I feel like I relate to this article. I have always struggled with math. My family has always struggled with math. This article shows the possible reasons why math has always been a struggle for me. It also shows how math anxiety doesn't just affect people in American it is very prevalent throughout the entire world.
A math tutor suggested playing Monopoly as a way of getting my son to practice math skills. It is amazing how much math there is every turn. Adding the dice, counting the paces forward or back, learning to jump 5 or 10 at a time, buying and selling, paying and collecting rent, giving change, calculation of interest...I highly recommend it!
Have you ever heard of the series "MathStart" by Stuart J. Murphy? The books teach a math skill by telling a story about everyday life skills that children recognize.
Kids learn from their parents at an early age that they don't like math. Mom says, "I hated math." Dad says, "I was never good at math." This social modeling tells kids that it's ok to dislike and avoid math. New low-cost programs such as That's Math aimed at making math part of our daily lives can help reduce math anxiety, leading to better student engagement and eventually better outcomes.
1
I have a PhD, am a published author, speak French fluently and have a some proficiency in Spanish and Dutch. Yet, a simple math calculation is difficult to perform. I have been studying neuroscience for about 20 years to understand the minds of poets. My conclusion is that people with atypically lateralized brains, that is, the right hemisphere is enhanced to the detriment to the left hemisphere, are more likely to experience this problem. Everyone understands the term "dyslexia" but what about "dyscalculia," which is equally difficult to deal with? French neuroscientist Stanislas Dehaene, explains that calculations that require holding one number in mind are performed in the right hemisphere, while multiplication tables are memorized are in the left and easily recalled. If you have right-hemispheric lateralization for language, you may have very little territory remaining for math calculations or visuo-spatial reasoning. I have spent many years studying C.G. Jung and only recently learned that he had difficulty in math too, despite his genius in psychology. His Red Book (2009) was written in meticulous calligraphy, both in Latin and German, with accompanying paintings depicting visions he saw using an active imagination method. We need different types of minds for original contributions to humanity's store of knowledge. Thankfully, calculators exist to balance the checkbook.
3
I've never suffered from "math anxiety". While many of the comments praised this article for validating their discomfort as real, I think it might only help to make "hating math" seem "more normal". People who enjoy math, like those who enjoy that other "nerdy" subject---science, and the subjects themselves, are often the favorite target for derision. I think that "hating math and science" is behind a lot of the "anxiety"---people from their earliest days are taught to NOT like them. Some normal parents probably refrain from helping their children to learn math through everyday experiences because....goodness, talking about how numbers relate to each other might cause their kids to be "abnormal" socially. The social stigma around math needs to change in order for people to allow themselves---and their children---to be comfortable with math, even finding it fun. At this point, people, especially young women, go out of their way to proclaim their "normalcy" by embracing math aversion, and parents reward this by paying for things like T-shirts that say "I'm too cute to do math", etc. What do you expect young children to think when their big sisters advertise anti-math sentiments?
1
This article was very fascinating and relatable to my life. All my life I've struggled on math (yet I still decided to take two AP math courses all in one school year)and the author said his mom wasn't able to always help him and I can relate to that a lot since both of my parents were immigrants and barely spoke English. However, I enjoyed this article because it stated that math anxiety is real, however people who want to do well in school tend to have more anxiety and I feel like that's me since I want to succeed to much. Growing up with math anxiety was stressful but it's good to know one of the causes of it is because I want to do well in school. It was also relieving to read that even in other countries (countries that perform better than us) suffer from math anxiety. A quote from the article "is being bad at math caused by math anxiety, or does being anxious make you bad at math? Very fascinating study, and another study I found interesting was how on all subjects math was the only subject that recorded high blood pressure/heart rate. Very cool studies in this article and I could relate to many of it.
I have a similar experience with math as well. My mother really thought it was up to the school to teach me math. Today, the methods teachers use to teach math are different from those in the past. This made it difficult for my mother to help with math homework or practice. I definitely grew up with math anxiety. It is to the point where now I just use a calculator for the simplest math problems, just to make sure I do it correctly. One thing to think about to relieve some math anxiety is to know that most of the content we learn in math class won't be used after school. That is of course if you don't go into a job field that requires the use of higher level math equations and formulas.
1
I like math better as an adult because I'm not betrayed by it. It doesn't affect my life like it did in school where tests were controlling factors of my fate. Arithmetic is pretty simple for all of us.
However, once one leaves the arena of concrete numbers, it changes. I still find higher math difficult - even algebra. My brain just does not connect with anything higher than x+y = whatever. I lose it. It becomes a foreign language. So it isn't the same as reading and writing. A better comparison might be learning English and Mandarin Chinese but not reading and writing all using the same phonetic sounds and symbols.
However, once one leaves the arena of concrete numbers, it changes. I still find higher math difficult - even algebra. My brain just does not connect with anything higher than x+y = whatever. I lose it. It becomes a foreign language. So it isn't the same as reading and writing. A better comparison might be learning English and Mandarin Chinese but not reading and writing all using the same phonetic sounds and symbols.
1
An interesting and 'potentially' useful article. But it unfortunately does not give adequately clear, practical and readily usable ideas on how to 'Fend Off Math Anxiety'.
2
I've been a volunteer Math Tutor working with several hundred students, ranging from 5th graders through adults at a community college. During that time, there have not been twelve students who fully knew their Multiplication Tables - a most fundamental underpinning for Math success. Of course, virtually all knew their 2x, 5x & 10x tables but beyond that they virtually hit a wall. At that point, many would immediately whip out their calculators to find the answer!
I showed these students the benefits of mastering the Tables - today and for the rest of their lives. Once they saw it, they were excited and agreed. I then devised a very simple and highly effective way to teach these students the complete Tables in a half-day.
The only resistance to any of this was from the "educators". They objected to what they called memorization. They had "no time" for teaching the Tables because it wasn't part of their syllabus, schedule or upcoming test. They wanted their students to "think", not do "drill and kill."
Once they have passed their Math exams/regents, most students will rarely use those subjects in real life. Some will go on to do so in the sciences, engineering and business related areas. However, each and every student can use their mastery of the Tables in everyday life from supermarket shopping to most every other transaction.
Failure is not an acceptable option in this endeavor.
[email protected]
I showed these students the benefits of mastering the Tables - today and for the rest of their lives. Once they saw it, they were excited and agreed. I then devised a very simple and highly effective way to teach these students the complete Tables in a half-day.
The only resistance to any of this was from the "educators". They objected to what they called memorization. They had "no time" for teaching the Tables because it wasn't part of their syllabus, schedule or upcoming test. They wanted their students to "think", not do "drill and kill."
Once they have passed their Math exams/regents, most students will rarely use those subjects in real life. Some will go on to do so in the sciences, engineering and business related areas. However, each and every student can use their mastery of the Tables in everyday life from supermarket shopping to most every other transaction.
Failure is not an acceptable option in this endeavor.
[email protected]
11
I find the same thing at the community college where I tutor GED math. Only in my case the students have not memorized the addition or subtraction tables as well. It is impossible to do any kind of math if one is still counting on one's fingers or guessing. And I have run into the same opposition from the "educators" most of whom not only oppose the "waste of time" and paper (yes, a certain amount of drill is necessary) but who are not particularly good at math themselves.
4
I have another resource for teachers and parents that helps alleviate math anxiety in young children: the Brick Math Series that teaches kids elementary math skills using LEGO bricks. The program helps kids get a deep understanding of the math concepts, and because they are using LEGO bricks, they are having fun and less anxious about the whole learning process. It, too, has shown in early research that students' math test scores improve.
1
Assuming this essay is about the mathematical sciences in addition to arithmetic, one thing that would be especially disconcerting about it is its making such disjoint from reading.
There are three distinct-but-related branches of the mathematical sciences: mathematics, computer science, and statistics.
Consider, for example, consider the following in-a-nutshell pairs of words:
correctly and exactly = primarily mathematics
accurately and precisely = primarily statistics
For those who took a good Stat 101 course this should all be clear. For example, in testing a hypothesis, there are essentially three steps:
1. Formulating the hypotheses (frequently in Words)
2. Finding the P-value (which, if understood and done correctly, could be found off a calculator)
3. Making a decision and Writing a conclusion.
Thus, Reading and Writing are just as important as the third "R".
(BTW, in other articles in ScienceTimes, this three-step test of significance is what's behind what's involved, with the design of the experiment being the key, as well as care with the calculation and interpretation of the P-value, which was something the American Statistical Association wrote about last March and was something receiving a good deal of press coverage.)
There are three distinct-but-related branches of the mathematical sciences: mathematics, computer science, and statistics.
Consider, for example, consider the following in-a-nutshell pairs of words:
correctly and exactly = primarily mathematics
accurately and precisely = primarily statistics
For those who took a good Stat 101 course this should all be clear. For example, in testing a hypothesis, there are essentially three steps:
1. Formulating the hypotheses (frequently in Words)
2. Finding the P-value (which, if understood and done correctly, could be found off a calculator)
3. Making a decision and Writing a conclusion.
Thus, Reading and Writing are just as important as the third "R".
(BTW, in other articles in ScienceTimes, this three-step test of significance is what's behind what's involved, with the design of the experiment being the key, as well as care with the calculation and interpretation of the P-value, which was something the American Statistical Association wrote about last March and was something receiving a good deal of press coverage.)
Many of the posted comments did bring up that Dr. Klass's essay concerns arithmetic, not mathematics, and so such doesn't have to be rehashed here.
However, let me give an example of a good connection between the two in the form of a five-step game:
1. Choose a number: ____
2. Add two: ____
3. Multiply by six: ____
4. Subtract three: ____
5. Divide by three: ___
Suppose someone ended up with, say 73. Quickly --- what number did he/she have in the first step?
Welcome to algebra, the language of mathematics!
However, let me give an example of a good connection between the two in the form of a five-step game:
1. Choose a number: ____
2. Add two: ____
3. Multiply by six: ____
4. Subtract three: ____
5. Divide by three: ___
Suppose someone ended up with, say 73. Quickly --- what number did he/she have in the first step?
Welcome to algebra, the language of mathematics!
3
That's not like the algebra I took. BTW, is the answer 35? I think we are afraid to ask...
Mary and Others:
Such could be done quickly if done algebraically first.
Here are the steps:
1. Choose a number: x
2. Add two: x + 2
3. Multiply by six: 6x + 12
4. Subtract three: 6x + 9
5. Divide by three: 2x + 3
Thus, since the last number was given as 73, then that means that
2x + 3 = 73.
Now most people should be able to subtract 3 from 73 mentally, and so
2x = 70.
Also, most people know what half of 70 is, and so x = 35. (I think you'll agree this is much faster and easier than doing it "backwards".)
Let's try another. Suppose someone ends up with, say, 9.4. With what number did he/she start?
We have 2x + 3 = 9.4 ---> 2x = 6.4 ---> x = 3.2 as the beginning number.
HTH
Such could be done quickly if done algebraically first.
Here are the steps:
1. Choose a number: x
2. Add two: x + 2
3. Multiply by six: 6x + 12
4. Subtract three: 6x + 9
5. Divide by three: 2x + 3
Thus, since the last number was given as 73, then that means that
2x + 3 = 73.
Now most people should be able to subtract 3 from 73 mentally, and so
2x = 70.
Also, most people know what half of 70 is, and so x = 35. (I think you'll agree this is much faster and easier than doing it "backwards".)
Let's try another. Suppose someone ends up with, say, 9.4. With what number did he/she start?
We have 2x + 3 = 9.4 ---> 2x = 6.4 ---> x = 3.2 as the beginning number.
HTH
It can be done just as quickly by just reversing the operations in reverse order. Algebra is more fun than this.
Well over a hundred years ago, Maria Montessori had figured out in her 'Casa dei Bambini' in Rome, just how to prevent 'math anxiety' from developing in children.
It's remarkable that in 2017, the 'educational wisdom' she brought to the world haven't become more or less standard practice in schools! There are, of course, a great many schools that claim they're following the 'Montessori Method' - but in many such schools this is just for the cachet of the Montessori name, and no doubt for the financial rewards it brings!
It's remarkable that in 2017, the 'educational wisdom' she brought to the world haven't become more or less standard practice in schools! There are, of course, a great many schools that claim they're following the 'Montessori Method' - but in many such schools this is just for the cachet of the Montessori name, and no doubt for the financial rewards it brings!
I have watched my son in the fourth grade be systemmaticlly attacked about his math skills. He is gifted. He started counting to ten before the age of two. Stangers would stop and ask in disbelief how old he was. This teacher had eroded his confidence so severly he belived he was bad at math. I told him all the stories he did not remember. Repeatly, we talked about how to deal with this teacher. He is fine now but I say this because I experienced a similar thing. I agree that children have a concept of being good or bad at something far earlier . For me it was kindergarten. I belive all humans are litterally hard wired for math. This article is very important. I have found many math teachers have a combative weeding out mentality. I believe it is a form of scocial abuse. I think the author could go deeper and conect to articles about when girls fall behind in math. Keep going! This is good stuff! Thanks.
4
It is not ok to be "bad at math." Yet, this is a pervasive attitude in this country. You will not be able to watch a tv comedy without hearing a negative comment. Pay attention. It comes up over and over. There will never be one person who smilingly admits to being "bad at reading." Why is this ok, cool even? When you start to pay attention, you will be shocked how often this attitude is displayed. Watch one Disney channel show (the ones that are just on in the background of every house with elementary school children). This attitude needs to be understood - why is it funny?
Dr. Klass, I too have an MD, as well as an MS in biochemistry, yet I continue to have this persistent nightmare into the advent of my 5th decade that I'm about to graduate from high school or college only there's this one class that I've never attended and it's invariably math! Which is peculiar since I wasn't a bad math student (though I knew I'd hit my glass ceiling at the end of AP calculus).
A while ago I had the idea that there were two sorts of people: those of us who think we're smart and those who are good at math.
A while ago I had the idea that there were two sorts of people: those of us who think we're smart and those who are good at math.
1
I always had math anxiety which was exacerbated by a bullying math teacher I had for several years as a youngster.
In retrospect I think I would have tried harder to master math had I seen how it fits into daily life. To me it was just dry and rigid and I had no clue how much poor math skills would hamper me in later life.
(For the record I have a master's degree. )
In retrospect I think I would have tried harder to master math had I seen how it fits into daily life. To me it was just dry and rigid and I had no clue how much poor math skills would hamper me in later life.
(For the record I have a master's degree. )
2
http://www.nytimes.com/2008/09/18/health/18iht-snangier.1.16200080.html ran interesting essay "Intuition and math: A powerful correlation" , highly relevant to the present discussion, interesting to read along side.
Reading the piece it instantly came to my mind the amazing gift of numbers unlettered vegetable vendor's have perfected, a computational skill of these uneducated have developed over the years for their very survival. It's just not about money transaction, experienced vendors can tell by mere sight with uncanny accuracy the weight of a chaotic heap of vegetable or fruits.
Reading the piece it instantly came to my mind the amazing gift of numbers unlettered vegetable vendor's have perfected, a computational skill of these uneducated have developed over the years for their very survival. It's just not about money transaction, experienced vendors can tell by mere sight with uncanny accuracy the weight of a chaotic heap of vegetable or fruits.
2
It's probably a matter of making efforts to make the subject of mathematics intuitive.
There's a world of difference between being able to solve a problem in 30 seconds and solving it instantly- to think that as soon as there is basic, conscious understanding, should be enough.
The reason this is so important is because humans have a finite amount of short-term/working memory. One can only remember a few things at a time. The maximum amount of items we can store in our conscious mind, in what's called our working memory, and a study puts the limit at three or four.
The same concept applies to Math. If a vector space occupies 8 units of memory, then you have no room to understand anything else. But if it occupies 1 unit, you have plenty of room to build from. And how do you reduce the amount of memory something takes up? By practice. This part of the learning process is called chunking.
"Chunking is a method of presenting information which splits concepts into small pieces or "chunks" of information to make reading and understanding faster and easier".
So the reason why math geniuses learn so quickly is because they've studied the basic material so thoroughly that it's become intuitive to them. The average high schooler will have to use up several units of memory to expand a complex expression , whereas a more advanced one will be able to solve that with 1 or 0 units of memory. That makes it easier to learn advanced concepts that build upon the simpler ones.
There's a world of difference between being able to solve a problem in 30 seconds and solving it instantly- to think that as soon as there is basic, conscious understanding, should be enough.
The reason this is so important is because humans have a finite amount of short-term/working memory. One can only remember a few things at a time. The maximum amount of items we can store in our conscious mind, in what's called our working memory, and a study puts the limit at three or four.
The same concept applies to Math. If a vector space occupies 8 units of memory, then you have no room to understand anything else. But if it occupies 1 unit, you have plenty of room to build from. And how do you reduce the amount of memory something takes up? By practice. This part of the learning process is called chunking.
"Chunking is a method of presenting information which splits concepts into small pieces or "chunks" of information to make reading and understanding faster and easier".
So the reason why math geniuses learn so quickly is because they've studied the basic material so thoroughly that it's become intuitive to them. The average high schooler will have to use up several units of memory to expand a complex expression , whereas a more advanced one will be able to solve that with 1 or 0 units of memory. That makes it easier to learn advanced concepts that build upon the simpler ones.
2
Those were two most interesting responses of yours, Mr Krishnan.
As to your ideas about "math geniuses", it is, I believe, only partly because the material is 'near-intuitive' to them: there is, I'm sure, a lot more than that underlying. (No doubt you're aware of Mark Kac's remarks about 'geniuses' and 'magicians', which were published in a biography of Richard Feynman by Richard Poundstone).
I'd like very much to discuss these issues in more detail with you at any convenient time. As we live in the same city, it should be possible to arrange.
As to your ideas about "math geniuses", it is, I believe, only partly because the material is 'near-intuitive' to them: there is, I'm sure, a lot more than that underlying. (No doubt you're aware of Mark Kac's remarks about 'geniuses' and 'magicians', which were published in a biography of Richard Feynman by Richard Poundstone).
I'd like very much to discuss these issues in more detail with you at any convenient time. As we live in the same city, it should be possible to arrange.
All people have some type of anxiety. For some its math. For others its public speaking or writing. That's part of life. We learn to deal with what we've been dealt. Labelling kids as "math phobic" doesn't help the situation and probably makes the situation worse.
2
The way math is taught today, aligned with Common Core, it skips over practicing the fundamentals, like long division done by hand. Kids aren't playing board games or cooking, both of which can offer practice with numbers. I finally realized that the reason I was always solid with the times tables up to 6 x 6 but a bit fuzzy on the 7s and 8s was because I played a lot of Yahtzee on rainy days as a kid.
Administrators would do well to reassure teachers and parents, especially in grade school, that high math test scores shouldn't be the goal. Having a solid grasp of the fundamentals and number sense is far more important. Fooling yourself that a second grader with a calculator doing a higher math problem has mastery is a bad idea.
The best thing I did for my son was cross out half the Everyday Math assignments, refuse to give him a calculator until middle school, and teach him math fundamentals not just through drills but in fun ways. I didn't want him thinking the fastest way to count ceiling tiles was '1, 2, 3..." It's a skip counting and multiplication problem: "2, 4, 6, okay 7, times 2, 4, 6, 8, 9, 7x9." I didn't have math anxiety as a kid--but I did learn better the old fashioned way and could see that he would, too.
Administrators would do well to reassure teachers and parents, especially in grade school, that high math test scores shouldn't be the goal. Having a solid grasp of the fundamentals and number sense is far more important. Fooling yourself that a second grader with a calculator doing a higher math problem has mastery is a bad idea.
The best thing I did for my son was cross out half the Everyday Math assignments, refuse to give him a calculator until middle school, and teach him math fundamentals not just through drills but in fun ways. I didn't want him thinking the fastest way to count ceiling tiles was '1, 2, 3..." It's a skip counting and multiplication problem: "2, 4, 6, okay 7, times 2, 4, 6, 8, 9, 7x9." I didn't have math anxiety as a kid--but I did learn better the old fashioned way and could see that he would, too.
3
Bravo. Everyday math has no place in the everyday classroom. It should be reserved as an extra curricular activity. Everyone quoted in this article is from the University of Chicago which of course is the outfit that peddles Everyday Math. So it was hard to read the article without reliving my older son's anxiety-inducing experiences with Everyday Math and how that program completely failed to teach mastery of the basics. There will always be those who gravitate toward math and those who don't, but I believe that curricula like Everyday Math, along with the current obsession with teaching advanced topics at an earlier age regardless of appropriateness, contributes to anxiety.
1
Please tell all the parents you know to end the practice of counting to three before punishing a child. I believe this is at the root of much math phobia. Number experiences have to be positive to start kids right. Also problematic: high pressure in elementary school math classes.
What would be more helpful is use of hands-on toys that teach math concepts, and the use of fun activities such as dancing and music.
What would be more helpful is use of hands-on toys that teach math concepts, and the use of fun activities such as dancing and music.
7
Don't call them 'toys' - they are 'educational tools' that children enjoy using!!
I so agree with too much pressure and too much math too soon in elementary. I teach third and the amount of math we have to cover is way too much and too rushed. Everybody agrees the fundamentals are important but no one acts on keeping in fundamental. Go figure.
Math is a symbolic language. To understand it, requires that the brain development level is ready to "think" in this symbolic way. Not every child's brain is ready to begin studying this symbolic language when it is introduced in school. Add incompetent math teaching and that the subject is taught in an accumulative manner, there is no other structural response than for a student to always be behind in math, if the student's brain development was not ready at the beginning. No other subject is taught in such an accumulative fashion, so a student has great difficulty in catching up. We recognize that not all students develop language skills at the same time. Math is just another symbolic language, and thus this statement holds true for this subject area,as well. And no, there is no necessary relationship among grammer language (such as English), logic, and math acquisition. One can be quite versed in grammar language and logic thinking and still struggle with the symbolic language of math.
5
The sign that you understand a subject matter is that you
clearly know when you have the right answer and when you do not
have the right answer - you know what you know and you know that
you don't know what you do not know.
I think we rush through math too quickly for some people.
Perhaps they best respond to diagrams that are as clear as clear can be.
Hands on problems that allow them to manipulate the data in question.
Better to understand the basics deeply
than too many topics in a shallow manner.
clearly know when you have the right answer and when you do not
have the right answer - you know what you know and you know that
you don't know what you do not know.
I think we rush through math too quickly for some people.
Perhaps they best respond to diagrams that are as clear as clear can be.
Hands on problems that allow them to manipulate the data in question.
Better to understand the basics deeply
than too many topics in a shallow manner.
8
The first step in recovery is confronting the truth. The truth is that people are afraid of arithmetic, not math. Now that we live in a world where everything is pre-calculated, they've even more scope for being afraid. It used to be that only the rich (or intellectuals) could afford to be arithmophobic, since the average person had to be able to do rudimentary arithmetic just to get by. Now just about everybody can take pride in never having to figure numbers, unless it's their job.
And calculus to do med school? Please. That was algebra. Calculus starts when you learn to differentiate a function from first principles. Algebra is the tool you need to do it, but it also supports plenty of basic science.
And calculus to do med school? Please. That was algebra. Calculus starts when you learn to differentiate a function from first principles. Algebra is the tool you need to do it, but it also supports plenty of basic science.
3
It doesn't make sense to leave racial and socio-economic factors out of this conversation. I am an Indian-American, and have never met an Indian or other Asian who says they are math phobic. This is just not a universal issue.
Cultural expectations play a huge role. Immigrants are drawn to science and math because those domains offer a more level playing field. Literature and the arts are judged subjectively, and it is harder for racial minorities and immigrants to get a fair chance there.
Cultural expectations play a huge role. Immigrants are drawn to science and math because those domains offer a more level playing field. Literature and the arts are judged subjectively, and it is harder for racial minorities and immigrants to get a fair chance there.
19
Immigrant parents encourage their children to study math for several reasons, but the fact that the arts are judged subjectively is probably not one of them. Language plays a huge role, particularly for East Asian parents who don't speak English fluently. Immigrants often have a clear understanding of what the US values in the job market. Unless you are very lucky and very well connected, your love of literature and the arts is not going to get you a well paid career, but your excellence in math will.
3
I agree with you - I don't understand the argument in the article that people believe it to be an ability which you have or not. It was always tought like writing, swimming, reciting, etc in the schools I and my brother visited - it is something to study and practice. And if the teacher does not find the right words to explain it, one of the other students, or an older pupil or your parent, will.
1
Poor math readiness in Kindergarten predicts lower math performance later on plus reading difficulty.
Possible explanations - Lower IQ, poor early parental stimulation.
Precocious math excellence - Asperger's?
Possible explanations - Lower IQ, poor early parental stimulation.
Precocious math excellence - Asperger's?
1
This comment really demonstrates Americans' fear of math. It's gotten to the point that kids who understand, like, and do well in math are now accused of having a disability.
5
I strongly disagree with you that precocious math excellence may indicate Asperger's syndrome - but I agree with the rest of your comment.
Just took the anxiety test. I'm a psychologist and teach statistics (to sometimes very anxious students). Doesn't this test just pull for anxiety?
1
Math anxiety may begin at home but it gets amplified by early childhood and elementary teachers who have math anxiety themselves. When I was a student in an elementary education program at SUNY Cortland over thirty years ago I discovered that a very high percentage of my classmates were planning to be teachers because they loved kids (you'd better love kids if you plan on spending all day with them) and their math skills were too weak for most other majors. I was appalled that these young women (only one male classmate) were going to be responsible for turning kids on to math.
18
Very unfortunate but very true.
I have been math phobic since elementary school. I was reading ready in kindergarten and had a 3rd grade reading level in first grade even though I was only 5 then. Yes, parents should realize that it is important for them to inspire an appreciation for the "beauty and power of the subject" and the intrinsic sense of reward of being able to solve math problems, even at the most basic levels. My lack of math skills, possibly due to anxiety and lack of confidence, has limited me in being able to fulfill my career goals and remains an obstacle to this day. I retired after 32 years and returned to school in order to become a veterinary technician. Just knowing that I will be required to take chemistry and medical math in the fall is daunting, but I will fight to master these courses this time because I already know what it's like to give up on your dreams.
5
Mathematics has no natural constituency in the US: neither on the left --- where the obvious fact that some groups are much more proficient than others goes against the equality dogma. The combination of native endowment, preparation, and family support needed to succeed are anathema to left-wing people. Only a few dare provide their children with the proper instruction and environment, all the while looking over their shoulders lest the social justice police catch them in the act. See: reading bedtime stories as an unfair advantage; also see: the destruction of the Ivy League and of selective schools, a cherished long-term project of the left.
Nor on the right, unfortunately: jock culture is against Mathematics and science. Also, traditionalists sometimes lack the needed openness to new and foreign ideas.
Making progress in science or Mathematics can be hard. In the US, people are educated that the only field in which it is acceptable to put any effort is sports. Any other sort of effort smells of entitlement (on the left) or try-hard (on the right).
In the US most people are ignorant of Mathematics and science and proud of it. This is why they have to keep importing scientists and engineers. Fortunately, the US can afford it.
This is not to say the US don't have many good aspects --- they wouldn't be the richest and most powerful country otherwise --- but this is still a deficiency, in my opinion.
Nor on the right, unfortunately: jock culture is against Mathematics and science. Also, traditionalists sometimes lack the needed openness to new and foreign ideas.
Making progress in science or Mathematics can be hard. In the US, people are educated that the only field in which it is acceptable to put any effort is sports. Any other sort of effort smells of entitlement (on the left) or try-hard (on the right).
In the US most people are ignorant of Mathematics and science and proud of it. This is why they have to keep importing scientists and engineers. Fortunately, the US can afford it.
This is not to say the US don't have many good aspects --- they wouldn't be the richest and most powerful country otherwise --- but this is still a deficiency, in my opinion.
3
Is that why we refuse to adopt the metric system? We should be ashamed - every one of us! It would be so much easier for children to understand.
Mary, Do you remember that President Carter tried to get us on the metric system? That went the same way as the solar panels he put on the White House.
Being good at math is both extremely easy and extremely hard, like most things. It's easy in the sense that if you study hard and smart, and do lots and lots of math, you will be much, much better at math than people who do not do these things, which is of course almost everyone. On the other hand, studying for long hours in a focused and intelligent manner is itself very hard to do for most people.
I have a two-year old and I think the main thing I will try to impress on him about math, or anything else that may seem difficult, is that it's within your ability to be exceptional at that thing, but being exceptional at it will mean that you will have to work hard, relish challenges, learn from your mistakes and continue to improve over a very long period. You'll notice that nowhere did I say you need to be "smart", "gifted", or "talented". While I don't deny that students' natural abilities will differ and some may learn more quickly than others, the idea you need to be a natural to excel at math is in my view largely an artifact of a distinct part of Western culture that elevates and celebrates the achievements of supposedly gifted students over those of the hard workers.
I have a two-year old and I think the main thing I will try to impress on him about math, or anything else that may seem difficult, is that it's within your ability to be exceptional at that thing, but being exceptional at it will mean that you will have to work hard, relish challenges, learn from your mistakes and continue to improve over a very long period. You'll notice that nowhere did I say you need to be "smart", "gifted", or "talented". While I don't deny that students' natural abilities will differ and some may learn more quickly than others, the idea you need to be a natural to excel at math is in my view largely an artifact of a distinct part of Western culture that elevates and celebrates the achievements of supposedly gifted students over those of the hard workers.
13
I had a wonderful math teacher all through high school classes in the 60's. He was smart, humorous, liked kids, liked his subject. I could get extra help from him, wasn't afraid to ask, but anxiety still ruled. I worked hard. I really liked geometry because of its visual nature.
When it came to a decision whether to take calculus in my senior year, I declined in favor of another class called Math Analysis, which included more set theory etc., and which served me well later in life. I was just plain afraid of calculus -- and would have been the only female in that class back then.
I still wonder about professions like medicine -- is it really necessary to take calculus to understand other medical classes and to succeed as a physician?? Or is it a traditional way to weed out students?
Would love to hear from your readers about that.
When it came to a decision whether to take calculus in my senior year, I declined in favor of another class called Math Analysis, which included more set theory etc., and which served me well later in life. I was just plain afraid of calculus -- and would have been the only female in that class back then.
I still wonder about professions like medicine -- is it really necessary to take calculus to understand other medical classes and to succeed as a physician?? Or is it a traditional way to weed out students?
Would love to hear from your readers about that.
4
I'd find it hard to believe that calculus is not needed in medical classes. It'd certainly be nice to know about exponential decay of the amount of a drug in the bloodstream as a function of time. And about the half-life of radiopharmaceuticals (which is a facet of exponential decay). Exponential decay is a very basic and important application of calculus.
You also have statistics. You need to understand integrals of portions of the Gaussian function (more commonly known as the "bell curve".
Anything involving rates can probably be understood better using calculus. Life involves biochemical reactions, which involve rates, and therefore calculus.
To all of you who thought trig, algebra, geometry, etc. were pointless: While useful in and of themselves, calculus is the big prize. Calculus is critical in science. And science is critical in medical studies.
You also have statistics. You need to understand integrals of portions of the Gaussian function (more commonly known as the "bell curve".
Anything involving rates can probably be understood better using calculus. Life involves biochemical reactions, which involve rates, and therefore calculus.
To all of you who thought trig, algebra, geometry, etc. were pointless: While useful in and of themselves, calculus is the big prize. Calculus is critical in science. And science is critical in medical studies.
As a math person, I have often told people I'm not a word person. Although I do enjoy reading for pleasure, numbers speak to me in a way words do not. Grammar gives me anxiety and analyzing literature just seems to ruin it for me.
3
I count myself among those who have "math anxiety," except for the fact that I have no problem with practical math--i.e., balancing a checkbook, figuring out a check or a tip at a restaurant, figuring out what the price of an item that's marked down 20 percent is, etc. Where I have a problem is when I'm asked about major equations. When I got a call from my daughter's math teacher about 25 years ago, during which time he said she was having difficulty with "polynomials," I had no idea what he was talking about. When I asked him to explain, he just kept yelling the word POLYNOMIALS louder and louder into the phone. I asked a friend at work what that meant, and he explained it to me. So no wonder my daughter was having problems with that teacher.
I loved geometry in high school, because there was an element of detective work to it. But those equations (i.e., polynomials) just about did me in. And now, the way they're teaching math in concepts of ten doesn't make any sense to me at all. App or no app, when someone asks me how many toys someone has when eight are on the mantle and six are under the Christmas tree, I remember what I learned in school when I was six: 8 + 6 = 14, not the way they teach now, which is almost as if you're using an abacus. Learn times tables, learn 2+2=4. That's practical math. I have no anxiety over that, but when someone screams POLYNOMIALS over the phone, yup, that sends me to the chocolate.
I loved geometry in high school, because there was an element of detective work to it. But those equations (i.e., polynomials) just about did me in. And now, the way they're teaching math in concepts of ten doesn't make any sense to me at all. App or no app, when someone asks me how many toys someone has when eight are on the mantle and six are under the Christmas tree, I remember what I learned in school when I was six: 8 + 6 = 14, not the way they teach now, which is almost as if you're using an abacus. Learn times tables, learn 2+2=4. That's practical math. I have no anxiety over that, but when someone screams POLYNOMIALS over the phone, yup, that sends me to the chocolate.
4
I suspect you have no real limitations in math, but came under then influence of some "teachers" who discouraged you from learning math. I studied math all the way up to a PhD without being phobic because I never had a teacher who suggested I could not.
6
This is because mathematical equations are symbolic representations (i.e., a particular type of language). Turn the equation into a grammer/prose statement and the structure of the equation becomes clear to those of us who think in grammer/prose language, and don't think in the representational language of mathematical numbers and symbols. Written musical notation is another symbolic representational language of which some individuals intuitively "get" and others struggle with understanding. Math is no different. For some people the language of math needs to be translated into grammar/prose language until the brain slowly can do its own internal translations. Some individuals brains know how to do this easily while others do not, and they have to be taught how to think symbolically in this mathematical way.
Yes, learn the multiplication table. You only need to memorize 36 non-trivial products and a few simple rules: order doesn't matter, 1 x something is that same something, and zero times anything is zero. The same goes for the addition table. That's not asking a lot! Yes, you can _teach_ the _meaning_ via the "abacus method" or similar, but you must also teach the practical. There's no point in reinventing the wheel every time you do an arithmetic problem!
This "abacus method" as a practical method is probably an overreaction to the horrors of rote learning. You can overdo anything, but that doesn't mean the cure is to avoid it completely!
This "abacus method" as a practical method is probably an overreaction to the horrors of rote learning. You can overdo anything, but that doesn't mean the cure is to avoid it completely!
"Dr. Susan Levine, chairwoman of the department of psychology at the University of Chicago, agreed: “An educated person doesn’t go around saying, I’m not a reading person.”
What she probably means is that the average person would never brag about being illiterate, but a lot of people are perfectly fine with saying they're no good at math. As far as educated people go, I know quite a few (two in particular come to mind) who do not respect "reading" when that means reading fiction or literature. These two men read newspapers, journals, and business publications, but reading anyhting that doesn't deal with hard core facts is considered by them to be a waste of both their time and their intellectual energy.
I even have a female friend whose PhD is in History (she's a college professor). She never reads fiction or literature, not even and especially "historical fiction", because she says the tales told therein "never actually happened", ie, the stories are not true, therefore they have no value. So in that sense, yes, even highly educated people can proudly state at times that they don't like "reading".
What she probably means is that the average person would never brag about being illiterate, but a lot of people are perfectly fine with saying they're no good at math. As far as educated people go, I know quite a few (two in particular come to mind) who do not respect "reading" when that means reading fiction or literature. These two men read newspapers, journals, and business publications, but reading anyhting that doesn't deal with hard core facts is considered by them to be a waste of both their time and their intellectual energy.
I even have a female friend whose PhD is in History (she's a college professor). She never reads fiction or literature, not even and especially "historical fiction", because she says the tales told therein "never actually happened", ie, the stories are not true, therefore they have no value. So in that sense, yes, even highly educated people can proudly state at times that they don't like "reading".
8
An educated person doesn’t go around saying, I’m not a reading person.”
Unless you are our President!
Unless you are our President!
14
Actually, I disagree with Dr. Levine. Some engineers, certainly highly educated people, seem almost proud to declare that they aren't "reading" people. This happens among the young ones. Two generations ago, an engineer would not be proud that he/she got an 800 on the math portion of the SAT and 400 on the verbal (1600 was the maximum score back when.) They would see the 400 as evidence of a skills area that needed improvement.
There's been a strange shift in attitude over the years. If you're not good in something, instead of just acknowledging that it's an area that needs some work, declare yourself a genius at the thing you are good at and call it a day.
There's been a strange shift in attitude over the years. If you're not good in something, instead of just acknowledging that it's an area that needs some work, declare yourself a genius at the thing you are good at and call it a day.
4
I have only my high school math teachers to thank for my lifelong math phobia --boring people droning on and on at the board and showing zero enthusiasm for the beauty and elegance of mathematics. As a college freshman I was required to take two semesters of math and I was terrified. Luckily for me, my prof was a whiz-bang teacher and I completely "got" concepts that had long evaded me -- and ended up with very respectable grades. Great teaching goes a long, long way to connect the learner with, as another commenter noted, the "wow" moment.
14
Math isn't flexible ... there are rules and it requires accuracy... its right or wrong... something we seem to have some trouble asking of ourselves and our children these days. But to me that is the beauty of math, it is reassuring in that 2+2=4, every time. Sometimes learning can be satisfying and that is enough.
4
It's A Big Problem
Of course I agree with those who said this article is about arithmetic, not mathematics. What most American kids "learn" and practice up to their introduction to algebra is either arithmetic or closely related to it. What one might call "real mathematics" begins, for most youngsters, with Algebra I.
With it's early focus on the calculus, colleges and universities (and now a great many high schools) do a great disservice to post secondary mathematics learning.
I very strongly disagree that being a competent reader is essential to one being called an educated person, but being mathematically competent is not. In recent years, many post-secondary schools have made a transition from being liberal arts colleges to being something called comprehensive universities and in the process have discontinued the mathematics major. I hope that's only in America; e.g.
https://www.sa.edu/academics/available-majors
http://webber.edu/
Furthermore, you can take my word for the fact that I can pick university departments like this one ...
http://www.egr.vcu.edu/departments/electrical/people/faculty/
almost at random. Don't look at their PhD universities ... just see where these faculty got their undergraduate degrees. The study and teaching of STEM programs (and especially mathematics) in the U.S. has reached such a low point, we would be in dire straits were we not able to utilize the expertise of those who received their undergraduate educations outside this country.
Of course I agree with those who said this article is about arithmetic, not mathematics. What most American kids "learn" and practice up to their introduction to algebra is either arithmetic or closely related to it. What one might call "real mathematics" begins, for most youngsters, with Algebra I.
With it's early focus on the calculus, colleges and universities (and now a great many high schools) do a great disservice to post secondary mathematics learning.
I very strongly disagree that being a competent reader is essential to one being called an educated person, but being mathematically competent is not. In recent years, many post-secondary schools have made a transition from being liberal arts colleges to being something called comprehensive universities and in the process have discontinued the mathematics major. I hope that's only in America; e.g.
https://www.sa.edu/academics/available-majors
http://webber.edu/
Furthermore, you can take my word for the fact that I can pick university departments like this one ...
http://www.egr.vcu.edu/departments/electrical/people/faculty/
almost at random. Don't look at their PhD universities ... just see where these faculty got their undergraduate degrees. The study and teaching of STEM programs (and especially mathematics) in the U.S. has reached such a low point, we would be in dire straits were we not able to utilize the expertise of those who received their undergraduate educations outside this country.
6
Suggestion: 'Arithmethic', despite the lowly status you've given it, is surely a part of mathematics. CF Gauss (1777-1856, I believe), one of the truly great mathematicians of all times (called the@romce pf Mathematicians by E.T. Bell in his wonderful book, "Men of Mathematics") considered himself to be an 'arithmetician' first and foremost.
Sorry, typo in my earlier posting: Gauss was called "The Prince of Mathematics" - not what I had typed in error. My regrets.
What they don't explore is that math has to be fun (the WOW moment) and show how it can open/connect in our world before a child who is not "predisposed" will embrace math.
4
It's also important for people to have a sense of what proves something, in math. What makes something actually true, as opposed to just looking true?
And it does cause anxiety, to realize how easy it is to make mistakes. It creates a sense of humility in front of the truth.
So some amount of math anxiety is inevitable. It's the result of having to recognize one's human limitations, how one can be misled.
If someone learns how to spot the fallacy in a proof that doesn't work, that could help them critique other assertions that don't involve math. It teaches one to be watchful for mistakes.
And it does cause anxiety, to realize how easy it is to make mistakes. It creates a sense of humility in front of the truth.
So some amount of math anxiety is inevitable. It's the result of having to recognize one's human limitations, how one can be misled.
If someone learns how to spot the fallacy in a proof that doesn't work, that could help them critique other assertions that don't involve math. It teaches one to be watchful for mistakes.
4
Although I wasn't particularly talented at arithmetic, I suddenly noticed a new subject, algebra, in Grade 8. I found it interesting and easily got As. I ended up majoring in Math, with Physics sub speciality.
One thing that I greatly liked was that math proofs were impersonal and anyone could do them. This was in sharp contrast to classes like English, where it seemed you merely had to write something that the teacher agreed with.
In Math, you could conceivably prove the teacher wrong (happened twice in my career, although not in high school, I hasten to add). Marks in English seemed subjective.
One thing that I greatly liked was that math proofs were impersonal and anyone could do them. This was in sharp contrast to classes like English, where it seemed you merely had to write something that the teacher agreed with.
In Math, you could conceivably prove the teacher wrong (happened twice in my career, although not in high school, I hasten to add). Marks in English seemed subjective.
5
Yes, math is very democratic! Somebody who makes sense at math will be listened to, regardless of what their degrees are or their status.
3
The movie, Hidden Figures, is a good example of your point.
One has to wonder if some of math anxiety passed from generation to generation has, at least in some part, due to genetics.
I always found math very easy although I didn't have any great interest in it but my father was a math whiz. I've spoken to a number of other people who shared my competency at it but lack of interest and all had at least one parent who was extremely good at it.
We know that many disorders of the brain have some genetic component. It wouldn't be too far fetched to think that some beneficial aspects might have similar components. In the same way that there is little we can do to fend off Alzheimer's, Parkinson's Disease, or schizophrenia, there may be little we can do to prevent math anxiety.
I always found math very easy although I didn't have any great interest in it but my father was a math whiz. I've spoken to a number of other people who shared my competency at it but lack of interest and all had at least one parent who was extremely good at it.
We know that many disorders of the brain have some genetic component. It wouldn't be too far fetched to think that some beneficial aspects might have similar components. In the same way that there is little we can do to fend off Alzheimer's, Parkinson's Disease, or schizophrenia, there may be little we can do to prevent math anxiety.
6
I totally agree. In my family we have generations of math loving people of both sexes, and on both sides of my branch and my husband's branch. We produced three children, all highly competent in math, and one finishing as a PhD in Applied Mathematics, having worked in both the software and aeronautics industries. My mother was a medical researcher, my dad a pilot and an engineer. My grandmother could work out complicated arithmetic in her head, and usually just scanned long lists of numbers visually before writing the correct sum at the bottom of the list. My husband is a CPA and his mother was one of two female math majors her year in college. She developed her math passion from her father, as her mother hated math.
My guess is that in Dr. Klass's family, while her mother was math-phobic, her father was not. Her husband is probably math-friendly +/or highly math competent. Hence, the math major daughter. Apples don't usually fall far from the tree.
My guess is that in Dr. Klass's family, while her mother was math-phobic, her father was not. Her husband is probably math-friendly +/or highly math competent. Hence, the math major daughter. Apples don't usually fall far from the tree.
4
Correct: Ducks have ducks.
3
What I want to know is... how do we help adults that have the deep seated math anxiety (that this article suggests started at a very early age)? I work with small business owners to help them manage their financials. Too often, I work with clients who stick their heads in the sand when it comes to numbers. How can we help them get over their fears to be more successful business owners?
Something called 'systems thinking' (ST) could help considerably. ST done by 'people-at-large' developed from the approach to systems modeling from the late John N. Warfield. Unfortunately, most systems scientists and all management 'scientists' failed to understand implications of Warfield's developments in systems science, and his insights lie largely unknown and unused today.
Perhaps the New York Times could take the lead in checking out how to bring 'systems thinking' to its readers' notice. Such systems thinking may be contrasted with the systems science which 'experts' like to talk at each other about!
I observe that systems thinking is easy enough for a high-school student to understand within a couple of hours, and to learn how to apply it within a couple of weeks (at most). This systems thinking is in fact far easier to understand and to practice than 'the three Rs' that every high-school student is expected to master.
Perhaps the New York Times could take the lead in checking out how to bring 'systems thinking' to its readers' notice. Such systems thinking may be contrasted with the systems science which 'experts' like to talk at each other about!
I observe that systems thinking is easy enough for a high-school student to understand within a couple of hours, and to learn how to apply it within a couple of weeks (at most). This systems thinking is in fact far easier to understand and to practice than 'the three Rs' that every high-school student is expected to master.
Get your seven-year old a FlashMaster(R) and challenge yourself with it at the same time--and with a sense of humor.
1
As usual, this article purports to discuss something pertaining to mathematics, and instead the article and most of the comments are talking about arithmetic - which is a small set of algorithms derived from number theory.
When properly taught, reasonably bright children catch on quite rapidly because:
1. The children are a clean slate; they don't already think they know what numbers (or other mathematical constructs) are - unless they already have been loused up by hack teachers, or their parents bad mental habits.
2. The basic ideas of mathematics are not that difficult when properly explained. The clarity and precision of mathematical ideas invariably appeals to anyone who can confront them. The trick is to encourage a certain kind of literal-mindedness and to promote in the student a dissatisfaction with anything that isn't clear and straightforward.
Re the arithmetical confusion that the article refers to (as well as much of the commentary): If you ask any high school student, "What is a number?", maybe the smallest fraction could answer adequately. So how can you expect such students to properly use arithmetical algorithms? The complex process that creates the structure of the real number system should be taught first. The rest then turns out to be trivial detail.
The problem is that teachers of young people don't deeply understand the subject. You cannot expect someone to teach a complex and subtle subject properly if they don't comprehend it themselves.
When properly taught, reasonably bright children catch on quite rapidly because:
1. The children are a clean slate; they don't already think they know what numbers (or other mathematical constructs) are - unless they already have been loused up by hack teachers, or their parents bad mental habits.
2. The basic ideas of mathematics are not that difficult when properly explained. The clarity and precision of mathematical ideas invariably appeals to anyone who can confront them. The trick is to encourage a certain kind of literal-mindedness and to promote in the student a dissatisfaction with anything that isn't clear and straightforward.
Re the arithmetical confusion that the article refers to (as well as much of the commentary): If you ask any high school student, "What is a number?", maybe the smallest fraction could answer adequately. So how can you expect such students to properly use arithmetical algorithms? The complex process that creates the structure of the real number system should be taught first. The rest then turns out to be trivial detail.
The problem is that teachers of young people don't deeply understand the subject. You cannot expect someone to teach a complex and subtle subject properly if they don't comprehend it themselves.
16
Exactly. There was very little about mathematics in this article, besides counting.
10
The educational framework of old had got it just about right: we all know about 'the three Rs' - Readin-'Ritin-'Rithmetic.
Maria Montessori over a century ago figured out in her 'Casa dei Bambini' in Rome just how to enthuse young learners about the beauty and power of math. She had developed practical tools showing how shapes, numbers, sizes - and above all, IDEAS, could be connected up in the minds of young learners. Children who've been through Montessori education in their early schooldays rarely if ever develop 'math anxiety' (I believe). [Of course, that's assuming their subsequent school experiences do not instill 'math anxiety' into them].
But we seem to have lost the insights she had brought into education, alas!
Maria Montessori over a century ago figured out in her 'Casa dei Bambini' in Rome just how to enthuse young learners about the beauty and power of math. She had developed practical tools showing how shapes, numbers, sizes - and above all, IDEAS, could be connected up in the minds of young learners. Children who've been through Montessori education in their early schooldays rarely if ever develop 'math anxiety' (I believe). [Of course, that's assuming their subsequent school experiences do not instill 'math anxiety' into them].
But we seem to have lost the insights she had brought into education, alas!
So, what is a number? Just curious. How would you explain that to a child?
There is quite a difference between math and arithmetic. What is really important is to be comfortable with arithmetic. When you see a price for $5.99, you have to read it as $6.
Then if you need to calculate 10 percent off, you know it's 60 cents off or you can easily calculate that 90 percent is $5.40.
The whole idea is to confuse the consumer. When you see, $5.99, it's natural to think 5 dollars and change. Furthermore calculations with such numbers are cumbersome.
Of course, since we all have smart phones maybe none of this matters.
Then if you need to calculate 10 percent off, you know it's 60 cents off or you can easily calculate that 90 percent is $5.40.
The whole idea is to confuse the consumer. When you see, $5.99, it's natural to think 5 dollars and change. Furthermore calculations with such numbers are cumbersome.
Of course, since we all have smart phones maybe none of this matters.
5
That's part of the problem! How often have you gone to a store and the price is $6.15--you dig out 15 cents and give it to the clerk and the clerk doesn't know how to give you change for a 10-dollar bill, because the cash register has already said that he or she should give your $3.85 in change?
When my kids (now 39 and 37) were young, we would have little "contests" to figure out--without calculators--what things would cost when they were on sale while we walked through stores. But then again, I had taught them how to make change when they were quite young.
When my kids (now 39 and 37) were young, we would have little "contests" to figure out--without calculators--what things would cost when they were on sale while we walked through stores. But then again, I had taught them how to make change when they were quite young.
2
May I sugggest a small correction to your comment?
You should, perhaps, more accurately say it like this:
Mathematics INLCUDES Arithmetic - but Arithmetic is only a 'small part' of mathematics. This could be easily illustrated in a Venn Diagram.
You should, perhaps, more accurately say it like this:
Mathematics INLCUDES Arithmetic - but Arithmetic is only a 'small part' of mathematics. This could be easily illustrated in a Venn Diagram.
Oh, come on---- "my mother, a college English professor who was terrified by the idea of calculating a 10 percent tip, and desperately grateful to leave it to any grandchild at the fourth grade level or beyond.
'Terrified and desperately'? What exaggeration.
I don't believe any college professor, or even a h.s grad, would be 'terrified!' to figure a 10% tip.
I'm lousy at math but just removing zeros to figure 10% ---or even 1% ---and going from there is EASY for any elementary school kid, once told how to do it.
'Terrified and desperately'? What exaggeration.
I don't believe any college professor, or even a h.s grad, would be 'terrified!' to figure a 10% tip.
I'm lousy at math but just removing zeros to figure 10% ---or even 1% ---and going from there is EASY for any elementary school kid, once told how to do it.
15
You have not witnessed the strange phenomenon first hand: the person who cannot calculate 10% of a number. Many years ago, I worked with a fellow I called my “reverse engineer” friend. He scored 800 on the verbal portion of his SAT’s but had the most dismal score, I think the lowest you can get by random answers, on the math portion. He was in his thirties, and had gotten along quite well leaning on his impressive reading, speaking, and writing skills. He was also fantastic with computers, his skill with programming languages was amazing. I had previously thought only engineers with rigorous logic skills would be good coders, but I was wrong.
Anyway, a day came when I was telling him to increase a number by 10%. He started rummaging around. He panicked because he couldn’t find his calculator. I said, “oh, for heaven’s sake, just do it in your head.” I thought he was one of those annoying early computer adopters who decided nothing need be done in your head or with pencil and paper anymore. But I was wrong. He couldn’t do it. He was practically hyperventilating, so I slowed him down and got him to tell me all about how he’s been dealing with what he considered a mental handicap he was hiding from the world. I sat with him and told him about moving the decimal point and all that, but although he was nodding, I could see it was going in one ear and out the other.
Math anxiety is real. The earlier in life it’s identified and dealt with, the better.
Anyway, a day came when I was telling him to increase a number by 10%. He started rummaging around. He panicked because he couldn’t find his calculator. I said, “oh, for heaven’s sake, just do it in your head.” I thought he was one of those annoying early computer adopters who decided nothing need be done in your head or with pencil and paper anymore. But I was wrong. He couldn’t do it. He was practically hyperventilating, so I slowed him down and got him to tell me all about how he’s been dealing with what he considered a mental handicap he was hiding from the world. I sat with him and told him about moving the decimal point and all that, but although he was nodding, I could see it was going in one ear and out the other.
Math anxiety is real. The earlier in life it’s identified and dealt with, the better.
4
He probably failed to grasp the concept of place value in first grade and then was passed up through grade school, getting more and more confused each year. I actually saw this myself. I was trying to help a 7th grader who could not understand why it is easy to multiply by 10, 100, 1000, etc.... Right there in first grade, she had missed out on place value, and - this is the point of my anecdote - her teachers never noticed or did anything about it.
1
Hi, Kaleberg—It’s hard to say what my friend did or didn’t grasp. Sure, we were taught about ones, tens, hundreds, etc. columns in first grade, but we were mostly focused on rote learning back in the sixties—memorizing tables and performing simple addition and subtraction. Concepts took a back seat to memorization and learning step by step procedures.
But, you’re right, somewhere along the line my friend got lost and nobody caught it and helped him. His being off-the-charts smart verbally probably didn’t help because everybody, parents and teachers alike, just figured he wasn’t any good at math, but he was really good at most everything else, so no big deal. But it was a big deal to him, and the way he told it to me the whole math thing just snowballed year after year until he just freaked out if he had to do any math at all, even simple stuff. That’s why he would panic if his calculator was not at hand. It’s hard for people like me, people with zero math anxiety, to understand. My point to Meredith is that “terrified and desperately” is not an exaggeration. What she thinks should be easy, what I think is trivial, simply is not for people afflicted with severe math anxiety. And math anxiety is very preventable if not allowed to morph into a giant mental block.
But, you’re right, somewhere along the line my friend got lost and nobody caught it and helped him. His being off-the-charts smart verbally probably didn’t help because everybody, parents and teachers alike, just figured he wasn’t any good at math, but he was really good at most everything else, so no big deal. But it was a big deal to him, and the way he told it to me the whole math thing just snowballed year after year until he just freaked out if he had to do any math at all, even simple stuff. That’s why he would panic if his calculator was not at hand. It’s hard for people like me, people with zero math anxiety, to understand. My point to Meredith is that “terrified and desperately” is not an exaggeration. What she thinks should be easy, what I think is trivial, simply is not for people afflicted with severe math anxiety. And math anxiety is very preventable if not allowed to morph into a giant mental block.
2
The key to fending off math anxiety is proper education. If you understand it you will not have anxiety. There may be some instinctive fear of heights and snakes, but I don't think there is one of math.
One comment mentioned that most people don't know which is larger 1/3 or 1/4. If one does not know this, it indicates a fundamental educational flaw that goes back to 4th grade or so. There is no point in going on to higher level mathematics if this is not understood.
One comment mentioned that most people don't know which is larger 1/3 or 1/4. If one does not know this, it indicates a fundamental educational flaw that goes back to 4th grade or so. There is no point in going on to higher level mathematics if this is not understood.
9
Try telling that to the people who come out of the education department and not the math department of the university. It's the Ed Dept people who become the teachers. The Math Dept people become the college professors who are appalled that college freshmen can't do anything without a calculator, and even then don't understand the answers they get from their machines. Most K-5 teachers are math-phobic and math-incompetent. They don't really understand math, which is why they stick to the lower level grades, where they think understanding and being able to convey deep number concepts isn't necessary. Hence the subject is not taught in the schools.
8
I completely agree. It's all about education. A child might not go on to master higher level math, but a sense of numbers and what they represent in the real world must be taught well in younger ages. Parents or caretakers can really help by talking math with their kids. Kids can get mighty interested. Pizza slices can becomes fractions even as simple activities like gardening and cooking can talk about spacing and ratios respectively. Our kids used to spend some time playing math games on the web (http://www.mathblaster.com/parents/math-games or https://maya.nmai.si.edu/maya-sun/maya-math-game), and this helped too. Their grandmother used to take them on nature walks with math as an invisible companion. They counted stones, spotted patterns, pocketed pairs, sang counting songs and even ate food with a dollop of math. So our kids 'talked math' before they got down to math in a proper classroom, and I think this helped in getting our kids comfortable with mathematics.
2
Schools of Education should change their curriculum to require that their admits spend the first two to three years in their university's College of Liberal Arts, where they are required to take mathematics from mathematics teachers (as well as English, science, etc. I am aghast at how so many primary and secondary school teachers can't spell, let alone teach arithmetic.) Then the final year (or two) in their School of Ed should be devoted to intensive courses in classroom management, which is essential if students are to learn in a controlled, non-chaotic environment.
2
I once didn't understand how anyone could have a problem with math. Then I ran into number theory. My conclusion is that the limitation is not based on how much complexity you can hold in your mind, but how abstract your thinking can become.
3
As a former high school math teacher and now a volunteer tutor, I am particularly dismayed by 2 things:
1. High School math classes (at least in Texas) are totally focused on getting students ready for Calculus. Colleges require either Calculus or Statistics for their "basic" math class. Applied mathematics would make so much more sense for most people. Why aren't we teaching all students Statistics and Number Theory as well as Algebra and Geometry. Why aren't they doing real world problems instead of practicing pages of Algebra II manipulation that they will never use except on an SAT test.
2. Most students from middle school to college these days have no "fraction sense"; e.g. they don't know which is larger 1/3 or 1/4, they can't add 1/3 + 1/3. Nor can they connect simple fractions to decimals and percents and money (cents). Not sure how this got lost in the Core Math, but it has. Time to start cooking and measuring, cutting pies, and counting money with all students again.
1. High School math classes (at least in Texas) are totally focused on getting students ready for Calculus. Colleges require either Calculus or Statistics for their "basic" math class. Applied mathematics would make so much more sense for most people. Why aren't we teaching all students Statistics and Number Theory as well as Algebra and Geometry. Why aren't they doing real world problems instead of practicing pages of Algebra II manipulation that they will never use except on an SAT test.
2. Most students from middle school to college these days have no "fraction sense"; e.g. they don't know which is larger 1/3 or 1/4, they can't add 1/3 + 1/3. Nor can they connect simple fractions to decimals and percents and money (cents). Not sure how this got lost in the Core Math, but it has. Time to start cooking and measuring, cutting pies, and counting money with all students again.
40
Perhaps this, like much of 'education' today, is merely separating the wheat from the chaff.
2
My late husband spent his entire career teaching secondary school math. He said about a third of his students (mostly this was in independent schools) "got" the theory about math and understood numbers. For the other 2/3's it might as well have been Sanskrit.
His method was to explain the entire concept. Those who got it, loved this and quickly became engaged and asked questions. He then spoke to those who looked gobsmacked and told them that if they didn't understand the number theory, just to memorize what they needed and buy a good calculator and they would be fine. No one who ever took any of his classes (and most of his years teaching was in all-girl's schools), ever left his class with math anxiety as he gave them a way to succeed. Teachers simply must realize that this is the way people just are!
Two of his grandchildren love math and totally get it. It's a shame he didn't live long enough to have some very long, totally geeky math conversations with them. He would have loved that and so would they!
His method was to explain the entire concept. Those who got it, loved this and quickly became engaged and asked questions. He then spoke to those who looked gobsmacked and told them that if they didn't understand the number theory, just to memorize what they needed and buy a good calculator and they would be fine. No one who ever took any of his classes (and most of his years teaching was in all-girl's schools), ever left his class with math anxiety as he gave them a way to succeed. Teachers simply must realize that this is the way people just are!
Two of his grandchildren love math and totally get it. It's a shame he didn't live long enough to have some very long, totally geeky math conversations with them. He would have loved that and so would they!
22
Innumeracy - not having a good sense of quantities - is the cause of a lot of the irrational thinking that plagues our society.
For example, Muslim immigration. Hillary Clinton proposed admitting 65,000 Syrian refugees per year. The population of the USA is 300+ million, so that's one Syrian refugee per 5000 people already here. Other Muslims immigrate into the USA in similarly tiny numbers.
Yet, some people are convinced that Muslim immigration is a risk to Western culture.
It matters a lot that people be comfortable with thinking numerically, otherwise our country will continue to be run by illusions.
For example, Muslim immigration. Hillary Clinton proposed admitting 65,000 Syrian refugees per year. The population of the USA is 300+ million, so that's one Syrian refugee per 5000 people already here. Other Muslims immigrate into the USA in similarly tiny numbers.
Yet, some people are convinced that Muslim immigration is a risk to Western culture.
It matters a lot that people be comfortable with thinking numerically, otherwise our country will continue to be run by illusions.
45
One of my personal favorite examples of innumeracy is when Tobacco Industry advocates talk about the thousands of jobs that would be lost if cigarettes were made illegal. Thousands of jobs vs. millions of lives, hmmm . . . .
2
I'm in my late twenties and suffered from math anxiety as a student. Now that I'm out of school and working, I have a goal of reteaching myself the math I loathed in school to understand what I was missing out on all of those years. Thank goodness for online courses.
7
You deserve a lot of credit (no pun intended). It's wonderful that you are taking responsibility for your own education and that you aren't letting early setbacks stop you.
5
Everything in the physical world can be represented mathematically, but many things in the mathematical world are not represented physically. Mathematically speaking, the real world is a letdown, but that’s what makes math an intellectual playground for some kids.
2
Most interesting ideas, Ed, thank you!
I still wake up sometimes from a nightmare that involves the horror scenario of my PhD being revoked because I failed calculus in high school. I did not fail calculus but I struggled with extreme math anxiety thanks to a high school teacher who belittled and berated students who "did not get it" right away. Only years later did I discover that I am actually quite good in math. I wonder what role teachers play in all of this.
15
A lot.
3
Teachers play a HUGE role - or, as President Donald 'G' (for Groper) Trump might say, a YUUGE role...
Math is intrinsically competitive because there is almost always a single correct answer. People who struggle, therefore, develop anxiety for the subject because being wrong all of the time is not enjoyable.
Reading, on the other hand, is exploratory and emphasizes balancing different perspectives. Without a single correct answer, people are less likely to feel inadequate and develop anxiety.
In other words, your reading ability may be just as bad as your math ability, but you are less likely to notice it and develop subsequent anxiety because there is no reading equivalent to calculating a 10% tip in everyday life.
Reading, on the other hand, is exploratory and emphasizes balancing different perspectives. Without a single correct answer, people are less likely to feel inadequate and develop anxiety.
In other words, your reading ability may be just as bad as your math ability, but you are less likely to notice it and develop subsequent anxiety because there is no reading equivalent to calculating a 10% tip in everyday life.
23
While there may be "one answer", there are often many ways to get to that answer, and where mathematicians get competitive is finding new, creative, approaches to the same problem, or finding the "minimal proof".
Teachers need to be flexible in grading math assignments and tests. Partial credit for "almost" answers, and adding corrections or hints in the grading process (combined with requesting a corrected answer from the student) works well. That can only be done with small class sizes.
One of the most devastatingly effective methods of grading that I've ever encountered, however, is simply putting a red check mark at the top of the page, when everything is done right. If the check is not there, the student has to figure out why by redoing each problem. Then resubmitting it for grading again. It's brutal, but as they say in the Marine Corps, "Pain is weakness leaving the body".
Teachers need to be flexible in grading math assignments and tests. Partial credit for "almost" answers, and adding corrections or hints in the grading process (combined with requesting a corrected answer from the student) works well. That can only be done with small class sizes.
One of the most devastatingly effective methods of grading that I've ever encountered, however, is simply putting a red check mark at the top of the page, when everything is done right. If the check is not there, the student has to figure out why by redoing each problem. Then resubmitting it for grading again. It's brutal, but as they say in the Marine Corps, "Pain is weakness leaving the body".
9
Good documentary from PBS, "The Great Math Mystery" and it tells you when "close" is good enough, as in how "close" put an unmanned vehicle on Mars. Precision has to be sacrificed for productivity and today folks have lost sight of that factor, it's a balance. Promoting this app here is a joke as don't trust app algorithms if you don't know where they came from, who wrote them and so on. Look at the trouble Uber is in with Apple, as we now have "code wars" on the map, even the best math engineers at Apple were duped with computer code from Uber. Computer code is math with all the functions that execute.
http://ducknetweb.blogspot.com/2015/05/the-great-math-mysterysoftware.html
Actual Plos One study shows the fear of math is what keeps folks gullible with biting on apps and other web software as well.
http://ducknetweb.blogspot.com/2012/10/algo-duping-plos-one-journal.html
http://ducknetweb.blogspot.com/2015/05/the-great-math-mysterysoftware.html
Actual Plos One study shows the fear of math is what keeps folks gullible with biting on apps and other web software as well.
http://ducknetweb.blogspot.com/2012/10/algo-duping-plos-one-journal.html
1
I question whether these "Psychology Professors and Graduate Students" at The University of Chicago, are: 1) actually good at Math themselves, and 2) even understand what Math is! What they are studying/observing is only a certain style, Lower Level Math, where the experts in this "kind-of-Math" cannot compete at the Top Level in World Competitions. This Human Trick Math that they are basing conclusions on, only goes so far, and does not go all the way. These Professors and Graduate Students look like Novices. Plus, do they have Brilliant IQ Scores on their Resumes? I know they don't, because of the uneducated statements in this article.
Are these Psychology Professors and Graduate Students, aware that students in some Traditional Schools, start with the Equals Sign, The Concept of Equality? First, they learn Equality. They don't learn "Numbers". If a person cannot grasp and maintain/hold the Thought of Equality, the = in the Mathematical Equation, then they do not advance to numbers, 1,2,3 etcetera. ---- I will not be purchasing the book, Choke, as well as the Mobile App.
Are these Psychology Professors and Graduate Students, aware that students in some Traditional Schools, start with the Equals Sign, The Concept of Equality? First, they learn Equality. They don't learn "Numbers". If a person cannot grasp and maintain/hold the Thought of Equality, the = in the Mathematical Equation, then they do not advance to numbers, 1,2,3 etcetera. ---- I will not be purchasing the book, Choke, as well as the Mobile App.
Given that only 15% of US college degrees are in STEM fields, it would seem that "math literacy" is in short supply. So how does a generation that is weak in math raise a generation that is stronger?
Worse still is that most of us have "bad math habits" and if we are going to promote learning math in the home we are just going to pass these habits on. Take adding a column of numbers. Most of us know one "standard" way of doing this, but there are many.
My favorite is grouping lines of numbers that sum to some fixed value, like 10, 20, etc.; then sum the resulting set of grouped numbers. There are many other approaches, and just "palying" with how to add numbers, 1-10, can introduce a lot of fundamental math concepts.
If you can make a child curious about math, they will stick with it. On my way to graduation as a math major, my experience was that individuals advance at different rates in different areas of math, and you will revisit areas that you do or do not know many times; gaining insight.
I skipped the grades where addition, times tables, and basic division were taught. I hate doing these tasks as an adult, nevertheless, you can be a practicing mathematician, without doing them. One thing you learn at the graduate level is just how "personal" math is. You won't succeed at a high level without a mentor. Find one for your child.
Worse still is that most of us have "bad math habits" and if we are going to promote learning math in the home we are just going to pass these habits on. Take adding a column of numbers. Most of us know one "standard" way of doing this, but there are many.
My favorite is grouping lines of numbers that sum to some fixed value, like 10, 20, etc.; then sum the resulting set of grouped numbers. There are many other approaches, and just "palying" with how to add numbers, 1-10, can introduce a lot of fundamental math concepts.
If you can make a child curious about math, they will stick with it. On my way to graduation as a math major, my experience was that individuals advance at different rates in different areas of math, and you will revisit areas that you do or do not know many times; gaining insight.
I skipped the grades where addition, times tables, and basic division were taught. I hate doing these tasks as an adult, nevertheless, you can be a practicing mathematician, without doing them. One thing you learn at the graduate level is just how "personal" math is. You won't succeed at a high level without a mentor. Find one for your child.
2
"With math, you have to learn the "how," "what," and "why" over and over again. What is one-third of 2/3? You learn to multiply the two fractions in your head--and in a split second you arrive at 2/9. But why? Well, a third of 1/3 is 1/9--so one-third of 2/3 will be twice as much. ... the point is that you get used to doing so while also making rapid calculations."
Whaattt!!? That's a prime example of people who "get" math trying to talk to those of us who don't. Even when you TRY to simplify it, you might as well be speaking Amharic (fewer speakers than Greek). Even the most math anxious adults can do everyday math given enough time, a calculator or pencil & paper, and the confidence TO NOT CARE what anybody thinks of them. Between the ages of 55 and 60, I took algebra at my local community college 3 times. Why? I was tired of hiding from math. I didn't care whether I passed or failed the course. I learned that I CAN do math given some conditions: 1st, ENOUGH TIME. I take as long as I need. Others just have to wait until I finish. 2nd, the right tools. I'll whip out my phone with the built-in calculator in the middle of a sales pitch to check the veracity of the sales person. 3rd, a thick skin that enables me to assert my need for more time while others huff & puff their impatience. Rapidity is not needed, just accuracy. Btw, by the 3rd final, I received a B+ (88%).
Whaattt!!? That's a prime example of people who "get" math trying to talk to those of us who don't. Even when you TRY to simplify it, you might as well be speaking Amharic (fewer speakers than Greek). Even the most math anxious adults can do everyday math given enough time, a calculator or pencil & paper, and the confidence TO NOT CARE what anybody thinks of them. Between the ages of 55 and 60, I took algebra at my local community college 3 times. Why? I was tired of hiding from math. I didn't care whether I passed or failed the course. I learned that I CAN do math given some conditions: 1st, ENOUGH TIME. I take as long as I need. Others just have to wait until I finish. 2nd, the right tools. I'll whip out my phone with the built-in calculator in the middle of a sales pitch to check the veracity of the sales person. 3rd, a thick skin that enables me to assert my need for more time while others huff & puff their impatience. Rapidity is not needed, just accuracy. Btw, by the 3rd final, I received a B+ (88%).
19
I published a paper in 2013 demonstrating that college students with good high school skills did incrementally better in math courses than those who did not. In fact elementary math skills predicted the grade in the college math course whether introductory or advanced math. The author and the psychologist Dr. Levine omit the most interesting advance in overcoming the largest stumbling block to graduating from college, elementary algebra skills. Khan, a retired investment banker with graduate degrees in engineering and business from MIT and Harvard started making youtube.com exercises for his niece who had math anxiety because she did poor on an exam for advanced placement in a high school math course. It is easy for many to miss one lesson in math and then another since the basic skills all build upon each otehr. What Khan Academy offers online and free are the basic math courses in 10 and 20 minute bites (bytes, if you prefer computer language) so that anyone with math anxiety or math deficits can catch up. Khan's book on the one room school house also challenges the current elementary and high school and college Prussian based model of education. That's probably why the Gates Foundation and Eric Schmidt have funded the Khan Academy free online internet testing approach to education not only in math, but science including economics, physics, chemistry, biology and many other fields.
5
What's brilliant about Khan's approach is you can direct your studies according to what concepts and areas of math are a) interesting, or b) easier to understand. In my own case, I learned concepts in calculus before, I understood algebra. Geometry was fantastic because it's where you learn to do proofs, and I loved doing proofs, as a child.
Learning math, for me, was a process of jumping all over the place. I had to do it via books in the library, but watching lecturers on line is a massive improvement. Lectures show the "framework" of the solution, but classic math texts just show the results, and hide how a proof, etc, was actually arrived at.
Math is a part of the process of "unlearning" how 90% of the people on the planet understand "reality", and "re-seeing" via STEM disciplines. Perhaps a better analogy would be, once you study music, you hear any piece, and song, in a fundamentally different way than anyone (non-musician) around you.
Learning math, for me, was a process of jumping all over the place. I had to do it via books in the library, but watching lecturers on line is a massive improvement. Lectures show the "framework" of the solution, but classic math texts just show the results, and hide how a proof, etc, was actually arrived at.
Math is a part of the process of "unlearning" how 90% of the people on the planet understand "reality", and "re-seeing" via STEM disciplines. Perhaps a better analogy would be, once you study music, you hear any piece, and song, in a fundamentally different way than anyone (non-musician) around you.
3
Math anxiety hits girls especially hard. Even girls who enjoy math tend to underrate their ability, while boys tend to overrate it. Unfortunately, cultural support for the notion that girls are innately weak in math is everywhere, from Talking Barbie (Thanks a lot, Mattel) to female elementary school teachers who model their discomfort with fractions in front of their female students. Timed tests make things worse. Many math anxious kids choke for the first few minutes of the test and never make up the lost time, even after they pull themselves together and start working the problems.
I don't think the early childhood strategies discussed in this article will help at all because the problem is not one of math preparation before school starts. We need to change the way we teach math. Even in the early grades, math instruction should be handled by specialists who work with several classrooms. Children should be explicitly taught that most achievement is the result of hard work, not innate genius, and this lesson should be driven home throughout the grade school years. Timed tests should rarely be used, and kids should be taught strategies to calm and focus themselves as they approach math problems.
I don't think the early childhood strategies discussed in this article will help at all because the problem is not one of math preparation before school starts. We need to change the way we teach math. Even in the early grades, math instruction should be handled by specialists who work with several classrooms. Children should be explicitly taught that most achievement is the result of hard work, not innate genius, and this lesson should be driven home throughout the grade school years. Timed tests should rarely be used, and kids should be taught strategies to calm and focus themselves as they approach math problems.
28
Math should absolutely be taught by specialists at all grade levels. The earlier the better. Studies show girls do better than boys in the early grades, and math contrary to the public perception is a very 1-1 learning process, so girls actually have a continual advantage in collaboration, and helping each other to learn.
My grammar school used a lot of "girls vs boys" class and homework exercises; which reinforced in-group cooperation, and across group competition (each group presented one exercise on the chalk board).
My grammar school used a lot of "girls vs boys" class and homework exercises; which reinforced in-group cooperation, and across group competition (each group presented one exercise on the chalk board).
6
I got nervous as soon as I read the headline for this article!
19
Dr. Levine is right that people don't go around saying "I'm not a reading person." Yet there are subjects besides math--languages and music, for instance--that can provoke anxiety. With them, as with math, fluency and understanding will lessen the anxiety. (But how do you develop fluency and understanding if you're afraid of the subject? You immerse yourself in the subject. You walk in it, think in it, dream in it.)
With math, you have to learn the "how," "what," and "why" over and over again. What is one-third of 2/3? You learn to multiply the two fractions in your head--and in a split second you arrive at 2/9. But why? Well, a third of 1/3 is 1/9--so one-third of 2/3 will be twice as much. There are many other ways of looking at it; the point is that you get used to doing so while also making rapid calculations.
With language, it's a similar matter. A new language can seem intimidating because of the combination of vocabulary, pronunciation, and grammar. But you start learning to say things--quickly and easily--while also learning the grammatical principles. Something analogous happens with music.
So then the question is: How to give children the opportunity to develop both fluency and understanding in these subjects? Practice, practice, practice, discussion, discussion, discussion. Not one at the expense of the other--not "concepts" over so-called "rote knowledge" or vice versa. Both together.
This combination brings not only confidence but enjoyment.
With math, you have to learn the "how," "what," and "why" over and over again. What is one-third of 2/3? You learn to multiply the two fractions in your head--and in a split second you arrive at 2/9. But why? Well, a third of 1/3 is 1/9--so one-third of 2/3 will be twice as much. There are many other ways of looking at it; the point is that you get used to doing so while also making rapid calculations.
With language, it's a similar matter. A new language can seem intimidating because of the combination of vocabulary, pronunciation, and grammar. But you start learning to say things--quickly and easily--while also learning the grammatical principles. Something analogous happens with music.
So then the question is: How to give children the opportunity to develop both fluency and understanding in these subjects? Practice, practice, practice, discussion, discussion, discussion. Not one at the expense of the other--not "concepts" over so-called "rote knowledge" or vice versa. Both together.
This combination brings not only confidence but enjoyment.
9
Take 10% and then add half to get 15%?....I'm sorry, but in recent years a 15% tip from a wealthy doctor (or anyone) for a waitperson or similar tip-compensated laborer is cheap and demeaning, whether or not your parents lived through the depression. Teach your kids to give an easy 20% tip, and find some other way to teach the more sophisticated arithmetic.
18
"Dr. Susan Levine, chairwoman of the department of psychology at the University of Chicago, agreed: “An educated person doesn’t go around saying, I’m not a reading person.”
That is because it is almost impossible to be highly educated without good reading skills. It is however very possible to be highly educated without good math skills. It is also very possible to work in a profession requiring a high degree of education without any math skills. The opposite is rarely true.
As a discalculate who cannot "hold on" numbers in my head And will transpose numbers on a regular basis, but who is more than capable of writing a sophisticated Appellate brief, I can assure you that math skills are unnecessary for many educated people. My guess is your mother was one of those people. And as a mother of a daughter who is a double engineering major I can attest to just how naturally math does indeed come to some people in a way it would never for me.
As for the inability to perform necessary daily mathematical functions like figuring a tip: in the words of some Twitter wit I have seen: "Remember when teachers told us that we wouldn't always have a calculator with us, well we showed them!"
That is because it is almost impossible to be highly educated without good reading skills. It is however very possible to be highly educated without good math skills. It is also very possible to work in a profession requiring a high degree of education without any math skills. The opposite is rarely true.
As a discalculate who cannot "hold on" numbers in my head And will transpose numbers on a regular basis, but who is more than capable of writing a sophisticated Appellate brief, I can assure you that math skills are unnecessary for many educated people. My guess is your mother was one of those people. And as a mother of a daughter who is a double engineering major I can attest to just how naturally math does indeed come to some people in a way it would never for me.
As for the inability to perform necessary daily mathematical functions like figuring a tip: in the words of some Twitter wit I have seen: "Remember when teachers told us that we wouldn't always have a calculator with us, well we showed them!"
17
Actually I have met people who are highly intelligent and well educated who have very poor verbal and writing skills. So a person may not go around saying, "I'm not a reading person", but they very well may be.
In "The Magic Mountain" by Thomas Mann, there is a character named Mynheer Peeperkorn who is a Dionysian figure of emotion. He can rarely finish a statement because of his poor verbal skills.
There are clearly differences between people in language or mathematical aptitude. I think anyone can learn to do simple arithmetic like add and subtract and multiply. But that is not mathematics.
In "The Magic Mountain" by Thomas Mann, there is a character named Mynheer Peeperkorn who is a Dionysian figure of emotion. He can rarely finish a statement because of his poor verbal skills.
There are clearly differences between people in language or mathematical aptitude. I think anyone can learn to do simple arithmetic like add and subtract and multiply. But that is not mathematics.
5
CF Gauss, one of the great mathematicians of all times, considered himself first and foremost to be an 'arithmetiician', I believe.
GSC:
Whatever the case actually is, you probably believe this because of the widely told story (probably within your country too) of Gauss as a five-year-old, which goes something like this:
The teacher wrote on the board for the pupils to find the sum of all the numbers [integers, of course] from 1 to 100. As soon as the teacher turned around, little Gauss had the answer on the teacher's desk, 5050.
This generalizes nicely to a simple formula from 1 to n by using algebra, the language of mathematics...
Whatever the case actually is, you probably believe this because of the widely told story (probably within your country too) of Gauss as a five-year-old, which goes something like this:
The teacher wrote on the board for the pupils to find the sum of all the numbers [integers, of course] from 1 to 100. As soon as the teacher turned around, little Gauss had the answer on the teacher's desk, 5050.
This generalizes nicely to a simple formula from 1 to n by using algebra, the language of mathematics...