Guest Post by Dan Haiem, @danclasscalcco1
“But teacher, why do I need to learn this math stuff if I can just use my calculator app?”
Every teacher has heard this at some point. Most students have asked this at some point. Up until around 2007, teachers had a great answer that kinda-sorta worked, which went something like “You’re not going to always have a calculator everywhere you go, are you?”
The opposite of that, of course, became a reality when the iPhone became the most common tool found in any student’s pocket, and gave them access to powerful calculators like Desmos, Geogebra and our very own ClassCalc.
Here’s my opinion though – the old answer was never a good answer, and the ubiquity of iPhones has given us a golden opportunity to re-evaluate a very valid question. What’s the point of learning math if we have calculators that do math for us?
In the words of NYC’s (possibly) most high-energy math teacher, José Vilson, “Math shouldn’t be limited to a disconnected set of rules and jargon that doesn’t seem to mean much of anything.” If math really was about the rules and jargon, then a calculator could truly replace the need for learning it. Fortunately (for humans), it’s not.
For the sake of simplicity, and because this is my first blog post ever, here’s a short roadmap of this post’s approach to this topic:
- First, we’ll discuss how math helps students build tools and skills, and define the difference between the two:
- Tools: Spreadsheets, running an analysis, doing taxes.
- Skills: Good communication, emotional intelligence, problem-solving.
- Next, we’ll discuss how math has shifted from serving us as a tool, to helping us sharpen our skills – most important of which is problem-solving.
- Finally, we’ll take a look at an example of how a specific math problem helps us build a specific problem-solving skill called mental-triage, and how we might help students make that connection as well.
Math serves two primary purposes in education: it gives students the tools to play with numbers, and it serves as practice to sharpen certain mental skills that are important in life. To define the two:
- Tools: Concrete things a person “knows.” Examples include: spreadsheets, coding, writing blog posts (a tool I clearly lack), social media advertisements, taxes, etc.
- Skills: More abstract, broad abilities that are not particularly associated with executing a specific task – the kinds you always see in leadership charts. Examples include: Hard working, communicative, optimistic, honesty with self, problem-solving, etc.
*Credits to Business Simulations
In this sense, math falls into an interesting crossroads as both 1) part of the abstract skillset a person has, (ie: “problem-solving”) and 2) a tool that can be put to work (ie: part of your “toolkit” – like running a statistical analysis on two data sets).
It’s important to keep in mind though that:
Math will play a fundamentally different role for different students, and we need to bring that understanding into the classroom.
An engineer will likely benefit from math as both 1) an exercise in problem-solving (skill) and 2) a tool to accomplish certain tasks.
An artist might use the abstract side of problem-solving (skill) but – and this is especially applicable today, with all the calculators in our pockets – they probably will not have much use for math as a tool
Here’s a screenshot I grabbed off the might internet that summarizes the point (albeit, aggressively):
I believe it’s important and ok to tell our students that not all of them will be using math as a tool. At this point, most people don’t need math to do taxes (here’s a calculator for that), split a tab (here’s a calculator for that) or accomplish any of the other tasks that might have required math as a “tool” before calculators were built.
- Math confers two types of skills: Abstract problem solving (skill) and an actual tool in your toolkit (tool)
- Everyone can (probably) benefit from the problem-solving (skill) aspect. While some people (engineers, some scientists) benefit from having math in their toolkit (tool), most can get by with their super powerful pocket calculator.
- So, for students not interested in pursuing a career that would require a math toolkit, we must focus on the abstract problem-solving (skill) aspect of math.
As brilliantly stated by Baltimore Ravens lineman and MIT mathematician John Urschel, we need “students to see that math extends far past the confines of the classroom and into everyday life.” What’s more “everyday life” in the 21st century than problem-solving?
Now the questions shifts to: How do we show students the relationship between learning math and developing this abstract ability to “problem-solve”?
I have two thoughts on this:
- One thing to consider is that skills and tools are actually mutually conducive. Google is a tool you learn to use. Being able to learn stuff on your own is a very important skill in today’s workforce. Knowing how to use the Google tool will help you build the learn-stuff-on-you-own skill. I think the same applies to math.
Although students may never use calculus directly, the mental exercises they go through in solving calculus problems might help improve the mental muscles required for peripherally related skills.
- We need to find good examples to demonstrate the above. I mean good examples. Not the “Well, don’t you want to know how to do this without your calculator?” type of answer and not the “Being able to do your taxes is very important” type of answer and not the “here’s an example of Timmy calculating the volume of the Earth by standing on a ladder and looking at the horizon (although that’s super cool)” type of example. These examples all focus on the tool aspect of math, which we know won’t be as relevant to all students. We need to focus on the skill aspect of mathematics.
So now the challenge becomes being able to demonstrate to students a link between learning math and learning how to problem-solve. A good approach might be to 1) Have your students break down what sub-skills are required to succeed in math 2) Have your students break down what sub-skills are required to problem-solve 3) Discuss the cross-overs.
One of my favorite examples is mental triage: the abstract skill of quickly finding the most efficient path through a challenge given a limited toolkit.
Here’s an example of a math problem that helps sharpen the sub-skill of mental triage:
- Math itself is a limited toolkit. You learn how to move numbers around. How to draw graphs etc. Each time you learn one of these new tools, you’re essentially learning a new way to play with numbers. When we approach a math problem, we subconsciously run an analysis that goes something like this: What do I want to make these numbers do? What tools do I have to move these numbers around? What tools am I not allowed to use? What is the most efficient tool path to an answer?
- As an example, let’s take the following problem, a favorite of the SAT:
- Here’s how my brain runs through my math toolkit.
- I gotta solve for x.
- Problem: x is in the exponent.
- Do I have any tools to get rid of an exponent?
- I can raise both sides to ^(1/4x) which would lead to:
- No good. Back to step c.
- How else can I get rid of the exponent? Logarithms, let’s try that:
- which simplifies to:
- Great, we got rid of x in the exponent. Onwards! Divide both sides by 4log(2):
- Plug into by handy dandy ClassCalc Calculator to get:
There we have it: mental triage in math.
Finally, we’ll bring it full circle with a real-world example of mental-triage as a sub-skill of problem solving.
Teaching my high school students how to pick a college
It was the last day of my physics class last year, and my students were just about done with school. They had already taken the AP test and were ready for summer. Instead of squeezing in another physics lesson, I decided to tackle a more pressing concern of theirs – choosing a university.
In retrospect, my method for picking a university was suboptimal – I just asked my good friends where they were going and what they thought a good college was, and ended up at UCLA. Lucky for me, I met good people there and had an awesome experience, but many others who take the same approach are not. I wanted to teach my students how to be proactive and problem-solving-oriented in making life choices.
Rather than start with “I want to go to college” I wanted to help each one of them hear their inner voice, and begin a dialogue with it. Start for the bottom. Here was my approach:
Student: *Says something*
And this is what the conversation ended up sounding like:
- Student: What college should I go to?
- Me: Why do you want to go to college?
- Student: I need to get an edu-
- Me: Yes, but why do you need an education? What’s your goal?
- Student: I want to make money. Goal number 1: Make money!
- Me: Honest, but fair. What else? A lot of jobs will make you money.
- Student: I want to become a doctor.
- Me: Do you for sure 100% want to become a doctor? Have you had real exposure to medicine? Or is it alluring to you for other reasons?
- Student: I’m not sure. I want to figure out what I want to become. Goal number 2: Explore career options!
- Me: Ok, what else?
- Student: I want to make good friends and party. Goal number 3: Have fun
- And so on..
By the end, we put together a list of priorities for each student. I could see their perspective change drastically. Rather than listen to a parent’s friend’s suggestion, they were determined to go online and research.
Now, I am not necessarily saying that a better mathematician is going to be better at selecting a college, but certainly, the tools we learn in math can inform our decision-making process for the important choices we must all make in life, especially if we are aware that there is a problem-solving oriented approach to making these decisions. Our jobs, as teachers, is to help students form that awareness.
A good method for cultivating that awareness is with Miyagi Moment every so often. What’s a Miyagi Moment moment, you ask?
It’s a metaphor for when a teacher (or sensei) helps a student develop a crucial skill by practicing adjacently related skills that at first do not seem connected.
In the first gif below, we see the legendary Mr. Miyagi teaching Daniel San how to…wax a car. Not really relevant to fighting karate.
In a later scene, Daniel san gets angry, accusing Mr. Miyagi of wasting his time with chores, when he should be learning super cool action moves to take down the big bully Johnny. Right then and there, Mr. Miyagi throws a couple of HYAH punches and BAM. Daniel san blocks them – all the while shocked in disbelief that he had developed the skills to do so. That moment of disbelief in the newly developed skill is the Miyagi Moment!
In math, students will often be practicing skills that seem almost irrelevant to them in life. It is up to us teachers to remind our students every so often that that is not the case. The best way to show them that is with a Miyagi Moment. It is time away from teaching the next chapter or lesson, but it is time well-spent.
I think blogger and math teacher John Trout McCrann put it beautifully in writing “Deep understanding about the process of solving an equation helps everyone understand how to create systems to solve problems at work, in their families, in our world. The kinds of problem-solving strategies you might use to tackle a big project, develop a more efficient engine, or address an issue that’s arisen between you and your partner. Deep understandings about shapes help everyone understand how to reason spatially, a skill that you may one day apply as a designer or as you lay out the furniture in your first house or apartment.”
About Me (Daniel Haiem):
- I love math and education.
- I’m an ex-physics teacher
- I founded and lead a company called ClassCalc – the lockdown calculator app that lets teachers lock students out of all outside distractions such as instagram, calls and texts, keeping students focused in class, and preventing cheating on tests. Our goal is 100% access to calculators for students across the planet by 2025.
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