Students failing their first course of Algebra

[Astronuc]

That's not entirely fair. As you wrote:

There's plenty wrong with education in the US, don't get me wrong. From my perspective, there are too many competing interests.

I think my assessment is fair, although I agree that there are many competing interests. With regard to the second point and the statement "It's difficult to focus on academics when one's world is in chaos," I was reflecting on other reasons why students might perform poorly in school, but those reasons may apply to something like 10 to 20% of typical schools, but might represent higher percentages in schools in areas with greater poverty. My comments are based on personal observation going back 50 years.

Reflecting on 50 years, I don't see much improvement or advancement in academics, and if anything, there seems to be a decline.

I have always wondered why I did so well and others seem to struggle with varying degrees. I often assisted peers with assignments.

 

[symbolipoint]

So, I'm now wondering with respect to the OP, does the question about algebra pertain to all students, or those in STEM, or those who pursue course work in the humanities (arts and letters)? I ask this because, algebra is absolutely necessary for those in STEM. I use algebra almost every day since it is a basic part of science and engineering. This week I used it to review some work on the calculation of stresses in tubing, and previously, I've used it in describing various material properties (I also have to use statistics in data analysis), which is part of design and performance analysis of systems and components, as well as in design of experiments. The work I do also involves integral calculus, and solving ODEs, PDEs and systems of ODEs and PDEs, which can often be nonlinear. Various codes that I use or review involve finite difference or finite element methods.

The question is about Algebra for all students.

 

[Isaac0427]

When I did my 4-year academic plan in the latter part of 8th grade, I signed up for all the math and science (chemistry and physics) courses I could. One of the counselors thought I was overdoing it (and seemed to discourage me in this respect), and my peers thought I was just showing off. I simply wanted to take advantage of the opportunity to do as much as I could as early as possible, since I expected to go to university to study mathematics, physics and chemistry.

We just scheduled for high school a few days ago, and I have had pretty much the exact same experience (although I was able to convince my counselor I was doing the right things for me).

But I digress. While I am a little bias with being a physics and math supernerd, I feel like there is hardly anything more useful than algebra, not only because it is necessary for the survival of our society (see my original post), but it shows up everywhere. While I have many things against the low standards of common core (and I don't wish to open up a discussion about this), it does do something extremely well; it gives examples of when you will need the math. I am still surprised that there are professionals who don't understand the importance of algebra.

 

[Astronuc]

But I digress. While I am a little bias with being a physics and math supernerd, I feel like there is hardly anything more useful than algebra, not only because it is necessary for the survival of our society (see my original post), but it shows up everywhere. While I have many things against the low standards of common core (and I don't wish to open up a discussion about this), it does do something extremely well; it gives examples of when you will need the math. I am still surprised that there are professionals who don't understand the importance of algebra.

I too believe that algebra is a basic necessity in life. Perhaps it's a matter of perception. Arithmetic and numeracy are basic necessities, but one can only apply that so far. I think it is important to know some algebra. It can apply to consumer mathematics, e.g., financing in terms of budgets, household expenses, mortgages, financing an automobile purchase. It can also apply to practical skills, e.g., plumbing, carpentry or electrical work where one might have to purchase various supplies for a project as an example. One might need to purchase so many pieces of lumber (2x4's or 4x4's) and fasteners (nails and/or screws), or so many sections/lengths of pipes and fittings, or so many lengths of wire and connectors/switches, etc.

 

[Andy Resnick]

<snip>Reflecting on 50 years, I don't see much improvement or advancement in academics, and if anything, there seems to be a decline.

Aha! I've solved the mystery- the original source of this quote is Astronuc! :)

I have always wondered why I did so well and others seem to struggle with varying degrees. I often assisted peers with assignments.

Again, if you think that a high-school diploma should only be awarded through mastering Astronuc-approved content, you should at least offer a thought regarding those students who, for whatever reason, can't 'get it'. Should they be kept in school until they pass?

I'll also brag that my prediction (paragraph 2, post #4) came true on this thread.

 

[Astronuc]

Again, if you think that a high-school diploma should only be awarded through mastering Astronuc-approved content, you should at least offer a thought regarding those students who, for whatever reason, can't 'get it'. Should they be kept in school until they pass?

I don't believe I have indicated any idea that students should 'be kept in school until they pass', nor have I suggested a specific curriculum to be mastered.

I have reflected on why little or no advancement in broader public education. Clearly there are plenty of kids who get it - maybe 5% or so, and a few percent who do extraordinarily well, e.g., math/science fairs/Olympiads.

As for those who don't seem to get it, then the obvious question is, "How can we improve the teaching of the particular subject (and precursors/prerequisites) so more kids 'get it'?" Or how can we make dry or abstract subjects, less dry or less abstract, i.e., more relevant. I certainly recognize that it is difficult to address all of the issues, such as distractions involving one's peers or situations outside the classroom.

 

[mpresic]

I too came from a school where in general: 9th grade: Algebra, 10: Geometry 11: Algebra 2 and Trigonometry. 12: Solid Geometry, assorted topics, or for the best students 12: Calculus. With one exception. The best students in 6th grade were selected out for a high level integrated program and these students even had access to a time-shared computer terminal (and this was in the 1960's and early 1970's). I wasn't in the chosen group.

One reason for the compartmentalization is that when the students go to college, it is easier for the academic advisors to assess, the student has algebra 1 and geometry (for example), than to say, the student has 2/3 geometry, 2/3 algebra, 2/3 trigonometry or some such. Likewise in a science curriculum, it is a lot easier to say the student comes with biology and chemistry, no physics, than 2/3 biology, 2/3 chemistry and 2/3 physics.

Now the interesting thing about the group of kids chosen early for math aptitude is this. I had a friend in the special group and he helped me study for a MAA exam. He told me while studying, his program treated geometry rather lightly. He knew he was at a disadvantage in the exam, and sure enough, the top 5 scores were all obtained from the lower group. That should suggest the folly of selecting kids for special programs too early.

I found in teaching physics in college that geometry is important, and many students need to be better at it. I have read ideas from young students to take geometry in summer school, or advance without geometry to get to something advanced and I entreat you, "Don't do this". Better you learn the basics thoroughly.

 

[Astronuc]

One reason for the compartmentalization is that when the students go to college, it is easier for the academic advisors to assess, the student has algebra 1 and geometry (for example), than to say, the student has 2/3 geometry, 2/3 algebra, 2/3 trigonometry or some such. Likewise in a science curriculum, it is a lot easier to say the student comes with biology and chemistry, no physics, than 2/3 biology, 2/3 chemistry and 2/3 physics.

Please explain 2/3's in a subject.

 

[mpresic]

This is what I am saying. If I see a student transcript and I see (s)he took mathematics up to but not including the 11th grade. In the old way, I could reasonably believe the grades are in algebra 1, and geometry. These course listings also appear in his transcript. On the other hand suppose I see two courses; Integrated high school mathematics I and II (of a three year math sequence) in secondary school. Can I assume (s)he has been exposed to synthetic division, (an 11th grade algebra 2 topic, the old way) or not. Can (s)he multiply complex numbers (I believe this is an eleventh grade topic the old way) or not. I know his grades are good for his 2/3 of a three year sequence, but I do not know how much of the algebra I, how much of the geometry, or how much of the algebra 2 he or she knows.

 

[symbolipoint]

This is what I am saying. If I see a student transcript and I see (s)he took mathematics up to but not including the 11th grade. In the old way, I could reasonably believe the grades are in algebra 1, and geometry. These course listings also appear in his transcript. On the other hand suppose I see two courses; Integrated high school mathematics I and II (of a three year math sequence) in secondary school. Can I assume (s)he has been exposed to synthetic division, (an 11th grade algebra 2 topic, the old way) or not. Can (s)he multiply complex numbers (I believe this is an eleventh grade topic the old way) or not. I know his grades are good for his 2/3 of a three year sequence, but I do not know how much of the algebra I, how much of the geometry, or how much of the algebra 2 he or she knows.

Your basic concept of a fraction of a course is very understandable. Clear enough. Someone who passes a "pre-algebra" course may be understood as having maybe 1/2 of Algebra 1. Someone who takes one semester of Geometry in high school but not the second semester, but has a Mathematical Analysis Plus Trigonometry course might be understood as having something between 1/2 and 2/3 of a course of Geometry. He might not necessarily be seen as having a full 100% course on Trigonometry, but that depends on actual content passed and the way the institutions view the course taken.

 

[ProfuselyQuarky]

Yes, let's get rid of algebra 1. And then you can't have algebra 2 without algebra 1, right? There's precalc without algebra 2, and no calc without precalc. Let's just get rid of all the math that is viewed as hard by society and cause people's GPA to stumble. I mean, who needs logarithms, derivatives and integrals

Well said! I entirely agree with you. This is an interesting thread. The math structure in schools I’ve been to is the same as listed in earlier posts (although, @jtbell, we have to generally geometry before algebra 2). However, I’m in 10th grade and my high school is really lax and flexible with all of that. You can meet ninth graders taking calculus and seniors taking algebra 2 . . . you’ve got the smart and the motivated and the equivalent opposites of these students. I was able to finish two years of math in the same time it would take a normal-paced student half a year. My goal is to be done with all calculus before junior year ends and then take either high school statistics or a college level math course come 12th grade. Anyway, there is no reason why any form of algebra to abolished. It should remain standard. That’s like asking a child to learn multiplication without learning addition.

I do believe, however, that pre-algebra was not necessary, for myself at least, and that there were hardly any new concepts learned in geometry, aside from learning to complete proofs.

 

[ProfuselyQuarky]

To add to that, I am constantly disappointed that my school is increasingly adding courses reflecting the “arts” such as drama and film and music instead of paying more attention the sciences.

(but that’s a bit off topic)

 

[Isaac0427]

I entirely agree with you.

You agree with my sarcasm, right?

I do believe, however, that pre-algebra was not necessary, for myself at least, and that there were hardly any new concepts learned in geometry, aside from learning to complete proofs.

You sound exactly like me!

Math classes I skipped:
Kindergarten math
Pre-algebra (8th grade)
Geometry

Well, I would have skipped 7th instead of 8th but they didn't let me take the test. I wouldn't be allowed to skip both and if I skipped 7th and went right to 8th, I would have been just as bored in 8th as in 7th (for some reason, I felt like other than Pythagorean theorem and y=mx+b, which were covered in 8th grade, the two taught the same exact things, as their final exams closely resembled each other).

 

[ProfuselyQuarky]

You agree with my sarcasm, right?

Of course with your sarcasm. Is there anyone in a physics forums that would say that integrals are not important? Or that we should get rid of math?

Well, I would have skipped 7th instead of 8th but they didn't let me take the test. I wouldn't be allowed to skip both and if I skipped 7th and went right to 8th, I would have been just as bored in 8th as in 7th (for some reason, I felt like other than Pythagorean theorem and y=mx+b, which were covered in 8th grade, the two taught the same exact things).

I understand the "being bored" part. All the math I took seventh grade to ninth grade felt like forced review. This became a slight issue. For example, in 7th grade, the first lesson on trig ratios only taught sine, cosine, and tangent. Can you imagine how confused I was to not hear anyone say anything about secant, cosecant, and cotangent? Instead of letting students who aren't interested in math do whatever they want (i.e. skip algebra), schools should give that same freedom to students who actually want to learn.

Kindergarten math

What's that even supposed to be? My parents tell me that my pre-K teachers constantly commented that I didn't pay attention. Maybe that was because I had had enough of counting straws and adding apples

My conclusion: schools systems are at least partly corrupt which is why self-study is a necessity to anyone who cares.

 

[micromass]

I understand the "being bored" part. All the math I took seventh grade to ninth grade felt like forced review. This became a slight issue. For example, in 7th grade, the first lesson on trig ratios only taught sine, cosine, and tangent. Can you imagine how confused I was to not hear anyone say anything about secant, cosecant, and cotangent? Instead of letting students who aren't interested in math do whatever they want (i.e. skip algebra), schools should give that same freedom to students who actually want to learn.

To be fair, nobody uses secant and cosecant nowadays anymore. They also could have mentioned the versine

 

[ProfuselyQuarky]

To be fair, nobody uses secant and cosecant nowadays anymore. They also could have mentioned the versine

Yeah, sure, but teaching three without the others is still incomplete, nonetheless. I use those trig ratios, though. It still gives you the right answers and the word cosecant sounds lovely in my mouth: cooooosecant

 

[micromass]

Yeah, sure, but teaching three without the others is still complete, nonetheless. I use those trig ratios, though. It still gives you the right answers and the word cosecant sounds lovely in my mouth: cooooosecant

Did they at least mention what tangent and cotangent have to do with tangent lines?

 

[ProfuselyQuarky]

Did they at least mention what tangent and cotangent have to do with tangent lines?

Nope. The words "tangent lines" did not even appear in the textbook. I learned that on my own during the summer between middle school and high school. I learned it officially in geometry.

 

[micromass]

Nope. The words "tangent lines" did not even appear in the textbook. I learned that on my own during the summer between middle school and high school. I learned it officially in geometry.

Sigh... And then they wonder why students hate math...

 

[ProfuselyQuarky]

Sigh... And then they wonder why students hate math...

Yes, it's not entirely students' fault. Lucky for me, my current school is independent study! I stay home almost everyday with the Internet and parents who love science and math as much as I do!

(No matter how amazing the class is, however, sometimes there is just no desire to learn. That's why you can now download video games onto graphing calculators. Who needs cell phones now? These guys are con artists!)

 

[Isaac0427]

7th grade, the first lesson on trig ratios only taught sine, cosine, and tangent.

That's all they teach in 10th grade...

Instead of letting students who aren't interested in math do whatever they want (i.e. skip algebra), schools should give that same freedom to students who actually want to learn.

Agreed.

What's that even supposed to be?

How to count. They figured out that I knew how to count up to 1,000 (they only required 500) and how to count down to -500 (which they didn't require until 5th grade) and let me move right into 1st grade math.

I'm pretty sure that anyone who can write inteligent things on PF has had a very similar experience and would find my school system to have unnecessarily low standards and bad curriculum (especially general 7th grade science. Oh man, you all would hate that).

 

[ProfuselyQuarky]

That's all they teach in 10th grade...

Well at least I'm going to start abstract algebra soon. Yeah, baby! Most of pre-calc is just teaching how to read graphs (what is the limit of this sine function? how do you find domain?), which isn't bad, though.

How to count

especially general 7th grade science. Oh man, you all would hate that

My 7th grade science was Life Science. You know, like the organelles within a cell, basic anatomy, reproduction, etc. That was a lot of knew stuff for me, and I really liked it . . .

. . . but then come 8th grade, I got a Physical Science textbook that was so basic, it devoted an entire chapter on "What is an atom?"

 

[atyy]

Did they at least mention what tangent and cotangent have to do with tangent lines?

Nope. The words "tangent lines" did not even appear in the textbook. I learned that on my own during the summer between middle school and high school. I learned it officially in geometry.

Sigh... And then they wonder why students hate math...

Should I confess that this is the first time I've heard that tangent and cotangent have anything to do with tangent lines?

Yup, one learns something new everyday!

 

[Isaac0427]

. . . but then come 8th grade, I got a Physical Science textbook that was so basic, it devoted an entire chapter on "What is an atom?"

That's our 7th grade science. We had what is an atom, how to read the periodic table (we spent 2 weeks on that) and F=ma took another 2 weeks a little later. Earlier that year we spent at least a month or two (I am not exaggerating) on unit conversions within the metric system (like from grams to kilograms). I thought my teacher got the worksheets out of a second grade science book.

 

[ProfuselyQuarky]

That's our 7th grade science. We had what is an atom, how to read the periodic table (we spent 2 weeks on that) and F=ma took another 2 weeks a little later. Earlier that year we spent at least a month or two (I am not exaggerating) on unit conversions within the metric system (like from grams to kilograms). I thought my teacher got the worksheets out of a second grade science book.

To be fair, most average second graders now are only just learning multiplication, so teaching math-based science (like F=ma) is probably too much. But, yeah, that sounds pretty bad! A month on unit conversions when it could've been tackled in less than a day . . .

 

[JorisL]

To be fair, most average second graders now are only just learning multiplication, so teaching math-based science (like F=ma) is probably too much. But, yeah, that sounds pretty bad! A month on unit conversions when it could've been tackled in less than a day . . .

Unit conversions should be repeated time and time again. A lot can be deduced just from the units.
More important is to teach students how to use units as a heuristic way to check their results.

Ideally one would spiral back to units every time a new "unit" is treated. (mechanics, gas laws, ...)

 

[ProfuselyQuarky]

Unit conversions should be repeated time and time again. A lot can be deduced just from the units.
More important is to teach students how to use units as a heuristic way to check their results.

Ideally one would spiral back to units every time a new "unit" is treated. (mechanics, gas laws, ...)

The entire focus of an entire class should not be devoted to unit conversion for over a month, though. Units are used all the time, so regardless on the specific subject, students will constantly be practicing it anyway.

 

[ProfuselyQuarky]

The actual lesson on how to convert is not difficult at all and I believe that that could be tackled in a day.

 

[JorisL]

A month can mean a lot of things. So does a day, you can hammer at it for 8 hours straight or just mean half an hour.
In what's called "technical education" in Belgium students that are learning a trade (woodwork, electrical installations etc) students have 1 hour of physics a week (50 min hours at that).
While a month is still a lot I think 2 weeks (=2 periods) wouldn't be overdoing it. Of course you have to ground it in applications for example converting metric wrench sizes to inches and vice versa to show they aren't interchangeable.

The point I'm trying to make is that not all students are very comfortable with these kind of things. A cliché example are the units of area.

 

[ProfuselyQuarky]

point taken :)

@Isaac0427 how long is one class period for you?