What most parents want in providing math tutoring for their child is to improve their grades, or perhaps to provide support for them in an accelerated math program.  And with the immediacy of an ongoing math class, this certainly makes sense.

And of course everyone looks forward to the improved self confidence and the positive attitude that comes with higher scores on homework and tests.

In addition, every one is aware that math skills can be useful in a variety of real life situations, from the most basic tasks like balancing a checkbook, to those undertakings requiring advanced mathematical abilities such as theoretical physics, or designing the next rover for Mars.

So improved grades, enhanced self confidence, and real world math skills are often the items that are at the top of the list when someone signs up for tutoring.

However, there are also some very big potential long term habits and skills that can be gained when working with a tutor or a knowledgeable parent.  These include:

• Learning to be careful, systematic, and methodical
• Learning and understanding how logically some things follow from other things (this is especially needed for geometry proofs) 
• Learning to think critically and ask “Does this make sense?” or “Is this true?”
• Learning to see a problem from more than one perspective and to use multiple approaches to check one’s work.

Students are less likely to develop these skills when working on their own, because their focus tends to be on simply getting their homework done so that they can go on to something else.  A tutor can encourage a student to slow down a little and take the time and extra thought needed to develop these skills.

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Developing the ability to be careful, systematic, and methodical

The most fundamental skill is simply learning to be careful and methodical.  As one gets into Algebra and as math problems start to become more complex – often requiring multiple lines of calculations – it becomes increasingly necessary to work in a systematic and orderly fashion.

Skipping steps, or doing too much of the work in one’s head will often result in incorrect answers, especially for the beginner.   So becoming really good at math means adopting the discipline of writing things down and being careful and rigorously systematic and methodical when working a problem.

Learning to think logically

Then of course math and logical thinking go together.  The student learns that if one thing is true, then mathematically, another truth follows from that.   While this kind of reasoning occurs in Algebra,  it is the heart and soul of the proofs that one does in Geometry.

For example,

If two angles A, and B are both right angles, then you know that it follows that angles A and B are congruent.

Getting good at math means getting good at following a logical train of thought..

Developing the skill of Critical Thinking

Critical thinking is not quite the same as logical thinking.  With logical thinking, you understand when one thing implies another,  so that being at point ‘a’ means that you will end up at point ‘b’.

With critical thinking you are stopping to ask “Does this make sense?” or “How do I know this is true?”. These are very useful questions to ask as you are doing math.  It can save you from basic arithmetic errors that will throw off a whole problem, as well as other more complex mistakes.

For example,

If you multiple 35 x 47 and get 783,  you can do a quick mental check to see whether or not this makes sense.   

Rounding the 35 and 47 down and then up, one can see that the answer should be in between a lower limit of 30×40 (1200)  an upper limit of 40×50 (2000).

So 783 is undoubtedly wrong — it isn’t big enough.   Asking “Does this make sense?” and making the estimates keeps you from blindly accepting the 783 as being right, and lets you know that you need to go back and rework the problem — hopefully more carefully and methodically. the second time around.

If on a second try you get an answer of 1645,  you won’t know for certain that the answer is correct, but at least you know that you have an answer that makes sense since it is in the right range.

Students who are struggling just to get their homework done usually don’t take the time for this kind of thinking.  A tutor can ask these kinds of questions and help teach the discipline of looking at a problem critically.

Learning to consider Multiple Approaches to Solving a Problem

Learning to use multiple approaches in order to check one’s work is also incredibly valuable.  As problems get more complex, one of the best ways to check one’s work is to look at it from another angle.

For example,

If you created the graph of a line by plugging in points for x and y and then plotting the results, what happens if you put the equation in slope intercept form and then graph it?

Do you get the same line?  Is the slope the same?  Is the line going through the y axis at the correct point?  Or is there a mismatch between the two lines?

A tutor can help a student to understand what they are learning from more than one perspective, and help to ingrain the habit of checking one’s work by looking at it from another angle.

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So in the short run, the goal of math tutoring tends to be about test scores and grades.  In the long run however, math tutoring can help instill habits and skill sets that will be of benefit for a lifetime — with these gains being far more valuable than any knowledge accrued in a given math course.
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Math can be difficult.  After all, it requires a certain amount of repetition and dedication.   You have to spend time working a lot of problems to get good at it.  Math takes time.   

And for students that have gaps in their understanding, all of the time in the world spent working problems may end up yielding only frustration rather than the desired results.  One missing piece can doom an entire homework assignment.

However, if a student is able to get help so that the missing pieces can be identified, and the needed explanations given, then a student’s time and efforts can yield success and mastery rather than confusion and frustration.

Math’s reputation for difficulty also stems from the fact that many students are not used to having to consistently apply themselves to a subject in order to do well in it, but they find that that is not true for math.

This especially can be the case for bright students who may have gotten through primary and middle school math just fine without ever really worrying about completing their homework.   Somewhere in high school this lackadaisical attitude often tends to create less-than-desirable results.   So by comparison to the easier days of primary school, math starts to seem hard.

So math does require diligence.  And all the diligence in the world is of limited use without sufficient understanding.   Lack of either hard work or lack of understanding will tend to create problems for the student, thus giving math the reputation for being “hard”.

But the flip side is also true.   With both hard work and understanding, mastery can be achieved and math can feel like a superpower.   This can be especially true for students for whom math once seemed utterly impossible.

Frustration and self-doubt can be turned around into a solid sense of confidence and self-esteem.  And for many students, that confidence and self-esteem is going to be of inestimable value going forward in life.   And every time they are able to confidently do a math problem or succeed in some real world math application, there will be that little burst of pride for a “difficult” subject that was taken on and conquered.

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