This is an excellent question and one that teachers struggle with. I'll be interested to see what others think.
Having been a math, science, and technology teacher for many years, I always tried to do what was best for each student.
For some math students that was quick memorization and execution of arithmetic skills. For some students that was giving them a tool to survive, i.e., a multiplication table, a calculator, whatever was useful to them at that point in their learning. For some that meant they just learned their facts differently but equally well in time. (I think Spell Check can do the same for spelling. The more I use Spell Check the better my spelling gets. :-)
In science I wanted the math to be done quickly so calculators were approved.
In all cases, I taught students to look at their answer and determine if it was reasonable. They were asked to estimate their answer in advance. Then they were to determine if their answer was reasonable.
I was never able to boil down my response to this question to a simple yes or no, but more often looked at what helped the student be successful.
For high school students, some equations are important to know but more important to understand. They can always look up an equation but do they understand how the equation was derived, how it is implemented, what the answer means when they get it. Common Core Math standards will put much more emphasis on critical thinking and understanding the process to arrive at an answer.
I did know of one math teacher who (ok this was several years ago) had a series of shirts with one equation on the back of each of his shirts. He said, he spent so much time facing the blackboard (remember what those were) and with them watching his back that he figured he might as will imprint important formulas on their brains.
Thanks for starting this discussion. I hope to see others respond.
I have primarily been a high school math teacher, but I do have several observations from over the years.
First, as a tutor for high school math, I am constantly amazed at the number of students who cannot do simple arithmetic calculations (fractions, integer arithmetic etcetera) manually and are dependant upon their calculators. I will often work with Alegebra II or College Algebra students who are grappling with, and understanding a fairly complex concept, but are stymied by the arithmetic portion. At this point, I make sure they can use their calculator to correctly compute and determine a reasonable answer. These students are usually able to get good grades, but muddled through their middle school years when they didn't quite "get" the arithmetic. However, unless these same students score highly on their ACT, they then usually need to take a placement test in college, and it is the foundational arithmetic skills which they cannot pass, and they often are placed in a much lower math class than they had progressed to in High school.
With this in mind, I believe that every math student should be assessed at every level, for knowledge of foundational mathematics computation skills, and they should be given student-specific remedial non-calculator supported practice until they have obtained mastery. Once mastery has occurred (and it should be tested, and remediated each year), calculators should be the norm. Their ability to enhance learning is phenomenal!
As a 9th grade Math teacher, I find the students only know the order to enter the numbers, not understand the concept. I usually have to start each concept using small numbers and no calculators in order for students to aquire any understanding. They also have no comprehension of the language of Algebra. I have tried to promote requirements for each grade level and offered assistance in teaching to understanding for years.
Do you think that the Common Core strategies for Math will improve the understanding by students of the language of math?
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Jane Brown, Thinkfinity Community Manager
Yes but it depends on good implementation in the lower grade levels. The problem is that many elementary math teachers struggle with math as well. Hopefully the new standards and new resources will help them feel more confident. I know decomposing and composing numbers is the most important skill for number sense in the lower grades...
I made everything about math at home with the 3 year old... how many, how many more, what else do we need, how many books should we read tonight, how many have we already read etc... he loves numbers and can count over 20.