The most common question I think many teachers receive is "When will I ever use this in real life???" I get this question from students, from my own kids, my wife (when have I ever used...)
First, I always find it funny that math is generally the main target for this question. The one subject that arguably is needed for any profession. So let's look at math skills and where they are used in real life.
"When will I ever need to know that a^2 + b^2 = c^2?"
Let's look at the skills needed to solve this problem:
1) Following Steps in Order.
In order to solve this problem you have to follow steps in the correct order. Assuming you are solving for a leg, you need to square then subtract then take the square root. If you are solving for the hypotenuse you need to square then add then take the square root. If you don't follow the steps in order, you end up with an incorrect answer.
Now, reflect on the times in your profession or in your life you had to follow steps in the correct order to complete a task. Recently I was tasked with putting together a squat rack with a lat pull-down attachment. There were over 20 steps and hundreds of different pieces of hardware. Following the steps in the correct order was key to putting it together.
2) Attention to Detail
Looking at the Pythagorean problem, you need to know if you are solving for a leg or a hypotenuse. You need to label your values correctly and put them in the correct part of the equation. You have to notice if you are adding or subtracting a constant. 25 + 36 = c^2 is a different problem than 25 + b^2 = 36. Students get training on noticing small details and the importance of those details. How many times have you heard someone say, "I only forgot the negative sign, why did it get marked wrong?" That one symbol is the difference between MAKING $40 and LOSING $40.
Obviously when putting together the squat rack, this skill was vital. There were bolts of different sizes, including a 72mm bolt and a 76 mm bolt. These are almost identical in size, but different enough that putting the wrong one in the wrong place would make the equipment non-functional.
3) Building Social Skills (especially to ask for help!)
The math classroom is the prime place for students to practice social skills and language to ask for help - not only from an adult but also from peers. Students solving for the missing value of a right triangle can check in with each other, practicing good social questions such as, "What did you get for the answer - I got 15." They can work on respectful dialogue when answers do not match "Oh, I got 9." From there they can use respectful, responsive language to determine who made a mistake.
This language is a skill and needs to be taught. Students do not inherently know how to handle conflict, especially if they are the one that made an error.
You can bet I was asking for help when I needed to attach the two vertical sides of the cage.
4) Using Resources
Sometimes you don't have the answer in your brain. Sometimes you need to use a resources. In the math classroom this can be a tool card, a notebook, a digital reference... each classroom has their own system. I can't think of a single profession where this skill isn't valued. Doctors have scores of medical text, mechanics have diagrams and reference cards for various vehicles. No profession requires you to know all of the answers to all of the questions off the top of your head. In teaching, like other professions, our greatest resource is each other - getting ideas on how to help students from other teachers that have had similar situations is vital, which connects right back to having the social skills to talk to others.
So many skills in one skill!
This is just a small selection of thoughts that came to mind as I have thought about this question. I've left off other skills such as patience and perseverance. Math is more than a list of algorithms. The math classroom should be a fluid, open classroom with dialogue and discourse. Students should be engaged in discussion and asked to defend their answers. Incorrect answers and appropriate-levels of struggle should occur. Those skills translate to the real world in many important ways.
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