Menu
Jump to content

What's with the ads?

Archived

This topic is now archived and is closed to further replies.

Dotwithaperiod

Do Students Need Algebra?

Recommended Posts

Does Germany not have any students like this?  If they do, where are they?  Are they really passing linear algebra in 8th grade and trig by 10th?  Are they just required to take these classes, and then fail them?  Can students with the 3rd type of diploma (if available) get a job?  If they can't pass quadratics can they still get the 3rd type of diploma?

 

Of course there are those students, too, mainly in large cities with a high immigrant population. Language issues are a big problem in those schools; there are second and third generation immigrant students who are not fluent in German because they socialize exclusively with students from the same background and their families are not encouraging German language learning. And of course there are also home life issues, drug abuse.

Students who are severely disturbed, have issues with violence and PTSD, can attend special schools (Foerderschule fuer Erziehungshilfe). They would have very small classes, more personnel, counseling, behavioral training, but I am not personally familiar and can not give details. I do not know which portion of students with issues are attending those schools. The curriculum guidelines for those schools, however, are the same as for regular schools, and attending is usually temporary with the goal of returning to a main stream school.

 

The other students attend the non-college bound school in states where there are two tracks, or the lowest tracking 9th grade school in states where there are three. The will spend a number of years reaching 9th grade, would typically repeat grades, and hopefully will receive the 9 year diploma ("Hauptschule"). As I said elsewhere: some will fail math, but they can still graduate if they pass in enough other subjects.

 

Their job perspectives are not good. In Germany, you don't just go work somewhere, you have to train for pretty much any job through vocational training or apprenticeship, and that would be the normal path for students graduating from 9th and 10th grade. There are very few jobs that require no qualification whatsoever, much fewer than in the US (As I explained once in another post: you go to school to be a secretary, or to work in retail.). It is difficult to find an apprenticeship with a 9th grade diploma, and there are a lot of initiatives and special programs to get these young people into some kind of training.

Students who dropped out of school without even completing the 9th grade diploma are a pretty hopeless case, with hardly any prospects. Some are just not employable and will fall through the cracks.

 

ETA: Just to clarify: the point of my earlier post was solely that there are differences in what is considered developmentally appropriate in math for average students. I did not mean to imply that all German students magically can fulfill these requirements - but that what is expected and considered achievable for a student of a certain age seems to be a lot more than in the US.

Share this post


Link to post
Share on other sites

We used to have a much better vo-tech system and a lot more tracking. We also had a lot of classism/racism where students who should have been college-bound were stuffed into a vocational diploma or a lower track because 'you know how those people are'. This still goes on today in some corners of the country.

 

 

I had a chat a few months ago with a retired dean from a community college.  He began his career there when the school was ____ Tech as opposed to ____ Community College.  Interestingly, he feels that the push the NC CC system has made to AA/AS degrees over certifications in trades has been to the detriment of students and citizens. The economy then forced the CC to increase the number of academic courses for students using the CCs as feeders to four year universities. 

 

In my area, there has been an increase in health related programs. But the old electronics program is not what it used to be which is too bad.  I taught tech math to students who went on to become repair people for cash machines, computers, copiers, etc.  There are always jobs in this field.  By the way, my electronics students took a two semester Algebra/Trig course with lots of application problems.  Logarithms were necessary.

 

Share this post


Link to post
Share on other sites

For comparison, the basic maths qualification in the UK (for college-bound and non-college-bound students) is the GCSE, usually taken at age 16 but often taken at 15 (I took it early, as did Calvin; Hobbes is heading that way too).  Until recently you could legally leave education at 16; now you have to stay in education or training (which can be combined with work) until 18.  Most jobs require GCSE English and Maths.  60% of entrants pass each of the exams in any one sitting, but many pupils will retake the exams (sometimes repeatedly) until they pass.

 

According to the writer of LOF (who had a cursory look at the international version of the GCSE, which has a couple of extra topics) to pass the GCSE you would need

 

"...the Life of Fred series (from the first book on fractions to the calculus and statistics books)....

         There is more in the Life of Fred Calculus and in LOF: Statistics than will be needed for this exam, but there is material in each of those books that will be needed."

 

L

Share this post


Link to post
Share on other sites

 

There is just a disconnect between the standard math curriculum and a large number of students.  I would argue that 20% of kids will really struggle with Algebra 1, and a good 40% will struggle with algebra 2.  It is not that these students do not need math instruction, they just don't need algebra.  What I would argue that they do need is math that they will actually encounter in life.  I am not talking about shopping and taxes, I am talking about problem solving and statistics.  Students need to be able to understand the math that they see in the news - which is the statistics behind medical claims, environmental claims, polls, etc.  And they can learn to interpret and understand this material with a minimum of algebra.  Plus any algebra that would be required could be taught from the point of view of "you actually need to know this to be able to read this newspaper that I am holding." 

 

I think that requiring Algebra 2 for graduation is absolutely nuts. 

 

---- ducking now ----

 

Ruth in NZ

 

:iagree:

 

Question: It has been my experience that people "forget" math. People who have been taught to memorize a specific algorithm will most likely not remember a longtime later what to do. Simple example: dividing by fractions. Many adults vaguely remember "there was something being flipped". Ouch. In contrast, something that has been understood once will never be really forgotten.

With the students in your example, I'd be worried that the algorithmic procedure ("first I do this and then i do that" ) will not be retained in the long term -whereas conceptual understanding is virtually indestructible and can still be retrieved decades later.

So, to extend the question: is there value in teaching students something they will have forgotten after one semester? What good does it do that they can do stoichiometry (or something else) precisely now, but won't retain the concept in the long term?

 

I still wonder if we could not train children to  a better abstract critical thinking if we incorporated it differently in early education. All small children are curious and want to know why things are the way they are. It is only when they have made the experience that those questions will not be answered and will be seen as a nuisance, something that happens mainly in schools, but also some families, that kids develop the intellectual laziness of "just tell me the procedure so I can work the problem and get my points". I think this is learned behavior. Maybe if formal education did more to encourage children's curiosity and foster critical thinking, kids would still want to know "WHY?" when they are teens.

 

Excellent points, Regentrude.  It's true - my method may results in less-than-permanent knowledge.  But I guess I'm looking at it as the lesser of two evils.  Ideally, I'd love it if we could teach all students to understand math conceptually and never have to worry about providing algorithms for kids to "plug and chug".  Realistically, though, what usually results from an all-conceptual approach to math is that the kids who can't grasp that approach (which, with the way math is taught in elementary schools, is most of them) end up never learning ANY math.  When only conceptual math is taught, then many, many students end up learning virtually no math because they simply can't grasp it.  This may be for a variety of reasons - their early math education was poor, they aren't ready for it from a maturity standpoint, or their brains just aren't hard-wired to comprehend conceptual math.  Whatever the reason, we end up with large numbers of students who don't understand math AND can't use it.  I see those two things as separate - not necessarily for everyone but for some.  With a conceptual-first and then an algorithmic approach, we at least end up with some students who understand conceptually, some students who may obtain a partial conceptual understanding but who are more comfortable with an algorithmic approach, and some students who will use the algorithms exclusively because they just aren't capable (either at that point in their lives or maybe never) of understanding the concepts.  The first set of students can understand and use math, the second set may have partial understanding but can also use math, and the third set will just use the algorithms they are given.  If that last set actually uses the math they learned (hence my push for math that based on a student's end-of-school goals), they will remember the algorithms.  Research on memory has shown that retention is partially based on how meaningful the information is but also how often it's accessed.  True - if the student never uses the algorithm, they likely will forget.  But if they never need it, does it matter that they forgot it? :D

 

Probably each one of us in this thread understands math conceptually so it's difficult to understand how so many students "just don't get it".  We tend to assume that if they were just provided with clear, precise explanations and opportunities for meaningful problem solving from the time they started school, then our math education problem would be solved.  I just don't think it would.  Some students will never "get it".  The conceptual-only approach puts those students in an impossible position.

 

I also agree that it's sometimes difficult to separate the lazy students from the struggling ones.  What I've found is that the lazy ones who actually do have partial understanding but prefer to use the algorithms will either end up moving away from the algorithms and towards conceptual understanding if their chosen career path requires them to continue to use those concepts or they will continue to use the algorithms just to make it to the end of the course and follow a career path that doesn't involve knowledge of those concepts.  The struggling ones who aren't capable of conceptual understanding at least get something (algorithmic knowledge) instead of nothing.

 

I was thinking about your question above (bolding was mine).  I'll have to think more on it.  The idealist in me wants to say, "Of course not!" but I've been in too many classrooms and worked with too many public school students to be that idealist - I'm jaded.  Because of all the factors that come in to play, if we're only going to teach things that all students are capable of retaining for long periods of time, I'm afraid that there wouldn't be much left to teach at the high school level.  This leads to the thought that perhaps modern public education is simply irrevocably broken and a massive waste of time - and I don't know how to counter those thoughts. :(

 

I can understand the argument that not all students need alg2 or capable of achieving that level of mathematical competence......but I don't really accept the argument that it isn't necessary bc is irrelevant to everyone other than STEM majors. Independent tradesmen would be well-served understanding basic quadratic functions/relationships in terms of their business costs/profits.

 

I think one area where math education fails is that math is taught simply as solving equations instead applied concepts.

 

I apologize, 8 - you are right. :)  I tend to think of "trades" as a homogenous group and it's not.  I do think that the same concepts could be taught to STEM bound students and trade bound students but the concepts should probably be approached differently for each group.

 

In Germany, every university bound student must take calculus in high school, irrespective of planned major.

 

Students on the non-college bound track cover:

linear equations and systems of linear equations in 8th grade

quadratic equations in 9th grade

the exponential function and trigonometric functions in 10th grade.

Geometry is taught interspersed, every year.

Those students graduate after 10th grade with a non-college-bound high school diploma. And did not start school until first grade at age 6 or 7.

 

I find it remarkable how different countries have so different expectations about student abilities:

stuff that is considered "advanced" here in the US because students do not have the "maturity" is taught in the lowest track high school to kids of the same age back home. I think we are expecting far too little in this country. I wonder what we have to show for *13* years of schooling....

 

 

 

Having gone through the Singapore equivalent of what every student must do, I must say they have similar list for their noncollege bound students as well. The Singapore math folks say you can count what they do as an Algebra II course. They even dabble in a bit of calculus, and this all by tenth grade. 

 

 

Regentrude, I am still a bit confounded as to where some of my students would go in the education system if they were in Germany.

 

Like I said before, I taught the 2nd and 3rd streams out of 4.  The lowest stream were students with intellectual disabilities and did not take standard classes, but were still in the school (I did not teach these students).  Let me paint a picture for the lower stream of the two streams of 10th grade math that I taught.  They were all considered capable of taking standard 10th grade math, which finishes up Algebra 1, Geometry, and basic statistics:

 

Class #1 (24 students):

8 students were medicated for ADD/ADHD with varing success

1 student was a 'coke' baby and was 'normal' in intelligence but had some *extreme* behaviours

1 student tried to commit suicide the year I taught her

1 student was was a refuge from Africa, and clearly had some PTSD

 

Class #2 (25 students) (this class was specially created to cluster these types of kids that had be kicked out of other schools):

3 students were so violent that they had personal body guards that attended my class everyday and protected the other students from their outbursts.  I did observe 1 of these incidents when the student started throwing *desks* across the room

1 student was cutting himself, and was later murdered that year

2 students were clearly on drugs everyday

3 students had a homelife which not conducive to any study (I will not say more, but you can use your imagination and you won't be far off)

2 students where English was their second language, and they were not very proficient.

1 student was in foster care and was so much trouble that he kept loosing his foster family.  Some days he would come to school and tell me that he had to go back to the social worker's office that night to find a place to sleep.

 

 

I completely agree with you that the bar should not be lowered for most students. I just don't know how to help the students that I have taught if they have to reach a bar set impossibly high for them.

 

I'm wondering if some countries have a more homogenous student population and that changes how public education is approached?  I've had similar experiences to Ruth - I had one Grade 9 science class where half of the students had some degree of FAS.  It wasn't a spec ed class - it was a class where the students could go on into the trade school stream.  One of the students would sleep through most of my class - it was the only place she felt safe enough to sleep.  She had watched her father murder her mother at age 4 and had bounced around from relative to relative ever since.  Another student had been expelled from Grade 8 for assaulting teachers and ed assistants.  Another student went into a school washroom after his end of semester exams in Grade 9 and proceeded to smear his feces all over the walls.  He also went on to murder his adoptive mother when he was 18.  I'm not saying this to shock and I'm sorry if it did.  I'm just trying to show that this is the student body we encounter in non-spec ed classrooms in Canada and the States.  Most of the mental energies of these kids are spent on surviving and coping with everyday life - even if they were capable of conceptual math, they haven't got the mental energy for it.  I teach the understanding - if they get it, great.  Then I teach the algorithms - they can use them right away even with no or partial understanding.  Sometimes working with the algorithms for awhile actually triggers the conceptual understanding - also great.  When I teach AP Chem, you better believe that I teach conceptual understanding! :)  Two completely different scenarios requiring two completely different teaching approaches.

 

Whew.  That was long - sorry about the novel!  I also feel as though I rambled and didn't really say what I wanted to say.  Maybe I'll post this and then come back later to reread and edit.  I'm loving this discussion, though, and everyone's contributions to it.  Sometimes just thinking about the state of our public education system makes me want to run and stick my head in the sand.  It just all seems so hopeless.  It keeps pulling me back in, though.  I truly do love teaching.  When I first started out, I tended to only want to teach the university-bound stream - probably because I felt I could relate to those kids.  As I started teaching the trade-school stream, though, I found that, in some ways, it was more gratifying.  The class I mentioned above may sound like a teacher's worst nightmare but I enjoyed those kids.  When even partial conceptual understanding occurred, it was a huge victory.  When it didn't occur at all, the algorithms were there so they could pass the class and just passing was a huge victory for some of those kids.

 

My idea of conceptual-first-algorithm-second teaching is probably just a band-aid solution over a gaping wound.  But until the entire public school system is completely rethought and overhauled, I figure it's better than nothing. :)

 

ETA:  I just wanted to say that I agree wholeheartedly with Ruth's statement above (the bolding is mine).  It sometimes feels like teachers in the public system are stuck between a rock and a hard place.  My method of conceptual-first-algorithmic-second is the only way I can think of that keeps the bar high but provides an alternative method for some students to reach for the bar.  I still teach limiting reagents in Reg Chem (I know some Reg Chem programs have dropped it) but the two-pronged approach to teaching it means that the bar is kept high (I haven't dropped the concept from the curriculum) while giving more students the opportunity to reach for that bar.  I know, I know - not ideal - just the lesser of two evils. ;)

Share this post


Link to post
Share on other sites

This is an interesting discussion.

 

I agree with Regentrude. Our expectations in the US are way too low. Students can do much more than what they are asked to do. I was one of the kids on the third track (Austria) Regentrude mentioned. And I am here to tell you that we covered most of what is taught in Pre-Algebra, Algebra I and II and Geometry. We did not cover Trig; at least not much. All this was done by 8th grade. The reason why is obvious. Most of us would never take a math class again. And you know what everyone in my class (30 kids) made it. Not everyone had A's or even B's. So yes, the expectation is higher in Germany and Austria. It can be done. Like Ruth, I also know that we did not have anyone in class with drug problems and other behavioral issures. These issues almost always seemed to get addressed much earlier in the school career. These kids do not stay in the main stream school. They go to Special Ed type schools which are in no way attached to the regular school.

 

All this to say that I believe in an effort to provide 'everyone' with the same opportunity we also cause everyone's level of education to be lowered.

Share this post


Link to post
Share on other sites
 I'm just trying to show that this is the student body we encounter in non-spec ed classrooms in Canada and the States.  Most of the mental energies of these kids is spent on surviving and coping with everyday life - even if they were capable of conceptual math, they haven't got the mental energy for it.

 

Yes, these are some of the kids that I have worked with.  It is just so sad. 

 

Definitely sounds like Canada, the USA, and NZ mainstream students that are sent to special schools in Germany.

Share this post


Link to post
Share on other sites

So, my next question. For the non-university bound students, why algebra?  Why not statistics? Not statistics as it is currently taught in 12th grade, but rather a large overview of all the different methods.  So focusing the on the bigger picture with the intent of understanding current events, rather than on doing the tests by hand. This does not have to be an easy class, but it would be incredibly useful.  If a student is only going to get up to 10th grade math and never take math again, why not teach them material that is needed to be an informed voter.  A dancer, musician, writer, craftsman is not going to use algebra, but is still going to vote. And right now many of them are voting in ignorance, because the math that they have been taught in school does not help them to understand the issues and the probabilistic nature of knowledge.

 

"The cornerstone of democracy rests on the foundation of an educated electorate." - Thomas Jefferson

 

People cannot make decent voting decisions without understanding the issues, and the issues are statistical. 

 

So what exactly is society's end-goal for a math education for a student going into a non-mathematical career.  I just can't see how the end goal should be manipulating algebraic equations.

Share this post


Link to post
Share on other sites

So, my next question. For the non-university bound students, why algebra?  Why not statistics? Not statistics as it is currently taught in 12th grade, but rather a large overview of all the different methods.  So focusing the on the bigger picture with the intent of understanding current events, rather than on doing the tests by hand. This does not have to be an easy class, but it would be incredibly useful.  If a student is only going to get up to 10th grade math and never take math again, why not teach them material that is needed to be an informed voter.  A dancer, musician, writer, craftsman is not going to use algebra, but is still going to vote. And right now many of them are voting in ignorance, because the math that they have been taught in school does not help them to understand the issues and the probabilistic nature of knowledge.

 

"The cornerstone of democracy rests on the foundation of an educated electorate." - Thomas Jefferson

 

People cannot make decent voting decisions without understanding the issues, and the issues are statistical. 

 

So what exactly is society's end-goal for a math education for a student going into a non-mathematical career.  I just can't see how the end goal should be manipulating algebraic equations.

 

This is a very good question. 

 

Here is how it works in Austria (Germany may be similar).

 

If you are a non-college bound student in Austria after you have finished 8th or 9th grade you still continue on with an education. In most cases you will get into some specialized field. What is now taught is directly related to what you are studying and this may include statistics or other types of business maths. I take myself as an example. I ended up going back onto a highschool track, however, the school I went to was now a specialized school. Instead of Calculus I studied Accounting and Economics. At the end of 4 years at this school I could either find work right away or go to College.

Many kids actually do apply for apprenticeships. As Regentrude pointed out you need some type of specific training for almost all kinds of work. During the apprenticeship years which last somewhere between 3 and 5 years depending on the field the student actually works for 4 days and goes to school for one day. At the end of this period they take an exam and become a Master of a trade. Most kids finish this by the time they are 19.

 

Now in the US people always say that there are apprentice opportunities available. One of the main differences I see, however, is that the student is still required to finish highschool first. Then you can go and find work as an apprentice. In Austria you do that instead of the regular highschool track.

Share this post


Link to post
Share on other sites

So, my next question. For the non-university bound students, why algebra?  Why not statistics?

 

How can a student understand statistics if they can't do algebra???

 

Algebra introduces the idea of a function, plotting of data, etc. How can they understand a standard deviation without knowing what a square root is?

 

Besides, as I pointed out earlier, algebra is fundamental for financial literacy, something that should be an educational goal of society as well.

 

I agree that statistics is very important, but it should not be statistics instead of algebra, but rather in addition to it.

Share this post


Link to post
Share on other sites

How can a student understand statistics if they can't do algebra???

 

Algebra introduces the idea of a function, plotting of data, etc. How can they understand a standard deviation without knowing what a square root is?

 

Besides, as I pointed out earlier, algebra is fundamental for financial literacy, something that should be an educational goal of society as well.

 

I agree that statistics is very important, but it should not be statistics instead of algebra, but rather in addition to it.

 

Calvin took GCSE maths at 15, then did a kind of extension course for another year (IGCSE course, but no exam at the end - his choice).  When it came to choosing which maths element to study for the IB, he chose the lowest stream - he has no interest in maths and wanted to concentrate on his favourite subjects.  The 'maths studies' course seems to involve a lot of statistics.  So the GCSE course gives sufficient algebra, then the 'maths studies' gives the 'in addition' statistics.

 

L

Share this post


Link to post
Share on other sites

So, my next question. For the non-university bound students, why algebra?  Why not statistics? Not statistics as it is currently taught in 12th grade, but rather a large overview of all the different methods.  So focusing the on the bigger picture with the intent of understanding current events, rather than on doing the tests by hand. This does not have to be an easy class, but it would be incredibly useful.  If a student is only going to get up to 10th grade math and never take math again, why not teach them material that is needed to be an informed voter.  A dancer, musician, writer, craftsman is not going to use algebra, but is still going to vote. And right now many of them are voting in ignorance, because the math that they have been taught in school does not help them to understand the issues and the probabilistic nature of knowledge.

 

"The cornerstone of democracy rests on the foundation of an educated electorate." - Thomas Jefferson

 

People cannot make decent voting decisions without understanding the issues, and the issues are statistical. 

 

So what exactly is society's end-goal for a math education for a student going into a non-mathematical career.  I just can't see how the end goal should be manipulating algebraic equations.

 

 

Why literature, if they're never going to read anything more than Danielle Steele or celebrity tweets? Why essays if they don't need writing skills beyond filling out forms at the DMV? Why lab science if they will never see another test tube after high school? Or history when they need only live in the present? Having a high school education and having earned a diploma should mean something, and I think it's reasonable that a passing familiarity with these subjects of secondary education is part of what it should mean.

Share this post


Link to post
Share on other sites

How can a student understand statistics if they can't do algebra???

.....

I agree that statistics is very important, but it should not be statistics instead of algebra, but rather in addition to it.

A student who is weak in algebra can still pass a statistics exam at high school level by using a calculator or the statistical tables. I'm not taking about a good grade but a decent pass grade of C or D.

In the GCSE system, algebra 1 topics are long covered in the years leading to 8th grade. It is hard to directly compare intergrated math to the US system. However 9th grade intergrated math would be like algebra 2 + starting at trigonometry part of geometry + statistics and/or mechanics option. By the time chi-squared, anova, central limit theorem and the rest comes around, students have enough algebra to understand the statistics.

Share this post


Link to post
Share on other sites

A student who is weak in algebra can still pass a statistics exam at high school level by using a calculator or the statistical tables. I'm not taking about a good grade but a decent pass grade of C or D.

 

But does the student understand what he is doing? If all he can do is plug stuff into a calculator, how does passing the course translate into being able to evaluate statistics in real life in order to be an informed citizen, which, I believe, was the intended goal?

Share this post


Link to post
Share on other sites

Let me talk about the population of Singapore because this has come up on other boards I have been on. I think even the Singapore math board. They have a more diverse population than most US locations with three different main languages and many smaller languages. They also have large multiple religious groups. If you think about their geographic location and their occupation by various European groups, this will make sense to you.

 

BUT they do have an extremely controlling government. In one of the elementary science text, in a section on water rationing there was this quote: 

 

 

You may have done this during a water rationing exercise. During the exercise, water supply to homes is cut off temporarily. The Singapore government conducts these exercises to let you experience the inconvenience of a water shortage. It hopes by doing so, you will be reminded to use water more carefully.

 

 

I'm pretty sure I read somewhere in the last ten years about someone being caned for littering. I think they also have homogeneous thrust to getting ahead with a huge cultural work ethic. 

 

So they are diverse in some ways but they have an overall cultural commitment to their society that is not American in feel. 

Share this post


Link to post
Share on other sites

But does the student understand what he is doing? If all he can do is plug stuff into a calculator, how does passing the course translate into being able to evaluate statistics in real life in order to be an informed citizen, which, I believe, was the intended goal?

 

If we are talking about work post-GCSE: one of the two GCSE papers is non-calculator, so the understanding has to be there at that level.

 

L

Share this post


Link to post
Share on other sites
If you are a non-college bound student in Austria after you have finished 8th or 9th grade you still continue on with an education. In most cases you will get into some specialized field. What is now taught is directly related to what you are studying and this may include statistics or other types of business maths. I take myself as an example. I ended up going back onto a highschool track, however, the school I went to was now a specialized school. Instead of Calculus I studied Accounting and Economics. At the end of 4 years at this school I could either find work right away or go to College.

Many kids actually do apply for apprenticeships. As Regentrude pointed out you need some type of specific training for almost all kinds of work. During the apprenticeship years which last somewhere between 3 and 5 years depending on the field the student actually works for 4 days and goes to school for one day. At the end of this period they take an exam and become a Master of a trade. Most kids finish this by the time they are 19.

 

I love this!!  Sounds like it would fix the problems that I have seen with students without either the aptitude or interest in math.  Many kids (but not all as I explained above) can rise to meet a challenging math course if they see some purpose in it.

Share this post


Link to post
Share on other sites

 

 

Why literature, if they're never going to read anything more than Danielle Steele or celebrity tweets? Why essays if they don't need writing skills beyond filling out forms at the DMV? Why lab science if they will never see another test tube after high school? Or history when they need only live in the present? Having a high school education and having earned a diploma should mean something, and I think it's reasonable that a passing familiarity with these subjects of secondary education is part of what it should mean.

 

I am *not* saying that kids should not have math proficiency in order to graduate.  I *am* saying that math is a very large field and that I believe that the current focus is wrong and in some ways archaic.

Share this post


Link to post
Share on other sites

I am *not* saying that kids should not have math proficiency in order to graduate.  I *am* saying that math is a very large field and that I believe that the current focus is wrong and in some ways archaic.

 

:iagree:  

Share this post


Link to post
Share on other sites

How can a student understand statistics if they can't do algebra???

 

This is what I posted up thread. It would be for the slower students or those truly not interested in math:

 

I have thought long and hard about this 'statistics in the news' class that I would like to teach. I would spend 4 months teaching the kids to thinking statistically and probabilistically by teaching them non-parametric statistics. These stats can be done with no algebra and done on either a spreadsheet or by hand. You are just ranking and counting. Once my student understood the purpose of statistics and how to actually do nonparametric statistics, I would spend 5 months going over the basics of distributions, generally what you are testing for with means vs distributions, and how to *understand* and *interpret* (not do) parametric statistics. They need to have a large overview of the possibilities of statistics in order to understand a newspaper or the statistician at their job. A black box for the details is ok. Any algebraic manipulation that needed to be taught would be taught in context of a larger goal. This is key.

 

I agree that statistics is very important, but it should not be statistics instead of algebra, but rather in addition to it.

At some point you have to choose. There is just not time for everything. I think that the current focus of math education for the non-mathematical is wrong. Really wrong. Basically, they are put on track to calculus (from linear algebra, quadratics, geometry, trig, pre calc, to calculus), but then they just stop this math sequence at some point when it gets too hard.  Math is a very large field, there are other things that can be taught.  I do not see a good argument to have low end students or kids completely not interested in math (like someone going to be a professional dancer or musician) focus in this type of math.  It is NOT relevant to their lives and never will be.  They need to understand math that they will see and use. 

 

To be clear.  I have NEVER needed to know how to factor a quadratic equation in real life, nor have I ever done a geometric proof, in fact I have never solved an inequality, or used vectors, or done transformations.  I have used my trig knowledge *once* when placing an angled light on a wall; and I have:

 

A PhD in statistical modelling of biological systems

Published in the top journals in my field

Worked at Statistics NZ and Ministry of Health as a multivariate statistician

Acted as statistician for 4 PhDs and 2 masters projects

 

So even with my very mathematical career, I have not used much of the math that is required to graduate from high school.  And we, as a society, are not just encouraging but *requiring* non-mathematical people to take this math?

 

Having been a public high school math teacher, I am telling you that many kids need math to be RELEVANT.

 

Share this post


Link to post
Share on other sites

I am *not* saying that kids should not have math proficiency in order to graduate.  I *am* saying that math is a very large field and that I believe that the current focus is wrong and in some ways archaic.

 

Per Keith Devlin, the instructor of the Coursera Mathematical Thinking course I'm taking: "Most of the mathematics used in present-day science and engineering is no more than three- or four hundred years old, much of it less than a century old. Yet the typical high school curriculum comprises mathematics at least three-hundred years old -- some of it over two-thousand years old!"

 

So I'm with you on the arithmetic-algebra-geometry-calculus arc being archaic. But if we use "algebra" as a shorthand for "mathematics above an elementary school level of conceptual understanding", I think something like it needs to be in the curriculum, and if students aren't adequately being prepared, then something needs to change in the foundational work that goes on in the early grades, and/or we need to revise our expectation of how the needs of college-bound and non-college-bound students differ. What we don't need to do is to somehow make this vast, mystical field of mathematics "relevant or die".

Share this post


Link to post
Share on other sites

So, for non-mathematical students, here is what should be taken out IMHO to make more room for a conceptual statistics course as described in my previous post:

 

1) The entire American geometry course  - I mean really, give me an argument for why this is more important for ALL students to learn than statistics, which is critical to making informed voting decisions.

 

2) The second half of algebra 1 - simultaneous equations, polynomials, factoring, fractional equations, quadratic equations, inequalities, number sequences  (if any of this is required for a financial course or statistics course it can be taught at the time.)

 

This gives me 1.5 years of time for statistics.  And as far as I am concerned ALL students should have to take a conceptual statistics/probability course in order to graduate.  If they are more interested or more skilled, a mathematical statistics course could be substituted.

 

Share this post


Link to post
Share on other sites

But does the student understand what he is doing? If all he can do is plug stuff into a calculator, how does passing the course translate into being able to evaluate statistics in real life in order to be an informed citizen, which, I believe, was the intended goal?

They don't generally but those who go for the statistics option and are hoping for a C or D needs that pass grade to get their high school diploma. So the intended goal is honestly different. The "joke" was that the statistics option was created for people who won't be able to pass the mechanics option for GCSE level math.

There are people who are great in languages but suffering in math. These people need the pass grade to move on to higher education. I have a few lawyer friends who barely pass high school math.

 

ETA:

Singapore does have a non academic track after 10th grade. My youngest nephew is now doing his AA in engineering. One of my niece did an AA in nursing after 10th grade than went on to do a Degree in nursing. Some cousins did their AA in accountancy before doing their degree in accountancy.

Share this post


Link to post
Share on other sites

My 11 year olds constant question these days is, "Why do I need this?"  So the other day I said well ok I'll build a small shack in the back yard.  I'll pass you a bowl of gruel and water a couple of times a day.  You'll have all you NEED....shelter...food...water.  See how it will go with your "needs" met.

 

 

Share this post


Link to post
Share on other sites

Yes, these are some of the kids that I have worked with.  It is just so sad. 

 

Definitely sounds like Canada, the USA, and NZ mainstream students that are sent to special schools in Germany.

 

In my area, there are many "special schools."  They are called "alternative" schools for the most part, or sometimes magnets or other names.  Also within the regular public schools, there are many different classrooms.  My oldest ds needed a geometry class and the only one our local public school could offer him was a geometry class where the kids were coloring shapes and just needed a math credit to graduate.  (Their other math was "integrated"; therefore he just didn't take geometry and moved on.)

 

 

Excellent points, Regentrude.  It's true - my method may results in less-than-permanent knowledge.  But I guess I'm looking at it as the lesser of two evils.  Ideally, I'd love it if we could teach all students to understand math conceptually and never have to worry about providing algorithms for kids to "plug and chug".  Realistically, though, what usually results from an all-conceptual approach to math is that the kids who can't grasp that approach (which, with the way math is taught in elementary schools, is most of them) end up never learning ANY math.  When only conceptual math is taught, then many, many students end up learning virtually no math because they simply can't grasp it. 

 

 

I think this is more complex than just conceptual/mathy and algorithm/don't get it.

 

My youngest is very mathy and he is not conceptual.  He totally spaces out with long "real life examples" and seems to think "I get it already" or something. It annoys him greatly when folks use a lot of words in math, as well as when they don't use a lot of words and don't say what they want clearly and efficiently.  I guess that doesn't make sense, but I'm just feeling this two-rut description of "conceptual/mathy" vs. "algorithm/don't get math"  isn't reflecting what I've seen in my kids as they failed or succeeded with various math programs.

 

I've seen those kids who were behind in math.  I basically think they didn't get a foundation.  The schools for whatever reason didn't have the time or the energy or the authority to get these kids immersed in the basics.  It doesn't mean the kids aren't able or shouldn't learn higher maths.  It might mean it's impossible, but that's a totally other subject, to me.

 

Julie

Share this post


Link to post
Share on other sites

 At some point you have to choose. There is just not time for everything. I think that the current focus of math education for the non-mathematical is wrong. Really wrong. Basically, they are put on track to calculus (from linear algebra, quadratics, geometry, trig, pre calc, to calculus), but then they just stop this math sequence at some point when it gets too hard.  Math is a very large field, there are other things that can be taught.  I do not see a good argument to have low end students or kids completely not interested in math (like someone going to be a professional dancer or musician) focus in this type of math.  It is NOT relevant to their lives and never will be.  They need to understand math that they will see and use. 

 

I understand your argument and agree to a certain degree. I am, however, concerned at what point one would decide to label a child as "not mathematical". If math instruction were better in the US, students would start working on algebra in 6th or 7th grade. Do we really want to label 12 year olds as "not mathematical" and channel them into an "alternate" math track that makes it practically impossible to become not just scientists and engineers, but also doctors or veterinarians (who have to take rigorous chemistry and hence need math) if they should so choose later? Do we want to close doors at such a young age and label them as incapable of understanding math?

 

To me, an important goal of all education is to keep doors open and not to eliminate choices too soon (the tracking in German schools does not eliminate subjects, it just slows down the pace of instruction and stops earlier). I do not believe that most children can be accurately labeled "future dancer" or "future engineer" at age 12. I also do not think a kid who is channeled into "math for non-mathematical students" will be easily able to catch up on the five or six years of math he missed, should he, at some later point, decide that he has an interest in a field that requires more math.

 

Share this post


Link to post
Share on other sites

So, for non-mathematical students, here is what should be taken out IMHO to make more room for a conceptual statistics course as described in my previous post:

 

1) The entire American geometry course

 

I am very grateful that my floor installer who did a large, expensive job in my house, was good at geometry, could calculate the amount of tile and wood he needed for the areas, could compose my kitchen floor of various triangular and square portions.

I sure hope the people who will install our new HVAC system have some basic geometrical understanding.

Architects and construction workers and surveyors use geometry and trig.

If any math is needed in practical life for people who you'd call "non mathematical", it would be geometry.

 

While they may not need proofs, I do not think any other branch of mathematics is at the same time so easily accessible and trains logical thinking as well as that. And I think this country is suffering a critical shortage of logical thinking.

 

ETA: There would be plenty of room in the curriculum for two years of statistics if schools did not waste time to teach fractions over and over and over again for three entire years of middle school. Finish arithmetic with positive integers by the end of 4th grade, spend the entire 5th grade year on arithmetic with fractions and decimals and negative numbers, and be ready for algebra and geometry in 6th grade. Two years easily saved.

Share this post


Link to post
Share on other sites

 To be clear.  I have NEVER needed to know how to factor a quadratic equation in real life, nor have I ever done a geometric proof, in fact I have never solved an inequality, or used vectors, or done transformations.  I have used my trig knowledge *once* when placing an angled light on a wall; and I have:

 

A PhD in statistical modelling of biological systems

Published in the top journals in my field

Worked at Statistics NZ and Ministry of Health as a multivariate statistician

Acted as statistician for 4 PhDs and 2 masters projects

 

So even with my very mathematical career, I have not used much of the math that is required to graduate from high school.

 

OK, and I have, in my short career as a physicist (a few years before I became a mom and then went into just teaching), used vectors, matrices, functions, calculus, probability distributions, trigonometry, proved things... I use all of those when I teach my students. (And I did not even do any advanced theoretical research; those guys use math that is not even taught in college.)

 

Not that this is in any way relevant for the discussion about math eduction in school. But there are people who use this in their "real life".

 

Share this post


Link to post
Share on other sites

 A dancer, musician, writer, craftsman is not going to use algebra, but is still going to vote. And right now many of them are voting in ignorance, because the math that they have been taught in school does not help them to understand the issues and the probabilistic nature of knowledge.

 

"The cornerstone of democracy rests on the foundation of an educated electorate." - Thomas Jefferson

 

People cannot make decent voting decisions without understanding the issues, and the issues are statistical. 

 

I completely agree!

But not all issues are statistical - some of the issues are scientific, so scientific literacy is absolutely essential. (there is a limit as to how much science can be taught without math) Some issues require an understanding of history, not just of the US but of the world, and of current events and things happening elsewhere.

 

Basically, first and foremost voters should WANT to be informed. People could do a lot to be more informed even with their limited knowledge of math... alas, it seems to me that being informed about issues is not what many people base their decisions on. As long as votes are decided by certain gut-feeling-trigger arguments that are very emotional, and not based on rational evaluation of available information, any attempt to educate the population remains futile. Yes, I am cynical.

Share this post


Link to post
Share on other sites

On the topic of math needed by "non-mathematical" people: I found an interesting summary of math skills needed by oil well drillers - a profession one would not automatically associate with math":

 

 

 

Oil and Gas Well Drilling Workers and Services Operators:
  • may prepare invoices, including pricing quantities of cement and additives and calculating rental fees for tools using a rate. (Money Math), (2)
  • may weigh out quantities of chemicals to add to the mud mixture. (Measurement and Calculation Math), (1)
  • may measure and tally the lengths of pipes being tripped down the well to tell the driller the depth they have reached. (Measurement and Calculation Math), (1)
  • may determine the fluid volume in a tank by measuring the depth of fluid in the tank and multiplying by a coefficient; also calculate the amount of fluid increase and decrease. (Measurement and Calculation Math), (2)
  • may calculate the volume in a 5-inch casing at a depth of 2150 metres using a chart that gives the volume in cubic meters per 1000 metres of depth. (Measurement and Calculation Math), (2)
  • may calculate the number of minutes it will take to pump a volume of fluid into the well, based on the volume per pump stroke and the number of strokes per minute. (Measurement and Calculation Math), (3)
  • may read oil and water pressure gauges to compare variations in readings at different points in order to know when to adjust valves. (Data Analysis Math), (1)
  • may determine rates of flow of fluids into and out of the well to report unexpected changes. (Data Analysis Math), (2)
  • may monitor gauges and analyze the relationships between pressures, weights and rates of flow to make sure levels are according to the prescribed program and to watch for unexpected changes which may mean shutting a system down. (Data Analysis Math), (3)
  • may estimate the amount of mud being lost in the drilling hole. (Numerical Estimation), (1)
  • may estimate the percentage of oil, water and sand in a sample taken from the drill to record in the swab report. (Numerical Estimation), (1)
  • may estimate the volume of fluid at a prescribed depth to determine pumping speed and to control whether the pump will end with a downstroke or an upstroke. (Numerical Estimation), (2)

 

Share this post


Link to post
Share on other sites

So, for non-mathematical students, here is what should be taken out IMHO to make more room for a conceptual statistics course as described in my previous post:

 

1) The entire American geometry course  - I mean really, give me an argument for why this is more important for ALL students to learn than statistics, which is critical to making informed voting decisions.

 

2) The second half of algebra 1 - simultaneous equations, polynomials, factoring, fractional equations, quadratic equations, inequalities, number sequences  (if any of this is required for a financial course or statistics course it can be taught at the time.)

 

This gives me 1.5 years of time for statistics.  And as far as I am concerned ALL students should have to take a conceptual statistics/probability course in order to graduate.  If they are more interested or more skilled, a mathematical statistics course could be substituted.

 

 

While I would agree about the algebra, I do not think I'd cut the entire geometry course. I would go ahead and cut the proofs and integrate the rest of geometry into the algebra course -- you'd then have a year to do half of algebra 1 and half of geometry, together. For the second year, you could do statistics and any algebra that was also needed. I think I would also include logical reasoning -- honestly, I don't think 1.5 years is necessary for statistics. The 'mathy' students could concurrently take the second half of algebra 1 and the proofs section of geometry.

 

FTR, the 'math for liberal arts' course I taught as a graduate student essentially covered a third of a semester each of statistics, logic, and consumer math. Very little algebra was necessary -- most of it focused on being able to do elementary probability calculations and detect statistical jibberish in the media.

 

Here is an example of statistical jibberish I have seen recently that I would expect an educated person to be able to easily debunk. "About 3/4 of the people who got measles had been vaccinated, so the vaccine has a failure rate of one in three. That doesn't seem very effective!"

 

Training people to be able to see through this reasoning doesn't take a lot of algebra or calculus, but some basic knowledge of how probability works.

Share this post


Link to post
Share on other sites

Per Keith Devlin, the instructor of the Coursera Mathematical Thinking course I'm taking: "Most of the mathematics used in present-day science and engineering is no more than three- or four hundred years old, much of it less than a century old. Yet the typical high school curriculum comprises mathematics at least three-hundred years old -- some of it over two-thousand years old!"

 

So I'm with you on the arithmetic-algebra-geometry-calculus arc being archaic. But if we use "algebra" as a shorthand for "mathematics above an elementary school level of conceptual understanding", I think something like it needs to be in the curriculum, and if students aren't adequately being prepared, then something needs to change in the foundational work that goes on in the early grades, and/or we need to revise our expectation of how the needs of college-bound and non-college-bound students differ. What we don't need to do is to somehow make this vast, mystical field of mathematics "relevant or die".

 

I think the bolded above is a large part of the problem. (Not you, sunnyday!  I just wanted to use your statement to further the discussion. :))  What exactly does each one of us mean when we say "Algebra 1" or "Algebra 2"?  In Canada, we don't have those designations - all of our high school math (except calc) is integrated.  Here's a link to documents that outline the Ontario high school curriculum for math:

http://www.edu.gov.on.ca/eng/curriculum/secondary/math.html

To graduate high school, a student must have 3 math credits - Grade 9, Grade 10, and one math from Grade 11.  University bound students will take the "Academic" math stream in Grades 9 and 10 and the "University" stream math in Grade 11 (and 12).  College or workforce bound students will take the "Applied" math stream in Grades 9 and 10 and the "College" stream in Grade 11 (and 12).  (College here in Canada refers to trade school - sort of.  Our colleges can only award 1 or 2 year diplomas - we don't have anything equivalent to an "Associates Degree" here.)  There are Grade 11 and 12 math courses labeled "Mathematics for Work and Everyday Life" but, in reality, those courses are reserved for spec ed students.  Students who are not university bound but who are not spec ed take the "College" stream - whether they are intending on going to college or straight into the workforce.

 

I'm not saying the Ontario system is wonderful - it isn't and it needs to be overhauled just as badly as anywhere else.  I just thought someone might want more detail of what an integrated math curriculum looks like. :)

 

 

I think this is more complex than just conceptual/mathy and algorithm/don't get it.

 

 

It definitely is, Julie - sorry if I gave that impression!  I think of it more as a continuum.  And I definitely don't always equate kids with conceptual understanding with "mathy" or that algorithms are just for kids who don't get it.  I use algorithms all the time.  I may know how to derive the Henderson-Hasselbach equation used for buffer calculations but I don't do the derivation every time I need it - I have it memorized.  Having it memorized is efficient and a better use of my time than constantly deriving it.

 

I think students fall all along the continuum - no understanding, partial understanding, full understanding, no use of algorithms, partial use of algorithms, constant use of algorithms - with all possible combinations.  Students will also move along the continuum - just because a student doesn't "get" something this week doesn't mean he/she won't "get" it next week.  As I stated up-thread, continued use of an algorithm can sometimes jump-start conceptual understanding.

 

I completely agree!

But not all issues are statistical - some of the issues are scientific, so scientific literacy is absolutely essential. (there is a limit as to how much science can be taught without math) Some issues require an understanding of history, not just of the US but of the world, and of current events and things happening elsewhere.

 

Basically, first and foremost voters should WANT to be informed. People could do a lot to be more informed even with their limited knowledge of math... alas, it seems to me that being informed about issues is not what many people base their decisions on. As long as votes are decided by certain gut-feeling-trigger arguments that are very emotional, and not based on rational evaluation of available information, any attempt to educate the population remains futile. Yes, I am cynical.

 

I'll join you in the cynical camp if you'll have me, Regentrude. :)  I was having a conversation the other day with my dh about who we thought would win the next federal election in Canada.  I feel Justin Trudeau will be our next Prime Minister - not because he's the best candidate but because he's Canada's version of political royalty.  Elections seem to have become no better than junior-high popularity contests.  You're absolutely right - voters should want to be informed.  But I doubt that they do unless the information can somehow pass painlessly and effortlessly into their brains through osmosis while they sit and watch prime-time TV. ;)

Share this post


Link to post
Share on other sites

"The cornerstone of democracy rests on the foundation of an educated electorate." - Thomas Jefferson

 

People cannot make decent voting decisions without understanding the issues, and the issues are statistical.

What I see is that you need an electorate that is interested in educating themselves about the issues. Statistics can always be misused by politicians and newsmakers. I think that middle schoolers need a good grounding in understanding logical fallacies. The statistics covered all the way to prealgebra is adequate to understand newspaper reports.

Mortgage and Certificate of Deposits (Fixed Deposits) was taught in my boy's 6th grade public school math textbook. He did more than 30 questions on those for class work. Still adults can miscalculate their own mortgage calculations. We need people to be able to use their learned skills to help themselves whether it is budgeting, 401k, to voting wisely.

Share this post


Link to post
Share on other sites

I understand your argument and agree to a certain degree. I am, however, concerned at what point one would decide to label a child as "not mathematical". If math instruction were better in the US, students would start working on algebra in 6th or 7th grade. Do we really want to label 12 year olds as "not mathematical" and channel them into an "alternate" math track that makes it practically impossible to become not just scientists and engineers, but also doctors or veterinarians (who have to take rigorous chemistry and hence need math) if they should so choose later? Do we want to close doors at such a young age and label them as incapable of understanding math?

 

To me, an important goal of all education is to keep doors open and not to eliminate choices too soon (the tracking in German schools does not eliminate subjects, it just slows down the pace of instruction and stops earlier). I do not believe that most children can be accurately labeled "future dancer" or "future engineer" at age 12. I also do not think a kid who is channeled into "math for non-mathematical students" will be easily able to catch up on the five or six years of math he missed, should he, at some later point, decide that he has an interest in a field that requires more math.

I am so glad to see this said.  Our system sends a dubious dual message.  On the one hand we claim not to track students, on the other math tends to define their placement and cohort groups as early as late elementary.  Many don't consider a math related field rather early on because they do not consider themselves mathematical.  It becomes a self fufilling prophecy.  Some may be able, but need more time to gain mastery of  the skills and understanding.  This is one reason I don't look down on those who take what we often call remedial math in college.  The logical thinking and executive neurological functions that often allow some to excel earlier in math, may develop slower in others who see excellerated development in other cognitve areas in the tweens/early teens.  Too often, they turn away from a math orientation before they should because the path is tainted by the assumption they are not "mathy".

 

I may be scoffed, but I am far less concerned about a 17/18 year old who enters college needing to begin with what we call College Algebra being able to progress to a STEM career, than I am about the student requiring remediation in composition or reading.

Share this post


Link to post
Share on other sites

I may be scoffed, but I am far less concerned about a 17/18 year old who enters college needing to begin with what we call College Algebra being able to progress to a STEM career, than I am about the student requiring remediation in composition or reading.

 

This is why I have said frequently, on this board, that I would rather see someone enter with a SOLID understanding of algebra 1, than a weak understanding of precalculus with some splinter skills in calculus.

 

A student who passes and genuinely understands pre-algebra, algebra and geometry can enter college and place into either intermediate or college algebra (depending on the strength of their algebra 1 course -- most would place into intermediate). They can take intermediate in the fall, college in the spring, precalculus in the summer, and be on track with a one-year delay. But they have to UNDERSTAND the previous levels in order to be able to catch up. It does absolutely no good to rush through algebra in order to get to precalculus and still not understand it.

Share this post


Link to post
Share on other sites

While I would agree about the algebra, I do not think I'd cut the entire geometry course. I would go ahead and cut the proofs and integrate the rest of geometry into the algebra course -- you'd then have a year to do half of algebra 1 and half of geometry, together. For the second year, you could do statistics and any algebra that was also needed. I think I would also include logical reasoning -- honestly, I don't think 1.5 years is necessary for statistics. The 'mathy' students could concurrently take the second half of algebra 1 and the proofs section of geometry.

 

Wowzers, I'd disagree completely. Geometric proof is the first example of *actual* mathematical thinking most kids are going to get. It tends to be bungled -- I detested my geometry class and never did "get" proofs despite taking upper-level math from Linear Algebra to Complex Analysis -- but it *is* *math*, and it's important. You can't tell kids they're no good at math before they've ever really seen any of it. Arithmetic and algebraic manipulation isn't *really* math.

 

My 6yo, in the car on the way to a friend's birthday yesterday: "I can cut a rectangle into two triangles. Each triangle will have some skinny angles and also a right angle." Me: "Can you change how skinny the skinny angles are by changing the shape of the rectangle? Can you change the fact that there is a right angle in each?" Him: "Yes, I can! Hm...did you know, that a triangle with a really really wide angle on the top will be very short?" Me: "That is a great observation. There is definitely a relationship between the angles of triangles and the lengths of their sides. The Ancient Greeks noticed this, and cared a great deal about puzzling out all the details of how that triangle shape really goes together."

 

And you're going to cut out the bit of Geometry that confirms my little son's observation, letting him wallow in the "I wonder if it's true" stage forever?? ;) I know, I know, if he's mathy under this scheme he will take Proper Math and Preparation for College Math and what have you. But I still think that if my kid can be motivated to ponder on this stuff, an 18 year old (even one who's destined to be a cook or a mechanic like the other graduates of my local high school that I know) who's received a *complete* secondary education can either be motivated and taught to do it properly or convinced to go through the motions.

Share this post


Link to post
Share on other sites

Geometric proof is the first example of *actual* mathematical thinking most kids are going to get. It tends to be bungled -- I detested my geometry class and never did "get" proofs despite taking upper-level math from Linear Algebra to Complex Analysis -- but it *is* *math*, and it's important. You can't tell kids they're no good at math before they've ever really seen any of it. Arithmetic and algebraic manipulation isn't *really* math.

 

:iagree:  There are kids with sequential weaknesses and spatial strengths who may not realize how "mathy" they really are until they get to geometry, if it isn't too late by then.

 

More importantly, geometry is the best logic course most kids will ever see, albeit in disguise.  Proofs are essential for that.  Understanding the details (statistics, for example) may not mean much without a logical framework, a big picture, in which to place such details.

 

I had a truly pathetic geometry teacher and my friend and I taught ourselves the course the weekend before the Regents exam; while I did well on the test, my geometry education wasn't the greatest and I hated proofs.  Yet years later, the day I realized that a written argument was really a proof, everything changed - what a lightbulb moment.

Share this post


Link to post
Share on other sites

But I still think that if my kid can be motivated to ponder on this stuff, an 18 year old (even one who's destined to be a cook or a mechanic like the other graduates of my local high school that I know) who's received a *complete* secondary education can either be motivated and taught to do it properly or convinced to go through the motions.

 

Grrrrr. My whole reply just got deleted and I have to remember what I was saying.

 

The whole thing I'm trying to get away from is "going through the motions" being necessary for a minimum diploma. I absolutely would agree that this should be necessary for any sort of college-prep diploma and recommended for all. I just don't think that students who are still struggling to understand why x + x is 2x and not x^2 are going to be able to do a serious geometry course in any fashion other than rote memorization, and I see little purpose in rote memorization of proofs (which, FWIW, is how a lot of courses were taught, even in the 50s and 60s).

 

I realize that I erred earlier when I said "mathy". What I meant was that college-prep students would take the second course which would do geometry and algebra, both with proofs. Very mathy students who are aiming at taking college math early could take that and statistics concurrently -- other students would take the two courses consecutively.

 

I think there should be a qualification between "received a complete secondary education", which is what I would consider a college-prep diploma to be, and nothing at all. And I just don't see proofs as necessary for this qualification. Removing them from "everyone must take this course to graduate" will allow increased rigor for the college-bound students -- we could actually do some algebra proofs as well -- proofs like the fact that sqrt 2 is irrational -- which are easily comprehensible to many students but not currently part of most curricula. Furthermore, integrating them will allow us to discuss geometric construction of sqrt 2 and the algebraic proof of the irrationality thereof at the same time.

 

If someone doesn't take the second math course because they're aiming at a general diploma, then they'll have to go to the community college for it, just as they do now when they didn't understand algebra.

Share this post


Link to post
Share on other sites

Yeah, so I do think I'd support separate tracks in some way. My heart breaks that my neighbor's son pretty much coasted from the time he dropped out of high school until he got his act together, finished his GED, and got into a vocational school to become a professional mechanic. I think he's 23 or 24 now, and his voice was SO full of pride last summer when he was explaining about how prestigious the program is that he was about to enter. You're right that going through the motions didn't help this kid at all.
 
However, what I'm saying is that before we just scrap algebra and geometry and replace them with "consumer math" or the like being the standard of achievement for a high school diploma, as long as there *is* just one high school diploma, we should re-evaluate what we're scrapping and why. Because by the same argument we could scrap all of high school foreign language and world history and English language arts. None of it is *directly* applicable to the career of the entry-level job-seeker.
 
IMO, geometry with proofs, as mentioned, is a way to dabble with applied logic. It's also putting us in touch with the questions that have been asked and answered for thousands of years. And it's exercise for the mind. Same with algebra, functions, equations, graphing. It's about patterns, and abstraction. Many many adults are concrete thinkers, but is that inherent? Or is it from lack of exposure and lack of practice in flexing those mental muscles? If someone can't get x + x = 2x, is that a deficiency in some inherent algebra ability and therefore math ability? Or is it just inadequate preparation from educators who themselves were terrified of symbolic notation or proud of their ability to conquer symbolic notation (all the while thinking that the *notation is the math*?)

 

My standard for secondary education is influenced by Elnora Comstock in The Girl of the Limberlost, who struggled so hard to get to attend high school, and when she got there on her first day in Freshman Algebra she had to write up a proposition on the blackboard and then defend her process with a recitation. THAT is math, and THAT is high school education. Complex thinking that goes above and beyond eighth grade level. Something to strive for, not to passively accept.

Share this post


Link to post
Share on other sites

This is a great discussion and I love hearing all the thoughtful and educated answers.  My thoughtful but not overly educated answer that I give to my DD is this:

 

Whether you will use the things you are learning in your everyday life is not the issue.  You are learning HOW to learn.  Your brain is developing and forming connections as it will at no other time in your life.  The more types of information you feed into it, the more connections you will form.  You will know as an adult that you are capable of learning whatever you need to do whatever you want.  Not that you will already know it, but that you will be capable of learning it and have confidence in that ability.

 

In addition, to be a well-rounded, responsible and capable adult, you need to be able to have at least a passing familiarity with the concepts of the world around you.

 

So far, she has accepted that explanation both for Algebra II and Shakespeare... :)

 

 

That said, I do agree with not requiring advanced math for all students.  By advanced, I would consider anything past Alg I and basic Geometry.

Share this post


Link to post
Share on other sites

I am so glad to see this said.  Our system sends a dubious dual message.  On the one hand we claim not to track students, on the other math tends to define their placement and cohort groups as early as late elementary.  Many don't consider a math related field rather early on because they do not consider themselves mathematical.  It becomes a self fufilling prophecy.  Some may be able, but need more time to gain mastery of  the skills and understanding.  This is one reason I don't look down on those who take what we often call remedial math in college.  The logical thinking and executive neurological functions that often allow some to excel earlier in math, may develop slower in others who see excellerated development in other cognitve areas in the tweens/early teens.  Too often, they turn away from a math orientation before they should because the path is tainted by the assumption they are not "mathy".

 

I may be scoffed, but I am far less concerned about a 17/18 year old who enters college needing to begin with what we call College Algebra being able to progress to a STEM career, than I am about the student requiring remediation in composition or reading.

I definitely needed more time to learn math.  I had a pretty good foundation and made As and Bs in math through elementary but it wasn't a subject I was comfortable with and I had to work a lot harder at it than other kids.  I hit pre-algebra in 8th grade and almost didn't even pass.  It was awful.  I was considered a good student and everyone treated my bad grade as this horror.  I was so embarrassed.  

 

In High School Freshman year they put me in remedial math which was basically Algebra 1 Pt.1 and Sophomore year I took Algebra 1 part 2.  My teacher had plenty of time to really review concepts in class, provide lots of scaffolding and give us the extra explanation we needed.  It was great.  I got A's.  I enjoyed the class. I started to understand at a deeper level.  Only now, to graduate, I had to take Geometry at normal speed, not at slow speed.  It was a struggle.  I enjoyed geometry but trying to keep up my grades was really hard.  I managed Bs and Cs and passed.  Only at this point I am a senior and I have to have Algebra II and Trigonometry (taught as separate years) so they put me in the accelerated math class for advanced 10th graders wanted to cover more advanced math before graduation.  I was told I had no choice if I wanted to graduate.  I knew there wasn't a prayer of my passing that class and felt miserable.

 

The only reason I passed that class was because my very sympathetic teacher realized right away that I was over my head.  He paired me with an awesome 10th grade student who was brilliant at math and sat next to me in every class quietly whispering additional explanations, who worked with me every lunch period on her own time and frequently, on her own time, stayed with me after school or talked to me over the phone to help me through.  Did I understand what I was doing?  Not at all.  But I passed, which I had to do to graduate because our school required it.  And I came away having lost my budding sense of enjoyment at doing math, and every ounce of confidence I had once had was completely gone.

 

I hit college, looked at the various areas I had interest (sciences, etc.) and chucked half of them because of the math requirements.   I ended up in a career I enjoyed, thankfully, but I went years not feeling I could handle anything but basic math.  I feared even doing any investing with my personal savings.  My employer was offering to match funds in their voluntary retirement plan and I was so fearful of math I chose not to invest for years because I was afraid I wasn't understanding the numbers I was looking at on the forms.  I just tossed that money away.  However,as time passed,  I had to learn how an amortization table works, how to handle statistical information, how to understand investment finances, etc.   I started forcing myself to try to understand the math I needed every day.  I started to realize that maybe i wasn't so bad at math after all.  

 

I went back to college for the express purpose of taking Algebra 1 and 2 again.  I ended up tied for highest grade in the class.  I also ended up taking over the finances for my dad's company when his health deteriorated.  I realized I COULD understand higher math concepts.  I just needed to have the time to do it.  I wasn't ready in High School but pushing me into it nearly undermined my ability to function as an adult.

 

My husband did terrible in math in school and actually almost didn't graduate High School.  And yet, as an adult, he is a brilliant engineer and highly successful in his field.  He is also a terrific pilot and very good with computers (repairing and operations and programming).  All require math, and sometimes quite a bit.  If he had been tracked in school, who knows where he would have ended up?  The school DID have a broadcast television class, and that is what kept him in school and trying to get through his classes; practical application math for the engineering side of broadcast tv.  He loved the class, so he didn't give up.  He saw a REASON for the math he was trying to get through.  I wish my school had had something like that, something I could anchor to.  Math just didn't make much sense to me until I was experiencing it in the world at large.

 

I don't know exactly what the right answer is, but I do think there needs to be a change.

Share this post


Link to post
Share on other sites

×
×
  • Create New...