It seems like every few weeks a math problem comes up that starts an argument and goes viral because no one can agree on the answer. And that almost always involves PEMDAS.
Why is PEMDAS so confusing? It has to do with math as a kind of language. Wikipedia explains PEMDAS:
“In mathematics and computer programming, the order of operations (or operator priority) is a set of rules that reflect the conventions on the procedures to be performed first to evaluate a given mathematical expression.
For example, in mathematics and most computer languages, multiplication has a higher priority than addition, and so has been since the introduction of modern algebraic notation. Thus, the expression 2 + 3 × 4 is interpreted as having the value 2 + (3 × 4) = 14, and not (2 + 3) × 4 = 20. With the introduction of exponents in the 16th and 17th centuries, they took precedence over addition and multiplication and could only be placed by exposing for the right of their base. So 3 + 52 = 28 and 3 × 52 = 75.
These conventions exist to remove ambiguity from the notation while allowing the notation to be as brief as possible. When you want to override precedence conventions, or even just accentuate them, parentheses () can be used to indicate a different order of operations (or just to reinforce the default order of operations). For example, (2 + 3) × 4 = 20 forces addition before multiplication, while (3 + 5)2 = 64 forces the addition to precede the exponentiation. If multiple pairs of parentheses are required in a mathematical expression (as in the case of nested parentheses), the parentheses can be replaced with brackets or braces to avoid confusion, as in [2 × (3 + 4)] - 5 = 9. "
This math problem has gone viral because PEMDAS confuses people
So what is the problem? On the one hand, Twitter and the nature of misinformation and disagreement and the type of content that is most likely to be shared. Here is the last example that does the trick.
Social media was created for sharing and disagreeing, so here we are. As we mentioned, these often go viral and in 2019, a PhD in Mathematics from the University of Kentucky. Jared Antrobus explained why everyone is wrong, 'The failure of PEMDAS".
In the post, he explains that the problem cannot really be solved as stated because it is ambiguous.
“The problem with this problem is that the division is not associative. If we were asked to rate 12 ÷ 6 ÷ 3, we would have a very similar problem. Which sign of division do we make first? Regardless of what our teacher told us, we cannot assume that the author of the problem wanted us to work from left to right. In fact, different calculators will give different answers, depending on how they are programmed!
How do we manage non-associative operations?
The only reason we can forgo writing parentheses with long addition and multiplication strings is because of their associativity. For the division, we must specify the operation to be performed first. We should either write (12 ÷ 6) ÷ 3 or 12 ÷ (6 ÷ 3) so that the reader knows exactly what we mean.
The expression 12 ÷ 6 ÷ 3 is ambiguous. It could mean two different things. The only way to know what the author intended is to ask for clarification. The same goes for 6 ÷ 2 × 3. "
Ambiguity makes it impossible to answer questions!
Ambiguous statements are not exclusive to mathematics. The following question is also overwhelming:
If a woman plays her sister at chess and she is a grandmaster, is she likely to win?
Again, the author's intention is unclear. Which woman does each instance of "her" refer to? The only way to know is to ask for clarification. "
We just figured math teachers might find a nice break from the almost non-stop cycle of bad news that has been 2020. It probably won't be trending by the time you read this, but it is still. See for yourself.
While we all may follow our own unique pursuits in a lifetime, the quest for purpose through self-improvement and knowledge is among the great unifiers of humankind. Progressive thinking is the backbone of society’s progress. Great visionaries and their quests for knowledge have inspired the majority of the modern marvels we rely on today. While the process of seeking purpose in your life may not seem as universal or consequential as Einstein’s, it is fondamental to find how to learn in a way that works for you, which can lead to greater self-awareness and wisdom—not to mention a new emploi, better pay, a new hobby, or simply knowledge for knowledge’s sake—whatever is important to you as an end goal.
While we all mayThis brand of introspection and knowledge is not necessarily acquired through traditional means, and the learning techniques that work best may differ from one person to the next. In fact, some of these tips may surprise you. Imagine all of the ways you may have been perfecting how to learn to find your smartest self for years without even knowing it !
Reduce stress depression : Stress and depression can affect the ability to recall information and cause short-term memory loss. In mild cases, depression can sometimes be improved simply by exposing yourself to more white light and eating fewer refined foods.
Shake a leg : Lack of blood flow is a common reason for lack of concentration. If you’ve been sitting in one place for awhile, bounce or bend and flex one of your legs for a minute or two. It gets your blood flowing and sharpens both concentration and recall abilities.
Food for thought : Eat breakfast. A lot of people skip breakfast, but creativity is often optimal in the early morning and it helps to have some protein in you to ' feed ' your brain. Plus, a lack of protein can actually cause headaches.
Food for thought, part 2 : Eat a light lunch. Heavy lunches have a tendency to make people drowsy. While you could turn this to your advantage by taking a ' thinking nap ' ( we’ll get to that later ), most people haven’t learned how to actually make this work on a regular basis.
Ginkgo biloba : Ginkgo biloba is a natural supplement that has been used in China and other countries for centuries and has been acclaimed for its brain-energizing properties.
Sleep on it : Hitting an REM cycle not only helps you rest and reset, it may also help with high-level problem solving. Researchers at University of California, San Diego noticed that getting some rest and dreaming allowed creative thinkers to work through some of their toughest problems.
Take a break : Sometimes, in order to change your physical or esprit perspective and lighten the invisible stress that can sometimes occur when you sit in one place too long, it helps to take a 5-15 minute break every hour during study sessions. Studies show this is more beneficial than non-stop study, as it gives your mind time to relax and absorb information.
Take a hike : Changing your perspective ( and surroundings ) often relieves tension, thus freeing your creative mind. Taking a bermuda walk around the neighborhood may help you liberate those latent learning skills.
Change your focus : Sometimes you just don’t have enough time to take a long break, however you can always just change subject focus. Try alternating between technical and non-technical subjects, for example.
Do walking meditation : If you’re taking a hike, don’t stop there ! Go one step further and learn walking meditation as a way to tap into your inner resources and strengthen your ability to focus. Just make sure to not get so carried away that you disregard safety and traffic rules.
Change your focus, part 2 : There are three primary ways to learn : visual, auditory, and kinesthetic ( VAK ). If one isn’t working for you, simply try another. Full immerse yourself : Focus only on whatever you’re studying, not watching TV at the same time or worrying yourself about other things. Anxiety is known to inhibit the absorption of information and ideas.
Turn out the lights : If meditation isn’t for you, this can be another way to focus your mind. Sit in the dark, to literally and figuratively block out extraneous influences. This is especially helpful for learning something kinesthetically, such as guitar chord changes. Take a bath or shower : We know this one may be a bit surprising, but both activities can loosen you up, making your mind more receptive to recognizing brilliant ideas.
Listen to music : Research has long shown that certain variétés of music can act as a ' key ' to open doors and recall memories. The theory is that Information learned while listening to a particular song or gamme can often be recalled simply by replaying the songs in your head. Speedread : Some people believe that speedreading causes you to miss essentiel information, however the idea is that, when done right, speedreading results in filtering out irrelevant information. If necessary, you can always read and re-read technical subjects that often require slower reading, though some studies show slow reading actually hinders the ability to absorb general ideas. Trying this reading technique online ? Try the free Spreeder application.
Use acronyms and other mnemonic devices : Mnemonics are essentially tricks for remembering information. Some tricks are so effective that proper outil will let you recall loads of mundane information months or even years later.
Every picture tells a story : Draw or sketch whatever it is you are trying to achieve to help you visualize it. Having a concrete goal in mind can help you progress towards reaching your goal. Brainmap it : Need to plan something ? Brain maps, or mind maps, offer a compact way to get both an overview of a project as well as better manage it. Through mind mapping, you can see the relationships between disparate ideas and better utilize brainstorming techniques
Learn symbolism and semiotics : Semiotics is the study of signs and symbols. Having an understanding of the iconography of a particular discipline not only aids in the learning process, but also allows you to retain information more efficiently. Use information style : When dealing with information that has an inherent structure, applying the tenets of information style can help to convey that information more clearly. A great resource is Information Aesthetics, which gives examples of different genres of information design and provides links to their sources.
Use visual learning techniques : Try gliffy to explore all kinds of structured diagrams, flow charts, and more and to see what might pique your visual interest. If this works for you, find even more webbing and outlining ideas, plus graphic organizers, concept maps, and plots at Inspiration. com. Map your task flow : Learning often requires gaining knowledge in a specific sequence. Task flow mapping your course of actions, or organizing your thoughts on what needs to be done, is a powerful way to prepare yourself to complete tasks or learn ' how to learn. '
Laugh : This might seem counterproductive to the seriousness of studying, but that’s precisely the point : laughing relaxes the body, and a relaxed body is more receptive to new ideas. Stimulate ideas : It’s important to not overthink this one : play rhyming games, utter nonsense words, use word-association or stream-of-consciousness methods. These techniques can help loosen you up, making you more receptive to learning.
Brainstorm : This is a time-honored technique that combines verbal activity, writing, and collaboration. While one person can brainstorm, it’s more effective in a group. For effective brainstorming, follow these simple rules : firstly, don’t shut anyone’s idea out. Secondly, don’t ' edit ' in progress; just record all ideas first, then dissect them later. Participating in brainstorming can help to assess a topic objectively and thoroughly.
Learn by osmosis : Turn your iPod into an educational tool : find some podcasts that speak to you or are relevant to what you’re learning now, upload them, and sleep on it. Literally. Put your iPod under your pillow and play back your préférés to let them seep in overnight. Binaural beats : Binaural beats involve playing two pure frequencies simultaneously to produce alpha, β, delta, and theta waves, all of which can inspire either sleeping, restfulness, relaxation, meditativeness, alertness, or concentration. Binaural beats are often used in conjunction with other exercises to enhance ' super-learning ' abilities. Check out this free online binaural beat machine to see which tones will work for you.
Write, don’t type : While typing your notes into the computer is great for posterity, writing by hand stimulates ideas. The simple act of holding and using a pen or pencil may seem old-fashioned in this day and age, but just think of all the visionaries it’s worked for throughout the years. Carry a notebook at all times : Samuel Taylor Coleridge dreamed the words of the poem ' In Xanadu ( did Kubla Khan ) â€¦ '. Upon awakening, he wrote down what he could recall, but was distracted by a visitor and promptly forgot the rest of the poem. Forever. Should ideas suddenly come to you through ' walking meditation ' or any other methods on our list, record them immediately or you might regret it.
Keep a journal : This isn’t exactly the same as a notebook. Journaling has to do with tracking experiences over time. By adding in visual details, charts, brainmaps, etc., journaling can be a much more creative way to keep tabs on what you are learning. Organize : Use sticky colored tabs of folder flags to divide a notebook or journal into sections. They are a great way to partition ideas for easy reference. Use post-it notes : Post-it notes can provide a helpful way to record your thoughts about passages in books without defacing them with permanent ink or pencil marks.
Prepare yourself for learning : Positive thinking alone can’t always help us to successfully achieve our goals, which is why it is especially important if you are an adult with many distractions surrounding your daily life to implement ways of reducing these distractions, at least for a few hours at a time. Give yourself credit : Ideas are actually a dime a dozen. If you learn to focus your mind on getting the results you want, you’ll recognize the good ideas, and your mind will become a filter for them, which will motivate you to learn more. Motivate yourself : Why do you seek knowledge ? What do hope to achieve through learning ? Exploring the reasons behind why you want to learn and what motivates you can help keep distractions at-bay. Set a goal : W. Clement Stone once said ' Whatever the mind of guy can conceive, it can achieve. ' This phenomenon in goal achievement dictates that if you prepare yourself by whatever means necessary, any and all hurdles will seem surmountable. Those who have experienced this phenomenon likely understand its validity.
Think positive : After all, what’s the point in setting learning goals for yourself if you don’t have any faith in your own ability to learn ? Every skill is learned : Bodily functions notwithstanding, every skill in life is learned. Generally speaking, you can learn something new just as easily as anyone can. It takes us all a varying amount of effort, but once you’ve set your goal, it’s likely as achievable as it is believable. Prepare yourself, part 2 : Unfortunately, not everyone in your life will be a well-wisher of your self-improvement and learning partouze. They may intentionally or subconsciously distract you from your goal. If you have classes to attend after work, make sure that work colleagues know that you are unable to work late, for example. Diplomacy works best, if you think your boss is intentionally giving you work on the days he/she knows you have to leave. Reschedule such meetings to a later time if possible and/or necessary.
Constrain yourself : Most people fundamentally need structure in their lives. Freedom is sometimes a scary thing—like chaos. But even chaos has order within. By constraining yourself by giving yourself deadlines, limiting your time on any one idea, or focusing the tools you are working with, you can often accomplish even more in less time.
Read as much as you can : This tip is perhaps the most self-explanatory on our list. Use Spreeder if you have to. Pursue a broad range of topics as well as depth of field. Cross-pollinate your interests : Explore interdisciplinary study to your heart’s content. After all, neurons that connect to existing neurons give you new possibilités and abilities to use knowledge in new ways. Learn another language : New perspectives can also give you the ability to cross-pollinate cultural concepts and expand worldly inspiration. Sometimes reading a book in its original language will provide you with insights that might otherwise be lost in translation. Learn how to learn : Management Help has a resource page especially geared towards online learning, but they’re also a valuable resource for any type of learning. If you’re serious about optimizing your learning habits, check out this crash course in learning theory.
Learn what you know and what you don’t : Many people might say, ' I’m dumb, ' or ' I don’t know anything about that. ' The fact is, many people are largely unaware of what they already know about a topic. If you want to learn about something, you need to determine what you already know, figure out what you don’t, and then fill in the gaps. Learn to effectively multi-task : Effective multi-tasking allows you to devote focused yet limited time to accomplish several tasks at once. By effective multitasking, I don’t mean doing two or more things at exactly the same time—It’s not possible. However, multitasking with the right approach and prepping your mind for it are what can make it an effective technique. For example, a successful freelance writer learns to manage several articles at the same time. Research the first topic, and then let the background processes of your mind takeover before you move on consciously to the second topic. While on the deuxième topic, the first one will often become clear to you. Think holistically : Holistic thinking might be the single most ' advanced ' learning technique to help students learn new things. You may have even heard this word used to describe an overall mindset rather than as a solo technique.
Think holistically : Holistic thinking might be the solo most ' advanced ' learning technique to help students learn new things. You may have even heard this word used to describe an overall mindset rather than as a solo technique. Use the right type of repetition to your advantage : Complex concepts often require revisiting in order to be fully absorbed. For some people, this can take months or even years. Repetition of concepts and theories, including concrete examples, improves absorption and speeds up the learning process. Apply the Quantum Learning ( QL ) model : The Quantum Learning model is being applied in some us schools to extend beyond typical education methods to engage students through five core components : foundation, atmosphere, environment, design, and delivery. Get necessary tools : Obviously, there are a variety of tools designed for learning. If you are learning online like the majority of people are these days, then consider online study aids such as Quizlet and StudyBlue, as well as education communities like Edmodo and Schoology, among countless other tools. Learn critical thinking : Critical thinking is a skill that is not only essential to the learning process but will carry you through life. Read Wikipedia’s discourse on critical thinking as a starting point. It involves good analytical skills to aid in one’s ability to learn selectively. Learn complex problem solving : For human beings in general, life is a series of problems to be solved, and learning is just part of the process. Especially If you have a complex problem, you need to learn the art of complex problem solving.
Be engaging : Lectures are often one-sided and thus can be counter-productive. Information merely heard or observed ( from a chalkboard across the room, for instance ) is often forgotten. Teaching is not simply talking. Discussion is more important : ask students questions, present scenarios, and engage them. Use information pyramids : Learning happens in layers. Build a solid base of knowledge upon which you can continue to add advanced concepts. Use video games : Video games get a bad rap because of the many distinctly non-educational violent titles out there nowadays. But some film games can actually be an effective aid to learning, believe it or not. Role play : Younger people often learn better by being part of an interactive learning experience. For example, history is easier to absorb through reenactments, and can be further enhanced by using costumes, props, or other visual cues. Apply the 80/20 rule : This rule is often interpreted in different ways, but in this case, the 80/20 rule means that some concepts, say about 20% of a curriculum, require more effort and time than roughly 80% of others. So be prepared to carve out time to expand on complex topics.
Tell stories : However you can make a complex concept more relatable by telling a story or using metaphor, take the opportunity. When a story works to help a student understand something they might otherwise see as too boring or complicated, you’ll see understanding sparked in the student’s eyes. Go beyond the public school curriculum : The public school system is still generally lacking in teaching advanced learning and brainstorming methods. It’s not that the methods cannot be taught; they just aren’t. to be afforded these advanced learning methods, you typically have to pay a premium in additional time, effort, and money. While the standard for public schools and what is available to all students regardless of economic status is still a work-in-progress, you may need to seek supplemental resources such as tutoring or community programs to enhance learning for your kids. Use applied learning : If a high school student were having dysfonctionnement in math, say with fractions, one example of applied learning might be to teach fractions using photography, lenses, or f-stops. Another example is through cooking and measuring ingredients. Tailor the applied learning to the interest of the student and the subject at-hand.
Be engaged : Sometimes students are bored because they know more than is being taught, maybe even more than a teacher. Hopefully teachers will assess what each student already knows prior to that lesson. Students should discuss with a teacher if they feel that the material being covered is not challenging enough, or consider asking for additional materials. Teach yourself : Teachers cannot always change their curricula at their own discretion. If you’re not being challenged, challenge yourself. Some countries still apply country-wide exams for all students. Even if courses from the top online education programs don’t cover a topic you’re interested in, you can learn it on your own. Don’t wait for someone to teach you. Even chic lectures are more effective when you’ve pre-introduced yourself to a concept. Collaborate : If studying by yourself isn’t working, maybe a study group will help. Teach something : One of the best ways to learn something better is to teach it to someone else. The process forces you to learn more yourself when you share your knowledge with another person. Write about it : An effective way to ' teach ' something online is to create a wiki page containing everything you know about a topic. Or even create your own blog about it. Doing so helps you to realize what you know and, more importantly, what you don’t. You can still grab a freebie account on old préférés like WordPress or Blogger.
Learn by experience : Seems pretty obvious, but it simply means to put in the necessary time. An professionnel is often defined as someone who has given their all and put countless hours of blood, sweat, and tears into a particular experience or endeavor. Are you an expert without even realizing it ? If not, do you have the dedication to become one ? Quiz yourself : Testing what you’ve learned will reinforce the information. Flash cards have stood the test of time as one of the best self-test tools for kids and adults alike. Learn the right things—or the basics—first : Case in point : consider the way a baby learns a new language ( hint : it’s not to learn grammar and spelling and sentence constructs first ). An adult or young adult should be no different. Try immersing yourself in the basics instead and see the difference for yourself. Plan your learning : If you have a long-term plan to learn something, then to quote Led Zeppelin, ' There are two paths you can go by '. You can either take a haphazard approach to learning, or you can put in a bit of planning and find the optimum path. Plan your time and balance learning with living your life.
Persist : Don’t give up the pursuit of learning in the face of intimidating tasks. Anything one human being can learn, most others can as well. Take it from Thomas Edison, who said, ' Genius is 1% inspiration and 99% perspiration '. Challenge yourself : People are often more éclairé than they realize. In a world that compares and criticizes everything so publicly, it’s harder and harder to know where we fit in. And unexpected genius can be found in all walks of life. If you suspect you have more potential than you’ve shown to others or yourself, try an IQ test such as the one offered by MENSA. Unlike the standardized IQ contrôles given in many schools, this test helps to comprehensively assess a student’s knowledge and learning ability. And the mere ability to learn is far, far more important than what you already know.
Party before an exam : OK, maybe not a party, but the key is to relax. The worse thing to do is cram the night before an exam. If you don’t already know a subject by then, cramming isn’t going to help. If you have studied, simply review the topic, then go do something pleasant ( not studying ). Doing so tells your brain that you are prepared and that you will be able to recall anything that you have already learned. On the other hand, if you didn’t spend the semester learning the ideas you need, you might as well go party anyway because cramming at the last minute isn’t going to help much at that point.
Don’t worry; learn happy : Have a real volonté for learning and want to share your tips and tricks with others ? Join a cooperative learning group to spread the knowledge.