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Calculators, Brain Atrophy, and the New SAT

calculators-brain-atrophy-and-the-new-sat-title-image

Disclaimer: I am not anti-calculator. Calculators are genuinely useful and they have an important role to play in academics as well as human progress. Furthermore, for various reasons, such as a dyscalculia diagnosis, some students have been granted an accommodation that allows them to always have access to a calculator for schoolwork, tests, and standardized exams. I am not advocating for these students to abandon the accommodation they are entitled to. I am simply making the case that any student will benefit from regularly practicing math without a calculator.

Math classes have changed during the past few decades. Boring, black-and-white textbooks have been replaced with more colorful ones. Rote memorization has been replaced with exploration and intrigue whenever possible. The newer textbooks favor real-world numbers for which the arithmetic is difficult. Calculators have become more complicated and much more prevalent.

All of this is well and good so long as nothing needs to be memorized and so long as calculators are always allowed. Since we now live in a high-tech world, complete with Google, smartphones, and fancy graphing calculators, it’s easy to see why math classes have shifted in this direction.

But in the spring of 2016, the College Board launched the newest version of the SAT. For over one third of the math questions on the new SAT, no calculators are allowed. Given the direction math classes have gone in recent years, it should come as no surprise that the #1 thing we hear from our students about the new SAT is that the no-calculator section is very hard.

To be fair, this isn’t entirely about calculators. Both of the new SAT’s math sections cover higher-level content than the old SAT did. Students are expected to know nearly everything from Algebra I, Geometry, and Algebra II in addition to the fundamentals from the lower levels of math’s inverted pyramid. In terms of math content, the bar has been raised. The new SAT, in short, is harder.

But it’s especially harder for students who don’t have their math facts memorized. If you have to struggle with arithmetic, the no-calculator section is brutal. After all, the exam isn’t testing arithmetic; it’s testing higher-level math skills and problem-solving. If you have to spend a great deal of mental energy crunching the numbers, you won’t have enough brainpower or time leftover to solve the test’s harder problems. You’ll be slower and less agile because you’re weighed down by excess cognitive load.

You might be surprised how common it is for a student to be doing well in high school math classes but to be simultaneously lacking knowledge of fraction operations, how to set up proportion problems, or how to decode a word problem. To a certain extent, the SAT has always punished students for forgetting these skills. Now it also punishes students for struggling with addition, subtraction, multiplication, and division. Many students have chosen not to memorize their math facts, and now they’re paying for it. Others once had their math facts memorized, but have since forgotten them through lack of practice, in a classic case of use it or lose it.

Because the brain is like a bunch of muscles, any skill we don’t use regularly weakens. If we never use the skill, it atrophies. If you always wear a brace on your wrist, your wrist will become weaker. If you always ride your bike with training wheels, you’ll never develop balance. If you always use a calculator, you’ll forget how to do math without one.

Students who wish to do well on the new SAT will have to go back to basics and put in some time working on the fundamentals. The prospect of devoting time to working on old math is particularly hard to stomach during the schoolyear because there is so much new content to learn, which makes summer the ideal time to work on this goal.

It is possible, however, to make significant progress on the basics while you’re learning higher-level content. You simply have to find opportunities to practice the basics while you’re doing your homework. Essentially, this means choosing to do problems the hard way. Not necessarily every time, but whenever you could do a problem without a calculator, that’s an opportunity to practice the basics. It will make your math homework take slightly longer, but you’ll be killing two birds with one stone. Do it the hard way.

The SAT isn’t likely to change again for many years, and it will take a long time for the school system to shift in such a way that helps students tackle the new exam. So that leaves you. You’ll have to change. You’ll have to choose to help yourself.

 

Image Credit

Title Image: Creative Commons Public Domain. Courtesy of Pixabay. https://pixabay.com/en/calculator-math-mathematics-988017/. Text added.

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Math Facts

math facts title

7+5=12

15-9=6

8×6=48

24÷8=3

72=49

¼=0.25=25%

These are math facts. They are simple, mathematical truths that all students are made to learn. At first glance, math facts are just like everything else students learn in school, but there is a difference between learning your math facts and, say, learning to write an essay. Math facts are meant to be memorized.

Now, you don’t have to memorize them. You can, through very slow methods, calculate the answers to each of the problems I just listed. But as you move up in the world of math, you’ll be increasingly expected to have them memorized, and for good reason. If you don’t memorize your math facts, you’ll spend your entire math career weighed down by excess cognitive load.

Solving higher-level math problems takes brainpower. Because the brain is like a bunch of muscles, your brainpower at any given moment is limited. Any mental energy you spend calculating a non-memorized math fact is energy that is not available for the more difficult math you’re trying to do. If, instead, you have your math facts deeply memorized, then you’ll expend no brainpower whatsoever to access them, leaving plenty leftover to think about the problem at hand.

Automaticity is the goal. When your math facts are deeply memorized, you see questions like 7×8 and react instinctively, instantly, and without thought. When it comes to knowing your math facts, understanding is not enough; you have to master them. Walk the mastery path. It’s worth it.

Remember, math is an upside-down pyramid. The simple topics at the bottom form a narrow foundation that supports a top-heavy structure containing all of the upper-level math content:

math pyramid

Because math facts are at the bottom of this inverted pyramid, you can’t afford to have weakness there. Start at the bottom and work your way up.

It’s never too late to memorize your math facts. Even if you’re taking AP Calculus or preparing for the SAT, you need your math facts. Both of those exams contain no-calculator sections, so you’ll be penalized if you don’t know them.

So how do you memorize your math facts?

The best method is to simply work with them on a regular basis. Devote two minutes each day to memorizing your math facts. Even though it would be more total time, doing 20 minutes once a week would be less effective because it would fail to utilize spaced repetition. You can work with your math facts on paper, with flashcards, on Khan Academy, or anywhere, anytime, using mental recall.

One neat trick is to stick a set of flashcards in between the salt and pepper shakers on your dinner table. At the beginning of one of your meals each day, do a two-minute drill with the flashcards. This method helps bring structure and routine into the effort, which is critical because you want the drills to become an automatic habit.

Self-development expert James Clear calls this method of attaching a new habit to a preexisting one “habit stacking,” and argues that it increases the likelihood of having the behavior stick.1 Other examples of habit stacking include flossing immediately after brushing your teeth, meditating right after you have your morning coffee, and writing in a gratitude journal as soon as you go to bed.

I would also like to point out that two minutes is just 0.2% of your waking hours. It’s hard to say you don’t have time for something that takes only two minutes. And I’m certain that you’ll wind up saving a great deal more time on your math homework. Memorizing your math facts is an investment that can pay enormous dividends.

Once you reach middle school, there are so many other concerns that memorizing your math facts will probably never seem urgent. Therefore, you’ll easily find justifications for putting it off or never doing it at all. Just please recognize that you could choose to memorize your math facts, and if you did, you’d be doing yourself a huge favor.

 

Works Cited

1 Clear, James. “Habit Stacking: How to Build New Habits by Taking Advantage of Old Ones.” http://jamesclear.com/habit-stacking.

Image Credits

Title Image: Creative Commons Public Domain. Courtesy of Pixabay. https://pixabay.com/en/maths-mathematics-maths-symbols-1426891/. Text added.

Math Pyramid: Loper, Chris. “Math is an Upside-Down Pyramid.” 2016.

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Math is an Upside-Down Pyramid

math pyramid title

Whoever designed the world of math wasn’t an architect. The nature of math curriculum is that it is fundamentally unstable because it is top-heavy. Nearly every math concept we learn is part of the foundation of many more concepts. Every door we open in math reveals three more. Math is an upside-down pyramid.

math pyramid

As students move up the inverted pyramid, frustration becomes increasingly common. Normally, this is attributed to the difficulty of the advanced content, and there is some truth to that. But it is much more common for the frustration to arise from holes in the base of the pyramid. The upper levels of the pyramid are very hard to build if you’ve forgotten the ideas upon which they are built. High level concepts are nearly impossible to learn without the proper foundation.

Frustration of this sort is especially common in the spring. The end of the schoolyear is when knowledge gaps get revealed most dramatically. For example, in the latter third of the year in high school geometry, most of the new concepts are combinations of several old ideas and a couple of new ones. If you don’t have the old ideas down, you’re going to be lost. Comprehensive finals can be particularly punishing for students who have holes in the lower levels of their math pyramid.

Math isn’t the only subject that is built this way. Science classes, especially chemistry, are upside-down pyramids, too. Learning a foreign language works this way as well. If you only did the bare minimum to keep up in Spanish I, you can expect Spanish II to be brutally hard. Do yourself a favor and get caught up.

You don’t want to wind up with a knowledge pyramid that looks like Swiss cheese:

swiss cheese

The solution to this issue is to walk the mastery path. As soon as new content arises, pursue understanding aggressively. Once understanding is achieved, keep practicing. Regularly revisit old concepts from earlier in the year and from previous years. If you don’t, you’ll forget.

The inverted pyramid of mathematics is constantly being eroded, so it needs constant upkeep. Patch holes in the base as soon as you notice them.

Returning to the fundamentals of math is never a waste of time. Content at the bottom of the pyramid is too important to ignore. We can prop up a weak foundation with a calculator, which is often an appropriate accommodation. This comes back to bite you, however, when you run into no-calculator tests. The new SAT, by the way, includes a no-calculator section.

When you know the fundamentals very well, moving up the pyramid takes much less effort. A good target to aim for is automaticity, which means knowing something so deeply that you can do it without thought, as a reflex. For example, you know the answer to 2+2 instinctively; that sort of arithmetic takes no effort at all. For every foundational math problem you can solve in this automatic way, you’ll have extra brain power leftover to figure out new math concepts.

In other words, learning and practicing the fundamentals of mathematics dramatically reduces cognitive load, the amount of mental effort required for the task at hand. Climbing up the pyramid is hard, so taking the time to lighten your load is smart. You’ll work harder now, but you’ll save energy later. Solving harder math problems takes a profound effort if you don’t have the basics down. For many students, that is the primary challenge they face when moving forward in math.

It’s never too late to get caught up. Summer is an excellent time to patch holes in your foundation. You can’t afford to have a weak base. Too much rests upon it.

 

Image Credits

Title Image: Loper, Chris. 2016.

Swiss Cheese: Loper, Chris. 2016.

Math Pyramid: Loper, Chris. “Math is an Upside-Down Pyramid.” 2016.