Learning and Memorizing Math. Lesson 11

There was a time that I thought that understanding math was more useful than memorizing it. A certain amount of memorization is essential, especially for lower-level math. For example, you need to memorize the multiplication table up to 9 x 9 in order to be able to multiply two four-digit numbers.

In elementary and high school, this was not an issue for me, because I did not have to work very hard to understand or memorize it. In college, however, they threw engineering calculus at me, and that was a very different story, not so easy. This was in second semester of my freshman year at the University of Tennessee. I struggled to understand the formulas and going into the final exam, my grade was an F. In desperation, I gave up trying to understand the calculus concepts. Instead, I memorized all the formulas we had covered in the course and the kinds of problems that each could solve. I made a 100 on the final exam, which converted my F to a C. While the memorization served a useful purpose, my lack of understanding would surely have doomed me if I had been in a curriculum that required more advanced classes later.

Now, many decades later, I discover that a similar experience happened to the author of an article published by the Dana Center. The author, Monette McIver, liked math before she went to college, and it was relatively easy. But college math was too challenging, and she too realized that maybe she should just focus on memorizing what was necessary rather than trying to understand. She actually got an undergraduate degree in math that way. She laments having to do this, however, because she really wanted a career as an engineer, and she did not end up with needed level of understanding.

So, what should you do when faced with overwhelming math challenges? I say, memorize it, in order not to be overwhelmed, but as you move along, keep striving for understanding. And seek out the teachers who are good at explaining the math. While the modern focus on understanding math, as in “New Math” and “Common Core,” is commendable, it too often leaves students confused, and they don’t learn the basics either. There is a certain utility in memorizing multiplication tables, even if you understand the principles well enough to build a table. The trick is in discovering what should be memorized to free you to apply that knowledge for higher-level problems. Here are some basic principles that might help:

1.     Use the math you do know to help you figure out the math you do not know. For beginners, work on developing number sense. For example, when asked to solve 7 x 8 question, someone with number sense may have memorized 56, but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56, or they may work out ten 7’s and subtract two 7’s (70-14). Alternatively, know that 7 x 8 means 7 rows of 8 or 8 rows of 7, to enable you to realize that 7 x 7 = 49 plus 7 makes 56. Mnemonics are very helpful with the memorization: for example: 56 = 7x8 is the numbers 5 6 7 8 in order.

2.     Solve many problems at the level you do understand. This will reinforce the memorization, and seeing the same issues in different contexts will gradually build up understanding.

3.     Try to identify, understand, and memorize the really crucial concepts and definitions that underlie many other math ideas. Examples include words like function, theorem, angle, tangent, sine, cosine, derivative, integral, matrix.

4.     Learn a special case of a math concept first and then generalize to more abstract extensions as your understanding improves.

5.     Whenever feasible, make drawings to illustrate a concept.

6.     Find good learning sources. Many are on the Internet. One of the best sites is Kahn Academy

7.     At the appropriate age, master algebra. It is fundamental to most of higher math.

8.     Develop mnemonic devices, but only for basic ideas and don’t overdo it. The point is to use memorization as an adjunct to mathematical thinking, not as a substitute for it.

In one lesson, I can’t cover much mathematics, but I will choose a fundamental of trigonometry to illustrate my point to identify what is useful to memorize and what you need to think through. To remember the core ideas of a triangle, the features you need to memorize are labeled in the figure below.

First, realize that the lines are measures of distance (a,b,c,). For future reference you could think of lines a and b as y and x axes of a graph, but this is not relevant here. The lines could be labeled anything. Notice also there are three angles (1,2,3). They also could be labeled anything. Mathematicians like to use Greek and for angles, they often use theta (θ). Angle #1 is a special angle, called a “right” angle. I don’t know why it is called “right,” but you might think of it being the right angle to focus on in a triangle: because it is 90^o by definition, the sum of the other two angles is also 90.^o Can you figure out why the other two add up to 90? On a scratch sheet, draw two triangles stacked against each other to form a square and a horizontal line running from top left to bottom right. The box has four 90^o angles if you remove the diagonal line. On the right you see that if you flip line b to form a straight line, you have created a 180^o angle, and if you keep on moving the line around, you create a circle of 360^o. I have no idea why a circle is defined as having 360 degrees. Some mathematician early knew it would be useful to carve a circle up into angular pieces and for what ever reason decided 360 would be a good number (not too small, not too big).

See that there are three boundaries, a, b, and the longest one, c, which is always defined as the hypotenuse. At about this point, you should be asking, “What is the point of all this? What is the ultimate objective?” The answer is that trigonometry allows you to calculate certain dimensions without actually having to memorize or even measure them.

At this point you must memorize that in a right triangle the longest line (c) is called the “hypotenuse.” Each of the two variable angles has a line opposite to it and the other non-hypotenuse line is called the adjacent side. Sine is arbitrarily defined as opposite/hypotenuse, cosine as adjacent/hypotenuse, and tangent as opposite/adjacent. In the diagram above, the sine of angle 3 is a/c, the cosine is b/c, and the tangent is a/b. With these ideas firmly cemented in memory, you are now free to explore the mathematical consequences and uses.

Sine, cosine, and tangent are just names; they could have been called anything. But they are useful because they are a way to label the ratios of the lengths of any two sides of the triangle. If you divide the denominator of the ratio into the numerator, you calculate the length of the third side without having to measure it.

You could, for example, measure on of the angles with a protractor and now instantly know the degree of the other angle that is not the right angle of 90.^o If angle 3, for example is 35^o, then angle 2 is 55.^o

Note also that the two non-right angles are complementary. They sum to 90^o and moreover, the sine of one is equal to the cosine of the other. The equations can be consolidated, as follows:

Sin 35^o = a/c, Cos 55^o = a/c, therefore, the sin of any angle, θ, = cos 90 - (θ).

With these few examples, you can see the usefulness for learning math of combining some memorization with reasoning.

                                 

Next Lesson: #11. Deliberate Practice

Sources:

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-complementary-angles/v/sine-and-cosine-of-compl

 

McIver, Monette. (2018). Memorization versus understanding. Better approaches to teaching mathematics. May 26. https://www.utdanacenter.org/blog/memorization-versus-understanding-better-approaches-teaching-mathematics

Learning from Lectures, Readings. Lesson 10

 Effective learning takes much less time if you “study smart.” To “study smart,” you need to approach learning in a deliberate way. To study smart, think about the strategies and tactics you need to be using to master a learning challenge. Be aware of any need to change strategies and tactics that are not working well for you.

Best learning occurs during lectures and videos if you make it a point to be alert and aware. The best approach is to think about what you are trying to memorize. Ask yourself questions about the information, such as:

·       What is missing that would be useful to know?

·       What do I not understand?

·       Where can I get this explained better?

·       How can I apply this information to what I already know, to other parts of the course, to other courses, and to different kinds of problems?

·       What new ideas does this give me?

Think about the information in different ways, in other contexts. Think about how the information relates to what you thought you already knew. What is new about it that you need to incorporate into your knowledge arsenal? 

Readings

Anybody old enough to be taking these lessons on improving learning and memory knows how to read. Right? Not necessarily. First, we have to address how students are taught the mechanics of reading. Significant numbers of people were not taught phonics, which was the traditional way of teaching literacy for hundreds of years in almost all languages. Then some educators thought learners could just skip the phonics stage and move directly to “whole-language.” The basic idea of whole language reading is to prevent learners from breaking down sounds in a word individually, but to fix the eyes on whole words and associate them with prior knowledge.

I think that the correct way to literacy is to begin first with phonics. Then as learners master the sounds of the alphabet, they can sound out strange words and decode their meaning. Once phonics is learned, whole language becomes a way to read words, rather than consciously sounding out each syllable. The International Reading Association (IRA) has supported the inclusion of phonics in the whole language approach to literacy.

Actually, this still leaves the problem of plodding along one word at a time. Optimal reading requires clusters of multiple words at a time, speeding the amount of material accessed. Thinking about word clusters imparts linguistic meaning faster and better than plodding through one word after another.

To see word clusters properly, you need to train you eyes to pop along from one fixation point in a line to the next point to the right, then the next, and so on. You might not know that everything the eyes see, whether it is text or nature scenes, results from quick snaps of eye movement from one fixation target to another. These quick jumps are called saccades. The trick is to expand the size of the visual target that is seen with each snap, that is, increase the number of words you see at each snap of the eyes from fixation point to the next fixation point. Just by trying to do this, you can increase the number of words seen at each fixation. At first it may just be one or two words. Soon, your eyes will take in four or five words with each snap of the eyes. This kind of training requires deliberate practice, but if you think hard about what you are trying to do, it starts to become automatic. Good readers take in a whole line of text in a book, for example, in two to three eye snaps. Tests show that readers with average reading speed can double or triple their reading speeds with no loss of comprehension.

Key points to remember:

·       Preview the reading material to get a feel for what it is like.  Note the heading and subheadings. Think about the overall scope of what is covered and not covered.

·       Think about your purpose ahead of time. Ask yourself, “What am I supposed to get out of this reading?” “What am I supposed to understand and remember.”

·       Skim first, looking for the paragraphs that matter the most. The first and last sentences in a paragraph usually provide the best clues as to which paragraph is most important.

·       Make yourself interested in what you must read. You punish yourself by allowing boredom.

·       Adjust your pace according to the denseness and difficulty of information.

·       Try to reduce the number of times you skip back to re-read. If this is a problem, work on your concentration and focus. Don’t let you mind wander when you read. Definitely, do not multi-task.

·       At first you may want to move your finger or a pointer underneath each line to guide your eye snaps. But as you practice and get better, try to eliminate this crutch.

·       Do not move tongue or lips to simulate saying the words inside your head. If you tend to do this, make it a point to hold the tip of your tongue against the roof of your mouth.

·       At each eye snap, THINK about what the words, as a group, mean.

·       Make sure you actually see all the words at each fixation point. If you can’t see all the words at each fixation, decrease the number of words you expect to register until you get better at this.

·       As you realize you are getting better at these eye snaps, increase the speed of snapping.

·       Pause from time to time to reflect on what you just read. Ask yourself to recall the information you just read. Ask yourself how you could and should use the information. Ask yourself how the information fits you existing knowledge and understanding. Ask yourself what you still do not understand? Ask yourself what information you need or want that has not been covered yet.

·       When you finish, DO SOMETHING with what you just read. Self-quiz. Write notes. Report to others what you just read. Use the information in a different way.

 

If you search on the web for “learning how to speed read,” you will find numerous explanations of how to improve reading mechanics. There are even computer apps that help train your eyes attend rapidly presented words, one at a time in rapid succession. See the review of 10 of these apps at https://bookriot.com/best-speed-reading-apps/. Many apps use the RSVP method in which words are presented at a preset speed. Sprint has a free browser based trainer that allows you to increase the number of words presented each time, which helps you learn how to expand the size of the visual field. However, this method fails to teach you how to snap your eyes across a minimum number of fixation points per line of text (see video at https://www.youtube.com/watch?v=kmDMrxUSXKY). I have not found any apps that train you to do what really matters: snap your eyes appropriately across each line of text and engage larger and larger visual fields with each snap.

Lectures

Many of the thinking aspects mentioned above for reading apply also to learning from lectures or on-line videos. Lectures and videos may demand more attentiveness that reading because it is not so easy to slow things down or pause or go back to reconsider information that did not register well. To help information register more effectively, it helps to do some advanced preparation. Good teachers may give you a reading assignment related to the lecture. The more you learn from this pre-reading material, the more you will comprehend and remember from the lecture.

This brings up the point that the goal for lectures or videos is to learn as much as you can at the time. You may not get a second exposure to an unrecorded lecture. A classroom environment presents a special challenge. Once there, you are more or less trapped and your time is pre-committed. As long as you are in class, you might as well bring your A-game so you get the most out of your time commitment. Students who are charged up, fully expecting to aim to remember everything presented in class, are the most likely to remember the most. Be as engaged in discussion if it is allowed. In my experience both as a student and an instructor, I have found the best kind of engagement is asking good questions silently to yourself or of the teacher when questions are solicited. Asking good questions requires deep thinking and deep thinking is the best kind of memory rehearsal. Such thinking and the Q&A that follow obviously can help understanding.

Everything learned in class is something you don’t have to study much after class. Besides, being fully engaged in classroom activities makes class more interesting —certainly more useful.

Get “up” for class, expecting to remember everything.

It should go without saying that you need to be rested. Sleep is vastly more important for learning than you probably realize. Not only does being rested keep you from wasting your time by dozing in class, but memory of what was presented in class is largely consolidated that night as you sleep.

Students should take notes during the lecture or watching a video. But in my experience, they get little good advice on how to take notes. Perhaps this is a good time to re-read lesson 5 on note taking. Note-taking is the standard process whereby information is transferred from the teacher’s notes to the student’s notes (sometimes without passing through the mind of either). The problem is that students are too busy writing notes and not busy enough thinking about what the teacher says and means. Good teachers hand out note outlines before class so students can pay attention to the lecture and get engaged with

Such “skeleton notes,” give the student freedom to leave out things they already know or can figure out. This approach really pays off when it comes time to study for exams.

Note taking should be minimal. Follow the principles given for reading. The idea is to think about what is being said, asking yourself or the teacher questions, expressing the ideas in your own terms, making mental images, and so on. What do you do in case you miss some key information while doing all this thinking?  If the teacher permits, use a tape recorder and use variable speed, so you can slow down for difficult parts and speed up through parts that are not particularly useful.

Next Lesson: Lesson 11. Learning and Memorizing Math Concepts

 

 

Lesson 9. Mnemonics

In prior lessons we learned some of the key principles of memorization:

·       Lesson 1: encoding, consolidation, retrieval, reconsolidation

·       Lesson 2: getting motivated

·       Lesson 3: paying attention

·       Lesson 8: making associations

in Lesson 12 we will learn how to implement these principles with Deliberate Practice. Here in this lesson, the objective is to teach you some mnemonic tricks that add to the effectiveness of using memory principles (Figure 1).

 Figure 1. Mnemonics assist all the stages of the memorization process.

The most basic feature of effective mnemonics is the association of what you are learning (the new) with what you already know (the old). Unfortunately, this association process does not always occur automatically. Mnemonics provide an explicit way to expedite the use of specific tactics designed to promote the creation of these associations.

“Memory athletes” are people who compete in memorization contests. For example, they compete to see how fast they can memorize the sequence of cards in as many as four decks). They compete to see who can memorize the longest string of random integers. Such feats are accomplished by people whose basic memorization ability is no better than yours or mine. They achieve such astonishing feats as memorizing four decks of cards in less than five minutes or memorizing strings of 80 or more digits. Nobody can do such things without mnemonics.

While you and I probably have no interest in such things as memorizing long strings of digits, we would like to be more effective at memorizing practical things we need to know. Mnemonics can empower us just as they do they memory athletes.

I identify seven known mnemonic approaches.

1.    Common-Sense Thinking

The word, “mnemonics” is defined as systems for improving and assisting the memory. The most powerful of all seven mnemonic approaches is just plain thinking about the material being memorized (Figure 2).

Figure 2. Thinking  hard in different ways about

 what you want to memorize is a very powerful memorization aid.

Every new bit of information has meaning. Thinking about that meaning improves your understanding and may give you new ideas. What is not so evident is that the process of thinking is actually promoting memorization in an automated way. Thinking embodies all the key principles of memorization. Thinking inevitably creates associations, strengthens encoding, and provides time for consolidation. As you think, you are paying special attention to memory targets, increasing your motivation as you realize practical applications, retrieving the information in working memory, and re-consolidating it with the new ideas as they emerge from thought. If you add one of more of the other mnemonic approaches during thinking, you are engaging the principle of deliberate practice.

2.    Acronym

Use the first letter of each word to create one set of letters, forming an  acronym. Ex:  U.S.A. for United States of America.

3.  Acrostic

Use the first letter of each word of a concept or item you are trying to remember to create a string of words. Ex:  All Cows Eat Grass (the white keys on a piano)

4.          Categorization

Group similar items together. Remembering any item in a category often will help you recall the others in that same category. Example: for a grocery shopping list, group vegetables (celery, potatoes, carrots), dairy (milk, cheese, ice cream) and so on.

5.          Subject-Object-Verb (SOV)

Imagine a picture of someone or something doing something to a target or object. Example: state where President Clinton grew up (picture political opponents throwing little rocks at Clinton … “Little Rock” is generally known as the capital of Arkansas).

6.          Memory Palace

Visualize (create a mental picture for) each item you want to remember and mentally place that item’s image on or in mental images of a place with which you are familiar (like objects in your room, parts of your bicycle, etc.). Example to memorize the first 10 amendments to the U.S. Constitution, using objects in your living room: #1 Freedom: Picture yourself oiling the hinges of your front door so it opens freely. If you need more detail, see yourself enter speaking about something controversial (speech) with a group of friends (assembly), each carrying a different religious symbol (religion). #2 Bear arms: See the chair next to the door filled with guns. #3. Protection from soldiers camping out in your house: See yourself moving the sofa to block the door so soldiers outside can’t come in. #4 Unreasonable search and seizure: See yourself turn off the lamp next to the sofa, so nobody can see what is on the end table.  #5 Criminal protections: See the sofa filled with jurors who see you enter the room (grand jury), See a double image of the sofa and jurors (double jeopardy). See a judge sitting on the floor  instructing the jury (due process, fair trial). #6 Jury trial: See the glass table in front of the sofa (public transparency). See your lawyer spin the table round and round rapidly (right to a lawyer and a speedy trial). #7: Jury trial rights for federal crimes: See the coaster on the coffee table that looks like the U.S. logo jump into the lounge chair next to the sofa (federal charges require a jury trial too). #8 Protection against unusual punishment: See yourself convicted, and the bailiff tries to throw you into the roaring fire in the nearby fireplace. #9 Other non-specified rights: See yourself being free to click through TV channels on your remote. #10 State rights: See TV screen showing map of U.S. red and blue states during a Presidential election to remind you that it is the states that have all other powers not specified for the federal government.

Story Chain

Imagine a picture for each item you want to remember and mentally place it into a story that you make up. Example, to memorize the order of planets: You could image having a Mercury car run over your toe (Mercury), having a vein burst on top of your foot (vein for Venus). Then you fall down on the ground (earth). A “first responder”drives up in a Jeep (Jupiter), not an ambulance, and gives you a Mars (Mars) candy bar to make you feel better. You need to urinate, and he gives you an urn to sit on so you won’t be so conspicuous (sat and urn for Saturn). The urine rains down from you know where (you and rain for Uranus).  A second responder dressed like the god Neptune takes the urn and empties it into the nearby sea (Neptune). Then a stray dog that looks like Disney’s Pluto (Pluto) comes over to lick your face and comfort you.

While each of these seven techniques can be applied using words or language, the most effective approach is to convert words into mental images (like the Clinton image mentioned above). Once you create your own mental images, memorization becomes easier and more reliable—even fun! Some of these tactics may seem dumb or a little silly, but often the dumber and sillier they are, the more powerfully they promote memorization.

Memory Palace and Story Chain techniques are especially powerful. Successful “memory athletes” use them to compete in international contests designed to see who can memorize the most in the shortest amount of time. Without these techniques, their memorization ability would be no better than your own.

Two advantages of these memory aid techniques are not so obvious. One advantage is that these techniques exercise and develop your creativity. You must imagine mental pictures that work for you. The more you use these mnemonics, the easier it becomes to imagine useful images.

The other advantage is that you have to think about memory targets in order to construct relevant mnemonic images. As mentioned, the thinking itself is a powerful memory aid.

For each given memory task, choose the technique that seems to work best for you and that best fits the task. Make up your own scheme based on what is easiest for you to remember and what works best for you.

Once you construct a mnemonic for a given learning task, force yourself to recall it three or four times the first day. Repeat that forced retrieval once a day for the next four or five days. Remember: forced recall is essential to effective memorization. Moreover, recall is most powerful if you say it aloud, write it down, or draw it.

 

Next Lesson: Lesson 10.Learning from reading, lectures

Source: the eBook, Better Grades, Less Effort 

Lesson 8. Making Associations

 

We have all heard that the key to memorization is associating new information to be memorized with related information that we already have memorized. Why does this work? How does it work to form memories?

Perhaps the first point we should make is that associations can be made passively and unintentionally or with deliberate attempt. Let us examine each way in turn.

Unintentional Associations

Ivan Pavlov was famous for discovering this kind of associative learning. He called it conditioned learning. The term was later modified to “classical conditioned learning,” because another form of passive associations was discovered that was called “operant conditioning.”

You may have heard about Pavlov’s study of dogs. He was initially studying digestion, and collected saliva and stomach juices to see how they responded to food. What he observed, not surprisingly, was that when hungry dogs smelled or saw food, they anticipated eating it by secreting saliva and stomach juices. This did not have to be learned—it’s an unconditioned response. It was just a natural, built-in response. What was surprising was that when the dog caretaker entered the room, the juices were released, even if the caretaker was not bringing food. They had learned to associate this person with food delivery. The two things went together. So, Pavlov made a more formal test of this “conditioned learning” by pairing a different kind of cue with food delivery. For example, he might ring a bell and then the food was brought in. If that situation were repeated several times, the dogs started salivating when they heard the bell. They were now learning a new association: bell ringing meant that food was likely to show up.  In general, the key is to pair an unlearned response with an association cue and repeat the cue often enough that the brain learns that the two things go together.

Operant conditioning was spawned by the discovery of Edward Thorndike, a contemporary of Pavlov. Thorndike observed that learning occurs from realization of the consequences of behavior. That is, behavior that is followed by pleasant consequences is likely to be repeated, and behavior followed by unpleasant consequences is less likely to be repeated. Procedures for optimizing this kind of conditioned were developed later by J. B. Watson and B. F. Skinner.

Several principles of operant conditioning have been discovered:

1. Rewards seem to be more effective than punishments.
2. The animal must do something that can be associated with subsequent reinforcement, even though the animal has no foreknowledge of what it is supposed to do.
3. Learning complex repertoires can be developed in which the final desired behavior can be shaped through a succession of small steps in which elements of the final behavior are sequentially produced. As each step is learned, the trainer builds on that by adding the next logical small step.
4. Reinforcement needs to be provided each time the desired behavior occurs.

Intentional Associations 

Associations are much more powerful if you make them consciously and intentionally. This approach puts the learner in charge of her own learning. The learner gets to choose which associations are the most powerful association cues, and that likely varies from person to person.

The reason that intentional associations work to improve memory is that memories are stored as a network of related items. These items are part of a shared whole. Any one item serves as a cue for retrieving other parts of the memory network. Dragging out one item in the network often drags the whole network of memory items into conscious awareness.

Even so, certain principles apply here.

Pick Relevant Associations

Associations can be made with a person, place, object, situation, or emotion. Pick whatever works best for the item you are trying to remember.

Use Images

The most important act is to use images rather than words as the associational cues. Images contain detail in a way that is automatically associated with other elements in the image. Thus, it is especially important to select images that clearly and rather directly capture the essence of what you are trying to remember. Note the image used here of overlapping circles of slightly different shades of color. The point of association and shared relatedness is obvious.

You can make up your own images or use images that are already established for certain mnemonic systems. Mnemonics will be explored in detail in Lesson 9.

Why do images make the best associations?  

 

“A picture is worth a thousand words for scientific reasons: The brain processes visual information 60,000 times faster than text. Forty percent of all nerve fibers connected to the brain are linked to the retina. Visual information comprises 90 percent of the data that comes to our brain, suggesting that our neurological pathways might even prefer pictures over text.”- Quoted from https://www.planview.com/resources/articles/what-is-kanban/.

Make the Association Concrete and Vivid

Vagueness won’t work well. Make associations that are clear, distinctive, and clearly relevant to what you are trying to memorize.

Tie the Association to the Key Item to be Remembered

Suppose you have to catch an airplane at 2 A.M. in the morning. The number two is the key element. How can you link that to flying in an airplane? You might think of airplanes as have two wings. Suppose the flight is 4 A.M? You might think of a big jet with four jet engines. Suppose the plane leaves at 3? You might imagine looking into the cockpit and seeing three people (perhaps pilot, co-pilot, and navigator).

Make the Association Personal. Add strong emotions.

Relevance is key. Making an association personal gives it more relevance. Because emotions are processed in the same part of the brain that forms memories (the hippocampus), emotional associations become strongly embedded in memory.

Repeat a Newly Created Association Right Away

An association has to be encoded, just like an original item to be remembered. So, once you have created the association, repeat it several times right away, and then a few more times later in the same day.

 

Next lesson from “Memory Medic”  to be posted soon: Lesson #9: Mnemonics

Follow my “neuro-education” group on Linkedin at https://www.linkedin.com/groups/4883556/

To check out my four books on learning and memory, see my web site: WRKlemm.com

 

Lesson 7. Strategic Approaches

 

Different learning tasks typically benefit from a prior analysis of how to approach the task. You wouldn’t want to approach learning mathematics the same way you would learn history. Because it is not feasible to specify an approach of all possible learning tasks in this short lesson, I will suggest some more general ideas.

Steps

Step 1. Assessment. The first thing to do is over-all assessment of the task. That is, identify how much you already know, which parts of the task are not even understood at the moment, which ideas and factoids you can figure out, and which you will have to memorize.

Step 2. Tackle the confusion. For the parts you don’t understand, go to whatever teaching resource needed to explain it. Why try to memorize something you don’t understand? In fact, the very process of trying to understand helps to form memory. Commonly, multiple explanations are needed in order to find the one that “clicks” with your capacity to understand. Sometimes asking a question in a web browser field will take you directly to a site that explains what you are looking for. Sometimes Wikipedia works. Sometimes Kahn Academy works. Use what works for you.

Step 3. Isolate what requires memorization. Why memorize trivia? Why memorize information that you can easily figure out? Save your brain energy and power for difficult memorization tasks.

Step 4. Develop a task-specific memorization tactic and/or mnemonic. When confronted with a learning challenge, ask yourself, “What is the best way for me to memorize this? What associations and cues will help? What that I already know makes this easier to understand and remember? Can I develop a mnemonic that makes this easy to recall?

Some General Tactics

Encoding.

Encoding refers to conscious registration of the information you are trying to memorize. Perhaps you have heard of the phenomenon known as “inattentional blindness,” wherein you can see things but not realize your eyes have seen them, and your conscious mind is unaware of them. The point is that you are only conscious of targets of attention. Conscious awareness strengthens encoding. Be aware. Focus, focus, focus.

Encoding is also strengthened by thinking about relationships and associated cues that can be attached to targets of attention. Recall what I said earlier in the lesson on concept maps.

Spaced Learning.

Students like to cram for exams. They often do this the night before an exam, studying late into the night and thus making themselves sleep deprived, which only compounds their mental performance on the exam because sleep deprivation makes it difficult to think straight.

The more fundamental problem is that memory formation takes time. The process, called consolidation, is like wet cement: impressions take a while to set up. After all, long-term memory is stored in the gene activation, protein synthesis, and growth of new neuronal membrane and synapses. Spreading out the study allows time for these processes to occur and complete robust consolidation. Also, the passage of time allows for relevant new thoughts and associations to attach to the memory and thus strengthen it.

Applications.

Using what you have just learned, especially soon after initial encoding, enriches the encoding, creates new associational cues, and allows the learned material to stay active long enough in neuronal circuitry to enhance the consolidation process. When application occurs in contexts other than that of initial exposure, it creates new frames of reference cues and may even broaden the depth of understanding. Furthermore, new handles for retrieval emerge.

Forced Retrieval.

Many students study by “looking over” their notes and learning resource materials. This creates the illusion that they are improving their memory. However, research has shown that strong memories require one to force retrieval. For example, if you are studying flash cards, you look at the question side of the card and mentally quiz yourself on what you think is on the answer side of the card, before you look at it to check for accuracy. The reason that forced retrieval works is that each time you retrieve a memory, its encoding can be strengthened by adding new associations and thoughts. In any case, once retrieved, the memory has to be reconsolidated, which adds to the strength of information storage.

Deliberate Practice.

In lesson 12, I will explain this more completely. For now, just realize that effective study requires not only forced retrieval, but also awareness of what you are retrieving and error checking and correction as needed.

Mnemonic Devices. 

These will be covered in Lesson 9. They can be extremely powerful. People who compete in memory contests rely exclusively of mnemonics. These contestants are just normal people who do not have a natural “photographic memory.”

Flash Cards. 

In a study by Katherine Rawsom at Williams College, students studied 35 Swahili-English word pairs on flash cards. The students were asked to practice until they got the vocabulary correct using either the entire stack or five stacks of seven cards each. Researchers instructed students to study the flashcards until they had gotten each translation correct either once, five, or 10 times, before taking a final quiz a week later. Getting the stack correct five times was three times more effective for the final quiz score than getting the stack correct only once. Also, study of one big stack was better than five little ones.

Students had predicted just the opposite. They expected studying smaller groups of flashcards would be more helpful than studying the big stack, and they expected no real benefit from studying cards more than once. Those who had studied the small stacks expected to remember nearly 60 percent of words, yet they recalled only 17 percent. In general, students were incorrect in two ways: 1) they give too little value to learning strategies that are difficult (using multiple sessions on the big stack), and 2) they give too much value to strategies that were later documented to be less effective.

 

Next Lesson from “Memory Medic:” Lesson 8. Making Associations

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To check out my four books on learning and memory, see my web site: WRKlemm.com

Lesson 6. Concept Maps.

Facts and ideas can be mapped in ways that show how they relate to each other. The map drawing usually begin with outlined notes, because few people can think fast enough to construct a map in real time during a lecture or video. In simple mind mapping, basic ideas are stated within circles, forming word clouds, and arrows are drawn from “parent” to “daughter” clouds. A useful addition is to write in brief text along the arrows that explain what the relationship is, as illustrated in Figure 1.

Fig. 1. Simple concept map for the relationship of cells and their organelles. Cross-linking is not shown because it is not particularly useful for this simple information cluster.

Each circle object in the map can be expanded to whatever level of detail is required. In the map above, for example, from “History” you could add a circle for “Hooke” with a labeled connecting arrow saying “the first pioneer was.”

Think with Concept Mapping

Recall lesson 4, where we made the point that thinking about what you are trying to memorize makes the memory process easier and more reliable. Memory becomes easier when you think about the context and ancillary information associated with your memory targets. If the material you are trying to learn is complex, it often helps to convert your notes into concept maps. In concept maps, you draw circles or other geometric shaped word clouds to act as containers for key information, and then you think about how the various items in the circles relate to other items to create concepts. You draw connections among the various circles and write in a few words to state the nature of each relationship.

This process is like so-called mind mapping, except that concept mapping captures information as nodes in an interconnected network, unlike the tree-like structure of mind maps that have one central idea with multiple branches. Concept maps allow multiple cross-connections among the various idea nodes and typically emphasize multiple inter-dependent relationships among the nodes.

The basic task is to think about the relationships among the linked word clouds. A good practical way to automate thinking is to make concept maps as you read, listen to lectures or watch education videos. With pencil and paper write down key words in different locations on the page for major facts and ideas as you encounter them in the learning material and draw a circle around them. Then, perhaps after the lecture, video, or reading, examine each item one at a time and draw a line to any of the other items to which it is associated. Along each line, write in a few words to state what the relationship is. For example, you might link idea A with idea B with the description “makes me ask,” “led to the wrong idea that,” “leads to the truly original idea of,” or whatever might be appropriate. Note that comments work best if they are based on active verbs. This learning strategy is useful for several reasons:

1. Maps give the learner a “bird’s eye view” of the big picture.

2. Learners must engage with the material (i.e. be especially mindful) in order to draw the  

map of key concepts.

3. Learners have to organize information in meaningful ways, a process that requires them to  

think, which facilitates memory storage and retrieval.

4. Information is displayed spatially, which in itself facilitates storage and retrieval. 

Memorizing things by mentally relating them to their location in space promotes remembering because the part of the brain that forms lasting memories (the hippocampus) is also the part of the brain that creates subconscious mental maps of objects in space.

How to Make the Maps

As with creating regular notes, doing it by hand is more engaging and more likely to be memorized easily. However, with maps created by hand, you can’t move objects around; you must erase and write back in. However, that is less of a problem if you have a computer with draw capability. Another option is to create an initial step of placing sticky notes on a wall and moving them around physically to see what is the best spatial layout.

Map construction can be facilitated by computer. There are many elegant computer programs, and some quite satisfactory programs are free (search Google for “free mind maps”) (I like X Mind). Most programs make it easy to move ideas around in the map and make multiple, non-linear links. Not all programs allow elaboration along linking lines, and you may have to write it in by hand. 

Actually, I think maps are a better memorization aid if they are hand-drawn, because that makes the process more personal, more flexible, and perhaps more engaging. If you change your mind about something you put in the map, you either have to erase it or re-draw the map. One option is to draw the map by hand at first and then re-do it later by computer.

Too much text annotation adds to clutter. Clutter is inevitable with broad topics that involve many ideas. Some computer programs create a map that requires a huge sheet of paper to get printed, and you can’t get it all on an 8.5 x 11 sheet without compressing the text so much it is unreadable. The solution here is to make multiple maps, one an overview of the whole thing (main ideas and first- or second-order sub-topics. Then each major sub-idea can have its own map.

Maps to Study By

Maps used for study purposes need to be kept compact and simple. Memorization is facilitated by using icons or drawings to represent ideas is more effective than a lot of text. Some computer programs even have a library of icons you can select. Just make sure the icons are effective representations of the text they substitute for. You might want to use text and a representative icon, but base your memorization rehearsals on the icon.

Concept maps not only direct you to think about and organize academic content, they also promote memorization because concepts are laid out in spatial arrays.

The study emphasis should be on the relationships. That will automatically help memorize the factoids in the word clouds and stimulate your thinking to develop new understanding and insights. Also, make it a point to note the spatial location of key word clouds.

Lesson 5. Taking Notes

If you have a serious need to memorize, you usually must make notes. Just what is it that I think is valuable about note taking? First and foremost is the requirement for engagement. Students must pay attention well enough to make decisions about the portion of the learning material that will need to be studied later. Paying attention is essential for encoding information, and nobody can remember anything that never registered in the first place.

Note taking requires thinking about the material to decide what needs to be captured for later study. This hopefully generates questions that can be raised and answered while working your way through the learning material. Sadly, in the last few years I notice that in my college classes, few students take notes. It is as if they think they can remember everything (they can’t). The cause may be that teachers tend to hand out prepackaged notes. I object to this practice, because it reduces the level of student engagement and thinking. If such notes are distributed, the notes should be in a skeleton form that just provides an organized framework for students to construct their own notes.

Handwritten notes have special advantages. If done in pencil, which I recommend, items can be erased or re-arranged. You can draw diagrams and pictures, which provide visual images to strengthen memorization and control spatial layouts. Using different layouts for each page gives each page a visual uniqueness that facilitates memory. A well-established fact about memory is that spatial location is an important cue for encoding and recall. Where information is provides important cues as to what the information is. The spatial layout of script and diagrams on a page allows the information to be visualized, creating an opportunity for a rudimentary form of photographic memory, where a learner can imagine in the mind's eye just where on the page certain information is, and that alone makes it easier to memorize and recall what the information is. This effect occurs because the hippocampus, which initiates memorization in the brain, also maps spatial location.

This brings me to another important point about visualization. Pictures are much easier to remember than words. Hand-written notes allow the student to represent verbalized ideas as drawings or diagrams. If you have ever had to learn the Kreb's cycle of cellular energy production, for example, you know how much easier it is to remember the cycle if it is drawn rather than described in paragraph form.

I will explain the several methods for note taking.

The type of note format should vary with the nature of the learning material and the level of expected memorization of the content from reading material, videos, and lectures. All these forms can be produced with computers, but I don’t recommend using a computer. Researchers Pam Meuller and Daniel Oppenheimer provide clear evidence that handwritten note taking produces better learning in college students. Remembering factoids was about the same in both groups, but memory for concepts was distinctly better in students who took notes by handwriting. In their report of three studies, cited over 1,000 times, learning efficacy was scored in two groups of students, one taking notes on a laptop computer and the other by handwriting.

When tested on learning material, students who used handwritten notes that they studied scored significantly higher than students using laptops, including agile touch-typists who took vastly more copious notes. Handwriters took fewer notes overall with less verbatim recording. There are many possible explanations for the superiority of handwriting, beginning with the "less is more" idea in which too much information produces cognitive overload.

For touch typists, taking notes on a laptop is a relatively mindless process in which letters are banged out on autopilot. A good typist does not have to think. Notably, when the typing students were told to avoid verbatim notes, they still did it. This suggests that there is something about typing that leads to mindless processing.  Handwritten notes involve more thought, re-framing, and re-organization, all of which promote better encoding, understanding and retention. The manual act of handwriting requires more engagement with the subject matter. Finally, handwritten notes capitalize on the use of drawings and of personalized spatial layout of the notes. Memorization involves not only what the information is, but where it is spatially located.

Outline
These notes are arranged in terms of topic, sub-topic, sub-subtopic, and so on. Each item is on a separate line and is indented. Each topic or sub-topic can be numbered and lettered. Here is an example for information on cell biology (Figure 1a):

Fig. 1a. Common numbering way of taking outline notes.

The numbering and lettering can become distracting. I prefer to use headings, sub-headings, sub-subheadings. This is readily automated in a word processing by using a styles menu (Heading 1, Heading 2, Heading 3, and so on). Here is an example (Figure 1b):

Fig 1b. Common way of taking notes using heads and subheads

Outline notes are most useful when you must capture large amounts of information quickly. If you don’t have much time to think, outlines are usually easy to construct because that is the way most information is presented in lectures, videos, and textbooks. A presenter typically presents a main thought, then explains it with some detail, and then moves on to the nest main idea.

For more understanding and to promote memory, it is important to think about the words that appear in an outline. Other note-taking methods may require reconstructing the initial outlined information in a different format, and this requires some thinking.

Charts

Here you create a table with separate columns for each topic (Figure 2). You typically have only one blank row in which you put the information you want to memorize. Depending on the subject matter, you might want to segregate facts and concepts.

Fig. 2. Chart format for note taking

Cornell Method

Here, you use a table that captures key facts or concepts in a different spatial layout (Figure 3).

Fig. 3. Cornell note format

Matrix Notes

Matrix notes place information in a table, where the columns might be categories of information and the rows contain items within each category. The columns represent one category of information (such as topics and the rows another, such as items. Here is the basic idea (Figure 4):

Fig. 4. Matrix note format

Matrix notes promote an overall “bird’s eye” view of relationships among ideas. This requires more thinking and may not be possible in the real time of watching a video or lecture. However, the method is very powerful, in large part because it requires you to think deeply. Such thinking may also provide insights that would otherwise not occur. Matrix notes can be more comprehensive and force thinking about content in a wide range of contexts. Matrix notes are most useful when cross-cutting relationships need to be recognized and clarified.

The advantages for learning are that the learner conceptualizes the ideas in the process of constructing the matrix. Because ideas are presented in one view, preferably in units of one page at a time, it is easy to see cross-cutting relationships that otherwise are not so apparent. Such organization is an aid to stimulating insight. In addition, the fixed spatial layout is a memory aid.

The process might usually begin with outlined notes, because few people can think fast enough to construct a matrix in real time during a lecture or video.

Next lesson: Lesson 6. Mind Mapping

Follow my “neuro-education” group on Linkedin at https://www.linkedin.com/groups/4883556/

To check out my four books on learning and memory, see my web site: WRKlemm.com

Sources:

Klemm, W. R. (2012). Memory Power 101. New York: Skyhorse.

Mueller, Pam A. and Oppenheimer, Daniel M. (2014). The pen is mightier than the keyboard: Advantages of longhand over laptop note taking. Psychological Science.  https://doi.org/10.1177/0956797614524581