Do You Like or Dislike Algebra? Why and Why not?

by Scott77 76 Replies latest social current

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  cognisonance

  I'm taking Calculus right now. . . .

  And you would think where I'd get hung up is in all that fancy calc stuff like derivatives, and derivatives of derivatives . . ..

  But nooooooo. . . . . . all of those concepts I've got down pat.

  Where do I always screw up? In the friggin algebra!

  Drives me absolutely mad!

  Same here! It's very annoying. My last test, where I thought I did really well (High B, Low A) turned out to be a low B (after a curve at that). I haven't gotten it back yet, but I suspect this is because of algebra mistakes. For me the issue with algebra is making small mistakes and not having enough time on exams to find the problems and fix them as in this last one, we had about 3.8 minutes a problem to solve and the bulk of the test is about Calculus concepts and algebra keeps getting in my way (and now the unit circle is beginning to as well).
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  GrreatTeacher

  Mathematics is the umbrella term and arithmetic is just one type of math. Algebra, geometry and statistics are others.

  Arithmetic is addition, subtraction, multiplication and division and is typically covered in elementary school. However, fractions and percentages are also arithmetic and these stump plenty of adults. Hmm, how much should I tip on this $25 check at 20%? If you know the algorithm, you can figure it out on the back of the check. So called "kill and drill" worksheets will get that result. However, mental math is quicker and it comes from understanding the concepts. Most easily, we could move the decimal in $25 to the left a space and know that $2.50 would be 10%. So, we can then double it to get $5 as your 20% tip. Or, you could know that 20%represents the fraction 1/5 and to find 1/5 of $25 you can just divide by 5. You still get $5. You can fugure these things out quickly if you have a good conceptual understanding and you are not a rigid thinker.

  Rigid thinkers know the algorithms and they will set up and solve the problem 25 x .2 and get the answer 5 after correctly placing the decimal in the answer. Worksheets and drills promote this type of learning. If this is the way that our students are able to do math, then we need to teach them that way. However, the goal is to get flexible thinkers who can understand and relate concepts from one math discipline to another. To teach to these students, a problem solving approach works best. Teachers need to help them make connections, but then they need to work to solve these problems.

  Instead of setting up a lesson in which we find a percentage of a number, I want to set up a problem in which we need to discover the amount of a tip for a restaurant bill. Would a tip be larger or smaller than the bill? They will say smaller. With a bill of $43 what do we know about a percent, say 20%? A percent is a part of a whole. Then, what is the whole? 43. Good, now when I say I want to know 20% of 43, take note of the word of because of means to multiply. So what are we multiplyi g? 43 x .20. Yes! A percent is a decimal. We only have to move the decimal point to the left two places. Now, usually we get a larger number as a product when we multiply. Do you think that will happen this time? No, when you multiply by a fraction or decimal we are only finding a part of the original number, which was 43 in this instance. So go ahead and multiply 43 x .20. You got 86? Don't forget to place your decimal point in the right place because we are looking for a smaller number and 86 is bigger. Yes, move the decimal point to the left and you get 8.6. Yes, now you remember that rule? So, what is our 20% tip on $43? Yes, $8.60. Can you think of another way to do that? What do you notice about the numbers 43 and 86? They're doubled. Good. Why might they be doubled? Because we multiplied by 2 in 20%. Good. What if we multipled by 10? Yes, we could just move the decimal to the left and get 4.30. Easy! That's 10 percent, so we can just double that to get $8.60

  Can you see the different ways of teaching. We want to teach like the latter example.This makes sure that kids get a good conceptual understanding. But, not all kids will and then we have to move to direct instruction of the algorithms and use worksheets to reinforce the raw memorization necessary for this approach. Often special ed students need this approach. And that's howcwe give students a passing understanding of math even though we really want to teach to deep conceptual understanding, but it doesn't always happen with every student. The difficulty is having both types of students in the same classes in heterogeneous grouping. Teachers complain about this problem a lot because it really is an inefficiency that would be easily solved by homogenous grouping.
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  Scott77

  GrreatTeacher,

  Sounds like you ar ea math wizzard. Just kidding. By the way, I recently purchased those three texts for my own fine tuning of math skills for everyday ulitilization as a consumer. I have not yet started reading them.

  Applied Business Mathematics: for consumer and business use Hardcover – January 1, 1965

  by Edwin and Joseph Gruber Piper (Author)

  Consumer Related Mathematics Hardcover – January 1, 1976

  by Kravitz/Brant (Author)

  Mathematics in Daily Use Hardcover

  by WALTER W. HART (Author) Please, did you use them before or ready them? Scott77
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  Quendi

  It has long been said that you don't actually understand algebra until you learn calculus and don't really grasp calculus until you learn differential equations. The experiences of Data-Dog and cognisonance would seem to confirm the first part of that observation. Much of the problem in math education today stems from the fact that many students are not exposed to different ways of thinking about and solving problems. For me, it started with drills. First were the drills in addition and substraction. Then came learning multiplication tables and applying them to learning not just how to multiply but to do division as well. Drills did more than teach me how to do mental arithmetic. I used them to discern relationships and principles that might have escaped my notice otherwise.

  I don't intend to turn this thread into a mathematics class, but I want to follow through on Scott77's idea of what to share. High school algebra revolves around solving for unknown quantities. You are given the clues and your job is to solve the mystery, in this case the mystery being the unknown quantity. The equations given to the student will have one or more unknowns which are represented by the variables. Think of yourself as a sleuth with enough information or clues in your hand to unravel the mystery.

  Geometry is the gateway to what mathematics is truly all about. For in geometry, the student is introduced to logic and how to use it to reach valid conclusions. Definitions, axioms/postulates and theorems are the building blocks to obtaining critical thinking skills and the ability to use reason. The introduction to the relatively simple theorems in high school geometry prepares the student to take on the concepts met in calculus and beyond because it is the ability to reason which will enable the student to see that mathematics is not about learning "formulas", manipulating numbers, and using them to solve problems, but to thinking both analytically and synthetically so that inferences and conclusions can be drawn which are not initially self-evident.

  I strove to teach my students thinking ability in my math classes. I was hampered by the fact that many were weak in the four basic operations of arithmetic and that weakness was what frustrated them when it came to learning about fractions, decimals and numerical relationships. My students were amazed at my ability to work with these things without ever using a calculator or computer. The key, I would say to them, was to use the best computer ever devised: the human brain. I would express confidence in their native ability to learn and use mathematics saying that if I could do it, anybody could. Thinking ability lifted the fog in their minds about things like the relationship between fractions and decimals, what an irrational number really was, and how to distinguish among real, imaginary and complex numbers.

  Let me also say this. Don't be deceived by the arithmetic and algebra you learned in elementary, middle and high school. What you were being taught in those early years was merely how to use these two branches of mathematics. You weren't taught how they really work. True arithmetic's and true algebra's workings lie on the other side of calculus and cannot be understood until calculus is mastered first. I don't say this to discourage anybody, but simply to put matters in their proper perspective. To those of you who are now studying calculus, I say hang in there. One piece of advice I will give when it comes to tackling the algebraic aspects of calculus is one I emphasize again and again to my students: TAKE YOUR TIME!!! Many mistakes can be eliminated by not being in a hurry to solve a problem. Also, look over the test when you get it and do the easy problems first. That will leave you a lot of time to pore over the more challenging ones and you won't feel yourself under as much pressure.

  Quendi
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  cognisonance

  I will give when it comes to tackling the algebraic aspects of calculus is one I emphasize again and again to my students: TAKE YOUR TIME!!! Many mistakes can be eliminated by not being in a hurry to solve a problem. Also, look over the test when you get it and do the easy problems first. That will leave you a lot of time to pore over the more challenging ones and you won't feel yourself under as much pressure.

  I tried doing this on my last exam (taking my time). I wasn't rushed, but I only had enough time to try each problem once without any time to double check. I felt goo about the test (and I actually did well on the previous quizs for the chapter gettings a 70, 100, 95, and 100). I thought I was going to get a high B, low A. My score? 81.5% after a curve (76.5 originally). The professor told us curves will be rare, so don't rely on them, even though both exams have been curved so far. I haven't gotten my test back yet even though I took it almost 2 weeks ago. We have two teacher teaching this class as they are dealing with vacation this summer and the teacher that gave us the last exam is now on vacation, but mailed the tests to the other teacher. I should have them tomorrow. My grades are online so I can at least see how I'm doing. In other exam where I went faster, I had time to double check my work. Either way it doesn't make much difference (that test grade: 70%, after curve 80%). The mistakes from going fast with double checking (and thus fixing some mistakes), are about the same as the mistakes from going slow without.

  I've always been a slow test taker in high school and college from over 10 years ago, and the same is true today. I'm the type of student that thinks really deeply about problems and typically spends on average, about 15 minutes a problem on calc 1 homework (which makes this take a long time, as I usually have about 50 problems to do each week). I do practice some problems before an exam to make sure I can solve them in say around 4 minutes, but it's always difficult. I'm just not fast, even if I understand the concepts and how to do things.

  I'm not a fast reader (even though I've bought a speed reading book and have tried to increse it, around 160-200 words/minute deepending on the material). I'm a softare engineer, and when I program, I'm also a slow thinker and want to fully understand what I'm about to do before I just jump in and get stuff done, but the quality of my work is very high.

  Essentially, If I had say, 2-3x the time that is typically devoted for a Calculus exam or test, I'm sure my grade would be a solid A, not hovering around the C/B range. If I go slowish on a test (that is only time to attempt each problem but not double check) it's still fast for me. I get to the answer and can tell it's not right, but have no time to fix it. If I try going faster, then I just make even more mistakes, and while I have some time to fix them, it just seems to be a wash with when I go a bit slower. I just am a student that needs more time!!! I try to devote more time to studying, but this is difficult working full time even though this is the only class I'm taking this summer. I'm spending about 15-20 a week on homework. And another 5-10 hours a week on supplimental review, quiz, and exam prep. So 40-hours of working, plus 20-25 hours of school. I don't really know how to devote even more time to become faster.

  I've also bought 3 supplimental books (humungous book of calculus problems, calculus lifesaver, and pre-calc for dummies). I only really have time for the pre-calc book for review looking up something I have forgotten (like the unit circle or factoring polynomials). I also use Khan academy to practice those concepts. I have test taking anxiety, which after 2 exams and a dozen quizes I'm just beginning to learn to deal with. As mentioned I'm a slow test taker. The classes are in the evenings after a long day of work, and my study time is mostly in the evenings as well. The best time for me is in the mornings to have my mental capacities running at optimum, but I work most of the time. I only have the weekend to work on class in the morning. I'm making sure I get 8-9 hours of sleep each night. I'm eating healthy. I take breaks when I get stuck on problems in the homework, but I am a bit stuborn in that I won't skip then until I finish them.

  I'm trying my best and I'm struggling to get a B- in this class. I never in my life struggled this hard with any class, including math (though I've always struggled somewhat with math but chaulked this up in the past to not really trying hard [because I feel I didn't try as hard as I could or really do my homework like I should]). This time around I'm giving in all the time I have. All the focus I can. I'm thinking about math so much that I often dream about equations most nights. It's all consuming and I still just can't get anything higher than a B- on tests. And quizes hover in the 60-100% range, averaging around 80%.

  I'm just frustrated!
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  nonjwspouse

  I was a product of the 60's-70's "new math" which put me at a big disadvantage most of my life. My core, base, rote math learning that makes math easier was not developed.

  Finally when in my late 30's I went back to the university and took remedial math, this time not the " new math" but the more traditional way, with alternatives presented of course, but lots and lots of rote.

  The boost to my ego was huge. I finally got through math all the way through calculus, statistics, etc with nothing less than a B.

  The ability to apply what I learned, use the ability to analyze with the math I had learned, was great.

  My daughter used her "traditional" math education to turn a nuclear plant mechanical department on it's head when she told them they had been doing something important wrong all these years, right after she was hired and studied the calculations. ( Kinda scary actually)

  I suffered from not being taught algebra appropriatly in high school. None of my kids have suffered from their algebra and higher math in high school. For each one it has been helpful, both in higher education and in their jobs.

  Oubliette,

  The number crunching is a base. It is an understanding that need to be held before relying on a calculator for the huge math problems. If the base is not lerned a base part of the math toolbox is missing. To be unable to calculate without a machine is to be handicapped.
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  Oubliette

  Quendi: What you were being taught in those early years was merely how to use these two branches of mathematics. You weren't taught how they really work.

  Amen to that!

  I find the same thing with my science students. They all seem to confuse the application of science with what science really is: a way of thinking and of understanding.

  Thanks for your very thoughtful comments and encouraging words for those that struggle with algebra.
• 

  Oubliette

  how much should I tip on this $25 check at 20%?

  $5. The service better have been pretty damn good for a 20% tip!

  BTW, I did that in my head.
• 

  Oubliette

  GrreatTeacher: The difficulty is having both types of students in the same classes in heterogeneous grouping. Teachers complain about this problem a lot because it really is an inefficiency that would be easily solved by homogenous grouping.

  The difference between how things are and how things should be.
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  Quendi

  cognisonance: You're on the right track, buddy. I know you are working real hard and I want to say that your approach is the right one. Keep up the good work. As the Bible says, if you sow bountifully, you will reap bountifully. In time, you will see your speed in doing problems increase with great benefits to you. Let me share this experience to encourage you. One of my students has Asperger's syndrome, a form of autism. When he took tests in my algebra class, he needed more than the alotted class time to finish. I always gave him whatever time he needed to do this (an advantage high school teachers have which college instructors don't). As the year progressed, he got faster and faster, eventually finishing his tests within the alotted class time. He turned out to be my best student, getting all A's with his lowest score on my tests being a 96! I have every confidence that your hard work will pay big dividends.

  nonjwspouse: Congratulations on getting a solid handle on learning mathematics!

  GreatTeacher: One of the biggest challenges in math education today is accommodating different learning styles. You have auditory learners, visual learners and kinesthetic learners. The good teachers learn how to reach all three kinds. I love to talk, but I quickly realized that some students simply tuned me out when all I did was talk and throw problems and equations on the whiteboard. So I employed diffeent strategies. Using computers to graph equations worked wonders for my visual learners. Employing origami and tactile puzzles and activities helped others. I had my students bring pies to class to teach how fractions work. Breaking the class into smaller groups and having the kids teach each other enabled other students to better grasp the principles being taught. Math became fun as well as interesting and my students definitely benefited.

  Oubliette: Thanks for sharing your experience as a science teacher. You know the value of making science come alive for your students and I'm sure they have deepened their appreciation for what science really means. There are many similarities between teaching science and teaching mathematics. My next door teaching neighbor last year was the science teacher and we collaborated a great deal in our teaching. Our students were better off for this.

  Quendi