by Karl Mattle | Dec 1, 2015 | Math
Mathematical Proof: What’s the Point?
The sum of two odd numbers is even. Two pictured triangles are similar. There are infinitely many prime numbers. Tell these facts to a student and the overwhelming response is “ok”; ask them to prove it, and the overwhelming response is “What’s the point?” In History class, my teacher doesn’t ask me to prove that Hannibal crossed the Alps. My literature teacher doesn’t ask me to prove that Herman Melville wrote Moby Dick. Even in Science, the subject most closely aligned with, and dependent upon Math, I’m not asked to prove Charles’ and Boyle’s laws. Why do I have to prove things in Math? Tell me what I need to learn, how to use it, and I’ll do it.
In a way, Mathematics has over-succeeded. It has been so successful in explaining, and, horror of horror, quantifying, so many natural, physical, economic, and yes, human relationships, that our modern Math and Science curriculum has become obsessed with conveying the many successes of Mathematics without conveying how those successes were achieved. Most students are well aware of the story of Isaac Newton observing an apple falling and being inspired to quantify gravity. But little or no time is devoted to how such a common observation could lead to the Law of Universal Gravitation, and its proof. Educators will lament, there is simply not enough time. We have to move on to the application. This argument is probably correct, reflecting the fact that we are trying to teach too little about too much. We should not be teaching, or trying to teach, our students every application of Mathematics. Rather, our Mathematics curriculum should emphasize the beauty of Mathematics with the emphasis on thinking, that is, proof.
Where in the busy Secondary Mathematical curriculum should proof be emphasized? The answer is in the first form of Mathematics in which rigorous proof was emphasized, Geometry. But the modern curricular movement is to teach practical application rather than proof. While Heron might be proud to know that modern High School students were being compelled to memorize his formula for the area of a triangle, in reality, few, if any, of the students are ever going to be required to calculate the area of a triangle using only the lengths of the sides. If they ever are confronted with such a problem, they will Google it, and get the procedure and formula from the internet. Heron would be far more proud if students were being instructed on the thought processes he employed to derive his formula and prove that it worked, every time.
Rare is the gainfully employed adult who is compelled to employ any of the Mathematical maneuvers, practical or otherwise, learned in High School. However, all productive citizens will, multiple times in their lives, find themselves having to make a point, or refute a point, using critical thinking and logic. In other words they will have to use proof!
by Rebecca Mayer | Nov 17, 2015 | Math, Teaching and Learning
Ther
e are many common math mistakes that I have noticed many students making. They are simple issues that are often overlooked, missed, or forgotten. For some of them, no matter how many times you mention them to a student, they seem to continue to be missed, usually out of bad habit. As a tutor, it is my job to continue to enforce correcting these mistakes through repetition. As a student, here are some of the most common mistakes that you may be able to look over, remember, and not make them in the future.
1. Overlooking or adding too many parentheses: parentheses are very important when solving any type of equation. They are, of course, part of our order of operations. Some students forget when to evaluate parentheses, don’t register their existence, or put too many in an equation when solving by steps, causing incorrect answers. For an equation such as 8=4(x+3), I have noticed some students attempt to put the parenthesis around the x, giving them 8=4(x)+(3), which would give the wrong answer.
2. Negatives: some students often make the mistake of not distributing a negative or forgetting that subtracting a negative number is actually just adding a number. For
5-(4+3), the negative can be distributed into the parenthesis giving 5-4-3. This is often rewritten by students as 5-4+3, in which they forget about the parenthesis and distribution.
3. Writing an equation incorrectly: some students like to rewrite equations on a separate piece of paper, and while there is nothing wrong with this, some students do not write it correctly and therefore result in an incorrect answer. Writing a fraction, such as x/2, as 2/x when rewriting would not be correct, because in the original expression, the x is in the numerator, .
4. Remembering formulas: when a teacher gives you formulas, they are important, use them! Some students ask for help over something they can’t solve, because they haven’t glanced down at the formula sheet that their teacher provided them. The problems aren’t solvable without them! Make sure to take a good look at your formula sheet, especially if one won’t be given to you on your test.
5. Not writing down all steps: Many very intelligent students are fully capable of solving equations in their head and just writing down the answer. A lot of times this is effective, but no matter how good the mathematician, they will most likely make mistakes if trying to solve equations all in their head. It is important to write down all steps when answering questions, first to be able to solve it mistake free, and second, in the event that a mistake was made or the correct answer was not found, to be able to look back at work to find the mistake. If you do the whole problem in your head and end up with the wrong answer, you won’t have any idea where the mistake was made.
by Nicole Dowling | Nov 2, 2015 | Math
Derivatives and Curviness
How to use Derivatives when Describing “
Curvy-ness” of Functions
Most students when they first learn about derivatives in Calculus are not exposed to the many uses of this important topic. However, students have heard the term “derivative” disguised as words like SLOPE and RATE OF CHANGE! So how do we use derivatives when describing how curvy a function is? Well, since derivative is just a fancy word for slope, how about finding the slope at select points to see if we change from a positive value to a negative value, or vice-versa, or if we even change signs at all!
For example, I was tutoring a student who had to find the maximum and minimum of various functions on some closed interval. Well we know that if we have a closed window we are looking into, then at some point the curve we are analyzing will have a highest peak and lowest trough somewhere, even if those points are the endpoints of the specified closed interval. Think about a linear function: the derivative of any linear function will be constant. What does that mean when describing the “curvy-ness” of linear functions? It means the slopes at each point in some closed interval is fixed, i.e. if the derivative is negative, then the slope is negative, then the maximum will be the leftmost end-point and the minimum will the rightmost endpoint. *Note the max is the rightmost endpoint and the min is the leftmost endpoint when the derivative is positive* Now what if we have an exponential or logarithmic function. Well, the same argument applies: if the derivative of either function is positive, then the rightmost point will be higher than the leftmost point; if the derivative of either function is negative, then the opposite is true. Why is this true? Think if someone pulled a linear function with some slope (other than 0) straight in either the x direction or the y direction; now we have a bend in our curve, yet we will still have a positive or negative slope AT EVERY POINT!
Now let’s talk about some more interesting functions. What about wiggly and curvy functions??? A wiggly function, like sin(x) or cos(x), will oscillate between some y-axis interval. Meaning we have a maximum y-value and a minimum y-value between some closed x-axis interval. The question is WHERE? We can no longer assume the max/min will be at the endpoints. Remember a term called “critical values”? I can almost hear the moans and groans bringing this term up again, but do not fret. Basically, we need to know when the signs of our slope changes. The method? You guessed it! Take the derivative, and this time set that derivative equal to zero. Why? We want to find out when our function has zero slope. Think about that moment at the very peak of your favorite roller coaster: there is a very brief moment when the machine no longer has to trek your car up a very steep slope, and you are motionless as you look upon the horizon straight in front of you. Or when your car starts its trudge down the other side of this steep slope, down the pits of gravity when all weight is thrusted perpendicular on top of your shoulders into the seat of your car. This is the very most bottom of the hill, at least for that section of the roller coaster, i.e. some interval of this curvy coaster! So the same argument applies for polynomials of some high degree, where the function wiggles many times in some interval. Just find the places where we have zero slope and calculate whether we have positive or negative slopes BETWEEN our critical values and voila! We know where our peaks and troughs are, and therefore we know the “curvy-ness” of any function.
All this can be analyzed WITHOUT graphing these functions at all! Pretty cool right?
by MaThCliX | Oct 20, 2015 | Math
I come across students daily who struggle in their academics, particularly in mathematics. For the students who are not trying or do not care, this message isn’t for you. Unfortunately, there is not much anyone can do to help someone who simply will not try or does not care. For all others, I see many students quietly failing their class as time continues to pass by. One bad grade after another accumulating. At some point, it may possibly be too late to undo! As I begin to understand and discover why this continues to happen to students who are “doing all the homework and still failing the tests”, what I have learned is the simple fact, you can do it, not always alone.
So, students, here are a few tips for you to win your math struggle…
- Do ALL of your homework. And furthermore, write it out on paper, step by step, even if that means you have to kill trees. Learning math means doing math and the paper is worth your learning!
- ALWAYS use answer keys while doing your homework. First, do the problem, then check your solution. Instant feedback is essential to correcting mistakes.
- After doing your homework, if there were types of problems that you didn’t understand or missed a lot of, ask for more practice problems from your teacher or find some on the internet or textbook.
- Okay, so you can do all of your homework? Does that really mean you are ready for that quiz or test? Not necessarily! Create a similar pre-quiz or test and TIME yourself and GRADE yourself. If you are going to make mistakes, make them on your practice quiz/test, not on the real one!
- Build a good working relationship with you teacher/professor. Visit tutorials and office hours. Let them know who you are and show them you are trying. If you are a college student, see if your university offers tutoring or a math lab.
- Create study groups with other students who care and are willing to work hard and want to succeed. Work problems together and check each other as you go.
- Write note cards with important notes/formulas as you go to keep everything in one place.
- And if you still need support, that’s okay! There’s MaThCliX!
You don’t “study” math, you DO it! Bottom line. Know your resources and use them. Others have gone before you and no one becomes a huge success all alone.
by Amanda Fagan | Oct 13, 2015 | Math
Change Your Mind, Change Your Grades
The other day, I was speaking to a friend who recently graduated from college with an Engineering degree. He worked full time during his whole time in school and still managed to graduate with exceptional grades. When I asked him how he did it, he said, “All of those people failed because they told themselves it was hard.” This made me think of all of the difficult classes I have taken over the years—and my attitude while I took them! I confess, I often did tell myself the work was difficult. I would find myself thinking, “This is too hard. I don’t feel like studying right now,” when in reality, those were the times when I needed to work even harder. How much better would my grades have been in those classes if, instead of feeling discouraged, I approached my work with feelings of excitement about the opportunity to learn so many new things? However, sometimes being positive is difficult. Here are some ways to make it a little bit easier:
- Set small goals.
Learning is a process. Sometimes the end goal seems completely unattainable. Try to take it one step at a time. Setting smaller, more attainable goals will give you the confidence boost you need to keep going. As you move towards each goal, you will find yourself improving more than you ever thought you could!
- Gain some perspective.
Try transforming all of your negative thoughts into positive ones. Instead of thinking, “This is too hard!” or “I just can’t do it,” try thinking about all of the progress you have made. Think of your mistakes as opportunities to learn even more. We learn far more from mistakes and failure than we do from doing everything perfectly the first time. Difficulty is simply another opportunity to practice. Try thinking about your homework as an exciting opportunity to learn and practice instead of a trial you must suffer through. Don’t ever think that you cannot do something. Instead try thinking something like, “I can do this, but it might take a little bit of work.”
- Never give up!
The most important thing to remember is to never give up. As long as you keep trying and keep working, eventually you will realize you learned something! Even if the steps you are taking are small, you will eventually have big results. Remember that every bit of practicing gets you a little closer to where you need to be. If you keep trying, you always have the possibility of someday reaching your goals. If you quit now, you have the certainty that you will not.
Remember that your thoughts and attitude have an enormous effect on your studies! Learning becomes much easier and much enjoyable when you treat it as something that can be fun instead of as a chore. It may be difficult at first to have a completely positive outlook. Start small. If you don’t honestly believe what you’re doing is fun, try at least pretending it is. You’ll be amazed what a difference your thoughts make!
by Ty Hilton | Oct 6, 2015 | Math
Dealing with Stress
Throughout my academic career, I have dealt with plenty of stress, whether it was from presenting a project, writing a paper, studying for a test, or writing a blog for work. In any of these situations, it is imperative to not succumb to the stress. Procrastination is a result of stress, but not the only one. Stress can motivate us to accomplish goals, but only if we have the right mindset. The difference between procrastination and completing an assignment promptly is positive thinking. Instead of overwhelming yourself with how much you have to do, try being thankful that at least you have something to do. Having free time is nice, but it can get really boring. Also, try to see the benefits of what you are doing. Try to see what you can learn from doing the assignment. In addition, Focus on how you will feel when you are done, and let that drive you. For example, walking away after completing a test is the best feeling for me. I look forward to that while studying for the test, and it usually motivates me. Lastly, if for any reason you procrastinate on an assignment, and find that you have more work to do than time, then I have one final piece of advice for you. Wing it, and don’t give up. Procrastination may be bad, but it can’t prohibit success. Do your best in any situation and you’ll probably do fine.
by Kenneth Griner | Sep 23, 2015 | Math
The Adult Struggle with Math…
I came across a video the other day that really got me thinking about the generation before me and where my generation might be heading. The video was entitled “Adults Try 5th Grade Math”. Since I enjoyed watching the show “Are You Smarter than a Fifth Grader?” I figured this would be an enjoyable video as well. As I sat there and watched all of the grown-ups who claim to have taken Calculus and other high end math courses that you need for a Master’s degree in some fields struggle adding fractions I began to laugh. This is no laughing matter though. There are adults all around us who cannot do math some elementary school kids find simple. So, why do adults struggle with 5th grade math?
These struggles make you wonder why. Why is it that when school is over with and in your rear view window you forget what you have “learned”? Is it because you focus on what you need for your career and forget the other things you saw as unimportant and not needed? Is it because you get so used to doing derivatives and solving some ridiculous equation that looks like hieroglyphics to others? I believe that it is because nobody has really been taught the lessons that elementary students are learning now.
Think about it. When you were in school it was all about making a good grade; not actually learning what was on that test. I know I have walked out of a test and not remembered how to do a single thing on it but that piece of paper had a big fat A on it. This brings up a whole issue I have with school systems and grading that I will have to bring up another time. Math is like a house. It starts with numbers and basic adding and subtracting which is the foundation and builds up from there. My wish is for the generations before me, my generation and the ones after mine to know that and not forget it. It is nice that you can find the derivative but if you cannot add fractions it is like owning just a window of the second floor of a house that is missing its ground floor. Go back and learn why something happens and you will truly understand it. If you cannot understand why you are already behind.
by Hanna Fleeman | Sep 22, 2015 | Math
We all know how important it is to learn good organizational skills and get organized. Imagine discovering a new law of physics and not remembering where you left your proof! Keeping organized is one of the most important ways to DO MaTh and DO it successfully, so here are some tips to keep you or your student organized.
- Keep everything together.
All of your MaTh assignments, notes, quizzes, tests, and homework should be kept together! Get a binder, and fill it up! Use dividers to separate different categories of assignments, and keep your assignments in order. All of this will help you to easily reference your materials, just in case you need to go back and find something.
2. Keep your notes in order.
A lot of MaTh builds on top of itself. Something you learned in chapter one could be used in chapter three. Keeping things in order will make it easier to go back and reference later, just in case you need to.
3. Organize the work itself.
We all know those lines on the paper are supposed to be used for something. Show your work step by step on your paper! This way, if you get an answer wrong, you can easily go back and see which step you messed up on. This is especially helpful when using order of operations. Finish a step? Write the next one on a new line. Simple, easy (and also very helpful to your teacher who is grading your work).
4. Keep your homework separate and check those answers.
Homework is the most important part of learning MaTh because it’s where you DO MaTh. Check your answers as you work. Did you get one wrong? Go back and explore to see why (look at those steps you’ve been writing out line by line). Ask questions! Whether in class or at MaThCliX, your teachers or tutors are there to help! It’s our job and we are thrilled to do it!
5. Organize your space and make time.
We all know MaTh takes time! And practice makes perfect, so be smart with the time you have. MaThCliX is a great place for you to come and DO MaTh, but it can’t all be done here. Set aside time every day to DO MaTh. Just 30 minutes a day can greatly improve your understanding and techniques. You should also have a specific space in which to DO MaTh. This space needs to be organized, neat, and distraction free. This will help you focus on your work and learn effectively.
Becoming more organized will help you or your student become less stressed and more confident in facing problems head on. Using these tips, make a plan. Organize a space, make time, work it out, and come to us with questions. Have your work neat so we can see exactly what you are doing, and overall be confident! Everyone can DO MaTh, and being organized is just one way to make it easier.
by Karl Mattle | Sep 5, 2015 | Math
Calculators: A gift from the angels or agents of evil?
Here is a true story: A high school student was working an algebra problem that required use of geometry to determine the value of x. He had reached an impasse and so he asked his teacher for help. His teacher assisted him to the point where x could be evaluated. The teacher then prodded the student with the question “Now what is 10 + 6?” The student got a befuddled look on his face and proceeded to punch the arithmetic in to his calculator. After a few seconds the student declared “60” and proceeded to write the erroneous answer on his paper. Besides the disgusted amazement of his teacher, what’s wrong here?
The error analysis is simple, the student accidentally multiplied 6 and 10 rather than adding them. But that is an understandable mistake. Is the problem that the student feels he needs a calculator to do simple arithmetic? In part. But far more alarming is the willingness of the student to accept as fact a woefully incorrect answer. The student had no “number sense”, no concept that the sum of 10 and 6 is nowhere near 60. Many brilliant mathematicians have made, and continue to make similar mistakes. But their sense of in what vicinity of the real number system the correct answer lies, tells them immediately that they have made an error, and they correct it. This may be an extreme example. But it should be an alert to the fact that we as a society, cannot let this tech-savvy generation pass in to adulthood without a sense of what’s greater than what.
Still, it’s not time for a calculator burning party. If the problem our geometry/algebra student was working required him to determine a distance that was miles, he would have to use a calculator to approximate the square root of 10. Telling a police dispatcher that you had witnessed a crime miles back, is likely to get you arrested yourself. But what our student and our crime witness, and indeed all citizens need is a sense that the square root of 10 is somewhere between the square root of 9, or 3, and the square root of 16, or 4. Further, they need to be able to instantly reason that is closer to than it is to because 10 is closer to 9 than it is to 16. Then, if our student, or witness, or good citizen accidentally punches in to a calculator and gets 10, they would immediately realize they had made a mistake and correct it.
So what’s to be done? Calculators do, in fact, make life more productive by freeing our minds for higher order thinking. But basic arithmetic is not higher order thinking. While memorization is anathema to modern educational philosophers, it does breed confidence. Our student should have been certain that 10 + 6 is 16, not because he reasoned it out, or because he counted on his fingers and toes, but because the fact had been burned in to his brain while still in Elementary School. This “number sense” is not acquired by doing Differential Calculus, or Linear Algebra, or Euclidean Geometry. Number sense should be established before tackling these subjects. Memorization is tedious. But its very tedium makes students appreciate the power of calculators once they are permitted use them. Until that number sense is established, calculator use should not be permitted.
We, as tutors, should be very reluctant to permit students to use calculators. However, when it comes to checking an answer, we should be enthusiastic about using calculators. If, for instance, a student from Booth Middle School, uses linear interpolation to approximate and gets 3.1, He or she should be shown that a better approximation is 3.1623, found by using a calculator. Now the student can see that they have done well (after all, they are just getting an approximation), they get a confirmation of their number sense, and they are compelled to think on a higher level in reasoning why their method works but is less precise than the calculator.
by Chuck Summers | Aug 31, 2015 | Math
I think most of us tend to look at the success of others and think, “Well, they are just talented.” I could never do what he does ”. How is it that the Michael Jordans, the Steve Jobs, the Tiger Woods of this world are so extremely successful at what they do, while a lot of others languish in mediocrity?
There has been a lot of research in this area in the last few decades, so the answer is not a mystery. The common view is that one person succeeds because of some innate ability they were born with, and unless you are born with that ability, you are not going to be as successful as they are. In a book titled, ‘Talent is Overrated’ Jeff Colvin outlines the key to achieving greatness: specific practice over time. It is not the practice that most of us do when we hit a few balls on the driving range, it is focused guided practice that has as it’s goal improvement in a specific area.
Key to this focus is feedback. It is necessary to receive feedback as quickly as possible as you practice. It is exceptionally helpful if you have a teacher, a parent, a mentor, or tutor who can help you see where you are failing. It is not necessarily 24/7 focus, but a couple of hours a day every day over time can make you exceptional in whatever you are working towards. Note that failure is necessary if you are working towards improvement in anything: it means you are pushing the edge of your capabilities and will grow as a result.
Most of us have a model of learning that limits our capabilities. We view our brain like a big box that can only hold so much. We are afraid that when we learn something that is not useful to us at a future date that we are taking up valuable space that we will need at a later date. Your brain is not in danger of getting full. The more connections you make, the better you are able to learn new ideas and make new connections. It has been shown that learning a new language helps in learning other things as well. Learning the much maligned quadratic equation (especially it derivation) helps your brain with the discipline of learning in general. When you memorize a poem or even memorize baseball statistics, it helps your brain learn to learn, and that is what you want to improve. Mental games that you can use when you are bored or driving can help your ability to focus. The ability to learn can be improved with practice, but if you don’t practice your brain will become lazy.
At MaThCliX, our goal is not simply to help you get good grades, but to help guide you towards exceptional results in math and in your life. If you come in and do your homework, get some encouragement and leave, then you are not taking advantage fully of what can really help you. Instead of being happy with just what is assigned, ask us to guide you to what the next step would be to improve your abilities and disciplines.
We have access to the curriculum of most students in our area and can help with more focused practice in areas you want to improve and once you have mastered a topic, we can show you what is next and help you excel. Don’t be afraid to learn something new and difficult: that is how you stretch your mind and disciplines.
The principle of focused practice over time works in every area of your life and can help you become the best you can be in the areas you are called to be successful.
