Math Musings: Integrated Math, Test Prep, Knowing when you are "done"

[Chrysalis Academy]

I have a few insomnia-fed math musings/questions for you all this morning;

 

1) Integrated Math.  Shannon and I are doing the EdX Algebra and Geometry classes right now.  Taking these concurrently is giving me a whole new appreciation for the concept of doing integrated math.  I like that by working on two subjects concurrently, there are no long gaps that lead to forgetting.  I like how working on concepts in the two concepts reinforce each other - i.e. the Pythagorean theorem was so much easier because we just reviewed exponents and square roots in Algebra - and I love how working on the topics together makes connections between them, and helps you understand what it's all for - i.e. working with the coordinate grid and making connections with geometric shapes & measurement.  I want to keep doing this.  How do other people handle integrated math? Have any good texts/curricula bubbled up from the CC Integrated Math classes I keep reading about? Do you use subject-text books concurrently?  Are there texts from other countries where IM is normal that I should be checking out?   Now that CC includes IM as a pathway, I'm not worried about how it would look on a transcript, I'm just curious about how to do it well.

 

2) Test Prep.  One of the reasons I like the idea of doing IM is that it keeps things fresh and active, which seems good for test prep purposes.  You need both geometry and algebra for PSAT and SAT, yes? What about the SAT subject test? ACT math? I haven't really delved into the differences between them.  Just thinking about timing math classes and tests so that tests are taken when the tested subjects are as fresh as possible, I guess.  Do you do specific math test prep outside of your regularly scheduled programming? When, how, with what?

 

3) Being "done" with a subject and ready to move on.  How do you tell? Are there good standardized tests you use to tell when your student has sufficiently mastered a subject? I know there is DOMA for Algebra, is there anything like that for Geometry?  She will complete Algebra and Geometry on Edx this year, but I'm hesitant to consider that "done" particularly with geometry, because I'm not strong in that subject.  EdX does cover the standard syllabus, but I don't think it includes enough practice for mastery, so we'll plan to keep going with a text or do an outsourced class, I think.  That will be my next question, no doubt.

[wapiti]

I just linked your earlier thread on another thread http://forums.welltrainedmind.com/topic/540226-sat-2subject-test-strategy/There might be something helpful in there.  IIRC, the SAT2 Math 2 test should be right after precalc and has a more forgiving curve than the Math 1.

 

The regular SAT that counts is, what, four years away for your dd?  Thinking out loud, I don't think an integrated schedule is likely to make a difference.  She will do some prep anyway, which will review what she needs.

 

It'll be interesting to see how much geometry is on the new SAT.  I'm not sure how clear it is right now.  Once in a while I check out this site for information and tips (this is likely to be the book that dd uses for prep, but she too has a few years):  http://pwnthesat.com/wp/

[regentrude]

I am all for integrated math, but I have not found that my kids forget their algebra while working on geometry, because even a separated geometry course requires the use of algebra all the time.

Actually, I do not think it is possible to "forget" algebra 1 once the student has truly mastered it. A student who "forgets" algebra 1 has not understood it conceptually, but has memorized procedures.

So, I would not be concerned about test prep. The student needs to do some targeted test prep anyway with a designated test prep book, so there will be review.

 

 

What we found on the SAT2 is that the topics from algebra 1 and geometry were not really an issue; the geometry that is tested is rather simple, and algebra 1 has been used for several years all.the.time in math and science. The issue were the more obscure theorems on polynomials from algebra 2 which are never applied outside of algebra 2, and certain things from trigonometry that had not been used in calc or physics.

I strongly recommend that any student interested in the math II SAT 2 take the test immediately upon completion of precalculus, and not after a year of calc. But forgetting algebra 1 should be impossible.

[Teachin'Mine]

It is easy to forget some of the terms used in Algebra 1 which aren't used in subsequent courses and do show up on standardized tests.  Prep books are good for reviewing them.  Solving the actual problems should be very easy after upper level math courses.  Integrating algebra II and geometry works nicely, but I'm not sure what should be integrated after that. Pre-calculus is usually a time consuming course, so it might be a better stand-alone courses.  Statistics and probability is often combined with calculus as the statistics is a lighter AP course - from what I've read.   Other than that, until higher level college courses, the progression needs to be followed as the calculus courses build on each other. 

[kiana]

A well-done precalculus course should review both algebra and informal geometry -- not to say that all do, but it should. Much of geometry is used in, for example, deriving values of trigonometric formulas or figuring out precisely which trigonometric function is useful to solve a given problem, while algebra is integrated in the more analytic aspects of trigonometry such as proving identities. The conic sections are also quite geometrically based.

[regentrude]

  Integrating algebra II and geometry works nicely, but I'm not sure what should be integrated after that. Pre-calculus is usually a time consuming course, so it might be a better stand-alone courses. 

 

Precalculus has intrinsically integrated geometry. Trigonometry is heavily applied to triangle and circle geometry. Vectors tie in to geometry. Trig functions in the complex form have a lot to do with circles.

 

Similarly, calculus can not really be divorced from geometrical interpretations and problems: derivative as slope, integral as area, application of integration to determine areas and volumes... there is no separation anymore.

 

[quark]

I have a few insomnia-fed math musings/questions for you all this morning;

 

1) Integrated Math.  Shannon and I are doing the EdX Algebra and Geometry classes right now.  Taking these concurrently is giving me a whole new appreciation for the concept of doing integrated math.  I like that by working on two subjects concurrently, there are no long gaps that lead to forgetting.  I like how working on concepts in the two concepts reinforce each other - i.e. the Pythagorean theorem was so much easier because we just reviewed exponents and square roots in Algebra - and I love how working on the topics together makes connections between them, and helps you understand what it's all for - i.e. working with the coordinate grid and making connections with geometric shapes & measurement.  I want to keep doing this.  How do other people handle integrated math? Have any good texts/curricula bubbled up from the CC Integrated Math classes I keep reading about? Do you use subject-text books concurrently?  Are there texts from other countries where IM is normal that I should be checking out?   Now that CC includes IM as a pathway, I'm not worried about how it would look on a transcript, I'm just curious about how to do it well.

 

Instead of IM books, kiddo did math in different strands and it was a great way to appreciate the link between the different areas of math...when he did abstract algebra, it was even more fun to see algebra, geometry, modular math and I think some set theory too collide. For texts that use IM, you could try the NCERT books from India that are linked in the Homeschool High School Math thread or something from MEP in the Secondary, GCSE, A Levels levels. We do have an A-Levels textbook by someone else (I will have to check author's name when I am home) that integrates math too but the problems are straightforward, better for practice only but not very interesting if a student wants proofs or to go deeper. No experience with common core materials.

 

2) Test Prep.  One of the reasons I like the idea of doing IM is that it keeps things fresh and active, which seems good for test prep purposes.  You need both geometry and algebra for PSAT and SAT, yes? What about the SAT subject test? ACT math? I haven't really delved into the differences between them.  Just thinking about timing math classes and tests so that tests are taken when the tested subjects are as fresh as possible, I guess.  Do you do specific math test prep outside of your regularly scheduled programming? When, how, with what?

 

When kiddo took the current SAT, we set aside about 4 hours a week for 5-6 weeks prior to the exam to work on the College Board blue book. The first week, he did a combination of a few subtests just to get used to the time/ speed because he doesn't usually like being timed. He was working on Algebra 2 + Trig concurrently and the math itself wasn't hard, only the speed required was a challenge. But after that first week he picked up speed and was able to complete one full practice test a week in conditions as close to the actual test as possible. He hasn't taken the SAT2 in Math although he has enough math for that level because he decided not to spend time on it just yet.

 

3) Being "done" with a subject and ready to move on.  How do you tell? Are there good standardized tests you use to tell when your student has sufficiently mastered a subject? I know there is DOMA for Algebra, is there anything like that for Geometry?  She will complete Algebra and Geometry on Edx this year, but I'm hesitant to consider that "done" particularly with geometry, because I'm not strong in that subject.  EdX does cover the standard syllabus, but I don't think it includes enough practice for mastery, so we'll plan to keep going with a text or do an outsourced class, I think.  That will be my next question, no doubt.

 

You will continue to carry the skills learned forward. It will be obvious if she doesn't understand something. We moved on after a textbook was completed and have never tested for mastery, but have only assumed that if he is doing well in the next level that concepts were mastered...a good program (and perhaps more so one that is proof based) should apply all those things previously learned or at least set a solid foundation for what's coming next. A good teacher will do that too...the tutor kiddo used customized his classes so that there was some integrated algebra, geometry and some abstract algebra at the algebra 2/trig level. Kiddo has not needed to review anything prior in order to move ahead. Every subsequent math has included the math that was done before...it just builds upon previous levels. Analytic geometry was also taught when he did algebra 2 and now as he is finishing up calculus.

 

[Teachin'Mine]

Precalculus has intrinsically integrated geometry. Trigonometry is heavily applied to triangle and circle geometry. Vectors tie in to geometry. Trig functions in the complex form have a lot to do with circles.

 

Similarly, calculus can not really be divorced from geometrical interpretations and problems: derivative as slope, integral as area, application of integration to determine areas and volumes... there is no separation anymore.

 

I answered the question which wasn't asked.  lol  I was thinking in terms of which math courses could or should be studied together and not which areas of math are combined in a specific course.