The Universal Solution

“Show your work”, “Show your work”, “Show your work”, etc.!  The all too often phrase said by math teachers to students. And the students reply is often like, “But, I got the right answer!”.  

So, why isn’t the answer good enough for most math teachers?  The answer to this question is that good mathematics is not about the answer, it is about reasoning, which will eventually lead to the correct answer.

Good mathematics is about reasoning.  Solutions in mathematics should never involve “leaps of faith”, guessing, or the idea that it just seemed right.  The beauty of mathematics is there is always a clear, logical argument as to why the solution is correct. The ideal should be that all students can learn how to present these arguments in such a way that anyone, anywhere can follow the solution.  This is what I am calling a universal solution.

What constitutes a universal solution?  

Well, first of all, it should be factually correct.  All information contained in a solution should be true and accurate.  Mathematics is built upon definitions and axioms. It is from there that we can launch forward to proving new ideas.  In classrooms, most students are not focusing on the proof of new ideas, but learning to explain clearly the ones that already exist.  Second, the solution should address the problem in a clear and coherent manner and not contain any irrelevant information. Finally, there should be a logical flow, from step to step, with sound reasoning as to why each claim can be made.  In summary, always ask yourself, could my solution be understood by anyone, anywhere, anytime?

Example of a Universal Solution

Explain which value is greater, 0.003 or 0.0006691?

When comparing numbers, the digit to the left of any other digit will always represent a higher place value.  For example, a 9 in the ones place will always be less than any digit that is in a higher place value, like even a 1 in the tens place.  For 0.003, the first non-zero digit, 3, occupies the thousandths place while in 0.0006691, the digit in the thousandths place is a 0. Although there is a 6 in the ten-thousandths position and 6 is greater than 3, thousandths are greater than ten-thousandths, so any digit in the thousandths position will always be more than a higher digit in a lesser position.  Therefore, 0.003 > 0.0006691.

 

When to Introduce the Calculator

When Should Students be Introduced to Calculators?  I am sure that this topic could spark an interesting debate. Well, I certainly have no interest in debating, but, I would like to bring to light, my professional belief on this topic, after working with math students for almost 20 years.

First, I would like to explain theoretical mathematics as doing math independent of the world.  It uses reasoning, proof, and abstract concepts to establish truth upon truth.  While things going on in the world around us may inspire a theoretical mathematician with new ideas, the study of theoretical mathematics does not depend on the world around us.  So, some might argue that if math does not have a direct “real-world application”, then it is useless.  This is not true since even applied mathematicians draw on theoretical mathematics to solve problems related to the the world.

Research has shown that students receiving more instruction in theoretical math do better overall because they build a mathematical foundation that will allow them to extrapolate math to other real-world situations.  If math is just taught for a specific application, then the student will not be able to transfer that knowledge to another context.  Theoretical mathematics does not change. It is simply, truth.

Now, let’s get back to the calculators! What is the purpose/role of calculators in mathematics?  To make it simple, calculators are used to speed up extensive calculations involved in real-world problems.  The problem is that students who haven’t developed a strong math foundation are using these calculators for much smaller calculations that they should be doing by hand, to strengthen their grasp of numbers, or even in their head in some cases.  When students do computations by hand, they develop a feel for number patterns and a respect for mathematics.  They build a foundation of mathematics that will be evergreen, and thus withstand the test of time and any changing technology.

Consider this example:  I watched an advanced 9th grader go to her calculator to compute 105-90.  A person with a strong sense of numbers would likely “mentally” compute this by knowing that from 90, it’s 10 more to 100 and then 5 more past 100 so the difference is just 10+5 =15.  Once this skill is developed, it is certainly quicker to state this difference without a calculator!  Just in case you think this example is too “complicated”, what about watching a student perform 13-9 on a calculator?  That is not what a calculator was invented to be used for!

So, what am I suggesting?  I am suggesting that calculators aren’t really needed until a student has a solid grasp of number sense.  So solid, they won’t “forget” how to do basic arithmetic on all numbers, including integers, fractions, and decimals. So, when is this?  Well, clearly it will vary from student to student, but in general, I would go as far as to suggest not allowing calculators until at least precalculus.  And even then, limiting their use.  Even the AP Calculus exam and the SAT have “no calculator” sections!

I say to all math teachers, let’s put the pencils in student’s hands and give them lots of paper and let them DO MaTh!  There is no shortcut or tricks to learning math.  Each student has to walk the road and allow their brain to make the connections.

 

Summer Learning

Summer Learning—An Enrichment Opportunity

Summer school!  What student or parent wants to hear that?    Not too many, and that is because the term “summer school” has a negative connotation associated with it.  Usually it insinuates that the student fell short during the school year and has to use the summer to make up their loss. But what about “Summer Learning”?  Does this sound more favorable?

What Do Studies Show?

According to a 2010 study by the Wallace Foundation, just 25 percent of school-age children participate in a summer learning program.  Why should only failing students use the summer to play catch up?  Why not all students use the summer to maintain and get ahead?  I suggest that parents and students see the summer downtime as an opportunity for enrichment and learning for fun, if you will.  Throughout the school year, our students experience a lot of pressures from grades and distractions from other students and activities and rarely get to learn for the sake of learning and to be enriched.

In addition, numerous studies have shown that students forget a portion of what they have learned during the school year over the summer.  This causes many teachers to spend a lot of time reviewing skills and delaying lessons at the beginning of the year.   Students just fall behind as they try to jump back into the school year and keep up. 

Nature of Learning Math

In particular, math is one subject that requires consistent practice and repetition.  Could you imagine if you interrupted your workout routine for 2 months and did nothing?  Surely you might experience, weight gain, muscle loss, and a decrease in cardiovascular strength.  It should make sense that the same thing happens to our math skills if we don’t stay in good practice.

There are many proven benefits to summer learning.  These may include students’ grades upon their return to school, their attendance, and even classroom behavior.  Summer enrichment and learning programs can be found in many places in the community.  MaThCliX®  offers summer math lab and several enrichment classes during the summer.  Math enrichment can even be practiced at home!  You can visit https://www.greatschools.org/gk/articles/build-math-skills/ to learn about some ways to build up math skills around the home.

The Department of Transpo-Repetition

Take a moment and think about your greatest talent or accomplishment.  Did it come naturally to you?  Regardless of your inherent skills and natural gifts, I’d wager there’s no fluke as to why you became proficient at it?  You probably practiced it.  Even those once-in-a-lifetime achievements are not flash-pan.  It takes consistent rehearsal to master any skill, conventional wisdom pointing towards about 10,000 hours of active repetition to completely develop any proficiency, skill, or craft.  

Cognitive sciences hold that the brain retains information through the creation of neural path via a process called “rehearsal”. Building a highway incorporates this rehearsal process into a simple analogy. The first time we see a new vocabulary word or equation, the brain begins the process of cataloging that information by making a very linear path to the idea. Prefatory manipulation of that same information soon develops that road into a two-lane highway, one could think of as an ellipse.  Over time, the more we use a given fact or review a certain concept, we keep elaborating on this highway and eventually our brain adds passing lanes and short-cuts along that same highway; instead of one to and fro’ route created to reach the destination, the brain builds a myriad of possibilities to aide in our scholarly journey.

As soon as you find something you enjoy, or something that really sparks your interest, “go with it”.  Do not hesitate to find sixth gear and “put the pedal to the metal”.  As we break down individual pieces of something we begin to not simply understand what, but why.  Several highly successful entrepreneurs and artists vigorously rehearse this truly simple process:  

  1. Write. It. Down.    

Get a small notebook, and keep it and a pen on you at all times.  Pockets are for things, not your hands.  Even a simple one-word note can lock it in your “thought processor”.  

  1. Re-write.  

Did you capture everything the first time?  What was it that you really wanted to say?  

  1. Re-work.  Re-build.  Re-do.  

Was that everything you wanted to know?  Is that the only reason you wanted to know it?  Is that the last time you want to visit this?  

  1. HAVE FUN WITH THE PROCESS

If at first you don’t succeed, try, try again. 

Many awesome things are nonsense the first time we experience them, right?  At some point we all could not talk or form sentences, we could not write, we could not read… In grammar school we constantly rehearsed these skills, and continue to rehearse these techniques the rest of our lives.   Mastering a skill takes no more effort than continuous performance and utilization of that skill. As the saying goes: “You know how you get to Carnegie Hall, don’t ya?”  Practice.  That first draft pales in comparison to that final, proofed, and re-worked draft.  Your first solo will probably not be the best, but your last will certainly NOT be the worst.  

Master the Digits of Pi

Three Strategies to Conquer MaThCliX Digits of Pi Contest

On March 14th, MaThCliX will be hosting our third annual Pi Day, which is filled with a variety of activities for students of all ages. The most anticipated event of the day is the “Digits of Pi” Contest. The rules are simple: whoever wants to participate merely has to recite as many digits of Pi, the famous irrational number, as they can (in order, of course), in other words, master the digits of Pi. The person that says the most digits of Pi accurately wins a Pi Day t-shirt and a pie/cake! Good luck to everyone competing; I hope you find these tips useful! (P.S. make sure you have the correct digits of Pi pulled up on your phone or computer while attempting to memorize it.)

Strategy #1:

One way to memorize the digits of Pi effectively is through auditory learning. Look at the first 5 digits of Pi and say each one of them out loud. Repeat the process four more times while still looking at the correct form of Pi to guide you. Then look away and try to say the five digits by memory. If you get it correct the first time, then repeat it four more times while looking away. However, if you get it wrong the first time, look back at the correct form and repeat the five digits five times while looking again. Next, attempt to say it five times without looking (successfully this time, hopefully). Repeat these steps until you feel like you have those five digits glued to your brain. If you can use this strategy every day for 10 days before the contest, you will have memorized the first 50 digits of Pi!

Strategy #2:

Carry around a piece of paper with Pi written on it. Whenever you have a minute to spare either in the classroom or at home, take out the piece of paper and begin writing the digits of Pi by memory, as many as you can do. Then look at the correct form of Pi and assess how you did. Next, write it again, maybe this time adding one or two digits on to the end. If you make this a habit for a week or two before the contest, you are bound for success.

Strategy #3:

This final strategy is based off the idea that it is easier to remember numbers that have a purpose rather than a random sea of numbers. What you do is assign phone numbers to each set of ten digits in Pi and then attempt to memorize each phone number. It helps to set patterns within the phone numbers to better remember them: make the first letter of the name for the first phone number an “A”, the first letter of the name for the second phone number a “B”, etc. Also, try making the numbers of letter in each name correspond with the first number in that phone number. Try memorizing one phone number every 2 days, and in 10 days you will know 50 digits.

Everyone is different, so a technique that works for one person might not work for another. Experiment with different memorization techniques and find which one works best for YOU. Also, just a reminder: last year’s winner recited 108 digits of Pi. Good luck, and we’ll see you on March 14th!

What is a Math Person Anyway?

What is a Math Person Anyway?

“I’m just not a math person!”  A phrase often said multiple times by students, but one that is quite false.  But, what is a math person anyway?  While math ability may be genetic to some, there are many different factors that could assist you in becoming a better math student all around. To showcase this, I dive into two factors that could help you become a more successful math student in the future:

  1. Preparation

One of the keys used to achieve the highest level of potential in a certain area of study is preparation. Athletes practice their craft everyday in order to become the best they can be. Musicians prepare their music and practice for numerous hours before ever thinking about stepping on to a live stage in front of their fans. The same principle applies when it comes to the study of mathematics. The most common quote could not be more true for math as practice truly does make perfect. Math requires being exposed to numerous problems in order to fully grasp all the ways a question could be asked to you. Without proper preparation, you are only hindering yourself from the success you could be seeing. Preparation helps take down the surprise effect whenever you’re sitting in class for a test and running through the questions. On top of that, preparation makes you feel more confident, which in turn could lead to more engagement in class or even with a tutor if you’re seeing one. This boosted confidence is just a reassurance to yourself that you’re starting to understand the concepts better and next thing you know, you yourself are slowly becoming that almighty “math person”!

  1. Organization

This principle is definitely overlooked, but ties in hand-in-hand with the skill of preparation. Students should attempt to become an organized person, as it helps in the studying process for tests and quizzes they will foresee in math. For example, a highly organized person always knows where to look back for practice problems, notes, and even homework assignments when preparing for upcoming tests and finals. This can come in handy as they will be able to not only follow along with their notes, but they’ll be able to utilize them in order to create practice tests and problems for a complete review. Organization is one of the key factors tied with any successful person in any industry, and so it certainly matters in the learning of mathematics!

All in all, I want it to be known that anybody can achieve the status of being a “math person”. These two factors are certainly not the only ones that can help benefit you in the long run, but they can be seen as the building blocks to beginning your successful math journey. It all starts with some preparation and organization and the future of your math career is all in your hands!  So next time you see that student next to you make that high math grade, remember, it’s more likely that he or she was prepared and organized and worked for that grade and is not just a “math person”.