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Apprehending Algebra by Daniel Vogt

Chapter 1 – Preparation and Practices for Success in Mathematics

 

A child asked, “Why do we go to school?” Another answered, “So we won’t be stupid.” And largely, people grow to equate stupidity with ignorance. Is this what they mean when they call themselves stupid? Is this one of the reasons why they are so ashamed of not knowing something? – John Holt

 

Ignorance is not a thing of itself, but only the absence of knowledge; and it is of a peculiar nature, once dispelled it is impossible to reestablish it. – Thomas Payne

 

A very wise and intelligent man was asked, “How do you know so much about everything?” He answered “By never being afraid or ashamed to ask questions about anything of which I was ignorant.”  - John Abbott

 

I. Understand how the learning process works for the apprehension (learning) of mathematics.

 

A serious student of mathematics will find that learning comes through exposure and reinforcement. In an academic setting an instructor will introduce and discuss concepts or processes that form a foundation to build upon with later concepts and processes. Generally, whenever you work with anything for the first time you encounter the most difficulty. And information might make sense in the class discussion, but on your own you realize you have not retained all the logic and details that were present when the instructor was involved. You can think of this as pouring a layer of concrete, and the more complete your understanding the more solid this layer; however, it is virtually impossible to gain 100% fullness of all the complexities and intricacies to each concept or process – and these missing details create holes in a layer of foundation (and it is these “holes” that cause confusion or uncertainty when a person is working problems applying these concepts or processes). One of the purposes of homework problems is to reveal these “holes” so a student will question, resolve these questions, and obtain a fuller understanding of the concepts and processes related to math (hence, some of the “holes” are filled). Additionally, as these concepts and processes are applied later to new ideas and applications the discussion subtly reviews and reinforces the details regarding these previous concepts and processes, sometimes including new insights about them as well, and this also fills in some of those “holes”; and yet, since the focus is on the new material with its own set of “holes” the student is often in a continual state of feeling a lack of comfort or confidence regarding the mathematics. Still, as a student moves onto subsequent classes the material from prior classes seems so much easier – this is the result of working through the initial exposure then reinforcing through repeated review of the concepts and processes (thus goes the process of learning). It’s unfortunate so many students feel a lack of talent in learning math because they are looking at where they are (with all its difficulties and uncertainties) instead of realizing how much they’re able to do (ie how capable they are) when they look back over the earlier places they have been (and how easy that math looks now).  

 

II. Accept the need to change one’s perspective of learning, methods of learning, and communication skills.

 

Humans are natural learners with innate curiosity, especially as children, but each of us must learn how to be a successful academic student; and since learning in a structured school setting is different than learning by life experiences many people struggle with acquiring all the skills required for getting the most out of the education offered from an institution of learning (and, in particular, maximizing the learning they can attain from instructors who have gained and provide expertise in various subject areas).

 

Take for example Allen - he attended secondary school and his only approach to learning was to watch and listen assuming information, concepts, and insights would then be learned solely by observation. In the math class, he followed the instruction and most things made sense; however, on the exams (especially as concepts became more complex and abstract) he often struggled – many times he’d have a problem about which he was uncertain or did not remember. And yet, from an educational perspective this predicament is understandable. In classroom instruction the math teacher does a majority of the thinking; steps in a process are presented in a logical sequence with explanation often provided orally as the presentation/lecture proceeds; and even when a student is asked to provide the direction that person has support because if he/she is incorrect or cannot remember the instructor is there to provide guidance and assistance. So, since the lecture makes sense the student leaves with the feeling of understanding. However, as a student later attempts to do the homework that person must recreate the logic of the process from his/her own memory and understanding - and if that student makes a mistake, begins going in the wrong direction, or doesn’t understand or has forgotten part of the process then that person can wander indefinitely the wrong way or in confusion/frustration. Still, some of this difficulty is to be expected - a student is in class for only a few hours each week (of which only a small fraction is dedicated to reinforcement of taught ideas), and he/she is bombarded by a multitude of distractions the rest of the week (a typical college class meets 4 hours a week which leaves 164 hours Apprehending Algebra by Daniel Vogt

outside of class, and even if a student works the suggested two hours on homework for every one hour of class that still only adds another 8 hours resulting in just 12 hours of the 168 hour week – that’s only about 7% of his/her time to perfect understanding and 93% of the time for the brain to be dwelling on other thoughts … if this was applied to war [we can consider this is a war against ignorance] and we were told we’d only be using 7% of our available resources in the battle, then we’d have to say that the odds were against us winning).  Moreover, homework exposes many of these gaps in understanding as the student discovers how much of the instruction has actually been retained, provides opportunity for the student to acquire any of the logic or definitions/laws that might be missing, and allows the student a means for attaining comprehension of the concepts and processes needed to do the problems. The student isn’t expected to just determine answers, but to communicate the method(s) for achieving those answers - just having a correct answer doesn’t necessarily indicate a person understands the logical processing or is applying the concepts correctly; take the following two examples (actual student work) to demonstrate.

E.G. #1  -  Simplify:  1 + 23

                                      = 33

                                      = 9

 

 
 

E.G. #2  -  Solve:  3 + 2x = 7

                                    5x = 7

                                  - 5    - 5

                                  x = 2

 

 
 
 

 

 

 

 

 

In both problems the final result is correct, but in each case the student has not applied concepts correctly (every step has been done incorrectly). Fortunately, each student managed to communicate the step by step thinking so the logic in their process (even though wrong) is obvious (in e.g. #1 the student replaced 1 + 2 with 3 and in the next line three 3’s gave the person 9, in e.g. #2 the student saw 3 + 2 and replaced it with 5 then in the next line subtracted 5 on both sides to cancel out the coefficient leaving just the variable x). Since the work was complete errors can be recognized, examined, and corrected.  Allowing students to skip steps can cheat a person out of being made aware of any errors or omissions in his/her understanding so these problems can be corrected and the student’s mathematical foundation can be made more secure before building upon it and proceeding; but it’s also unfair to the students working hard to master these skills to fully credit a person for getting “lucky” (often by making mistakes that counter prior errors). In math it’s not only important to be familiar with the ideas but also to have the skill to communicate them (where each written symbol and expression illustrates the logical transition from one step to the next, efficiently and effectively). Unfortunately, far too many students lack the confidence or effort to pursue such understanding until it is attained “Some people drink from the fountain of knowledge, others just gargle” (quoted from Robert Anthony). And this is not a slight on their intelligence; often these hindrances are the influence of environment rather than intellect - past struggles without adequate support have trained many to doubt their ability to succeed academically and conditioned them to give up when difficulties arise; sometimes the lack of effort is a matter of responding to life’s demands as they prioritize how they’ll apply their time and energy - just the demands of work and home life can consume most of our daily time, but successful students treat education as a job in which they conduct themselves professionally and realize the more they put into it the more they will get out of it … and their pay is the education, and grade, that they receive from the class). And, like Allen, many people struggle with math because of the way they approach the subject – they listen to concepts being presented and explained figuring that if they are present in class and pay attention then they should be able to use whatever is discussed, and if they can’t apply it or don’t understand then they surrender because they “just don’t have a math brain.” This is a false assumption. But we are not all processing at the same speed which is why it is important be open to help from various sources. Moreover, instructors should be treated as allies rather than adversaries. Yet, sadly, for too many people, classes outside their major(s) are too often treated as prerequisites to get past (and passed) rather than opportunities to broaden their knowledge and thinking skills. “We’ve brought into the idea that education is about training and “success” rather than learning to think critically and to challenge. We should not forget that the true purpose of education is to make minds, not careers” (Chris Hedges). “Education isn’t for getting a job. It’s about developing yourself as a human being” (Liz Berry).

 

Allen learned a huge lesson concerning this in what he believed would be an easy elective class (i.e. Biology 30 - Human Sexuality and Reproduction). Allen was trying to balance his course load (Astronomy, Calculus, Biology, Health Science, and Political Science), and his friends advised him Biology 30 would be a rather easy class on his schedule. However, his class was being taught by a new instructor, a practicing physician, who wanted to share her expertise with the class. The class began with 35 enrolled and 18 trying to add. A video was played the first class:  a three hour movie covering the name of every bone in the human body. The second class was an exploration of Latin roots that are found in the various anatomical names, and a twelve page handout of vocabulary to learn. The class finished with, including Allen, just five people – two A’s, a C, a D, and an F. After the final exam Allen was last to leave the class so the instructor stopped him and asked for his final impression of the class. Although he had earned one of the “A” grades he respectfully explained that her class had turned out to be almost as difficult as his other Apprehending Algebra by Daniel Vogt

classes combined. Her face became saddened with disappointment, and she asked him what he had hoped to gain from her class; why had he enrolled in college. The teacher continued by asking him if he realized the significance of education. He was silent so she went on to answer her rhetorical question stating that to relate to the class at the start she had to lower her level of communication (vocabulary, examples, etc) to meet theirs but through the class she was elevating their ability to communicate up towards hers. She concluded this was the point of education - to elevate each person’s ability to communicate; it lifts each person to a higher level of understanding. This implies all learning has a level of high expectation. “Nobody rises to low expectations” (Lloyd). This lesson Allen never forgot.

 

Thus, this book would do a disservice to the student to avoid the difficulty associated with understanding the “why & how” of concepts and their applications (a deeper level of thinking – see Bloom’s taxonomy [or Webb’s Depth of Knowledge]):

 

Bloom’s taxonomy categorizes thinking skills into six levels based on the mental demand (or exertion) required:

 (this is usually expressed in a pyramid with the lowest level of thinking skill most common and most used to the highest level of mental demand  being the least common and least used)

 

Knowledge / Information recall: The ability to memorize information.

 

Comprehension / Understanding: A realization of the overall meaning and importance of ideas and processes.

 

Application: An ability to use ideas and processes in meaningful ways (such as problem solving).

 

Analysis: Understanding processes so that concepts and patterns can be identified, organized, and analyzed.

 

Synthesis: Insightful regarding the inner workings and mechanics of processes and concepts so that they might be modified and adapted to different situations, combined with other ideas, or lead to the composition of new theories or processes. This stage includes predicting, inferring, and creating.

 

Evaluation: Assessing conjectures and theories, evaluating outcomes, as well as judging and recommending with logical support and reasoning.

 

 

 

 or “water down” the material in order to make topics easier on the student. Martin Luther King Jr. said, “The function of education is to teach one to think intensively and to think critically, intelligence plus character – that is the goal of true education.” Additionally, expecting students to memorize “steps” for doing a problem without understanding why those “steps” are applied limits the students to doing only problems similar to the given example; however, if a student understands the goal, and the mechanics involved in reaching the goal, then this person has the versatility to deal with that problem as well as the many variations of that type of problem (memorizing may be a life preserver to keep a struggling student afloat but understanding gives a person the ability to swim). An education in mathematics not only enables you to solve problems involving numeric values (counts, measures, etc) but it also develops your analytic thinking (which can benefit you in other areas of learning and life as well – such as strategic & organizational thinking). Admittedly, developing higher level thinking skills is not easy – as a matter of fact, it is often grueling and frustrating, but the rewards far outweigh the costs. Abigail Adams states: “Learning isn’t attained by chance, it must be sought for with ardor and diligence.” The intention of this text is to help students rise to higher levels of understanding, confidence, and ability to apply mathematics. And yet, like an instructor, this book can only provide the opportunity to learn – it is the duty of the student to actually achieve the learning. Lowering expectations for student learning is in essence handicapping them, and deteriorating the quality of education in general (because some of these students will go on to instruct others, and any weakness in their education will affect their ability to teach others – some of whom may include the next generation of students and educators; and the title instructor is not limited to the classroom – it also includes parents teaching their children, people tutoring their friends, etc). “I think what has happened is that people know less than their predecessors, so they’re trying to reshape the world – spellchecks, calculators, etc. – in attempts to compensate for the growing ignorance. And I believe that increasingly they will carry on reshaping the world to accommodate for a continual net loss of knowledge” (Sebastian Faulks).

 

Oddly, society places a high emphasis on language skills but does not put the same value on mathematical literacy. “It has become almost a cliché to note that nobody boasts about literary ignorance, but it is socially acceptable to boast of ignorance of science and proudly proclaim incompetence in mathematics” (Richard Dawkins). That is, Apprehending Algebra by Daniel Vogt

people find not being able to read or write unacceptable (a source of shame or recognized deficiency) – but being illiterate in math is viewed by many as acceptable, justifiable, and often popular. Imagine an employer received the following note on an application for employment from an adult who has no mental or physical impairments:

 

Hi. My naem is Chris. I am 21. I wood reely licke too work for you. I thank you half a good compunie. I hope you will envite me to meat you and get to knew me. I wont to work for you now and thank we could have a good fucher together. I promiss  I will work harter then the uthers  – I look froward to herring frum you.

 

Do you think this would affect that employer’s opinion of this potential candidate for the job? Does it affect your opinion of the unknown person? Does this suggest certain things about the person’s education, and about the person’s character (did the person not try, not have a good home life, not care)? When this is the best a person can show for twelve years of schooling this must reflect on his/her work ethic and ambition;  and why would anyone think that the inability to express ideas mathematically would be any less revealing about a person?

 

III. Develop questioning practices

 

Well, the next obstacle to overcome is how do we become proficient at understanding and communicating in math – and the answer is work and questions (lots of both). The student should work on topics as they are taught, but also continue reviewing these ideas after they are passed … and question anything and everything that is not completely understood. “I believe that we learn by practice. It is the performance of a dedicated precise set of acts, physical or intellectual, from which comes a shape of achievement, a sense of one’s being, a satisfaction of spirit. One becomes, in some area, an athlete of God. Practice means to perform, over and over again in the face of all obstacles, some act of vision, of faith, of desire. Practice is a means of inviting the perfection desired” (Martha Graham). “Learning is by nature curiosity – prying into everything, reluctant to leave anything, material or immaterial, unexplained” (Philo of Alexandria). If a student wants to learn then that person cannot be content with just the class time explanation, but should seek out deeper insights – “I believe in study. I believe that men learn much through study. As a matter of fact, it has been my observation that they learn little concerning things as they are, as they were, or as they are to come without study” (Marion Romney). Also, an avid learner cannot get discouraged by setbacks – mistakes are often a useful part of learning: “By seeking and blundering we learn” (Goethe) …“Mistakes are proof you are trying” (Jennifer Lim) … “I was hugely impressed with each person who realized what he or she didn’t know, was perfectly willing to admit it, and didn’t want to leave until it was understood. That’s heroic to me. I wish every student had that attitude” (Randy Pausch) … “Remember the two benefits of failure. First, if you do fail, you learn what doesn't work; and second, the failure gives you the opportunity to try a new approach.” (Roger Von Oech).

 

A few suggestions for enhanced learning, better performance, and longer retention: asking questions, group study (working with one or more others), research, & frequent review. A common problem with students is their reluctance to ask when they are not understanding – sometimes this is due to not wanting to appear less intelligent in front of others and other times it is in hope that if they wait long enough it will get resolved. But you are at a disadvantage by allowing a question to exist since it is a distraction (your mind ponders it, attempting to figure it out, so this point of confusion prevents you from giving full attention to instruction) taking away from your ability to follow the continued conversation and it prevents you from understanding any discussion based on that topic. Yet, here are seven advantages to asking the question (four personal and three for others): 1) it answers your question; thus, improving your learning, 2) it slows the instruction giving you time to catch up (especially since ideas are reviewed to answer your question), 3) it takes away the distraction allowing you to fully focus instruction continues, and 4) it strengthens your foundation (upon which further mathematical ideas may be constructed). Also, 5) it provides others the opportunity to gain a better understanding of the idea(s) [it is amazing how often others have the same question or confusion and if one person asks all benefit], 6) one person having the courage to ask a question emboldens others to ask questions, and the last (initially benefiting the instructor): 7) it indicates care and involvement from a student; this is motivational for the instructor and this can energize the atmosphere in the room enhancing the learning for all.

 

IV. Utilize the various resources available for research.

 

Occasionally some people after a class find themselves unfamiliar with vocabulary, rules/definitions, or processes when reviewing material. This might occur because certain concepts are assumed to have been acquired in previous education. Or, this could happen because when the vocabulary or processes were used during explanation they made sense in the context of the lecture but once students are attempting to assimilate the information on their own, from their notes and memory, uncertainty in these areas reveals itself. If you have ever left class feeling confident you had Apprehending Algebra by Daniel Vogt

learned practically everything in class only to discover you significant struggles when later doing the work on your own then this is the problem you have experienced. There is a logical reason for this – during the presentation the instructor is doing the vast majority of the critical thinking (even if the teacher incorporates student contributions while working on problems or discussing concepts)  Also, a person might desire practice work (in addition that which was done or assigned from class) to solidify the fullness of his/her comprehension. Whether a person is completely ignorant in regard to certain ideas or is just missing a few details to make sense of the material there is a wealth of resources available for people willing to research information. Now, the preferred order of reference for students is 1) the instructor [these people know how they’ve laid a foundation & how they’re going to build upon it in the future so their answers best complement the instruction you’ll receive in class] , 2) the teacher’s notes/book [this should be the next best thing to actually talking to the instructor and directly reflect the classroom instruction], 3) classmates [they receive the same instruction so discussing information with them provides the next most similar explanations as in class] , and then 4) outside references (learning centers, tutors, other books, internet research).

 

V. Reinforcement quickly, frequently, and continually.

 

Although questioning will benefit your understanding in class, it is reinforcement which will benefit your retention (as you move the information from short term to long term memory). Regarding retention without reinforcement, Hermann Ebbinghouse (German psychologist, 1913) claimed that after 20 minutes of hearing random information almost ½ (50%) was forgotten and after a day almost 2/3 (66%) was lost (and, of course, more time resulted in more loss). H.F. Spitzger (1939) did research that yielded similar results, but he also found that students who reinforced learning by reviewing material immediately after exposure retained almost 4/5 (80%) of the reviewed information. Successful students also tend to take more class notes (about 64% of the material discussed, on the average), but a good student is also discerning about which notes to write – you don’t always need to write information just because the instructor has it written but often you should write important details that are spoken even though the instructor does not actually express them in a written form (this is one aspect of being a good student that we all have to learn).

 

So, you should review class notes as soon as you possibly can (and then you should do so again about eight hours later). Copy example problems onto a blank sheet of paper and then work those problems out again on your own (using your notes as an answer key reference). You can also create a collection of problems from examples worked out in the textbook. Keep these sheets of review problems and retake them weekly/monthly as a means of perpetual review because the repeated reinforcement will improve your retention (moving information from short to long term memory). Make it your goal to “overlearn the material”. Too often students are content that they are able to do most of the work correctly (only getting a few problems wrong) rather than being concerned that the incorrect problems reveal a lack of full understanding then working to resolve these problems and attain complete comprehension. The more familiar and comfortable you are with the mathematical ideas the less you will find yourself uncertain and second-guessing yourself on the exams – the main exam struggles tend to be “silly” mistakes & lack of preparation.

 

There are many strategies to develop this familiarity. Another strategy for developing a working knowledge of the significant information in mathematics [vocabulary/definitions, symbols, properties, rules, formulas, theorems, and any other important concepts] is to create flashcards – to put a word, symbol, or even phrase on one side of a 3”x5” card (or any other practical size card or paper) and the definition or rule on the other side of the card; these cards can then be used as a means of repeatedly self-quizzing until you achieve “total recall”. Or you can write or type this information over & over until you know it exactly. You can also recopy your notes (writing them, when possible, in your own summarized wording). Importantly, these concepts need to become readily available for communicating.

 

VI. Build a network of support with classmates.

 

Another means of gaining deeper understanding and insights is working with classmates in small groups. This allows each person to practice communicating with the concepts and vocabulary in a safe setting. Don’t be afraid to share your work and thoughts with others for discussion (especially those who can offer you valuable input regarding how you are communicating your ideas) … and be receptive to their input. If resources were numbered by the order and importance they should be utilized this list would be: 1) the instructor [this person knows what has been discussed, how it has been taught, and how it is working as a foundation to prepare you for what is to be learned in the future], 2) the class notes [these remind the student of how the material was presented and should reinforce the practices and definitions demonstrated by the instructor], and 3) classmates [they are receiving the same instruction of the material so they should be able to provide explanations that complement the instructor’s presentation].  Group study can be a very helpful learning tool – a study group offers you immediate feedback to Apprehending Algebra by Daniel Vogt

errors and accomplishments, as well as inspiring deeper investigation into all facets of mathematical study. The stronger and weaker both benefit from this interaction: the stronger must delve deeper into understanding why things are done so it can be explained to the other causing that person to mentally strengthen; the weaker gains insights & understanding allowing that person to also strengthen cognitively. “No person ever reached excellence without having passed through the slow and painful process of study and practice” – Horace (65 – 27 BC).

 

Let’s examine two more anecdotes regarding Allen. The intention to these stories, as well as the quotes, is to learn from the experiences and wisdom of others. People tend to divide themselves into two opposite groups in many situations, and we find that dichotomy in this case too – one set of people can learn from the insights of others (if they walk into a kitchen perceiving the smell of burned skin and a grimacing face directs them to stay away from the cast iron skillet because it is very hot that group can comprehend the logic being shared and keep distant from the metal) while the other set of people seem to need to experience everything, good and bad, for themselves to make those connections (they are the ones who grab the cast iron handle to really understand why the warning is being issued). Still, according to Publilius Syrus (85 - 42 BC), “Many receive advice, only the wise profit from it.”

 

Allen had the dream of going to college but lacked the background preparing him to enter and successfully complete this goal – that is, his family wasn’t academically trained and he wasn’t involved in college preparatory classes (or guidance) in his youth. Still, he was granted money to go so he decided to leave his minimum wage job in hopes of “working” through college in pursuit of a career (a better job providing a better life). He was ill-prepared for success in this environment – he was unaware of the resources available or the process of navigating through the system and gaining a “higher education.” He obtained a list of course possibilities to satisfy a two or four year degree and, using that, began his journey into academia. Allen entered summer session, deciding to tackle math and English as his first classes. He decided to review algebra but the school offered beginning, intermediate, and college algebra courses so he figured since he intended to take algebra at the college he’d take the college algebra class. And yet, as it turned out, college algebra was actually a pre-calculus class; and he was in over his head. However, this now was his job – literally. He had that money allotted to him, but he had to perform well to get it. He realized this predicament and adjusted to it by doing something he had not done in his youth – he started asking questions … many and often. He knew most of the others already had the foundation he was trying to develop, but he couldn’t worry about their opinions of his questions or of him – he wanted to understand, and he also needed to survive. Additionally, he noted which classmates performed well or had insightful comments and sought them as study partners. The result of these changes: he not only survived but also grew strong in understanding – he had learned greatly. He didn’t realize how much he had gained until he moved on to his next math class (calculus) and discovered he had not only caught up with the others but, because he pursued understanding so intently, he had actually surpassed many; and he had developed a beneficial habit of questioning whenever things were unclear or confusing. He was rewarded with this dedication to understanding with an “A” in the course; but, more importantly, with the restructuring of his view and character that enabled him to succeed as a learner and mature as a person.

 

Furthermore, since young Allen (in secondary school) didn’t have anyone helping him to develop the skills to become academically successful (and he wasn’t learning to be a successful student just by attending school – not many do) as he progressed through the grades he found struggles became more common as he moved along. He even found himself demoted from one math class to a lower one, but as previously mentioned he didn’t WORK to gain learning (he thought his mere presence in the classroom along with watching the instruction was more than enough effort to attain and retain understanding). This demotion was linked to his seventh grade prealgebra class. His teacher was a short, burly, hairy man. His grade made a constant descent until his second quarter progress report declared he was in danger of failing. Allen’s dad decided physical intervention might correct the situation, but though Allen went to class the next day truly wanting to do well he did not know the process for correcting the situation. He sat and watched what looked like numerical fireworks on the board (the teacher was demonstrating how to use prime factorization to determine the LCM, after having taught how to find the GCF the day before; the numbers seemed to explode with factors streaming out in what was called a tree diagram). Allen sat in confusion. When the teacher finished and sat down Allen did something he had not done in that class before – he walked up and joined the line beside the teacher’s desk of students wanting to ask a question. As he stood there looking out at the other students busy at their desks he felt so “stupid” – they were all working away and he couldn’t even get started. Finally, he reached the teacher’s desk. He stated as quietly as he could (not wanting anyone to notice his inability), “I don’t understand how to do this.” His teacher glared up at him then began a ten minute tirade starting with, “Allen, what have you been doing the whole time I was explaining this? Thirty minutes I stood here teaching and now you tell me you don’t understand any of it?” He seemed to be so angry as he yelled, and Allen stood there with his peripheral vision noticing that everyone in the class was staring at him with open jaw. After the scolding the Apprehending Algebra by Daniel Vogt

teacher quickly pointed to the board and noted the high points of the problem in such a high speed blur the words just sounded like background noise. He finished by asking, “You got that, Allen?” And Allen answered, “Yes, I understand.” Then he walked back to his desk and gave up on the class (until the end of the semester when he was moved to a much lower “remedial” math class) – he didn’t take out paper for notes, he wouldn’t look up to watch the teaching, he just sat at his desk and entertained himself. As you’ve read, Allen changed in college and went on to excel in math (as well as his other classes); upon graduating Allen wished he could run across that teacher and gloat. But later he realized two important lessons from that incident. First, the teacher wasn’t really mad at Allen, he was upset (and frustrated) with the situation – Allen had waited far too long to ask questions, and as a result he made himself difficult to “fix” (to teach Allen the process to find an LCM the teacher would have to go back to explain prime factorization, but that would mean going back to explain primes and composites, which would imply a lesson on divisibility tests, and then the teacher would still have to explain the GCF and how it compares/contrasts with the LCM). Allen realized he should have got help before the concepts had moved so distant from him leaving him with so many holes he was virtually beyond repair (though it may have been possible it would have required much more sacrifice and commitment than Allen was willing or equipped to give). Secondly, he had held resentment against that teacher for all those years – and responded by refusing to try – now he discovered that anger had only served to hurt himself. Blaming his teacher gave him an excuse to fail, but it didn’t improve his life (and it didn’t affect the teacher’s life – that’s one problem when we hold anger or resentment, it only hurts ourselves). He realized (though he didn’t agree with teacher’s reaction) the fault was his for not being responsible about his education to begin with (waiting so long to seek help and not practicing through the homework); then for making things worse by allowing the teacher to become the scapegoat instead of taking responsibility and working to make the changes that would lead to improvement. Still, these lessons had changed him; now he works at fixing problems as soon as possible and taking responsibility for those situation that affect his life (no blame, no excuses). “In my parents’ time it was the person who fell short; now it’s the discipline. Reading a book is too difficult so the book is to blame. Today the student asserts his incapacity as a privilege. I can’t learn so there is something wrong with the subject. And there is something especially wrong with the bad teacher who wants to teach it” (Philip Roth). “It is wise to direct your anger towards problems - not people, to focus your energies on answers - not excuses” (William Arthur Ward).  “The person who really wants to do something finds a way; the other person finds an excuse” (Sir John Templeton).

 

VII. Maintain the proper relationship with instructors – they are allies not adversaries.

 

Ultimately, the education of each person is determined by his or her own self: “What I've found about it is that there are some folks you can talk to until you're blue in the face--they're never going to get it and they're never going to change. But every once in a while you'll run into someone who is eager to listen, eager to learn, and willing to try new things. Those are the people we need to reach. We have a responsibility as parents, older people, teachers, and people in the neighborhood to recognize that” (Tyler Perry). Perhaps one important aspect to learning is appreciation of our education: “This is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our check books; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges” (Ellen Kaplan) … “Some know the value of education by having it. I know its value by not having it” (Frederick Douglass, ex-slave). Appreciation might be an important component to real learning, and those that appreciate education are the people who desire it the greatest and seek it most persistently. We live in a time when we are not just allowed an education, but we have it forced upon us in our youth; so, many people have developed an indifference to this gift. Many view school as prison, and math (with its work and frustration) makes them feel its a life sentence “at hard labor.” Maybe if we were transported back to a time when we’d be denied learning and had to live in forced ignorance then we’d return with a very different perspective of this aspect of life - perhaps we’d all desire it, value it, and enjoy it. “If the stars should appear but one night every thousand years how many would marvel and adore them” (Emerson).

 

VIII. Realize that success in learning is measured by progress, not necessarily performance.

 

Still, to be fair, we don’t all learn at the same rate. Some have more favorable conditions allowing them to gain from Apprehending Algebra by Daniel Vogt

instruction while others have more obstacles to overcome to obtain education. And yet learning is not a competition. Learning is the development and improvement of our thinking skills applied in different areas; and though we may not all be at the same place in the development of our abilities we can all notice progress in these abilities. Consider two marathon runners – one gifted physically and the other born without a left leg. When the starting gun is shot the first runner disappears while the other hops fifty feet and then stumbles. This person has the choice to get up and continue going or to quit. If that person quits then in four hours, when the conditioned runners are crossing the finish line, this person will still have only gone fifty feet. If that person gets up and hops another eighty feet then falls but gets back up again and continues this process of stumbling, overcoming the difficulty, and pressing on towards the goal then in four hours that person might not be crossing the finish line but he or she will be considerably distant from where that person started. Moreover, if that runner continues moving forward then eventually that person will cross the same finish location as everyone else. And that, in this author’s opinion, is the true measure of success – the optimal distance between where you began and where you move on to (based on your potential). Thus, although success in a course might be measured by reaching a destination, success as a learner is determined by your own personal progress. A grade reflects the individual’s status based on goals defined for populations (that is, students that do this [goal] get that [grade]), but an education reflects the growth experienced based on personal potential. Now, friendly competition in the performance of objectives can push some individuals to reach higher results and gains, but in learning we do not compete against others but rather it is a personal contest each person has within him or her self to achieve maximum growth as a student (and in a general view we are all students for the entire span of our lives, “There is no shortcut through life. To the end of our days, life is a lesson imperfectly learned” – Salisbury).

 

IX. Think positive.

 

Our perspective will not only affect the way we view, but also how we feel – “We can complain because rose bushes have thorns, or rejoice because thorn bushes have roses” (Abraham Lincoln). And this is very true in math … it will determine how you view yourself in terms of success, and how you feel about the material as you work through it – you can either see math problems as burdensome chores or as interesting puzzles, as boring & frustrating or as entertaining & challenging, as beating you down or building you up. You determine your learning experience. As for this book, it is written with the belief in each and every person who reads it; the author believe in you – now it is just a matter of you believing in yourself! “Those who trust us and believe in us educate us” (George Eliot). 

 

This chapter has mentioned several tips to developing as a successful student such as utilizing instructors as allies (seeking out their help and being involved through questions/participation), study groups, and a diligent work ethic in homework and study; and this topic will continue to be addressed occasionally throughout the book. And yet, one more important factor will be mentioned here: Attitude is everything (“Attitude is a little thing that makes a big difference” – Winston Churchill; “Weakness of attitude becomes weakness of character” – Albert Einstein). You have to believe in yourself, and know that if you work through problems long enough you will overcome these difficulties and achieve learning (and work in this case means to continue asking and continue trying until you have resolved any misunderstandings or confusion). “Whether you think you can or you think you can’t, you’re right” – Henry Ford. Implied in this determination to work through difficulties is the dedicating of time to this goal, and that is where priorities come into consideration – it is a question of what do you want and what are you willing to give for it. Moreover, realize that the first time you encounter a new experience or concept is the most difficult time, but each subsequent involvement with it becomes easier and more understandable. And education is built upon these two elements: exposure (introduction to ideas) and reinforcement (repeated review of the ideas). So, believe you can succeed then don’t quit until you do.  “Your attitude, not your aptitude, will determine your altitude” – Zig Ziglar.

 

Calculator: Every person should be comfortable and competent working with a hand held calculator. If a person can work with that then it is an easy transition to work with the calculator function on a cell phone, the calculator on the computer, a graphing calculator, etc. Still, calculators should not replace a person’s ability to work mathematically – when a student is learning rules for arithmetic operations the calculator should only be used for verifying results. Still, when students are working on complex algebraic processes a calculator can allow them to focus on developing ability with the process by assisting them with the arithmetic. So, it is advantageous for each algebra student to own a calculator. Some functions that might be desirable on the calculator might be the fraction key [a b/c], the exponent key [xy] or [^] and root key [ or  ], the trigonometric functions [tan] / [sin] / [cos], the factorial key [!], the combination key [nCr] and permutation key [nPr]. Student need to take time to determine which will be most useful.

 

 

 

In conclusion of this preface, a brief examination of a couple words is appropriate. “Educate” comes from the Latin Apprehending Algebra by Daniel Vogt

word educere meaning to lead forth, draw out, raise up. Our course objective is to lead you to find understanding, draw out your cognitive abilities, and help you rise to a higher level of both communication & awareness. The word “apprehend” is defined as to seize and, as used in the book’s title refers to seizing hold of understanding and firmly grasping meaning. The goal of this book is found in this title, Apprehending Algebra – it is designed to assist you in obtaining meaning and understanding of the topics covered in this algebra book , and enable you to recognize and apply them efficiently and effectively using clear mathematical communication. Math not only enables us to solve numerical problems, but also develops a strategic (analytical) type of thinking that can be applied to many aspects of life. And now, you are encouraged to be filled with learning. Fulfill your potential. Embrace your opportunities.

Chapter 1 Exercise Set:

 

1)  The text stated, “instructors should be treated as allies rather than adversaries.” To develop that alliance establish communication with your instructor be sending an electronic letter to your instructor’s desired email stating:

a)  your preferred email address (also useful for a teacher who creates distribution lists to disperse information),

b)  a contact phone number (in the case of a necessary or urgent situation*),  c)  the name by which you prefer to be called (if different from the name appearing on the roster),  d)  college experience (eg “this is my first semester in college”, “I have just three more classes I’ll do next semester  to receive my AA”, “I did two semesters six years ago and now I’m returning to continue” etc),  e) your major and career goal(s),  f)  anything that you’d like to say, or should be known, about yourself as this course begins – any comments, questions, or concerns.

 

2)  List seven benefits to interjecting questions (which apply to all courses, and definitely to math classes).

 

3)  Several quotes have been included in the text,

a) how many quotes were used?  b) select the quote that you find most meaningful and explain its significance to you.

 

4)  If you are in class a couple days and outside of class several days that doesn’t allow much time for developing mathematical (analytical) thinking and abilities (this is more obvious when viewed in percentage of hours dedicated to working in this material) so learning can seem to be two steps forward and slide one step back.

a) Determine the number of hours spent in class each week and the total hours in a week then find the percentage of your week that is spent in classroom learning (reinforcement and instruction).

b) Assume you spend two hours in study and homework for each hour in class, determine the percentage of your week that is then dedicated to developing mastery of mathematical concepts and terminology.

c)  Consider and then list strategies you can (and presumably will) employ to overcome this time discrepancy so that time becomes an ally rather than an adversary too.

 

5)  Is successful learning based on personal growth or performance compared to others? That is, if a student worked hard in a math class, doubled his/her knowledge in the subject, but received a “D” based on testing in the class – did that person experience success or a failure? Or, if a student already knew most of the information so he/she did not really try nor really learned but received a “B” is this a successful education? Defend your opinion with persuasive reasoning. (Which person would have more reason to be proud of their education – which person was a more successful student?)

 

6) List the stated strategies for increasing successful learning/performance. Which is/are most meaningful to you? Explain.

 

7)  Woodrow Wilson (28th President) said, “We want one class of persons to have a liberal education, and we want another class of persons, a very much larger class of necessity in every society, to forego the privilege of a liberal education and fit themselves to perform specific difficult manual tasks.” Suppose that the budget problems of the government caused lawmakers to act on Wilson’s declaration and introduce a bill to restrict the number of people being given an education in college or even high school. Their logic is to make schooling competitive to push the “best and brightest” towards fulfilling their potential by a series of testings in which only the highest performing candidates will be allowed to continue through the educational system while the others will be denied academic schooling and given the option of a trade school or pursue work as manual labor. They also support this bill by stating that removing these individuals, that neither value education nor put the necessary work into utilizing it, will lighten the burden on the system and raise the educational atmosphere for those remaining. Thoughtfully consider this case study then explain (with clear and valid reasons) whether you’d support and oppose it.

 

8)  Look up “Webb’s Depth of Knowledge” and compare/contrast it to “Bloom’s Taxonomy”. How important is it to you to develop higher level thinking skills? Regarding responsibility for your education: in your opinion what percentage is yours? What percent, then, is your instructor’s responsibility for you learning? Do you agree with the quote: “A teacher cannot force a student to learn, he/she can only provide the opportunity and the rest is up to the student”? Explain.

 

9)  If you were an employer, or even as you read it as a student continuing in your education, what impressions do you form about Chris from his letter (page 3). And do you believe illiteracy in language skills or mathematics should be acceptable for society (explain)? Given the argument presented in this book, how does it affect your view of the importance of communicating correctly in mathematics (as you’ll demonstrate in homework & tests)?

 

Apprehending Algebra by Daniel Vogt

 

10) State the most significant insight(s) you gained from Allen’s experiences, and explain how the insight(s) will alter the way you see, or do, things. Explain.

 

11) About how much of the new information that is not reinforced immediately will be lost after 20 minutes? After a day? Approximately how much (and quite possibly more) will be retained if reviewed immediately & repeatedly?

 

12) You are posed with the same question as Allen, “Why are you here?” (To you, what’s the meaning & purpose of pursuing an education?) List the strategies you plan to employ to ensure you’ll achieve quality learning in math.

 

Apprehending Algebra by Daniel Vogt