told that I should use pencils for math.”

“But why?” I threw back at the student.

With some support and agreement from his classmates, the student argued that it is important to use a  pencil to have the ability to erase mistakes and to keep work organized. This became a teachable moment for a message that I always bring up with students: there is a difference between thinking work and presenting a solution.

Reason #1 for using pen to do math: solutions should be planned

The students were absolutely correct in saying that work in mathematics should be organized – but this is only when they are presenting a solution to be read by others (classmates, the teacher, friends, and family).

Mathematical solutions need to be organized because they follow a logical sequence – if this statement is true, it follows that the next statement is true. Notation and terminology are also important here to avoid any ambiguity.

Quite often, I see students merge their rough work with their solution leaving teachers to find an answer in the mess that is left behind. Other times, students expect teachers to be mind-readers and make assumptions about the students’ understanding in order to jump to their own logical conclusions.

Here was my argument to the students – these solutions should be planned out carefully. Solutions are the result of our thinking to be shared with others. I emphasize the order and logic of the presentation getting students to reflect on whether there are any gaps in jumping from one part of the solution to another. I also get them to consider any assumptions that are being made without being stated. What information does the audience need to know?

Framing a solution as having an audience has helped students gain a more holistic understanding of the problem in front of them as they navigate what information is important and how they can use it to craft a compelling argument towards an answer.

Here are students presenting their findings from a question they had and answered using statistics. Notice the relevant terminology used and the overall quality and organization of their solution.

Here are some students crafting a solution to be read by their classmates. Notice that while there is not a complete explanation of each step, they are presented chronologically (line by line) where the logical step from one to the next is clear.

Reason #2 for using pen to do math: it is faster

The thing is, once we distinguish between thinking work and solutions – we understanding that thinking (or rough work) should be messy! It should be fun, include doodles, drawings, numbers, diagrams, and mistakes! Glorious mistakes! It should be a safe space for people to just get down thoughts, ideas, play, and experiment.

There wouldn't be time during this process to stop and use an eraser.

If a graph isn’t looking right, scratch it out – or better yet, just move on to the next available blank space and keep going. Using a pen here is just faster. 

Once all of the thinking work is down on paper – the process of going through the thinking to write a solution is a valuable one.

If you follow me on Twitter, you may know that I structure my classes using a framework that gets the students thinking in groups and up on a whiteboard. While not quite the same as using a pen for rough work, take a look at what's happening here. The rough work is used as a talking point as the group thinks together. If you look carefully, notice that the student has crossed out some numbers instead of erasing them. Again, sometimes it is just faster to cross out rather than erase (even in this case of an erasable white board; imagine using a pencil!)

But there's more to this story. The reason the students have crossed out the number instead of erasing it is because they wanted to hold on to that information. Using their mistakes or steps as valuable learning points. Which leads to the next point: celebrating mistakes. 

Reason #3 for using a pen to do math: celebrate mistakes

The following day after this conversation with students, I was at an education conference and I kid you not, during one of the sessions, the presenter said “We should be getting kids to do math with pens rather than pencils.” Just perfect.

The rationale that followed is probably the most compelling of all – we should be celebrating mistakes and learning from them.

Lisa Anne Floyd shared this quote from Seymour Papert's book, Mindstorms.

This can help students re-frame their perception of mistakes. If we can learn to ‘fail fast’ in math class like people do in other contexts such as programming, students can see errors as portals of discovery that will lead to quality learning.

Ditch the pencil and eraser. Leave mistakes on the paper as evidence of thinking and use them to further your learning.