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.
by Rebecca Mayer | Aug 26, 2015 | Math
How do you know if it is time for a challenge?
I believe that a very important aspect of learning is knowing when to challenge yourself. Struggling on things helps make you better and m
aster topics rather than taking an easy route, which might allow you to learn the basics of something, just not grasp it fully. Accepting a challenge and at least trying to accomplish it is what will lead you to success, even if you pass a few failures along the way. Not trying at all will always result in failure. Some people like to coast through their education, but I liked to be pushed further than the basic standards of a class, so that I am stimulated through that subject and not bored.
I recently created a picture graph for my pre-calculus class, and I decided to create a personal challenge out of it. Instead of doing the twenty required equations to create the image, I decided to go further and I ended up making a picture with over 300 equations. I felt way more successful and proud of myself after completing it, more than if I had done the bare minimum of what was required. Rather than spending ten minutes on it, I spent countless hours that I believe paid off long term.
Overcoming a challenge can feel extremely rewarding and make you feel so much more successful than if you did the bare minimum of something. Even though challenges might take longer or more effort to accomplish, you will feel way better about succeeding at something difficult than something easy.
by Nicole Dowling | Aug 17, 2015 | Math
Mathematics IS Truth
When I first came to college, I went through an emotional culture shock. I had no idea what kind of box we were all in throughout high school, but once I took the next step toward supporting myself in the adult world, I see now that it’s a scary thing breaking away from the habit of high school. Ordered. Structured. Dates were set for you; your whole schedule seemed all planned out. Of course, personal success depended on each and every student outside of the classroom, but you were surrounded by support: teachers, other students, the atmosphere of it all.
College has the same feel, don’t get me wrong. But you are not in school 8 hours a day, 5 days a week anymore, from someone else’s design anyway. You are setting your own schedule, creating your own support groups and may even have time in between classes to do with how ever you please. Those who are diligent will continue that “all day at school” mentality in order to keep up with your own studies. Success can look like many things, sometimes it can just be encouragement from a friend to continue pushing through the hard studies. This is supposed to be a team effort.
The acceptance of change in one’s own life can seem like being on the tallest hill and down below you oversee the obstacles that lay ahead, as dangerous as they may appear. Sometimes you are on that hill and all you see when you look down is fog. Unknowing to all that is ahead of you. But you know in your heart it’s time to keep going. For something. Chest out. Head high. One foot at a time, till you’re on your path of finding your own truth. Spoiler alert: Mathematics IS Truth.
Mathematics uses proof and logic in order to state the truths about our universe. The use of a proof is simply a description of why something is true and uses all possibilities to obtain these proofs.
But we have a slight advantage: there was a curious spark, an omnipotent insight that pops in when truth is all around. It feels good. And if you pay attention, the world seems to flow in harmony.
So, if you ever find yourself on top of that tallest hill, delve into mathematics. There’s a branch for everyone’s interests and you might find that creative brainwave needed to take your first step. Every student, thus every human, has an influence within the world in which we live. Influenced by movement. Not necessarily which choices to take next, but the act of choosing as they come. This is a fast existence. Darwin should have called his studies: Survival of the Adaptive.
by Katie Boswell | Aug 6, 2015 | Math
The Answer is Blowin’ In the Wind
Wouldn’t it be great if all of the answers were actually blowing in the wind? Unfortunately Bob Dylan wasn’t speaking of the ACT and SAT when he wrote this fabulous song over 50 years ago. What Dylan did get right was the wisdom of the roads we must travel to actually get the answers. With major changes coming to the SAT, the answers may be even harder to find, but at MaThCliX we are working around the clock to demystify these changes and help your student score their best on standardized testing.
Many have asked me about the changes and how they will impact students. The test is changing the scoring system from 2400-1600, with the writing portion now being optional, but more importantly the content is changing significantly. The approach to the reading passages will be completely different and the math will now explore higher level content areas. The reading will now include more content based reading, but with more common vocabulary. I GREATLY encourage everyone to learn more about these changes and schedule a meeting with me to discuss how this may change things for your student.
The PSAT is also changing and even sooner! The PSAT will be debuting its changes, quite similar to the changes for the SAT, in October! The scoring will now more closely align with the scoring of the 1600 scale SAT and it is expected that PSAT scores will now be a better indicator of how a student will do on the SAT. In the past the PSAT was a good introduction to the SAT, but lacked transferability to SAT scoring. I think this is a great new feature of the new PSAT. I look forward to making my own comparisons once we have real world examples to use.
“The ACT isn’t changing too, is it?” is the other question I’m getting asked quite often. No, the ACT is not changing this year, except for its typical simple updates. In all reality, the SAT is changing to be quite a bit more like the ACT. This is good news for those who like the ACT already and it is also great for those who would like to prepare to take both tests.
Come join us this Fall to prepare for all of your test prep needs. Debuting this Fall, MaThCliX will be offering an unique opportunity to learn test prep skills in a book club. Test Prep Book Clubs will run for two months and will teach reading comprehension, vocabulary, writing and test taking strategies. We will also continue to offer our great one on one tutoring packages designed to see the greatest growth in test prep achievement.
Contact me today to create your perfect plan and learn more about all the changes!
by Aaron Sandlin | Aug 4, 2015 | Math
Break out of your cage!
Forget about the lines that have been placed around the equation in your mind! They aren’t real! You may have been taught a certain formula or equation that you hold sacred in your belief in the immutable structure of math. Please, do away with these structures that hold you tight. The only things that are not fluid in math are the basic rules; the properties inherently found. These are things like the commutative property, where two plus five equals five plus two and so on. The majority of mathematical discovery has been the result of people willing to play around with a known equation.
For example, the law of cosines is able to be turned into the dot product of vectors. These two formulas are separated in instruction by different classes; trigonometry and calculus 3, respectively. However they are really separated only by a little algebraic manipulation.
“The definition of the dot product incorporates the law of cosines, so that the length of the vector from X to Y is given by
|X-Y|^2 = (X-Y)·(X-Y)
= X·X-2X·Y+Y·Y
= |X|^2+|Y|^2-2|X||Y|cos(θ),
where θ is the angle between X and Y.”
~https://mathworld.wolfram.com/LawofCosines.html
We are taught to stay on the path that has been built by centuries of achievement prior to our instruction, but all of that achievement was accomplished exactly by people who wandered off of the beaten path. If you want to learn all that there has already been discovered, then by all means stay in your predetermined path. It is much safer there. If you want to truly discover the full power of math, though, then let your mind become fluid, only allowing the basic laws of mathematical nature to be your bounds. This is the difference between a painter and an artist. Even if you do not become the next Rembrandt and your work looks more like Picasso’s, you will have done one seemingly small but immensely important thing: You allowed yourself to question, to explore, to have a brain that sees things in a scientific light.
If you never discover a formula for anything new under the sun, you might, through the mere fact of having developed an inquisitive mind, have allowed yourself to see EVERYTHING in that same fluid manner. Art, business, sports, tactics, law, medicine….the sky is the limit.
See, math isn’t just for counting; Math is for THINKING and in a manner that literally restructures one’s mind to see all of the universe in a new light. So, cast off those chains of the rigid equations! Play with your numbers! Move things to the “wrong” side of that equal sign! Remake your mind.
Calculators: A gift from the angels or agents of evil?