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.

 

Back-to-School Success

Tips for Students and Parents

And just like that, another summer is over and a new school year begins! Here are some tips for both parents and students to work together to ensure a successful school year.

Students

1.  Set goals: Write them out clearly and display them somewhere that you see them everyday

                ex: I will complete my HW before I watch TV

2.  Get organized: This includes finding a way to organize papers going back and forth from subject to subject.  How are you going to know what your assignments are and when they are due? 

3. Plan: What HW, tests, and quizzes do you have this week?  How will you prepare for them?  Make sure you plan out your study time.

4. Practice: This is how you learn!  Make time each day to practice.

5. Get Help: Are you not understanding what you are supposed to be learning?  Ask!  Get help!  Go to your teacher, parent, and of course, MaThCliX!  That is what we are here for.

Parents

1.  Make sure that you know how to communicate with your student’s teacher.  Know when conferences are and plan to have a presence and be proactive in your student’s academics.

2. Check grades!  Even if your student is old enough to check their own grades, it never hurts to have a parent checking, too.  Know when progress reports and report cards are due.  If you see grades dropping, intervene quickly!

3.  Make sure your student is doing the success tips for students.  Ask them how they are doing each one.

4.  Find out about what student’s are learning each week so that you can help or get help, as needed.  Find out about tutorials, teacher websites, and recommended resources.

5. Bring your student to MaThCliX! 

Math Manipulatives

Math ManipulativesDo Math Manipulatives Help Our Students Learn?

What are they?

A math manipulative is an object that is used in the teaching of mathematics that allows students to perceive the idea or concept they are learning through touching and moving the object.  These manipulatives can range from anything like dice or money to pattern blocks, two-color counters, and even playing cards or dominoes. 

What age groups?

All ages can benefit from the use of manipulatives while learning math.  Math manipulatives are most commonly used in the early elementary ages or younger.  Once students become more capable of abstracting concepts (older elementary, middle, and high school), teachers seem to have students spend more time doing math with paper and pencil, and less with hands on methods.

What are the benefits?

The use of manipulatives in the learning of mathematics allows students to represent math in multiple ways.  More senses become engaged, including visual and tactile, which keeps a student more attentive.  They are able to “see” math, which reinforces the conceptual understanding.  This lays the groundwork for the mechanics that they will use later and allows the rules to be more meaningful and make sense, which in turn, will be less for them to “memorize”.  Seeing math allows students to expand on ideas and uses of math in the world around them.

Why aren’t teachers using them?

Three reasons that math manipulatives are not used as often as they could, is time, money, and lack of knowledge.  Developing the concept with a manipulative may require more time and so often, our teachers are burdened with getting through the material.  While many math manipulatives on the market can be costly, not all manipulatives are expensive, but having enough for a class set could get pricey.  Each math manipulative can be used to teach a variety of concepts.  Often teachers may not know how to teach various concepts with these tools, and so they just do not get used.  There are many companies out there that do trainings with their manipulative for teachers to learn.

This blog has an ultimate list of math manipulatives that can get you started!