Beginners Calculus: Remedial Student's Experience & Pedagogical Advice

[DeusAbscondus]

Hellow mathematical brethren and sisteren,

For the many who don't me, I am 52 year old lay-about sometime snake-catcher, educator manque and now, most recently, amateur of maths and physics.

I am currently going through a beginner's course in Calculus which has very low previous learning requirements.

My question:

Please have a look at this error of mine, kindly pointed out by Mark FL then Sudharaka, at: http://www.mathhelpboards.com/f10/another-maximum-min-problem-2113/
then consider the following:

Given that the error is arithmetical, it seems to me on reflection that I am in need of a great deal of remedial work in algebra and even pre-algebra. (Another area I have already identified as a weakness is: trigonometry and the identities. And I understand that weakness or ill-preparedness in these areas is the most commonly reported reason for failure at Calculus)

I would love to hear from you educators out there as to my predicament:
- did I go off half-cocked, therefore, ill-advised, when I took on Calculus when I did
- or, if not, how common is my predicament in your collective experience of adults trying to learn math(s)?
- finally, while there is no going back (I could not imagine turning my back on the little calculus which I have managed to get under my belt, in order to concentrate exclusively on remedial, or "back-fill" work on shoring up a hastily constructed calculus edifice) I would, nevertheless be open to constructive help as to how to design a course of study tailored to meet the needs of someone like me, or even develop a better pedagogical understanding of my predicament and what it entails: what pitfalls to avoid, what strengths and habits to cultivate etc given my acknowledged status as a remedial student (and that is no more than to observe that I have years of maths formation and gradual maths maturation missing)
Thanking you in advance for your considered input.
Deus Abscondus

 

[Jameson]

Hi DeusAbscondus,

I can't comment on all of your post but I want to say that the mistake you made in the other thread is one of the most common mistakes I've seen with people of all ages when working with algebra problems. Don't get too discouraged over it. It's good that you recognize that you need to fine tune of your pre-calculus skills and it is doable :)

One of our administrators, Ackbach, wrote a very nice post on http://www.mathhelpboards.com/f2/algebra-dos-donts-1385/. I highly recommend reading it through. I know you're aware of Paul's Online Notes already but it seems he has a section devoted to algebra and other pre-calculus topics which should be useful.

The theory of what you should be doing is interesting but I don't feel qualified to comment on it. I've seen others struggle way more than you so again please don't be discouraged too much. Keep combining theory with lots of practice. You need both to put the concepts to use.

Jameson

P.S. By the way, I'm heading back to school this spring to continue my own math studies so I'll be suffering along with you in the pursuit of higher knowledge.

 

[DeusAbscondus]

Hi DeusAbscondus,

I can't comment on all of your post but I want to say that the mistake you made in the other thread is one of the most common mistakes I've seen with people of all ages when working with algebra problems. Don't get too discouraged over it. It's good that you recognize that you need to fine tune of your pre-calculus skills and it is doable :)

One of our administrators, Ackbach, wrote a very nice post on http://www.mathhelpboards.com/f2/algebra-dos-donts-1385/. I highly recommend reading it through. I know you're aware of Paul's Online Notes already but it seems he has a section devoted to algebra and other pre-calculus topics which should be useful.

The theory of what you should be doing is interesting but I don't feel qualified to comment on it. I've seen others struggle way more than you so again please don't be discouraged too much. Keep combining theory with lots of practice. You need both to put the concepts to use.

Jameson

P.S. By the way, I'm heading back to school this spring to continue my own math studies so I'll be suffering along with you in the pursuit of higher knowledge.

Hey Jameson,
Thanks for the reply, which is both encouraging and instructive.
(And informative: ie: references)

I take to heart the injunction not to lose heart and to press on, with the confirmation that I *do* need to revise and hone the tools that I wish one day to become second-nature.

Just yesterday we pushed on into terra igcognito: the integral and anti-differentiation, and when notification came through about your response, I was working on antidifferentiating $$-\frac{3}{\sqrt{9x+1}}$$ and digesting sage councel from my notes, such as "since the denominator in this case is not linear, don't be tempted to look for a natural log in the primitive" and my head is hurting as its putative plasticity gets tested: command central, telling all the circuits to back off and come at the whole thing from the other side of the looking glass...

Though it is the biggest intellectual challenge of my life, and it positively hurts sometimes, it has got to be better than spending my life off the end of a bar telling bullshit stories about what I *might* have done, and drowning the neurons in alcohol, as nice as that can be from time to time, if one can do it safely, and this black duck cannot! ...

(I'm so glad to have found a passion that keeps me on the straight and narrow (I'm a reformed boozer) and which gives me a reason to cultivate my brain, quite apart from the fact that "professionally", it is always better when handling deadly snakes, as I do quite regularly, to have steady hands and undamaged reflexes)

Once again, Jameson, great to hear from you.
(I actually answered your last PM with a long and rambling response, but, just like the time before that, it went puff! into the cyber-ether, and i didn't have the coeur to rewrite it; must be something I'm habitually doing wrong when sending PMs)

All the best with the upcoming course of study too, though I'll talk with you well before then, I hope.

Warm regs,
Michael
DeusAbs

 

[SuperSonic4]

Hey Jameson,
Thanks for the reply, which is both encouraging and instructive.

I take to heart the injunction not to lose heart and to press on, with the confirmation that I *do* need to revise and hone the tools that I wish one day to become second-nature.

Just yesterday we pushed on into terra igcognito: the integral and anti-differentiation, and when notification came through about your response, I was working on antidifferentiating $$-\frac{3}{\sqrt{9x+1}}$$ and digesting sage councel from my notes, such as "since the denominator in this case is not linear, don't be tempted to look for a natural log in the primitive" and my head is hurting as its putative plasticity gets tested: command central, telling all the circuits to back off and come at the whole thing from the other side of the looking glass...

Gordon Bennett! That description would confuse me if I didn't know it was basically saying $$\int \frac{dx}{ax+b} = \frac{1}{a}\ln|ax+b| + C$$

For my part I found it easier to learn certain results of anti-derivatives. In your example it's easiest to use a substitution - let u be whatever the linear factor under the root is and you end up with the easier $\int \frac{k du}{sqrt{u}}$ where k is some constant

Though it is the biggest intellectual challenge of my life, and it positively hurts sometimes, it has got to be better than spending my life off the end of a bar telling bullshit stories about what I *might* have done, and drowning the neurons in alcohol, as nice as that can be from time to time, if one can do it safely, andthis black duck cannot! ...

(I'm so glad to have found a passion that keeps me on the straight and narrow (I'm a reformed boozer) and which gives me a reason to cultivate my brain, quite apart from the fact that "professionally", it is always better when handling deadly snakes, as I do quite regularly, to have steady hands and undamaged reflexes)

Glad to hear it and how stereotypical Australian is your job lol? Do you throw shrimps on barbies (Giggle)

 

[DeusAbscondus]

Gordon Bennett! That description would confuse me if I didn't know it was basically saying $$\int \frac{dx}{ax+b} = \frac{1}{a}\ln|ax+b| + C$$

For my part I found it easier to learn certain results of anti-derivatives. In your example it's easiest to use a substitution - let u be whatever the linear factor under the root is and you end up with the easier $\int \frac{k du}{sqrt{u}}$ where k is some constant
Glad to hear it and how stereotypical Australian is your job lol? Do you throw shrimps on barbies (Giggle)

ah, Now I'm really confused. I think you must have mis-read my response to Jameson, Super: I am quoting from my notes when I paraphrase: "since the denominator is not linear do not look for ln in the primitive and now you are telling me to thus construe it! sheesh...

I get: $$-\frac{2\sqrt{9x+1}}{3}+C$$ when I take the integral; where is the need for introducing a log?

No, I don't do B-B-Qs and I despise the memory of Steve Irwin; I hero worship the memories of homosexual genius Marcel Proust and the mildly misanthropic, retiring genius of Samuel Beckett, so, subtract the snake-handling, and I probably have more in common with characters in and writers of books than I do with my fellow-Australians, the middle strata of whom watch "Neighbours" and other cultural algae-blooms, and in their exeunt from wage-slavery, go to the "Colluseum" to watch the "big men fly" in our national "game"/religion: AFL football. (Tauri)

On the other hand, I do like my motor-bike, my horse, my dog and connecting with native fauna through WIRES, a wildlife rescue outfit for which I do volunteer work.
And I like the "s**t-stirrer" sub-theme in the best of our intellectual life, the irreverence and larikinism (only when mixed with intelligence), but these qualities are British-derived, so... in the end, I'm an apatriot I guess.
;)

 

[Jameson]

Differentiation is all about calculation. Once you memorize the rules then the difficulty of differentiating usually falls in the simplification, from my experience.

Integration is all about recognizing situations. You'll get to use quickly analyzing an expression and be able to automatically say what form it falls into. So in your head you'll immediate think "natural log, u substitution, integration by parts, trig substitution" among other forms you'll cover. Once you know what to look for you can practice recognizing these groups.

Whenever you see a fraction where the numerator resembles a possible derivative of the denominator, that's a big clue to consider the natural log since \(\displaystyle \frac{d}{dx} \ln(f(x))=\frac{f'(x)}{f(x)}\)

You integrated the expression correctly in your last post. Well done :)

 

[QuestForInsight]

I agree with Jameson. The mistakes you're making are common and most people make them when they're learning calculus for the first time. I would say continuous rectification of these mistakes and the algebra that integration and differentiation usually require build one's algebraic skills better than the usual algebra courses.

 

[DeusAbscondus]

I agree with Jameson. The mistakes you're making are common and most people make them when they're learning calculus for the first time. I would say continuous rectification of these mistakes and the algebra that integration and differentiation usually requires build one's algebraic skills better than the usual algebra courses.

Thanks Quest,
This also is encouraging: practise using it at the relatively high level of calculus, and make the skills one's own at the same time by constant self-monitoring and repetition of problems and problem-solving attempts.

I like it; makes sense and "girds my loins" in the sense that, viewed your way, it feels less like I'm batting at the wind and more like I'm building my arm muscles while working on my swing.

Cheers,
Deus Abscondus (and no-one knows where he is hiding(Giggle) )