“I don’t write it down to remember it later, I write it down to remember it now.” Umberto Eco
Here are a few of my ideas to help you study, take notes, and learn.
- Take copious and detailed notes, especially in math and science. The more detailed and clear, with diagrams, the better. The process of making the drawing/diagrams and taking the notes will cement the concepts in your brain. It is almost like you are writing on a whiteboard in your brain that you can refer back to when you need it. The reason this works is because you are using multiple senses to learn and you are focusing to do it correctly.
- If you have questions that arise, write them down in your notes and make sure you ask the questions as soon as possible. Interaction with the teacher will help you learn the material and how important the subject is to the teacher. If you can ask the questions during the class lecture time, do so: you will not be the only person with the question, and it will encourage others to ask helpful questions. It is my opinion that the only bad question is the one that is not asked.
- Dedicate a specific time and place for studying. A special quiet place without distractions is best. MaThCliX provides such a place and many of our best students take advantage of the environment to do their work. For many, even if they intend on doing their math work at home, there are a host of distractions awaiting in that familiar environment. In college, we called the distraction of our bed the ‘rack monster’, so we were more productive in the library away from our familiar spaces.
- This is controversial, but I believe, and many studies show, that music causes enough distraction to defocus and make you less productive in studying. It can be used when you take a break to recharge your brain, but during intense study, absolute quiet is best.
- You may not know this, but your attitude have a huge effect on your success in learning a subject/topic. If you go into a topic with eagerness, wanting to learn and master it, you will be a lot more successful than if you do it begrudgingly, not wanting to learn it, or hating the subject. You will do a lot better if you trick yourself into finding something to enjoy in the subject. Finding some other subject that is related can help you find a mental link to hang the new facts on.
There are a lot more suggestions, but practicing these will help you as you study.
I know very intelligent people who for whatever reason believe that they just can’t do math. While there may be some people who have suffered a brain injury of some sort where that part of the brain is damaged, for normal people, the brain is quite capable of learning and doing even complex math.
Brain researchers have discovered a section of the brain that is wired specifically for math. Studies show that part is activated when a person concentrates on math problems much like the speech center reacts when speaking or reading. See the references at the end.
So why is it that many people believe they cannot do math? I think it is partially due to mental laziness because learning math takes time and effort. Like learning to do any complex task, doing math requires practice and focus. Another reason people believe that they cannot do math is because they have been told by others (parents, teachers, friends) that mastering math is only for “special math people”, who have the talent of math. You have probably heard people remark that they can’t do math because they are not a math person, perpetuating the myth even further.
Another reason some don’t want to believe that they can do math is because they think that those who do math well are socially inept nerds. The stigma of being a smart person (in math, science or any other academic subject) is promoted in the media and is “uncool” among middle school and high school students. How many of us remember such a character in TV and or movies? That is a stereotype that needs to be demolished. I believe it is part of the belief that if you work hard to excel that you will miss out of fun that you would otherwise have. I have discovered a wise saying from a friend, “If you take the easy way out of life, life gets harder, if you take the hard way out, life gets easier”.. That’s why I take exception to the phrase “take it easy”.
Resist the temptation to excuse yourself from learning math. You are born with the ability, all you have to do is develop it. While you may not eventually do higher math in your vocation, studies have shown that exercising the math part of your brain strengthens the decision making part of your brain (prefrontal cortex) which will help you in mature relationships and the wise decisions you have to make.
References: About being bad at math:
Brain on math:
Innumeracy – https://en.m.wikipedia.org/wiki/Innumeracy_(book)
Should Students Memorize Multiplication Facts?
I have been struck recently on how many students (and adults) struggle with multiplication. This skill is so basic to math that lack of mastery makes progress in math very difficult, at best. I am not even concerned so much about multiplication of positive and negative numbers as knowing what 6×7 is, for example, because it is hindering many students in higher math.
I don’t believe that a person needs to know “what” multiplication is, for example 6×7 is 6 groups of 7. The multiplication facts are more like the basic phonemes or words of math. When you learn to speak your native language as a child, you don’t really understand the meaning of the words, you are learning and mimicking the sounds and patterns.
As a young pre-teen, I was determined to learn the Morse code in order to become a Ham Radio Operator. I spent time memorizing the code, like a is dot-dash, b is dash-dot-dot-dot, and so forth. I found that I was limited in the speed I could decode the sounds because of the way I was learning the code. I would hear dot-dash and count the dots and dashes.
Unfortunately the requirement for the General Class license was 13 words a minutes, well beyond the mere 5 words a minute I could muster by counting the dots and dashes. My breakthrough came when an older Ham told me, “you can’t just count the dots and dashes, you have to hear the patterns”.
So I started listening for the patterns, so that a was not simply dot-dash, it was a short beep followed by a longer beep. In listening for the patterns I was able to boost my speed up to 18 words per minute, well above the required speed. After a while, you tend to hear whole words and ending as a single unit, so you aren’t just decoding single letters but whole words, much like we learn to hear a language.
So what does all this have to do with learning to multiply? I propose that it is easier to learn the multiplication facts by seeing the patterns and learning the patterns without trying to understand what they mean. That will become clear as you use them. Young children are like sponges and have no trouble learning repetitious and sometime annoying things. Why not teach them the multiplication tables? In the past, memorization devices like catechisms were used to teach children profound truths that they didn’t understand but became part of their knowledge and understanding when they were older.
For older children, repetition is the key. Use all the senses: writing the facts, seeing the facts, hearing the facts spoken, speaking the facts (or singing the facts?). The more senses can be brought into the memorization process, the better.
It is never too late, even for adults, as the human brain is amazingly elastic. If you know what kind of learner you are (visual, audial, tactile) then use that to your advantage. The nice thing about memorization, especially for adults is that it expands the use you have of your mind. So make a point to learn and master the multiplication tables. You will be glad you did.
How to Reprogram Your Brain
You no doubt have heard the expression, “You can’t teach an old dog new tricks.” Well, it turns out you can. The human brain is quite elastic and can be molded and changed, even in older people.
Imagine a ski slope that has a fresh fall of snow. At first the slope is a dream come true because you have full control, and the slope is not too fast. As time goes on and more people use the slope, the grooves get deeper and faster because of ice build-up. You have less freedom to veer from the path, but you can go faster.
The Brain is similar in that you can purposely increase the speed of processing. The synapses that provide the connections within the pathways will shrink the gaps with more use. This will also add more synapses for the purposes you program your brain to do. This applies to whatever you want to learn, whether it be math, history or a language. Brain washing techniques use this to reinforce ideas that are being promoted, but you can purposefully do the same to yourself. It is no mystery that when you focus on a topic and repeat exposure to it over and over that you will master the subject.
In a study at Harvard Medical School it was found that not only was repetition of a particular task necessary to rewire the brain, just the mere thought of doing the task also had a similar result! Check out this article from Time: https://content.time.com/time/magazine/article/0,9171,1580438,00.html
“Mental practice resulted in a similar reorganization” of the brain, Pascual-Leone later wrote. If his results hold for other forms of movement (and there is no reason to think they don’t), then mentally practicing a golf swing or a forward pass or a swimming turn could lead to mastery with less physical practice. Even more profound, the discovery showed that mental training had the power to change the physical structure of the brain.
This is great news for those of us who think they aren’t smart or strong. Because we CAN teach “an old dog new tricks”. The neuroplasticity of the brain extends even into old age. Even with damage, such as a stroke:
“If a stroke knocks out, say, the neighborhood of motor cortex that moves the right arm, a new technique called constraint-induced movement therapy can coax next-door regions to take over the function of the damaged area. The brain can be rewired.“
There is a verse in the Bible that is useful in understanding this effect:
Philippians 4:8-9 (ESV)
8 Finally, brothers, whatever is true, whatever is honorable, whatever is just, whatever is pure, whatever is lovely, whatever is commendable, if there is any excellence, if there is anything worthy of praise, think about these things. 9 What you have learned and received and heard and seen in me—practice these things, and the God of peace will be with you.
The impact focused thought can have on your life is amazing. When you think on good things as the verse says, your brain rewires itself to be much more open to such things and less so towards others. If you want to have a positive outlook, fill your mind with positive thoughts. The inverse is also true, when you think “unhappy thoughts” you will bring these thoughts to fruition, and your brain will actually make it easier to have such thoughts in the future.
So what does this have to do with learning MaTh? For one, if you start training your mind to think in mathematical patterns, even simple ones, then your brain will rewire and accommodate your efforts. Practice with simple math puzzles and logical problems. If you don’t know where to start, we at MaThCliX can help. It will expand your mind and help you to learn other similarly patterned subjects. Attaching positive emotions to the process will also help your brain rewire It is more difficult to learn something you don’t like. There is a saying among motivational speakers: “If you think you can, or you think you can’t, either way you are right”. It seems that there is a physiological explanation for this phenomenon.
Reprogramming your brain for success is possible with focused effort, but you can do it!
Pinewood Derby Physics
Have you ever seen a pinewood derby race, a race that is held every year in many Cub Scout dens? I have worked with my grandson to enter one a couple of years, and being the engineer that I am, I wanted to use my knowledge of math and physics to perhaps win the race.
We tried to add weights to the car, decreasing the friction of the wheels (it is illegal to use ball bearings), and experimented with the shape of the pinewood derby car to give it a more aerodynamic shape. All of these things help to one degree or another, and all are well known in the Cub Scout community as a simple search will prove (https://www.wired.com/2010/12/pinewood-derby-physics/).
Tautochrone Curve (or Brachistrochrone Curve)
There is something that can affect the speed of an object down a pinewood derby track and that is the shape of the track. You would think that the shortest distance would give you the fastest time (We all know that the shortest distance between two points is a straight line). But in the real world with gravity pushing an object down a slope, the distance takes a peculiar shape, called the Tautochrone curve (also brachistrochrone curve), shown in the graphic above. The next simulation shows this.
Another strange property of the curve is that regardless of where an object is placed on the curve, the time required to reach the end is the same. The first illustration shows this, and is totally unintuitive.
The curve is a form of a cycloid (see https://en.wikipedia.org/wiki/Cycloid), that is the path a point on a wheel makes when it rolls on the ground. There is a lot of math involved in the description (but not too much for a student familiar with trigonometry).
The point to all this is that math is fascinating and has a lot of amazing applications for the student with an inquisitive mind. There is a lot of history behind the study of the cycloid, as noted from the wikipedia article cited above shows, so there is an endless supply of interesting applications of math and a lot of unsolved math mysteries for those who want to make a difference.