The semester is over , well, almost over . All that is left is for me to post the final grades.
At first glance this would seem like a simple task that I should be able to complete in a matter of minutes, except for one thing, I detest posting the final grades ! I do not like failing students, although I realize that not failing some of these students would be totally unfair to them and to the rest of the class.
Let me explain. I would actually love posting the final grades if all of my students had done well enough to pass. But that is not the case. This past semester I taught Math 098 which is a Developmental course in Algebra. Simple stuff, and yet, many of the students seem to struggle through this material that , lets face it, they already had in Middle School and High School.
So, the way I see it, it all comes down to me making a decision as to whether or not a student is "ready" to move on . And I do not enjoy making these decisions.
At the end of the semester I always struggle with this and I always end up blaming myself for some of the failings of my students. I should have done this, I should have done that. I should have presented the material in this way or that way or which ever way that would have helped the students do better.
But lets face it, not all students are mature enough to be in college. That is the way I feel . College work requires that students have a level of maturity and responsibility that is at least higher than High School. And some students lack this maturity.
Well, the semester is over , and next time I hope to do things better so that all of my students pass the course. Let's hope !
Friday, May 17, 2013
Monday, April 1, 2013
Blackboard or PowerPoint ?
When preparing a set of lectures for a college mathematics course one of the decisions that an instructor has to make is whether to prepare notes for a blackboard presentation or use more modern technology such as PowerPoint slides and a projector. In my opinion, it depends on the audience (i.e. the students ).
PowerPoint slides offer several advantages . For one thing , it is relatively easy to integrate color , sound and animation with these presentations. One can capture the attention of (some) students with fancy dynamic effects and a rainbow of colors.
While such presentations take some time to put together, once the slides are finished they can be reused , albeit with minor modifications, year after year.
Blackboard presentations on the other hand have obvious artistic /visual limitations and after a repetitive writing and erasing the writing on the board can become sort of cloudy and confusing. On the other hand , I believe , blackboard presentations are more flexible and the content can be changed at any moment.
But what is best for the audience ? I believe it depends on the level of the students.
After years of trying both, PowerPoint as well as Blackboard presentations, I have come to believe that for students taking remedial mathematics courses , PowerPoint presentations move much too fast. At a moments time a whole equation appears on the screen and this sudden emergence of algebraic notation is much too fast for many students to capture and understand.
When writing an equation on the board on the other hand, every part of the equation shows up relatively slowly. This gives the student's mind some time to absorb and understand what is being presented.
PowerPoint slides offer several advantages . For one thing , it is relatively easy to integrate color , sound and animation with these presentations. One can capture the attention of (some) students with fancy dynamic effects and a rainbow of colors.
While such presentations take some time to put together, once the slides are finished they can be reused , albeit with minor modifications, year after year.
Blackboard presentations on the other hand have obvious artistic /visual limitations and after a repetitive writing and erasing the writing on the board can become sort of cloudy and confusing. On the other hand , I believe , blackboard presentations are more flexible and the content can be changed at any moment.
But what is best for the audience ? I believe it depends on the level of the students.
After years of trying both, PowerPoint as well as Blackboard presentations, I have come to believe that for students taking remedial mathematics courses , PowerPoint presentations move much too fast. At a moments time a whole equation appears on the screen and this sudden emergence of algebraic notation is much too fast for many students to capture and understand.
When writing an equation on the board on the other hand, every part of the equation shows up relatively slowly. This gives the student's mind some time to absorb and understand what is being presented.
Saturday, March 23, 2013
Emphasize Proper Mathematical Syntax
After teaching remedial mathematics and algebra courses for a number of years I have come to
recognize a common problem among my
students that hinders their ability to read and understand simple mathematical concepts.
I call this problem : Mathematical Syntax
In other words, I believe that some of the main difficulties that my students have in understanding basic mathematics is because during their school years they were improperly trained in the correct use and meaning of basic mathematical symbols and components such as equal signs and parenthesis and the difference between expressions and equations and how to properly work with them.
These problems might explain why , quite often, students tell me that they can’t understand their textbooks, even though the textbooks we use are generally quite straight forward and they often contain many clearly written examples.
Students often fail to recognize the proper use and meaning of parenthesis as a way to clarify or emphasize the order of operations in expressions and so, quite often , I find them writing expressions like X+ - 5 without a parenthesis separating the addition and subtraction symbols.
I call this problem : Mathematical Syntax
In other words, I believe that some of the main difficulties that my students have in understanding basic mathematics is because during their school years they were improperly trained in the correct use and meaning of basic mathematical symbols and components such as equal signs and parenthesis and the difference between expressions and equations and how to properly work with them.
These problems might explain why , quite often, students tell me that they can’t understand their textbooks, even though the textbooks we use are generally quite straight forward and they often contain many clearly written examples.
Students often fail to recognize the proper use and meaning of parenthesis as a way to clarify or emphasize the order of operations in expressions and so, quite often , I find them writing expressions like X+ - 5 without a parenthesis separating the addition and subtraction symbols.
On a regular basis students also fail to make
proper use of the equal sign. Often leaving this important symbol out of
equations or expressions. Or students tend to write two or more
equal signs in the same equation. These tendencies often lead to total confusion
during the solution of an equation in that the original equation ends up
looking more like an expression rather than like an equation (expression 1=
expression 2) . Sometimes even the actual variable gets totally lost in their work !
I also find that quite often High School students learn to apply the addition/subtraction property of equations by writing the terms to be added or subtracted in a vertical format rather than in a horizontal format. I find this quite unfortunate given that most college mathematics books use the horizontal format. This , I believe might explain, in part, why many students get confused when reading their basic algebra college textbooks.
So, I am of the opinion that proper mathematical syntax has to be strongly emphasized in schools to prepare students for college mathematics courses. I also believe that High School mathematics teachers should emphasize the format that students will encounter in college rather than the vertical addition format used in High Schools.
I also find that quite often High School students learn to apply the addition/subtraction property of equations by writing the terms to be added or subtracted in a vertical format rather than in a horizontal format. I find this quite unfortunate given that most college mathematics books use the horizontal format. This , I believe might explain, in part, why many students get confused when reading their basic algebra college textbooks.
So, I am of the opinion that proper mathematical syntax has to be strongly emphasized in schools to prepare students for college mathematics courses. I also believe that High School mathematics teachers should emphasize the format that students will encounter in college rather than the vertical addition format used in High Schools.
Friday, January 25, 2013
The values in Math
Jan 22, 2013
Today is the start of the Spring semester. I have thought a great deal about what outcomes I want the students to take with them at the end of the course. Clearly one outcome is to “know” the basics of algebra. They will need some of this for their next course which is Statistics. But surely there must be other valuable knowledge that the students can take with them. I am thinking of outcomes that transcend their college studies and that they will find applicable after they leave college and enter the workforce regardless of the area of work that they decide to pursue.
So, here are some valuable outcomes that I believe will serve my students well in their lives :
(1) I would like my students to become self–learners . I would like them to be able to take a book, any book on just about any subject , and be able to make some reasonable sense out of it. I want them to not need someone to hold their hands all the time. This can be applied to any area in their professional lives !
(2) I would like them to question themselves . To realize when they do not understand something and to search for answers on their own. Again, this applies to any career paths they might decide to follow.
(3) I would like them to learn how to think “logically” . That is , if a series of steps lead to a solution , then step 2 must logically follow from step 1 , and step 3 must logically follow from step 2 and so on.
This applies in Math as well as in English. When writing a paper for example, paragraph 2 must somehow follow logically from paragraph 1 and so on !
(4) I would like them to learn to focus and concentrate on their work and to learn to pay attention to detail. In Math this means to pay attention to equal signs and units , etc.
This can be applied not only to Math but also to Business or to English or Landscaping Design for example. When writing a paper (on any subject !) the details are important !!!
(5) I would like them to put all of the above points together and to learn to question the final outcome or result, i.e. I would like them to do some critical thinking.
(6) Finally, I would like them to realize that objects such as calculators, printers, etc. are just tools.
That the important first steps in solving a problem are understanding the problem and coming up with a path that leads to a solution. A calculator will not do that for us. Our ideas, concepts or solutions will not become better just because we change from a typewriter to a word processor.
So, when I am teaching Mathematics, I like to think that I am actually teaching much more than just that I actually believe I am giving my students valuable tools to do better in their jobs and in their lives regardless of the career path they eventually select.
I
Today is the start of the Spring semester. I have thought a great deal about what outcomes I want the students to take with them at the end of the course. Clearly one outcome is to “know” the basics of algebra. They will need some of this for their next course which is Statistics. But surely there must be other valuable knowledge that the students can take with them. I am thinking of outcomes that transcend their college studies and that they will find applicable after they leave college and enter the workforce regardless of the area of work that they decide to pursue.
So, here are some valuable outcomes that I believe will serve my students well in their lives :
(1) I would like my students to become self–learners . I would like them to be able to take a book, any book on just about any subject , and be able to make some reasonable sense out of it. I want them to not need someone to hold their hands all the time. This can be applied to any area in their professional lives !
(2) I would like them to question themselves . To realize when they do not understand something and to search for answers on their own. Again, this applies to any career paths they might decide to follow.
(3) I would like them to learn how to think “logically” . That is , if a series of steps lead to a solution , then step 2 must logically follow from step 1 , and step 3 must logically follow from step 2 and so on.
This applies in Math as well as in English. When writing a paper for example, paragraph 2 must somehow follow logically from paragraph 1 and so on !
(4) I would like them to learn to focus and concentrate on their work and to learn to pay attention to detail. In Math this means to pay attention to equal signs and units , etc.
This can be applied not only to Math but also to Business or to English or Landscaping Design for example. When writing a paper (on any subject !) the details are important !!!
(5) I would like them to put all of the above points together and to learn to question the final outcome or result, i.e. I would like them to do some critical thinking.
(6) Finally, I would like them to realize that objects such as calculators, printers, etc. are just tools.
That the important first steps in solving a problem are understanding the problem and coming up with a path that leads to a solution. A calculator will not do that for us. Our ideas, concepts or solutions will not become better just because we change from a typewriter to a word processor.
So, when I am teaching Mathematics, I like to think that I am actually teaching much more than just that I actually believe I am giving my students valuable tools to do better in their jobs and in their lives regardless of the career path they eventually select.
I
Jordi
Friday, May 15, 2009
"This Is Not How I Learned it in High School "
May 15 , 2009
I have been teaching Algebra courses for several years now and I am still "surprised" (i.e. annoyed) when a student tells me , in reference to a topic that we covered in class - "This is not how I learned in high school, do I need to learn it your way ?"
First of all, I want my students to understand that "my way" is not my way , but one way in which the problem being discussed can be solved (most likely, the solution I presented follows the text we are using) . There can be other correct ways in which the problem can be solved .
What I think is really important - at the college level at least - is that students learn to recognize that , in general, there are several ways for solving a given a problem.
Furthermore, I would like students to realize that they should be stressing understanding of the various steps , rather than memorization.
Consider the following example: (x+1)/x can this be simplified ?
Some students will(probably) say yes, we can cancel the x's . Right ? ... WRONG !!! The x's cannot be cancelled in this case. But why ? Do we understand why ?
It is my experience that most [ non-science ] students can't give a correct explanation of why the x's can't be cancelled , they just memorized what they were taught in high school.
In other words, they fail to see cancelling as an the application of the identity property.
I think both , memorization and understanding , are important learning elements in mathematics and science, but in general, I believe that understanding is extremely important.
I have been teaching Algebra courses for several years now and I am still "surprised" (i.e. annoyed) when a student tells me , in reference to a topic that we covered in class - "This is not how I learned in high school, do I need to learn it your way ?"
First of all, I want my students to understand that "my way" is not my way , but one way in which the problem being discussed can be solved (most likely, the solution I presented follows the text we are using) . There can be other correct ways in which the problem can be solved .
What I think is really important - at the college level at least - is that students learn to recognize that , in general, there are several ways for solving a given a problem.
Furthermore, I would like students to realize that they should be stressing understanding of the various steps , rather than memorization.
Consider the following example: (x+1)/x can this be simplified ?
Some students will(probably) say yes, we can cancel the x's . Right ? ... WRONG !!! The x's cannot be cancelled in this case. But why ? Do we understand why ?
It is my experience that most [ non-science ] students can't give a correct explanation of why the x's can't be cancelled , they just memorized what they were taught in high school.
In other words, they fail to see cancelling as an the application of the identity property.
I think both , memorization and understanding , are important learning elements in mathematics and science, but in general, I believe that understanding is extremely important.
Thursday, February 19, 2009
To my students:
Don Marquis wrote " If you make people believe they are thinking , they will love you.
If you really make them think , they will hate you "
Algebra is like a game . Think of any game, soccer, football, Scrabble, Monopoly ...etc., what do they all have in common with algebra ? The answer is rules.
Without rules a game would have no meaning and no fun.
Imagine for a moment that we decide to play Scrabble but we agree to eliminate all the rules.
Any word , whether real or imaginary, properly or improperly spelled would be allowed in this Imaginary Scrabble. Scoring would be simple, just add the number of letters used in a "word".
Try playing this game with a friend. Soon after you start you will find that the game presents no challenge and you will soon stop playing .
Without rules a game has no purpose and no fun.
It is the rules of the game and how well we can play ,within the rules, that makes the game fun to play.Learning how to play the game means ,first, learning the rules and second, learning how to achieve the goals of the game (i.e. winning) without violating the rules.
How we use the rules to win is very much left to our creativity and imagination.
Mathematics, in this case algebra, is no different.
In an algebra class the instructor presents you with a set of rules. and shows you a few examples of how those rules can be used. The rest is up to you. You must use your creativity and imagination to "win" (i.e. to solve similar albeit more complex problems) without violating the basic rules of the game.
So, next time you take an algebra course, think of it as a game. Pay attention to the rules , understand them and know how to use them to your advantage .
And one more thing, try to work without your calculator. Using a calculator in a basic or intermediate algebra course is like playing volleyball wearing a football uniform, it will slow you down and you will miss the point of the game !
So, what is your instructor trying to accomplish ?
Your instructor is trying to challenge your imagination and creativity within the game of algebra.
Your instructor is trying to make you think !
Don't hate him/her . Instead, learn the rules of the game ! It can actually be fun.
jmarti
Don Marquis wrote " If you make people believe they are thinking , they will love you.
If you really make them think , they will hate you "
Algebra is like a game . Think of any game, soccer, football, Scrabble, Monopoly ...etc., what do they all have in common with algebra ? The answer is rules.
Without rules a game would have no meaning and no fun.
Imagine for a moment that we decide to play Scrabble but we agree to eliminate all the rules.
Any word , whether real or imaginary, properly or improperly spelled would be allowed in this Imaginary Scrabble. Scoring would be simple, just add the number of letters used in a "word".
Try playing this game with a friend. Soon after you start you will find that the game presents no challenge and you will soon stop playing .
Without rules a game has no purpose and no fun.
It is the rules of the game and how well we can play ,within the rules, that makes the game fun to play.Learning how to play the game means ,first, learning the rules and second, learning how to achieve the goals of the game (i.e. winning) without violating the rules.
How we use the rules to win is very much left to our creativity and imagination.
Mathematics, in this case algebra, is no different.
In an algebra class the instructor presents you with a set of rules. and shows you a few examples of how those rules can be used. The rest is up to you. You must use your creativity and imagination to "win" (i.e. to solve similar albeit more complex problems) without violating the basic rules of the game.
So, next time you take an algebra course, think of it as a game. Pay attention to the rules , understand them and know how to use them to your advantage .
And one more thing, try to work without your calculator. Using a calculator in a basic or intermediate algebra course is like playing volleyball wearing a football uniform, it will slow you down and you will miss the point of the game !
So, what is your instructor trying to accomplish ?
Your instructor is trying to challenge your imagination and creativity within the game of algebra.
Your instructor is trying to make you think !
Don't hate him/her . Instead, learn the rules of the game ! It can actually be fun.
jmarti
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