Algebra at our House

[Myrtle]

I know that there are at least one or two people out there who are using Frank Allen's Algebra because they emal me from time to time but I never hear back from them to find out how it's going, if something was too confusing, or maybe they had some clever approach to something I had difficulty teaching, or maybe they just didn't see the point and gave it up for something entirely different.

 

Nonetheless, in chapter 8 the switch is made from two column proofs to proofs in paragraph form and I tried to document how that that went and how I handled it along with scanned copies of my son's work that you can see

 

 

the lesson at my blog

 

 

 

In education,al rhetoric there is a lot of lip service paid to the fact that one "uses logic" in math, but the problem that my son did really illustrates this well, I think and there is also this direct connection between every day language and symbols that can be seen.

 

And here is a good article summarizing the role of logic in teaching proofs, this might be something useful to those who are trying to teach proofs in geometry right now. More specifically, this author brings up the issue of why "bridge courses" are needed before college students take upper level math classes as well as the sorts of things that happen in math classes that promote fallacious reasoning.

 

"...it is possible that the resulting de-emphasis on formal proof will lead to even fewer numbers of students emerging from secondary school with a real sense of deductive argument."

 

You may note that the author of the above article goes into the topic of quantification which is something that we haven't done yet, hence, the potential for ambiguity in the proof with the "k"...is it for some particular k or all k? That sort of thing will be straightened out by next year since we plan on doing Suppes First Course in Mathematical Logic. So I'm thinking, that when we get to Algebra II, we'll be ready to use those symbols and manipulate them even though that isn't something that is explicitly taught in Frank Allen's Algebra II.

 

Moving closer and closer to, "for every epsilon..."

[Nan in Mass]

This is great, Myrtle. Thank you very much. It is very helpful to see stuff like this.

[genie]

I'm glad your eldest is ahead of mine so that I can watch your processes and glean what I can before it's too late. I appreciate your openness in sharing these types of examples, because it helps to see the theory you discuss put into action.

 

Thanks!

[Kimber]

Somehow I missed this post. Thank you, Myrtle, for this glimpse of Allen's Algebra. This is really helpful.

 

 

Thanks,

 

Kimberly

[Caia]

Thanks Myrtle! I really appreciate your posts and the glimpses you share with us of your child's math schoolwork. It helps to keep us motivated.

[Myrtle]

Thanks Myrtle! :001_smile:I really appreciate your posts and the glimpses you share with us of your child's math schoolwork. It helps to keep us motivated.

 

:001_smile: Thanks. Without feedback it's never clear whether it's of interest or useful to anyone.

[Jane in NC]

 

And here is a good article summarizing the role of logic in teaching proofs, this might be something useful to those who are trying to teach proofs in geometry right now. More specifically, this author brings up the issue of why "bridge courses" are needed before college students take upper level math classes as well as the sorts of things that happen in math classes that promote fallacious reasoning.

 

.....

 

Moving closer and closer to, "for every epsilon..."

 

Several things struck me in the article that Myrtle mentioned. The first is that I realized my son will have the distinct advantage of understanding the meaning of mathematical language because it is incorporated into my husband's and my speech patterns. I actually use "if--then" and "if and only if" statements in real life. When my son jumps to the assumption that the truth of an "if--then" implies the truth of its converse, Mom is there to correct the potential fallacy.

 

As the author points out, words do take on different meanings. Consider the word "theory". In everyday parlance, sportscasters and talking heads offer their "theories" (hunches) on performance which may be based on nonsense (Eastern teams win the series when the stock market falls) or a few empirical facts. No wonder that students feel that a bit of handwaving can pass for a mathematical proof.

 

I am tutoring an adult taking Calc I through a correspondence school. Imagine my surprise when he recently pulled out his Larson text and asked me to tell him what this epsilon/delta stuff was all about. The guy is bright and caught on after I worked through several examples. Mastering the definition of the limit requires algebra skills, but also requires an understanding of language as the author of the article emphasized.

 

On a different note, my son has been graphing trigonometric functions lately. Yesterday he was graphing things like f(x) = cos (2x) + sin (-x) by first graphing cos (2x), then - sin(x) (noting sine is odd, of course), then adding ordinates. I was thinking that yes, this would be easier on the calculator but what would he learn? Last week he was graphing sine waves with assorted amplitudes, periods and phase shifts, again easier on the calculator but here is the inherent problem: you often have to adjust the calculator window or the axes scales when graphing. If you don't really have a clue what the outcome of the graph should be, how do you know what adjustments to make? Or is it all a guessing game? Is there a point to guessing games in high school mathematics courses?

 

The funny thing is that my son checks his graphs on the calculator and assumes that the calculator is wrong if the graphs do not coincide. This is sometimes the case. The calculator graph can be incorrect due to an entry error--yet I cannot tell you how many students have argued that their answers on a test were correct because "the calculator said so". The calculator has become the Grand Verifier of all truth.

 

Good post, Myrtle. Thanks.

 

Jane

[Guest Dennis In Va]

Hello Myrtle,

 

Do you have the ISBN number for the Frank Allen Algebra book you are referring to? As a math tutor, I would like to get a copy. Thanks!

 

Best,

 

Dennis In VA

[Charon]

Several things struck me in the article that Myrtle mentioned. The first is that I realized my son will have the distinct advantage of understanding the meaning of mathematical language because it is incorporated into my husband's and my speech patterns. I actually use "if--then" and "if and only if" statements in real life. When my son jumps to the assumption that the truth of an "if--then" implies the truth of its converse, Mom is there to correct the potential fallacy.

 

 

I don't know that there is any way around this phenomenon, and, in the end, it may be what matters most. My father was a mathematician, and he did not actually do virtually any math with me growing up. And yet, I have always had a philosophical approach to life -- a distinctly mathematical/"rigorous" mindset. It comes out in the language as you say and in the little things one does or doesn't say -- what's considered an important consideration on any given issue -- the way he questions assumptions all the time. I was just raised like that.

 

Myrtle has been doing a lot of math over the years -- not just computational or even intense computational stuff, but number theory, mathematical induction -- interesting problems and proofs. We talk about it all the time. And, we apply it to something the kids might be doing. Now, it looks like I may be doing some math, myself, at my level. (I have started working meticulously through Lang's Algebra, for instance.) And, we talk about it infront of the kids.

 

And, just beyond that we say lots of complictaed stuff. I am notorious for my multiple negations used to make subtle distinctions that I just expect everyone to automatically follow. (Those are probably moments of true obnoxious geekiness on my part.) Myrtle and I talk a lot about other things in the same kind of way we talk about math. I wonder if our real program isn't so much Frank Allen or Gelfand or Singapore but just doing math ourselves and becoming as personally preoccupied with it as we can to create a "math culture".

[Jane in NC]

 

I wonder if our real program isn't so much Frank Allen or Gelfand or Singapore but just doing math ourselves and becoming as personally preoccupied with it as we can to create a "math culture".

 

Let's tie this to Latin: English is not necessariy imprecise, but at least in the US we tend to use English imprecisely. In my opinion this contributes to students' difficulties in understanding the difference between verb forms (imperfect, perfect and pluperfect) or why the subjunctive is used. Herein may lie part of the reason that I am drawn to Latin in the educational equation since Latin contributes to the mindset that you mentioned.

 

We have developed a family conversational pattern that includes math, science, technology, the news, etc. It is what other posters like Gwen in VA and Nan in Mass have noted as well. I think you are right--that is the real program. (Although my son remains convinced that he is doing MPE--a Monty Python Education. One must read philosophy, literature, history, learn French, etc. to appreciate fully the nuances of Monty Python.)

 

Jane

[Kimber]

When I got to college, I made a habit of asking students what their parents did for a living. I found that most engineering students had parents who were engineers. I learned that most pre-med students had parents that were doctors.

 

I think you're right. It's starts in the home.

 

I think the best we can do for our kids, when we feel we don't know enough about a subject to teach it properly, is akin to what Myrtle is doing--immerse ourselves in a subject, conquer it, and pass on to our children that desire to succeed and that newfound passion for the subject.

 

The problem for me is that I've got so many areas to improve upon. It can be quite overwhelming.

[Charon]

When I got to college, I made a habit of asking students what their parents did for a living. I found that most engineering students had parents who were engineers. I learned that most pre-med students had parents that were doctors.

 

I think you're right. It's starts in the home.

 

I think the best we can do for our kids, when we feel we don't know enough about a subject to teach it properly, is akin to what Myrtle is doing--immerse ourselves in a subject, conquer it, and pass on to our children that desire to succeed and that newfound passion for the subject.

 

The problem for me is that I've got so many areas to improve upon. It can be quite overwhelming.

 

 

And, that is why, I personally think, you really can't do a whole bunch of subjects. Even though I am something of a subject matter expert in mathematics, I still feel like it is less an expertise issue and more a personal interest issue. Myrtle is doing a better job conveying math to the kids just because she is doing it, and that is what they pick up on -- her attitude and interest and priorities in life. You just can't fake that. And, how many things can you really genuinely be interested in -- one or two. It's not like you can't box check a whole lot of stuff, but to really do something -- to have an interest that goes "outisde the box", so to speak -- I think you just can't do it all.

 

I think we really do math, as in "whatever it takes". We also refuse to drop Latin but it isn't nearly the same as the infinite subject that mathematics is for us. After that, it rapidly gets to doing things very sporadically or doing our hour and what we get done we get done and that's it kind of thing and so on. In other words, it is kind of box checking or not even checking that box at all. And, it is mostly so that when we have the Joneses over for dinner, we aren't completely mortified when our kids don't know who the Egyptians were or something.

[Storm Bay]

We have developed a family conversational pattern that includes math, science, technology, the news, etc. It is what other posters like Gwen in VA and Nan in Mass have noted as well. I think you are right--that is the real program.

 

Jane

 

Yes, I think this is spot on. When I was growing up, my parents spoke to us and asked us questions designed to make us THINK. We haven't all grown up to think the same way about things (we vary from biblical Christians, to norminal Christians, to atheist, agnostics, and other things). That said, there were certain areas they assumed we could learn intuitively, which isn't always true.

 

However, what's interesting is that we don't all think a lot about the same things. My sister, for instance, thinks a great deal about medicine (she's an MD) and certain other areas, but has never spent an enormous time on introspection, philosophy, religion, the meaning of life, etc. From the time I could articulate the questions, I was asking a lot of questions in these latter areas.

 

In our house, some of our conversations are limited by having highly sensitive dc, but we do discuss a great many areas, and add more as our dc get older. However, dh is not a big talker, so a lot of times these discussions are between my dc and me or with other adults we are in contact with. One of our friends loves to have discussions like this with our dc, and it's helpful for them to do this with others, as well.

[Kimber]

You're absolutely right. Even though I had this gut feeling about my kids learning more from me than from the texts, I never reached the end result of "I need to focus in on what's really essential." There must be a hierarchy of subjects somewhere. I'd rather my dh and I decide what those are on the outset rather than it happen on the back end simply because we did what was easier rather than what was important.

 

Thanks,

 

Kimberly

 

(One example of lessons getting caught rather than taught--my dh needed me to purchase silly putty for him so he could do a demonstration at work on creeping. He's an engineer that focuses on materials and silly putty is supposed to to demonstration creeping really well. My kids now talk about creeping in normal everyday conversaion and they know what it means. I don't--that's why my explanation is so vague.)