Lesson 14a. Lifestyle Matters: Effects of Stress

School is stressful. Students also commonly must deal with emotions arising from boy-girl problems, over-bearing parent problems, bullies, worries about school, frustrated attempts to be popular, and their future. College students have enormous adjustments to make in the transition from leaving home and the cloistered environment they were used to in high school. Older adults returning to school have to cope with paying the costs, finding time for the family, and for the job.

The aspects of lifestyle that have the greatest direct effect on learning deal with stress, exercise, and sleep. This post explores stress.

Stress

 All learning at some point involves stress. Acute stress, which releases adrenalin, can be a good thing. It pumps you up and promotes focused attention. However, chronic stress impairs learning. When students struggle to improve, chronic stress ensues and creates a vicious cycle in which the stress impairs performance, which produces still more stress.

I have written numerous science-based posts on the bad effects of chronic stress on the brain. To review these and type “stress” into the search field of this blog.

Scientists have long known that chronic stress is bad for your brain. Adolescence seems to be an especially vulnerable time. Not only is the adolescent brain still being built, the brain is being re-built during the teen years.

Human imaging studies show the cerebral cortex shrinks during adolescence. In a recent study of "adolescent" rats, researchers found that the cortex shrinks in both males and females, and there is a loss of neurons in the ventral prefrontal cortex, the part of the brain humans use to make rational decisions and do higher-level thinking. More loss occurred in females than in males.

Until now, the reason for brain tissue loss has not been clear, but we may be able to explain it by considering the stresses that afflict teenagers, who routinely endure peer dismissiveness and bullying. Numerous animal studies show that stress impairs brain function and memory. Even single bullying exposures can be damaging.

Similar results have been reported in multiple humans studies. Here is a very real example of stress-related memory impairment from human bullying in schools that few may realize. In the study, young healthy men were tested for their ability to recall lists of 10 positive, 10 neutral, and 10 negative words. The men were given two minutes to learn the lists and were then tested immediately. Thirty minutes later they were given a psychosocial stress which included a fictitious job interview in front of live interviewers and counting backwards in steps of 17 in front of judges. Control groups did a five-minute speech and did the same counting, but not in the presence of judges. The next day, subjects were tested again (delayed recall) 10 minutes after cortisol injection. Other psychological tests were administered and the amount of cortisol in the saliva was measured at several key points in time.

The stressed group had elevated cortisol levels even though the stress was mild. Recall of both negative and positive emotionally arousing words was impaired, but there was no effect with neutral words. These effects could not be attributed to decreased attention or working memory span, which memory tests showed were not affected by the stress. Providing cues for recall eliminated signs of the stress effect on emotionally charged words.

So, this study shows recall of emotionally charged information can be impaired under stressed conditions. Imagine how great the deleterious effect of stress could be in situations where there is real stress, as in witnessing car accidents, crimes, or dangerous situations in which recalling what happened could be very important. Studies like this are consistent with many real-life observations with eye-witness accounts, where what is remembered may well be false.

The most obvious stressor for students comes from examinations. “Test anxiety” causes students to choke, typically failing to recall information they know they know, but just can’t dredge up under stressful test conditions.

A couple of studies have shown that test scores rise if anxious students are allowed to write about their test worries for 10 minutes just before a test. It also works to write down attributes of successful problem solving just before a test.

Why does this writing alleviate test anxiety? One study showed that such writing increased student engagement in the test. That is, students who were anxious were distracted by their anxiety and did not fully engage in solving the problems in the test.

Such writing before an exam is only beneficial for students who suffer from test anxiety. Other students don’t need this, and may in fact do worse by such a distraction. The best way to deal with choking on tests is to: 1) thoroughly know and understand the material being tested, and 2) develop long-term confidence by a string of test successes. If you truly “know your stuff” and believe in your ability as a student, there is no justification for test anxiety.

 Next Lesson: 14b. Lifestyle Matters: Exercise

Stress Sources:

Merz, Christian (2010). Stress impairs retrieval of socially relevant information. Behavioral Neuroscience. 124 (2), 288-293.

Newcomer, J. W., Selke, G., Melson, A. K., Hershey, T., Craft, S., Richards, K., et al. (1999). Decreased memory performance in healthy humans induced by stress-level cortisol treatment. Archives of General Psychiatry, 56, 527–533.

Vogel, Susanne (2016). Learning and memory under stress: implications for the classroom. Science of Learning 1(1), 16011. DOI: 10.1038/npjscilearn.2016.11

Young, A. H., Sahakian, B. J., Robbins, T. W., & Cowen, P. J. (1999). The effects of chronic administration of hydrocortisone on cognitive function in normal male volunteers. Psychopharmacology (Berl), 145, 260-266.

Zoladza, Phillip R., Ashlee J.Warneckea, Sarah A.Woelkea, Hanna M.Burkea, Rachael M.Frigoa, Julia M.Pisanskya, Sarah M, Lylea, Jeffery N.Talbotb (2013). Pre-learning stress that is temporally removed from acquisition exerts sex-specific effects on long-term memory. Neurobiology of Learning and Memory. 100, 77-87 https://doi.org/10.1016/j.nlm.2012.12.012

Make neuroscience religion’s ally

TriuneBrain Trailer Low Res

Especially Difficult Memorization Tasks. Lesson 13.

 

Some memory work seems more difficult than others. We have already covered learning math. Here we will consider such tasks as expanding vocabulary, learning foreign language, and remembering dates and places, and music.

Vocabulary

The obviously important factor in thinking is vocabulary (or, in math, number fluency). Everyday thought is conducted with words. A limited vocabulary limits the range and rigor of thought. Sadly, young people generally have limited vocabularies. I see this every year even with upper-division college students in a course where I require them to write essays. They frequently use words that don't match the ideas they are trying to convey and fail to use words precisely. Children who grow up in culturally deprived homes are handicapped because they are exposed to the limited vocabularies of their parents. Teachers and parents should make it a point to expose children to new works and to instill a commitment to using words precisely.

Research suggests that some review sessions need to be widely spaced by two months or more from the initial learning. The amount and spacing of foreign language vocabulary learning has an enormous effect on how well the material is retained years later.  In one study, 300 English-foreign word pairs were studied with either 13 or 26 relearning review sessions at intervals of 14, 28, or 56 days. Retention was tested for 1, 2, 3, or 5 years after the end of training. Although longer intervals between learning a review impaired acquisition slightly, this was offset by substantially higher long-term retention.  Thirteen review sessions spaced at 56 days yielded retention comparable to 26 sessions spaced at 14 days apart.

One experiment related to flash cards has examined the role that retrieval had on the ability to recall that same material after a delay of a week. College students were to learn a list of 40 foreign language vocabulary word pairs, which were manipulated so that the pairs either remained in the list (were repeatedly studied) or were dropped from the list once they were recalled. It is like studying flash cards: one way is to keep studying all the cards over and over again; the other way is to drop out a card from the stack every time you correctly recalled what was on the other side of the card.

Number of self-testing events and their spacing are influenced by the number of retrievals that are correct. It helps to repeat self-testing on cards already correctly recalled. In one study where students studied flash cards of 35 Swahili-English word pairs, 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 than getting the stack correct only once. Also, study of one big stack was better than five little ones.

A related study examined the correctness effect when college-aged students were asked to study for a week a pack of 48 paired Swahili vocabulary words with their English translations. To facilitate learning, students were taught to use a cue (word, phrase or concept) to link both words of a pair. Students controlled how many times they felt they needed to repeat study of a word pair until they reached 1, 3, 6, or 9 correct retrievals. Immediately after finishing the learning, students gave an estimated rating of how well they thought they had learned the material. On an examination given one week later, gains in correct answers were larger with more correct retrievals during the study period.

Student predicted judgments of their learning are important in real-world study because such judgments govern how long students will practice what their assigned learning. Judgment of learning success was found to depend on the number of self-testing events, their spacing, and the number of correct retrievals.

Another study established that best learning occurs when students can correctly recall items multiple times during the initial learning experience. In that study, students studied 70 Swahili-English word pairs either 1,3,5,7,8, or 10 times during encoding. For example, a cue and target pair were initially presented to study for 10 seconds. Then during practice the cue was presented and learners were given eight seconds to show they could recall the target. Incorrect recalled targets were given a four-second restudy opportunity before moving on to the next word pair. Pairs continued to be rehearsed until they reached the assigned level of correct remembering (1, 2, 3, etc.). The subjects were divided into two groups, one taking an exam within 25 min after study and rehearsal and the other one week later. Results revealed best final exam performance when the test-item presentation intervals were long and when the final test occurred 25 min after the study and rehearsal. Almost none of the word pairs were learned when the learning occurred when the intervals were short and the final test was delayed for a week. The best final learning occurred when the initial learning practice retrievals were correct and when more time elapsed between each recall attempt (6-minute lag versus 1-minute lag).

There are some other general strategies for building vocabulary. I have a couple of ideas based on memory principles.

            Learn word prefixes and suffixes. Prefixes and suffixes are great aids. “Pre” suggests before or ahead of time; “ism” suggests a state of being, “re” suggests back or again, and so on.      

            Learn word families. Many words come from the same family. If you know what one word in the group means, you can get the general idea for the others from the context in which they are used. The other words will be easier to remember because they are similar to the word you already know. Here are some word groups.

            despise, despicable, despot, despotism, despoil

            habit, habitat, habitation, habitual, habitué

            jet, jettison, jetstream, jetty, jet set, jetlag, jetsam

            line, liner, lineman, linear, line drive, line-up

            parent, paternal, pater, patriarch, paternity, patrician, patricide

Create images for strange new words. Examples:

            Gazebo: see yourself staring (gazing) at the ugliest boy (bo) you ever saw standing in a             building that only has a roof, no walls. Feel disgust.

            Adumbrate (meaning incomplete understanding or explanation): see a “dumb brat” with a dunce cap, sitting in the corner partially hidden by a screen. Sense the pain he must feel at being so ostracized and tries to hide.

            Daguerreotype (an early photographic process on metal plate): visualize a picture on a sheet of metal and you have stabbed it with a dagger because you hate it so much.

            Perspicacious (meaning especially insightful): see yourself working up a sweat (perspire), scratching your head with question marks around it, then jumping up with a eureka  moment when you realize you figured it out.

Foreign Language 

In this globally interconnected world, many people want to learn a foreign language. In my own case decades ago, I stumbled through Russian and French because they were required for my Ph.D. Most recently, I am trying Spanish, because I live in Texas.

            Some things I know that help learning foreign language is the use of flash cards for vocabulary (using images, not just the words).

            Another thing I know is the value of strategic approaches and planning. For example, I first have to confront my negative attitude (negative attitudes impair learning). I have a negative attitude about the irrational things in language. Take the gender business in Spanish and many other languages. Why does everything have to have a sex identification, like male and female endings for inanimate objects? That is just plain stupid! Irregular verbs are another problem. When I was in high school, I learned Latin, which wasn’t so bad, because it was a much more orderly language than the modern languages that “evolved” from it. Latin wasn’t broken. Why didn’t people leave it alone?

            My first strategic realization was that I had to get over my pique. Who was I punishing with my negative attitude? Certainly not the people who created the irrationalities in the language. No, my attitude would be a de-motivator for me to learn. So, I tell myself, “Get over it.”

            Next, I think about some basic principles that might expedite my learning. You don’t have to be a professor of modern languages to know that certain key components in language include the following:

1.       Meaning of words. Here, try to recognize cognates (words similar to English words you already know. For other words, try to think of mental images that represent the meaning.

2.       Gender identification. Fortunately, you can usually predict whether a word is male or female just from the meaning of the word. Most macho-type words are male; soft, feminine type words are usually female. Unfortunately, there are exceptions, which you just have to memorize by brute force.

3.       Verb conjugation. Look for patterns. All “regular” verbs have the same pattern. In Spanish, all verbs end in AR, ER, or IR. The conjugation pattern is similar. For each, you drop the infinitive ending and add endings to the stem of the word. For AR words, the ending is either o, as, a (singular) or amos, ais, or an (plural). For ER words, the endings are o, es, e (singular) or emos, eis, en (plural). For IR words, the endings are o, es, e (singular), or imos, is, en (plural). Even irregular verbs have generally predictable patterns, except for a couple of endings.

4.       Counting. Here, again, look for patterns. In Spanish you have to brute-force memorize the first 19 numbers, but thereafter predictable patterns emerge.

5.       Articles, like “a,” and “the.” In Spanish, you only have to remember “un” for “a.” But, since the article has to be a gender match to the word it refers to, you have to add an “a” (una) to refer to female words. If you are referring to a definite person, you must use “el” or “la,” depending on the person’s gender. Plural references add an “s” (as in “los/las”)

Pronouns, like I, you, he, etc. In Spanish, there is a definite pattern that includes gender recognition and singular vs. plural. It becomes easier to remember if you organize words in a table.

	Female	Male	Male or Male/?
This That These Those	esta      esa       estas esás	esto or este eso or ese	estos esos

 

Well, you may not want to learn Spanish, so I won’t expand further. My point here is that learning is greatly facilitated when you look for patterns. Memorize the patterns, and it is easier to memorize the specifics.

            The point is this: structure your learning material in ways that work best for you. Develop a strategy. Look for patterns. A strategic approach should also include developing ways to categorize things in the most useful way for memorization.

Dates and Places

There is a number coding system that makes it easier to remember numbers. This is especially useful for remembering dates. The code is as follows:

The principle is to first convert an integer (0 through 9) to a letter that is a consonant. Then insert vowels between the consonants to create a word that can be imaged. Words are constructed by insert vowels, which are neutral and do not affect the scheme. Think of vowels as wildcards. So, in the example above, 44 stands for “rower,” with the word constructed by inserting appropriate vowels between the two r’s that represent number four.

A set of rules determines how to construct number-associated images. The rules dictate what letters and sounds go with numbers 1 through 0, as follows:

 

Number — Letter or Sound — Mnemonic Aid

1 — t or d — each letter has one down stroke

2 — n or kg/gn — two down strokes (kn/gn have same sound)

3 — m — three down strokes

4 — r — last letter of word “four”

5 — L — Roman numeral  for the five in 50

6 — j,  ch, sh,   soft g — reverse J looks like 6,

7 — k, ck, hard g or c — attach a flipped 7 to a straight line on its left

8 — f, ph, v — F joined with another upside down F

9 — p or b — backwards p or a rotated b look like 9

0 — z, s, soft c — Z as in zero, soft c as in cent

 

It may seem like a lot of trouble to memorize these rules, but once done, it gives you a lot of memory power. You can construct all sorts of images based on these rules (see number-image list below for 1 to 100).

The words, all nouns, that you make up from these consonants can be a single word that includes all the consonants or several words that are sequentially linked. The number system can be useful for long-term remembering of some things, like dates, for example. If you wanted to remember that the Declaration of Independence was signed on July 4, 1776, you could build a memorable picture as follows: a number code for 1776 could be dog (17) and cage (76). Couple this with whatever image comes to mind for the Declaration, such as the liberty bell. Now picture your dog sleeping in its cage crate, being awakened by the ringing bell. For the link to July 4, you might want to add a firecracker image that goes off after the bell rings, like a Pavlov experiment where the bell rings, unconditioned firecracker stimulus goes off, and dog responds with jumping around in his cage.

The other use for this system is to create an easily remembered peg list. To convert this to a numbered peg list, you make up words using these rules. For example:

 

1  tie 2  knee 3 ma (mom) 4 rye 5 law 6 jaw` 7 key 8 ivy 9 bow 10 toes 11 tot 12 tin 13 dam 14 door 15 tail 16 dash (-) 17 tack 18 dove 19 tub 20 nose

21 nut 22 nun 23 Nero 25 nail 26 nacho 27 neck 28 knife 29 knob 30 mice 31 mat 32 moon 33 mummy 34 mower 35 mule 36 match 37 mug 38 movie 39 mop 40 rice

41 rat 42 rain 43 ram 44 rower 45 roll 46 roach 47 rock 48 roof 49 rope 50 lace 51 lot 52 lion 53 lime 54 lure 55 lily 56 leash 57 log 58 lava 59 lip 60 cheese

61 sheet 62 chain 63 jam 64 cherry 65 jail 66 judge 67 check 68 chef 69 ship 70 case 71 cot 72 coin 73 cam 74 car 75 coal 76 cage 77 cake 78 cave 79 cap 80 fez (hat)

81 foot 82 phone 83 foam 84 fur 85 file 86 fish 87 fog 88 fife 89 fob (watch fob) 90 bus 91 bar 92 bone 93 bum 94 bear 95 bell89 fob (watch fob) 90 bus 91 bar 92 bone 93 bum 94 bear 95 bell 96 beach 97 book 98 beef 99 pipe 100 daisy

 

Although the list has 100 pegs, they are relatively easy to memorize because they are constructed by a rule. If you know the rules for converting numbers to letters, you can even generate your own word peg in case you forget.

There are even more powerful uses of this code than remember numbers or dates. Suppose you are trying to memorize a textbook, page by page. Suppose on page 47 (rock) the page explains alpha rhythms, showing an EEG trace, and pointing out that they occur mostly when eyes are shut and that alcohol and sedatives lower the frequency. So, to remember this, you visualize a rock floating on ocean waves that look like the graphs you saw in the book. You only see this image when you shut your eyes. Imagine opening your eyes and the rock/wave image goes away. Shut your eyes, see the rock/waves again. Then imagine drinking a beer, and the waves get larger and slower (i.e., fewer of them).

You could go through a whole textbook like this. How would you deal with several textbooks? This problem, not easily solved, is how to un-remember what is on the pegs. After all, you use the pegs over and over again for different items. Actually, this did not seem to be a problem when I was using this system to memorize magazine content. Clearly, this system works best only for items you just want to remember for a short while.

Music

Music is hard for most people to learn. You not only have to memorize the notes, their timing, and sequence, but you have to train the body parts like fingers, lips, and tongue to execute the notes.

            “You’ve probably heard the old joke: “How do I get to Carnegie Hall?” “Practice, practice, practice!” Well music practice does take time, and it’s probably not a satisfying answer for people looking to learn music quickly. But surely there are techniques and strategies to expedite the process. Here’s one method I’ve created using related memory principles:

1.     Skim the whole score to identify hard and easy parts and phrases that repeat. Then start at the beginning, memorizing in chunks, one or a few bars at a time, depending on the capacity of your working memory. After memorizing a bar or phrase, see if you can play it without peeking. Musicians do not learn a new piece from beginning to end all at once. They often start at the beginning of a piece and learn a small section until they get it right. Then they learn the next piece. Then they practice stitching the pieces together. They repeat this process until they get to the end.

2.     Memorize the mechanical acts needed to play the notes (keys on a piano, valves on a clarinet, etc.). Learn one hand at a time. Look at the hands and keys while playing.

3.     Play what you have just memorized from memory and repeat until you feel it is mastered. Play one hand at a time, then play with both together. Don’t peek at the score until after you have played the section. Check for any errors in your recall.

4.     Play the chunk slowly at first, then test the tempo by playing with a metronome.

5.     Move to a different chunk and repeat steps 1-3. Add one bar or phrase at a time. Mark sections of the score as they are learned.

6.     Join the latest chunk with those previously learned and play from the beginning.

7.     In the next practice session, rehearse what was learned in the previous session before moving on to new material.

Sources:

Bahrick, H.P., Bahrick, L.E., Bahrick, A.S., & Bahrick, P.E. (1993). Maintenance of foreign language vocabulary and the spacing effect. Psychological Science, 4, 316–321.

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

 

 

Delibrate Practice. Lesson 12

 

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

Photo by Jonathan Chng on Unsplash

We will learn how to implement these principles in this lesson on Deliberate Practice. Anders Ericsson and his colleagues came up with the idea of deliberate practice during the 1990s, based on their study of musicians. The researchers saw that deliberate practice requires considerable time investment, but it is more than just repeating what you are trying to master. It is not “drill and kill.” It is practice in which you:

·       Have a clear long-term objective in mind.

·       Plan what you need to do in detail.

·       Monitor how you executed the plan, with attention to specific details.

·       Noticing what to avoid and what to repeat in the future.

·       Apply corrective feedback to adjust the plan if needed and remind yourself what you need to do different next time.

·       Affirm and reward yourself for progress.

·       Get coaching from an outside source, like a teacher.

·       Keep raising the standards for acceptable performance. 

When I transitioned from a D student in the 4^th grade to an A student in the 7^th grade, I think the change was made possible through deliberate practice. Though I did not understand much about deliberate practice, I did intuitively use some elements of it. Penmanship class was likely the turning point, because deliberate practice is baked into the learning. When I looked a drawing of a cursive “a” and tried to duplicate it, the results were tangible and immediate. I had to think about where my drawing missed the mark and what I needed to do to make it look better. I had to keep repeating the process until what I produced looked like the instructional example. Aside from the utility of learning cursive, this may be the most beneficial example of teaching penmanship in school. Few schools do that these days.

 

Until you have mastered the learning goal, deliberate practice is a cyclic process that is repeated again and again. Most everyone has had this kind of learning experience at some point, usually when they are trying to perfect some kind of physical action. If you were in school band, for example, you used deliberate practice to memorize sheet music and master your instrument. If you were in a sport, you used deliberate practice in perfecting the movement skill sets.

If you did not have these experiences, here is one you can try right now: To perfect learning how to stand on one foot in a yoga pose, you could just do it repeatedly without thinking about what you have to do to make it work. Try it. You will see that does not work well. Now try it again, focusing on a visual spot far away and think about what muscles in your foot you have to activate to keep you balanced. These deliberate tactics will train you much faster to master this task.

Deliberate practice is not limited to physical movements. It is equally applicable to mastering school work. The practice objective might likely be to perform better on exams or to develop competence you know will be required in later courses.

Study sessions need to be strategic. That is, at the time you are studying, you need to think about what you need to do to make your memorization better. You may need to pump up your motivation level. You will need focus and self-discipline to work outside your comfort zone. Perhaps you need to invent better mnemonics. You need to think about how often to repeat self-testing forced recall. You need to contrive ways to apply what you are trying to learn. You need to take practice quizzes and solve related problems. You need a way to check on the completeness of your understanding. You need to have a way to check on the correctness of your recall and establish criteria for satisfactory mastery.

Next Lesson:

Especially difficult learning tasks

Sources:

Clear, James  (n.d.) Deliberate practice and how to use it. https://jamesclear.com/deliberate-practice-theory

Keep, Ben (n.d.) Deliberate practice in the classroom. The Learning Curve. https://www.the-learning-agency-lab.com/the-learning-curve/deliberate-practice-in-the-classroom

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