• moja ~ 1
• mbili ~ 2
• tatu ~ 3
• nne ~ 4
• tano ~ 5
• sita ~ 6
• saba ~ 7
• nane ~ 8
• tisa ~ 9
• kumi ~ 10

If we want to say eleven in Kiswahili we’d say kumi na moja which is literally 10 + 1. Twelve becomes kumi na mbili (10 + 2), and thirteen is kumi na tatu (10 + 3). Sixty is sitini (6 x 10) and seventy is sabini (7 x 10). The pattern is only broken where we find Arabic influence enter into the language.

The number system that the English-speaking world uses is also a decimal (base 10) number system. But not all number systems in the world are base 10. The Yasayama of the DRC have a base 5 number system. Their number system is based on the numbers:

• omoko ~ 1
• bafe ~ 2
• basasu ~ 3
• bane ~ 4
• lioke ~ 5

So in the Yasayama language, the word for six is lioke lomoko (5 + 1), the word for seven is lioke lafe (5 + 2), the word for eight is lioke lasasu (5 + 3), and so on.

The Yorùbá of Nigeria and Benin have a vegesimal (base 20) number system. The Khoisan of southern Africa have a binary (base 2) number system. In fact, on the continent of Africa, base 5, base 10, base 20, and base 2 are the most common number systems. But why are these number systems so common?

The base 5 and base 10 number systems likely evolved from our anatomy. Humans have 10 fingers – five on each hand – and thus base 5 and base 10. If we include our total number of fingers and toes then base 20 also makes sense. But what about base 2? Why was this a common number system in antiquity?

The 20,000 year old Ishango Bone – one of the world’s oldest mathematical artifacts – has been found to contain markings indicating an early binary number system. The classical mathematics of Nile Valley civilizations would later build upon this binary foundation. If other number systems that developed in Africa were related to our human anatomy, then could this also be true for the binary number system? The answer is a resounding ndio!

The binary number system is a system of doubling. It is the ones and zeroes of binary code that underlies modern computing. Think of the place values of our decimal number system: the one’s place, the ten’s place, the hundred’s place, thousands, ten-thousands, and so on. Where do these numbers come from?

1, 10, 100, 1000, 10000… are all powers of 10. That is:

10⁰ = 1 (one’s place), 10¹ = 10 (ten’s place), 10² = 100 (hundred’s place), 10³ = 1,000, 10⁴ = 10,000

The 10 is the base of this number system. The binary system works the same way, except that the base is 2. So what are the place values in the binary number system?

• 2⁰ = 1
• 2¹ = 2
• 2² = 4
• 2³ = 8
• 2⁴ = 16
• 2⁵ = 32

These are all powers of 2! Therefore the “place values” of the binary number system are the one’s place, the two’s place, the four’s place, the eight’s place, and so on. We see that the place values are doubling: 1, 2, 4, 8, 16, 32, 64… A binary number system simply means doubling.

With that, can you guess what human biological function this relates to? Humans “double” by giving birth! The binary number system was born from the simple observation of our family ancestor tree.

There is 1 of us. Each of us has 2 parents, 4 grandparents, 8 great-grandparents, 16 great-great grandparents, 32, 64, 128, 256…We are all binary representatives of our family ancestor tree. The sequence 1, 2, 4, 8, 16… is known in mathematics as a geometric sequence with a common ratio of r = 2. This means that for each progressive generation we have twice the ancestors as the previous generation. How would you calculate the number of ancestors that you had in the generation 25 generations before the present? In Black Radical Algebra II we learn how to answer this question by finding the n^th term of a geometric sequence using exponential equations.

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In African philosophical understanding, numbers are understood to be symbolic and metaphorical.

Thus number is essential in the realization of the universe. Everything in the universe is assigned a number (or number of signs). In this way Amma used the principle of number to organize the creation. Indeed, the Dogon believe that the po will contain the entire universe conceived in Amma’s thought through the concept of number.

~Dr. Chukwunyere Kamalu, Word at Face Value

One number that is important in African cosmology is the number four, or as we call it, the fantastic 4. On the most basic level, the number four represents the the four elements (earth, wind, fire, water) and the four cardinal directions (north, south, east, west).

The fantastic 4 also relates to many traditional accounts of creation that we hear across the Continent that all seem to be attempting to tell some version of the same story… That is, in many traditional creation stories the original ancestors of human beings are described as beginning from four (or sometimes four pairs) of male/female twins.

We also see the fantastic 4 represented as four quadrants in the symbology of the ancient African world.

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The fantastic 4 can be observed in traditional divination systems. Mẹ́rìndínlógún (literally “sixteen”) cowrie shell divination – used by Olórìṣà in the Yorùbá traditional religion – uses 16 cowrie shells: 16 = 4². Diviners of Ifá in the Yorùbá traditional religion use the ọ̀pẹ̀lẹ̀ divining chain or the ikin Ifá (sacred Ifá palm nuts) and the ọpọ́n Ifá (divining tray) to mark out one of 256 possible Odù Ifá signs. Each sign contains a number of verses and collectively they represent the ẹsẹ Ifá corpus – a collection of oral narratives that represent all possible scenarios and possibilities of human existence and interaction as well as the prescription to overcome any of life’s challenges. There are 256 odù signs: 256 = 4⁴.

The Dogon of Mali have an alphabet and a divination system that is comprised of 266 signs. From Dr. Kamalu’s quote above, we know that in Dogon cosmology, Amma (the supreme being) organized the universe according to the concept of number. That is, Amma assigned everything in the universe one or more of the 266 signs mentioned earlier. For example, in Dogon cosmology the male number is 3 and the female number is 4. Together, their sum (7) represents life. The number 7 tends to hold special significance in traditional African religions.

The 266 signs that comprise the universe in Dogon cosmology are non-coincidentally similar to the 266 days between the conception and birth of a newborn child. Beyond that, the 266 signs of the Dogon actually break down further. There are two guide signs and eight master signs which serve as the keys to understanding the other 256 signs. Therefore it is: 2 + 8 + 256 = 266. This is not dissimilar to the 16 principle ojú odù of the Ifá (Yorùbá) system which are paired together to give birth to all of the 256 odù Ifá signs (256 = 16²).

The Bamana of Mali use a system of divination that is also similar to the Ifá system. Bamana specialists perform their divination in the sand in a process known as cenda. The process of cenda divination begins with the specialist drawing four dashed lines horizontally in the sand.

The dashes are then paired together. If a row has an odd number of dashes then the result of that row is a single vertical mark. If the row has an even number of dashes then the result is a double vertical mark.

This is repeated four times in a recursive process that results in a total of 16 symbols (16 = 4²) by the end. The 15th symbol is known as “this world” and the 16th symbol “the next world.” From there, the specialist dictates a narrative (reminiscent of the ẹsẹ Ifá oral narratives) from the symbols that addresses the original reason for the divination.

This is Afrikan math.

]]> https://newmoodle.nkalaeducation.com/number-nature-and-the-fantastic-4/feed/ 0 https://newmoodle.nkalaeducation.com/sacred-geometry/ https://newmoodle.nkalaeducation.com/sacred-geometry/#respond Tue, 03 Aug 2021 12:50:33 +0000 https://newmoodle.nkalaeducation.com/?p=1438 Sacred Geometry Read More »

• Lacking discipline
• Academically apathetic
• Zoned out
• Your spoiled ‘lil baby

If your child exhibits these symptoms then ask your doctor about Liveinthevillage^TM. Liveinthevillage is an all-natural supplement that works to cure LAZY by immersing your child in a village environment. Side effects include: going to fetch water, cooking their own meals over a wood/charcoal fire, sweeping the dirt-floor compound with a straw broom, running all types of errands, going to bed soon after sunset and rising just before sunrise, and generally being unplugged from all of the distractions that cause LAZY.

Ask your doctor about Liveinthevillage today!

A junk rep is a repetition of an exercise that has little training effect. That is, because of poor form or execution, it does not significantly contribute to muscle growth or strength gains. Muhammad Ali famously noted that:

I don’t count my sit-ups. I only start counting when it starts hurting…

The idea here is that when the reps start hurting is when he knows that they are effective reps. For a good example of a junk rep versus an effective rep watch a video of a crossfitter performing kipping or butterfly pullups and contrast it with an athlete performing strict pullups. The kipping pullup is minimally effective for strength gains, therefore, compared to the strict pullup we can call them junk reps. Junk reps are inefficient at making you strong.

Homework and study are very much analogous to strength training. Successful study requires intensity (appropriately difficult problems), volume (number of homework problems worked), frequency (how regularly you’re working homework problems), and recovery (adequate sleep and nutrition). Although many students consistently complete their homework assignments, often times their work is mostly junk reps!

You get your assignment and immediately dive into it with the goal of completing the assignment as fast as possible. You’re not focused and you do not commit a conscious effort to understanding what you’re doing. You’re on auto-pilot. The result is a completed homework assignment but a sub-optimal quiz score (training effect).

If you want to become intellectually ripped, swole, and jacked then you must focus on effective reps when completing your assignments. Follow these tips:

1. Ritualize your study time.
2. Digest the assignment before beginning. Do this by surveying the problems and taking note of the different types of problems that the assignment contains. For example, an assignment on graphing linear equations might contain problems focusing on plotting points on the coordinate grid, identifying the slope and y-intercept from an equation in slope-intercept form, graphing a line from an equation in slope-intercept form, and graphing a line from an equation in standard form. These are the main skills that the assignment is helping you to master. Make note of this.
3. Focus on accuracy and not speed. Speed comes from familiarity, and familiarity comes from volume and frequency. Do not try to rush through problems, instead sit with each problem and identify the potential challenge or lesson in the problem and then solve the problem accurately.
4. Check your answers once you’ve completed the assignment. If the assignment is not online and does not offer immediate feedback then get with a classmate and check your answers against theirs. Where you have differing answers go back and check to see if you can find a mistake.
5. Rework any missed problems and try to find your mistake. If after reworking the problem a couple of times you are still unable to arrive at the correct answer then ask your teacher or tutor for help.

Now get to work!

One fundamental tenant of the African worldview is the idea that what is first is sacred and by imitating what is sacred, we too become sacred. Given these ideas, we asked students to discuss the connection between our topic and the Wọlọf proverb below. The top student responses follow. Wohu no sɛn? How do you see it?

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Ganaawu yaay; garabu liir la.

Mother’s back is the baby’s medicine.

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I think this proverb means that anything that helps us grow is like medicine. For example doing math is like medicine because it makes our brain stronger. Exercising is like medicine because it helps strengthen our immune systems. Medicine is not only something given to us by a doctor. Medicine can be anything that helps us get stronger.

Doing math is like taking medicine. Math is like medicine because it helps us become smarter. We use math to solve problems. Math can help us think clearly. It is like medicine for our mind. Being smart is important because it helps us to make better decisions as we grow.

Exercising is like medicine because it helps strengthen our immune system. When our immune system is strong, we are less likely to get sick. If we do get sick, a strong immune system can help us heal faster. Medicine can also help us to feel better when we are sick . Exercise and medicine can work the same way.

Medicine is anything that helps us grow. A mother is a baby’s medicine because she helps the baby get stronger as he or she grows. The mother makes sure her baby has what he or she needs. A mother also carries a baby on her back to give support. Medicine can be many things.

~Niara M.
Black Radical Pre-Algebra

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The literal meaning of this proverb would mean the mother’s back helps the baby. The baby uses the mother’s back to get around since the baby itself cannot walk. This relates to medicine because you need medicine when your body cannot fight something off on its own. The proverb could also mean everything has a foundation.

Everything comes from something that came before it. It’s origins hold up that idea or thing. Without an origin, the thing would not exist. For example, without looking at the origins of math we would not have the math we use today nor would we know how to do it. The first to come paved the way for the later generations. Without the first generations, there would be nothing right now.

This proverb could be applied to life when you are having trouble with a topic, or that topic does not make sense. You could go back to its origins/basics, and possibly develop a better understanding of what you’re doing. You could also use this if there is a problem and you are trying to figure out how it happened. You could go back to the basis of the issue, and use what you find to help you prevent future issues of the type.

I think this proverb is important because if you want to understand something or be good at something you need to fully understand the basics/origins of what you’re doing.

~Jabea M.
Black Radical Algebra I

Ọgbọ́n ju ágbára

I’ve been reading Ìjàpá stories lately. Ìjàpá is the trickster tortoise among the Yorùbá who is always using his cunning intellect to outsmart the people and achieve his (usually greed-based) goals. Ọgbọ́n ju ágbára is most definitely Ìjàpá’s motto. Although this is generally a noble adage, for Ìjàpá there is another issue: Ìjàpá is extremely lazy, and he uses his intellect to avoid any and all work – and thus always finds himself embroiled in some conflict.

Ọ̀ràn kìí yẹ̀ l’órí alábaun (ìjàpá).

The responsibility for trouble never misses the head of the tortoise.

When it comes to our studies, many of us are like Ìjàpá. We are very quick to comprehend new topics and absorb new information, but we refuse to study and practice what we learn. The lecture was exciting, the class was engaging, the teacher laid everything out elegantly and gave us a homework assignment for extra practice. Well, since we understood it in class….it made perfect sense when the teacher explained it…there’s no real urgency to engage in these repetitive homework exercises…we’ve got this. Ọgbọ́n ju ágbára!

And like that, we outsmart ourselves. We become our own Ìjàpá. Of course the reality is that intelligence alone is not enough for mastery. We also need to develop or strengthen our base level of intelligence through study and practice. Jordan and Kobe were genetically gifted athletes with a natural talent for basketball. However, what made them legends of the game was their extraordinary commitment to training and practice.

But of course, we already know that. The issue is that if you want to see how fast a child can run, send him on an errand to the place where he wants to go. Simply put, we often neglect study and practice because it can be boring! This is where we can use ritual as our medicine against what is sometimes unexciting. We have our morning ritual of waking up, showering, dressing, and eating breakfast. We have our ritual related to going to the gym or working out at home. We may ritually pour libations every morning. Perhaps we perform religious rituals every five days, or every eight days, or every 42 days. A ritual is simply a series repetitive actions that we perform at regular intervals.

We can ritualize our study time as well to make it habitual and more purposeful. Here are some ideas that you can use to ritualize your study time. Make these steps a consistent part of your schedule for 30 days and pay attention to how you start to feel about study time.

• Put your phone in airplane mode and place it in another room
• Take a quick bath/shower
• Put on some comfortable clothes that you wear specifically for studying
• Burn a fragrance (incense, essential oil, etc.) – our sense of smell is strongly connected to our memory – that you use specifically for studying
• Drink a healthy, brain-stimulating beverage (coconut water, sorrel, sea moss, ginseng, herbal tea etc.) and treat that beverage like a “pre-workout” supplement for your study-time
• Play a particular type of soft music that helps you to focus
• Recite your list of academic affirmations (i.e. “I am focused, I catch every detail, I will put in the necessary hours to ace my exam,” etc.)
• Work in the same location, preferably a quiet and well-lit place. Use earplugs if necessary.
• Make a to-do check list of topics/assignments that you plan to complete
• Any other ritual elements that you like
• Do the work!

By implementing steps such as these you ritualize your study time and thereby increase its meaning. We train our brains to associate the drudgery of studying with the euphoria of acquiring new knowledge (and getting good grades). It is a lesson in voluntary hardship and delayed gratification that will serve you throughout your life.

Now get to work!

In many traditional Afrikan communities there were special trees under which the elders would sit and discuss the problems which the community faced. They not only discussed “problems,” but – talked about the current news of the day, both good and bad. It was a place where announcements were made, where griots and poet-singers often came to express themselves. We borrowed the phrase “The Chattering Tree” from a movie filmed in Senegambia in the year 1975 under the direction of Sate Faye, one of the few Afrikan women filmmakers. The film was called “The Peasant Letter.” We hope that this small monthly newsletter will serve our community in the same way that the thousands of “chattering trees” served our traditional communities in Afrika, a place where one can come and present his ideas, opinions and news of the day. Also, where one can come and hear the ideas and opinions of others. Remember, the “chattering trees” only served as places where people came to discuss. The trees themselves, never interrupted, never criticized and never contradicted. In like manner our monthly news letter is only a place where one can come by way of articles, letters, poems, short stories, ink drawings and news items about the “third world.”

We invite you to come and sit under NKALA’s chattering tree and learn from those who actually do it.

Well, since today is Labor Day in East Africa. I’ve decided to take a break from the teacher code and tell you all the truth:

Beyond foundational topics, you’ll likely never use most of what you learn in your mathematics classes. The same can be said for many other classes as well.

I once had a student who was interested in pursuing a career in medicine. We were studying calculus and he often lamented about how he would not need to use calculus as a doctor. Being a guardian of the sacred teacher code I knew that I couldn’t let that sentiment stand in my classroom. I approached a doctor friend of mine and asked her to give me examples of how she used mathematics in her day-to-day work. If I could get the backing of a whole doctor, then I’d be sure to eliminate the attack on my classroom’s integrity.

Sɛ ɔbaa no ka anyansasɛm bi ara a, fa w’aso kɔ fam. Ɛnyɛ Ntikuma na ɔkyerɛɛ ne se Ananse kwan a ɛsɛ sɛ ɔde nyansa kukuo no foro dua no? Sɛ mmɔfra no mpo de nyansadwen ba a, fa.

If a woman speaks some wise words, take your ears to the ground (listen humbly). Was it not Ntikuma that showed his father Ananse the Spider what he needed to do to be able to climb the tree with the pot of wisdom? If children even come with wise thoughts, then we should indeed take those wise thoughts.

To my surprise the doctor actually validated what the student said! She told me that, with the exception of simple arithmetic used in calculating patient dosages that they truly did not use much (if any) advanced mathematics in their day-to-day work. When I began to think about it further I realized that, even as an engineer, I did not use many of the advanced mathematics topics that I learned in school. The reality is that if you approach an engineer with 10 or 20+ years of experience and ask him about a calculus or differential equations problem, he will likely have no idea how to solve the problem without first researching it online or in a textbook.

So then what is the point? If we, as professionals in science, technology, or engineering, will likely never be asked to use partial integration to find the volume of cones or be tasked with writing a proof that shows that two angles are congruent according to the side-angle-side theorem, then why do we learn this stuff?

The answer is that in science, technology, or engineering you may actually need these skills in the context of solving more complex problems. And although professionals typically use softwares to perform complex calculations for us, knowledge of the underlying mathematics is still useful in troubleshooting or debugging.

But what if we’re not pursuing a career in STEM?

One thing that most do not understand about education is that the content knowledge – the math formulas and algorithms, the literature passages, the dates in history class, the grammar rules, and all of the chemical formulas – is not the only purpose of our education. The content knowledge is only half of the picture. Learning the content is obviously necessary, but we must also simultaneously learn how to learn.

New college and university students soon realize the difference between their teachers in secondary (middle and high) school and their college professors. One of the main differences is that in secondary school students are taught by their teachers while in post-secondary colleges and universities students must begin to teach themselves. This applies to STEM and non-STEM subjects alike. Secondary school has teachers, post-secondary colleges and universities have professors. A teacher’s job is to teach, while a professor’s job is to profess.

If we think back to who we consider to be our best teacher, it will likely be a teacher from secondary school. College professors tend to have greater content knowledge and more credentials related to their field. However, greater content knowledge and greater credentials do not translate into better teaching. Teaching is a separate skill from knowing. Having advanced knowledge in one’s head does not at all ensure that one will be able to effectively transfer that knowledge into someone else’s head.

The Most Important Skill

So then, the most important skill is not the content that we learn – the math, the science, the coding; the most important skill is knowing how to teach oneself – how to figure things out without instruction. This is one reason why homeschooled students tend to perform better in post-secondary institutions than students from private or public schools. Homeschool students, often due to their parents’ schedules, learn early on to become independent learners. In that way, they get a jump start in learning skills that other graduates will only begin to learn during their years in college or university.

Technology is constantly changing. New programs, new computer models, and new tools are always developing and replacing technologies that were once considered cutting-edge. Having a well-honed ability to teach oneself new things then becomes one of the most valuable skills for an engineer/scientist/technologist/mathematician. The same may be said of entrepreneurs.

Because of this, learning how to learn mathematics and science becomes just as valuable for the medical student or the liberal arts student as it does for the engineering student. Focusing on learning the process of how you learn is the best way to maximize the benefits of studying any subject.

So, when will you use this stuff? You’ll use the same thought processes that you used to learn the foreign concepts of arcsine, arccosine, and arctangent to figure out how to build your own website for your new business venture. You’ll use the same step-by-step method that you used to learn how to solve quadratic equations to figure out the rules for how to market your company’s products in that new country. You’ll use the same logic that you followed in creating that geometric proof to learn how to model using that new software platform.

Every person has two educations: the one which he is given, and the one that he gives to himself.

Mfomsoɔ kyerɛ nnipa nyansa, ɛno nti deɛ ɔse ɔnyɛɛ mfomsoɔ da no, ahwere adeɛ.
Mistakes teach people wisdom, that’s why the person who says he has not made mistakes before has indeed lost a thing (an opportunity to learn).

Akan Proverb

With over a decade of experience as a classroom educator and over two decades as experience as a tutor I have come to notice several trends among most of my students. Here are the top four mistakes students make in mathematics and math-based science classes and how to correct those mistakes today.

1. Arithmetic errors

We’ll be working on a complex calculus or physics problem and my student will be executing the procedure beautifully, and then…

7 x 4 = 11 …Doh!

Simple arithmetic errors are very common mistakes that can single-handedly render pages of good problem-solving technique incorrect. Mastery of arithmetic comes from practice. Building efficiency and consistency with respect to the recalling rules for multiplying negative numbers, or dividing fractions, or adding decimals is like building muscle memory. The mind is a muscle. The more you train a skill, the faster and more efficiently your brain is able to execute that skill.

I recommend to my students that they download an arithmetic app on their phone and dedicate a few minutes each day to practicing their arithmetic. Choose an app that includes the appropriate level of arithmetic, i.e. adding, subtracting, multiplying, and dividing with integers (positive and negative whole numbers), fractions, and decimals. Be sure to set a daily alert that prompts the student to complete his daily practice. Apps that feature rewards and incentives are best; I find that tokens or other recognition for consecutive days of practice and efficiency of practice are very motivating.

2. Not showing all steps

If I’m at a restaurant I’ll definitely calculate the tip for the waiter in my head. If I’m in the field and want to determine whether or not a design idea is feasible I’ll most certainly do a quick back-of-the-envelope calculation; and after decades of experience I might not bother to write each and every step down. But for homework, classwork, tests, or any assignment that I am submitting to another human being for review then I’ll definitely show each and every step.

The two solutions above are both correct answers to the same calculus problem. The solution on the left shows all steps, while the solution on the right does not. Again, both solutions are correct. So, what is the benefit from including the detail as shown on the left? Consider that this is just one of 20 other homework problems that this student is working on. In addition, this student has been learning about the chain rule for a week so he is very confident in how to solve this problem. The issue is that weeks or months after solving this problem the student may forget the chain rule. If the student refers back to his homework and the only thing that he finds is the solution on the right, will this solution be very helpful to him?

We show all of our steps because not only does it reduce the likelihood that we’ll make a mistake, but it also gives us a good study tool to refer to in the future when the topic is not so fresh on our minds. Notice how the solution on the left not only mentions the method required to solve the problem (the chain rule), but it also lists the formula for the chain rule and continues to list step by step each procedure required to solve the problem.

Finally, as an added bonus, listing all steps is a great test-taking strategy if the problems for an assignment or a test are graded using partial credit. Listing your steps shows the teacher that you know what you are doing; therefore, if you make a small error but follow the correct procedure you can still get credit for your work. But, if you do not show your steps and you make an error then there is less of an opportunity for your teacher to give you partial credit. Our solution above on the left explicitly shows knowledge of the chain rule, the power rule, and how to differentiate natural logs while the solution above on the right jumps directly to the answer without any indication of how we reached it.

3. Not reality-checking answers

A student is knee-deep in crunching numbers to solve a word problem until, finally, an answer: –51.6. Okay, circle the answer and quickly move on to the next problem… Whoa, not so fast slick! The problem asks how long (in minutes) will it take the boat to arrive and the answer that you got was –51.6. Negative minutes? Don’t think so.

Students know that they can often check their answers for correctness by substituting their answer back into the original problem or equation. Most will choose not to do this, however, since it is time consuming. But reality-checking answers is a much faster process. It simply takes a moment to look at the answer that you’ve gotten and ask if it makes sense given the context of the problem. There are no negative times or lengths, the rocket is not traveling at a speed of 4.5 m, and it’s unlikely that each of your friends would receive 2.42771 apples.

4. Not reviewing solutions after the test

Too many students see the test as the end of the learning on a topic. Whatever questions they got correct represent the totality of their knowledge on the subject from that point onward. Whatever questions they got incorrect will forever remain shrouded in mystery. This, of course, is missing out on the best opportunity for true learning. After the test, when you have the solutions available to you gives you a great opportunity for self-assessment and correction. Comparing the solutions with your (incorrect) thought process will give you a clear indication of exactly where there are flaws in your thinking as well as any other skills that you need to tighten up (such as those mentioned above).

The test is just one assessment of your capacity at a single point in time. It does not, and should not, represent the totality of your knowledge on a particular topic. Strive for mastery regardless of whether or not your grades officially recognize it.

That’s it!