kitchen table math, the sequel

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Any ideas on an online course that might be appropriate for a student who has been struggling with New York State Regents geometry? The course would have to prepare the student to pass the NYS geometry Regents exam. What about online tutoring options? While this student is self-directed in some ways, he is not one to “teach himself” from a book or other resources. He probably needs direct instruction, with targeted feedback and guidance on his progress. Should online even be an option in this case? Does he simply need a traditional tutor? Maybe a blended learning option would work well? Any other alternatives?

Any and all ideas would be welcome.

from California, JC writes:

UNDER THE HEADING OF THEY DO WHAT THEY DO

N will be entering the 9th grade this fall and will hopefully be enrolling in the local public high school (or maybe not). Currently in the 8th grade, N is taking his final today in Honors High School Geometry. N attained this level of competency by working year round on math ever since the 1st grade. He has not had to skip a year of instruction as the local public students have.

The local district, in its infinite wisdom, will not accept N’s transcripts showing straight A’s and Standardized test scores in the 99th percentile for mathematics as proof of his ability. In their defense, N does not have state administered test scores on the CST Algebra 1 or Geometry tests – those tests aren’t made available to home schools that operate independent of a school district. The local high school is requiring N to take a 50 question, calculation heavy, 2 hour, Geometry course challenge exam. The test will cover material from a classroom he’s never attended and a textbook he has never used. The test won’t be a neutral exam such as the standardized exams.

According to “district policy” my son must pass the challenge exam with a 90% grade or better in order to continue on in the honors track. We were not to be supplied with the answers to the study guide that was provided. We were also informed that the study guide was missing ¾ of the materials that could be included on the exam. I wouldn’t be able to correct the study guide questions, as N has passed me by mathematically speaking. Thankfully Barry G came to my rescue and worked the problem sets, and provided fantastic comments and notations which helped me to provide my son with a proper review where he was weak.

The district, county and state all refused to let N take the CST tests this spring because he wasn’t enrolled with the public schools. In May I gave N the CST retired questions for Geometry (posted on the CDE website) and he only missed 2 out of 64 questions. We also gave him the on line Algebra 1 test, and uncovered an area of weakness, which was remedied with one evening’s chalk and talk thanks to purple math - he then only missed 3 questions on the whole Algebra 1 exam.

Needless to say, I’m a bit worked up because our high school won’t accept N’s test scores and grades as proof of his abilities. It seems terribly unfair when you consider that the district students only need a B- grade on their report cards to advance to the next honors level class.

If you investigate further - our high school’s CST scores for the 9th grade single honors Geometry class indicate 75% of those honor students fail to score in the advanced category. That’s not a stellar record -- especially knowing how low the proficiency bar is set by the California Department of Education. N is trying to gain entry to the 9th grade Honors Algebra 2 course. The Algebra 2 double honors track has better CST scores with only 22% of the honors students failing to score at the advanced level. The improved performance of the upper track is likely a product of after schooling.

The administration is slamming the gate shut on students who transfer in from schools other than the local district. UPDATE: It isn’t “all” out of district transfer students who must take this exam as I had previously been informed. No, it’s just the students who come from “non traditional” high schools that must pass a challenge exam. So apparently, private home school students must out perform the majority of the local high school honors math students in order to gain admission to the honors math track. Perhaps, hypothetically speaking, it’s just the home school children of veteran math warriors who are expected to perform at this level.

This all seems so horribly unfair. N is an excellent student and has worked incredibly hard, and the standardized tests all place him at the very top. I wouldn’t be so bent out of shape if the local high school CST scores indicated that the honors classes were full of elite math stallions, but that just isn’t the case.

N finished the high school’s study guide exam and took 40 minutes longer than the 2 hour time limit they are setting on the exam. Hopefully the multiple choice exam he will be given next Tuesday will be easier to complete within the time limit otherwise, N will be getting the gate slammed shut in his face.

UPDATE: I’ve just learned the ruling we will get as a result of this test is not the final end – I can always appeal the decision! This is great! I can spend my summer vacation fighting with the school district -- and my son isn’t even in their classrooms yet.

THERE IS HOPE:

On a happier note, I do have some good news to report. N took the California High School Proficiency Exam last Saturday. The CHSPE allows students to exit high school early, and continue their education at the community college level. Though the exam is designed for students 16 years and older, N (only 14) was able to take the exam with the permission of his “non traditional” school (heh). We will get the results in mid July.

Previously parents have posted on KTM that skipping the high school math program and moving a student up to the college level for math isn’t perhaps the best move. I do remember Wayne Bishop having once written about saving a student from the public high school system and getting him placed into a college math program early. I would like to know what those of you who frequent KTM think about taking this approach.

With the CHSPE proficiency certificate in hand, N will be able to enroll in community college courses without permission from the high school and receive dual credit for his courses. Some local home school parents even send their middle school students to the community college. N will still be able to attend the HS and play sanctioned sports as long as he attends 4 classes. A few of the ambitious local students have managed to graduate high school having also completed an Associate Arts degree.

The CHSPE does NOT allow a student to stop attending school. Students must (by law) attend classes until they reach their 18th birthday. Additionally, it’s possible the HS will refuse to allow N to receive a HS diploma and be in the graduation ceremonies if we go this route (though there may be a way to work this out). The CHSPE certificate is supposed to suffice as an equivalent document in lieu of a high school diploma and is accepted by the state of California and, I believe, the armed services though I am not sure about that.

I would appreciate hearing what KTMers think about this opportunity and look forward to reading your comments.

In triangle PQR, PQ = 4, QR = 3, PR = 6, and the measure of angle PQR is x°. Which of the folllowing must be true about x?

(A) 45 ≤ x < 60
(B) x = 60
(C) 60 < x < 90
(D) x = 90
(E) x > 90

FedUpMom on remembering area vs circumference:

I had the following brainstorm at an embarrassingly advanced age:

For a long time, I knew there were two formulas that were somehow relevant to circles, namely 2πr and πr², but I could never remember which one was area and which one was circumference.

I finally realized that πr² must be the formula for area, because area is described in square units.

Tell your kids.

How do you solve this?

from The Math Page:

THE CLASSICAL LIBERAL ARTS included logic, grammar, rhetoric, and geometry. Just as today's liberal arts, they were not for the purpose of learning a trade. They served the purpose of education, which, as Albert Einstein once observed, "is not the learning of many facts but the training of the mind to think."

Geometry, moreover, embraced logic, grammar and rhetoric, because it was approached purely verbally. There was no algebra, no symbols for "angle" or "equals." What the student saw, he explained. For geometry is based on looking, and the sensitivity it develops is the essence of science.

In the 4th century B.C., Alexandria in Egypt was the center of culture and learning, and it was there that the Greek mathematician Euclid assembled the most remarkable textbook the world has ever seen: the Elements of geometry and arithmetic. Written in simple, straightforward language, the Elements has been translated the world over, and through the centuries it has been the model for clear and eloquent reasoning. It was the first written work to introduce what is called rigor into mathematics. That same rigor -- What gives us the right to say that we really know? -- is part of the culture of mathematics today, and it is the model followed in theoretical physics. Anyone truly interested in what mathematics is, can have no firmer foundation than Euclid.

Efforts have always been made to express the Elements in the language of each time and place. The pages that follow are adapted from the translation by Sir Thomas Heath (Dover) as well as the edition of Isaac Todhunter (Elibron Classics)...

Lawrence Spector

Back to Algebra 1.

I wonder if I'll ever get back to Fluenz?

I've just finished topic number 210 in ALEKS geometry. One topic left to go.

ALEKS lets you compare what you're learning from ALEKS to your state standards. According to ALEKS, New York state geometry standards includes 101 of ALEKS' 211 standards.

I don't know whether New York's geometry standards include topics ALEKS doesn't cover.

I posted about my son skipping out of geometry earlier on this blog. (Thanks to all who commented!) Both my 8th & 9th grade sons (削除) are (削除ここまで) were in geometry classes at different schools.

These were Wednesday's homework assignments. The first two pages are from the book: Geometry by McDougal Littell (2004 version). They were the 8th grader's homework assignments.

The second two pages are the first 2 of 3 created by the 9th grader's geometry teacher. The school uses the newest version of Prentice-Hall Geometry & Algebra 2 books. The pages were a take-home quiz and he could work with partners. Just the first 26 questions.

I see more challenging problems in the 6B Singapore Math workbook.

The older son starts Algebra 2 on Monday. He's already read through the first few chapters that he's missed and over the weekend we'll be working problems similar to what he did last year in Paul Foerster's Algebra.

(FYI- I didn't check my son's work, he did the assignment in 10 minutes in the car. He asked me to black his name out. - Anyone want to talk about years of failed handwriting instruction?)

On our next to last day in St. Romain en Viennois we went to the local Intermarché, where I purchased a Mini Kit Incassable in the school supplies section.





contents:
ruler
protractor
right triangle 45-45-90
right triangle 30-60-90

I'm pretty sure it is not possible to buy trigonometry paraphernalia in U.S. supermarkets.

I also scored a Travaux Pratiques for writing proofs and a Vocabulary Coach for memorizing Spanish & French vocabulary. Vocabulary Coach is very cool.

The Intermarché is a big-box Target-type store, like Meijer's in the Midwest.





When I get back to Springfield, IL in a couple of weeks, I'll see if Meijer's has right triangles idéal pour la trousse.

I'm guessing no.

"Let no one ignorant of geometry enter here."

- inscription above the entry to Plato's Academy

(thanks to MagisterGreen)

Voltaire, eager to undermine the claims of conventional religion by contrasting the infighting of the Catholic Church with the sedate purity, unity and rationality of the students of Euclid, boasted that there were no sects among geometers. This, with the arrival of non-Euclidean geometry in the next century, would prove to be overly optimistic, but more important, invoking geometry as some kind of antithesis of revealed religion was a rhetorical mistake. After all, some of the best geometers of the past two centuries (including Christopher Clavius, the author of the preeminent early-modern version of Euclid's Elements; including Fracois d'Auilon; including Giovanni Girolamo Saccheri, author of books which, after a period of long neglect, would help lay the basis for non-Euclidean geometry) were Jesuits.

God's Soldiers: Adventure Politics, Intrigue, and Power -- A History of the Jesuits
by Jonathan Wright
p. 193-194

In theory, C. is working through these books:

Megawords 6

Analyze, Organize, Write by Arthur Whimbey & Elizabeth Lynn Jenkins (text reconstruction)

Sentence Composing for High School: A Worktext on Sentence Variety and Maturity by Don Killgallon
Grammar for High School: A Sentence-Composing Approach---A Student Worktext by Don Killgallon & Jenny Killgallon
Sentence Combining Workbook by Pam Altman, et. al.

French by Association by Michael Gruenberg
Behind the Wheel French for the car^*
ALEKS Geometry

I have no idea whether the Killgallon books 'work,' whether Killgallon's exercises, in and of themselves, improve a person's writing. I don't care. I love them so much I'm insisting C. do them. I'm going to do the college book myself. Here's the Killgallons' web site.

As for the Whimbey/Jenkins book, I take Myra Linden's word for it:

"In a study of ‘before‘ and ‘after‘ papers of students who used text reconstruction, I concluded that text reconstruction was effective in helping students learn to organize their ideas into coherent paragraphs. The most marked improvement in their second samples was in content. Students were able to furnish specific details to support general statements. In addition, students showed consistent improvement in organization and style.

"I believe that this approach works because the constant practice in arrangement of content from the general to the specific provided by the text … fosters the most basic writing skills of presenting specific content and organizing it logically into a cohesive pattern. Students replace their faulty composing patterns with effective ones that become automatic skills. In addition, students who do significant amounts of text reconstruction learn indispensable techniques for effective study skills. It helps note-taking. It aids in learning outlining and summarizing skills, and it teaches the general-specific arrangement of ideas.

Why Johnny Can't Write by Myra J. Linden

^* Behind the Wheel Spanish & Spanish by Association for me -- also Fluenz

re: TERC on Establishing Truth in Geometry

Oh really .... I don't know where to begin!!!!

"establishing the validity of ideas is critical to mathematics"

I know straight away that my blood pressure will need to be checked by the time I get to the end of this!!!

But wait, there's more!!!

"Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms."

Of course, that should read "Most mathematics instruction and textbooks AND ALL MATHEMATICIANS ..."

"evidence for its validity in the form of a proof"

By this stage, it's pretty obvious to me that the author isn't a mathematician. "Validity in the form of a proof" ..... what other type of validity is there??

Of course, the trained mathematician should have their 'proof by obfuscation' alarm bells ringing by now.

"For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument."

Is this even English? [ed.: I've been asking myself the same question. When I finally learn how to diagram sentences, I'll be able to answer it.]

"However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?"

They'd better see it as (ahem) "a way to establish the validity of their ideas" or their teacher hasn't really communicated the difference between Science and Mathematics too clearly.

Worthy of mention is the desire to "convince students" .... you may accuse me of semantic nitpicking here, but it's important!!

"Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work."

Which roughly translates as "In order to save the village we had to destroy it!"

---------------------------------

I think Melanie Philips (British author) summed it up best in her book 'All Must Have Prizes':

"A fundamental shift in emphasis from knowledge transmitted by the teacher to skills and process 'discovered' by the child has undermined the fundamental premises of mathematics itself. The absolutes of exactness and proof on which the subject is based have been replaced by approximation, guesswork and context."

Melanie Phillips, All Must Have Prizes

No one would deny that establishing the validity of ideas is critical to mathematics, both for professional mathematicians and for students. But how do people establish "truth"; how can they prove things? According to Martin and Harel (1989), in everyday life, people consider "proof" to be "what convinces me." Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms.

But mathematicians most often "find" truth by methods that are intuitive or empirical in nature (Eves 1972). In fact, the process by which new mathematics is created is belied by the deductive format in which it is recorded (Lakatos 1976). In creating mathematics, problems are posed, examples analyzed, conjectures made, counterexamples offered, and conjectures revised; a theorem results when this refinement and validation of ideas answers a significant question. Hanna (1989) argues that because mathematical results are presented formally by mathematicians in the form of theorems and proofs, this rigorous practice is mistakenly seen by many as the core of mathematical practice. It is then assumed that "learning mathematics must involve training in the ability to create this form" (pp.22-23). The presentation obscures the mental activity that produced the results.

In fact, according to Bell (1976), personal conviction grows out of internal testing and forming a judgment about whether to accept or reject a conjecture. Later, one subjects this judgment to criticism by others, presenting not only the generalization formed but evidence for its validity in the form of a proof. For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument.

In sum, formally presenting the results of mathematical thought in terms of proofs is meaningful to mathematicians as a method for establishing the validity of ideas. However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?

Let me guess.

No?

No, students do not see proof as a way to establish the validity of their ideas?

Is that it?

Conclusion

Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work.

I had a feeling.

By focusing instead on justifying ideas while helping students build the visual and empirical foundations for higher levels of geometric thought, we can lead students to appreciate the need for formal proof. Only then will they be able to use it meaningfully as a mechanism for justifying ideas.
Geometry and Truth
by Michael T. Battista and Douglas Clements

Only then, after sophomore year has come to an end and so has geometry.

Here's a question.

How many sophomores in high school have mathematical ideas?

Vicky S sent me notice of a workbook Stephen Wilson has posted on his web site: SAT/ACT Math and Beyond: Problems Book by Qishen Huang.

The book is listed here. I've just ordered the solution manual, which Dr. Huang says is highly detailed (460 pages for the manual, 131 for the workbook). That's critical for those of us teaching ourselves, and not easy to find.

Dr. Huang estimates that 20% of Chinese high school graduates can work 90% of these problems, which he says are not as difficult as those on China's SAT equivalent.

And...on the subject of workbooks, I've emailed Myrtle, who is using the NEM Workbooks (New Elementary Mathematics Syllabus D 1 and New Elementary Mathematics Syllabus D 2).

Meanwhile, I have done no math at all this summer, because I am busy reading C's massive Summer Assignment list, all 2549 pages of it. As to that, please know that you are in the presence of a woman who has read every last word of Guns, Germs, & Steel. There are few amongst us who can say the same.

comment from lsquared re: Thursday

How discouraging. No wonder kids hate proofs--they get tested on them without ever having a chance to learn them (OK--this isn't true of all schools/teachers). I love proofs and love teaching them, but there's nothing I know that can be effectively done by Thursday.

There's just one word for this comment: droll.

I love droll!

Speaking of what can and cannot be done by Thursday, if I hurry up and order now I could, by Thursday, have 5 or 6 or 7 books about proofs.

Any suggestions?

Also, does anyone know anything about Math Dictionary With Solutions by Chris Kornegay? It looks and sounds fantastic (way expensive, but fantastic - exactly what a person at my stage of the game desperately wants and needs).

I have just spent 3 hours doing Chris’ new homework assignment. The problem set is drawn from two short lessons on parallelograms appearing in one short chapter of Glencoe Geometry New York. The two lessons cover NINE theorems about parallelograms, not one of which I’m able to prove although I am apparently expected to be able to prove all 9 now that I've seen them tidily numbered and listed in an attractive Glencoe Geometry Theorem Chart enhanced with a red-and-tan color scheme.^* One two-column proof of Theorem 8.4^** (Opp. angles of parallelogram are congruent) and one paragraph proof of Theorem 8.10 (If both pairs of opposite angles are congruent, quadrilateral is a parallelogram) and it's off to the races.

That's not all. Having read nine theorems about parallelograms & 2 proofs, I am now Glencoe-ready to solve homework problems involving PARALLELOGRAMS ON THE COORDINATE PLANE USING THE MIDPOINT FORMULA, etc.

This is a textbook written by math educators.^***

Speaking of math educators, thank God I have the Teacher Wraparound Edition. Unfortunately, what I really need now is the Teacher Solution Manual (ISBN: 0078602041). Too bad I didn't think of that.


^* I feel about tan the way I feel about beige, only more so.
^** And, yes, I did spend time Googling the known universe to find out whether these numbers are official: is Opp. angles of parallelogram are congruent always and everywhere Theorem 8.4? Apparently not.
^*** written by math educators, but sounding suspiciously like contemporary geometry textbooks authored by actual mathematicians...

What if Zig Engelmann set out intentionally to write the worst textbook he possibly could? ...If you think about it, Zig should be able to pull this off better than anyone alive. ...Extremely confusing concepts all would be introduced at the same time and/or in close approximation. Stuff would be "taught" and then dropped, or more accurately, "covered" and then dropped. New material being covered would logically require mastery of prerequisite knowledge that most of the students most likely wouldn't have.

[snip]

If Zig were to engage in this little heuristic exercise, the result, I believe, would be a textbook that would sell like crazy and generate a fortune in royalties, and it would take about one-twentieth of the time that it would take to write an instructionally sound textbook.... [T]his ... came to me while looking at my daughter's geometry textbook. The thought hit me that if Zig had tried to write the worst possible geometry textbook in the world, it would end up looking a lot like Emily's geometry text, published b one of the few major educational publishing companies still standing.

[snip]

Within two lessons of a single chapter of a best-selling geometry text, eight major, similar concepts are introduced, with eight more or less similar names or labels. Absolutely no one associated with this textbook ever took a single minute to look at all of this stuff through the eyes of the learner.

It's same ole, same ole. A few kids get A's on the tests, which passes as proof that the book is fine, but there is something wrong with the kids who flunk the test or get C's or D's. Zig could have written this book without having seen it just playing the game of trying to make everything as difficult and confusing as humanly possible. I've looked carefully for spots where Zig could, in fact, make the book worse than it is, but they are few and far between.

Isn't that strange? Really. You'd think that any best-selling textbook would get a few things right, would do a few things that actually take a student perspective into consideration. You'd think that Zig's horrible geometry text would be far worse than a best-selling geometry text. This is both astounding and depressing. Someone is making a fortune off a textbook that could be just slightly better than the worst textbook we could imagine, the worst textbook Zig could write. Get out the thesaurus: overwhelmed, dismayed, incredulous, confounded...

What Would You Get if You Set Out to Write the Worst Textbook You Could Possibly Imagine? A Best Seller
by Bob Dixon
Direct Instruction News
Spring 2008

Of course, I'd love to know what textbook he's talking about.

I was thinking Glencoe Geometry, but it can't be because the lead author on the textbook Dixon's talking about is a mathematician.

I just checked the author page for Glencoe Geometry & all 5 authors are math educators.


kitchen table math

Yes, occasionally I have some non-phonics thoughts.

The Well-Trained Mind has an excellent review of several math programs, here's a short excerpt:

Which method is better? In my opinion, the one which the student understands most clearly. In both cases, it is possible for the student to learn the mental trick without thoroughly understanding why it works, although the sheer amount of repetition in the Saxon method makes it easier for this lack of understanding to escape detection. But the strongest mathematical training of all would come from a combination of programs – in which the student is taught to do a mathematical process using several different methods and mental procedures.

Currently, Singapore and Math-U-See are “thought-oriented” math programs available to home school parents; Saxon and A Beka are “skill-oriented” programs. A combination of Saxon + Singapore, or Saxon + Math-U-See, or Singapore + A Beka, or A Beka + Math-U-See, may come closest to fulfilling the goals of classical education. Math-U-See + Singapore would also be an excellent combination, as long as you use MUS’s supplementary drill sheets. Treat one program as primary and the other as secondary; when you cover a concept in the primary program, look it up in the secondary program and see whether it is explained and illustrated differently.

I like her idea of using 2 programs. I do that all the time with phonics--I know what is best about each program and pick and choose accordingly, I have dozens of phonics programs from my tutoring. Also, if a student is struggling with a certain area, it's good to have a few books that explain it in slightly different ways. I hadn't thought to do the same with math.

There is also an interesting review at Sonlight, a basic review of their math choices from grade school through calculus, and then, they explain their choice for algebra and geometry--teaching textbooks:

• CD-ROM-based "whiteboard" lectures and step-by-step explanation of how to solve every practice problem taught by a tutor who has been teaching homeschoolers for years...since the days he tutored probability and statistics at Harvard . . . and . . .
• CD-ROM-based "whiteboard" solutions guide that works every step in every homework problem, so you see exactly how to solve each problem . . . and why you want to use the methods the instructor (or, more accurately, personal tutor) uses.

Their calculus choice, thinkwell math, also includes CD-ROM video tutorials.

We're currently using Math-U-See with our daughter for Kindergarten math. I personally like Singapore or Saxon better, but this is what is working for her. The good thing about Math-U-See is that they have a DVD with each lesson so that if you get that "blank stare," (yes, it is even possible with kindergarten level math!), you can just plop in the DVD and have an actual math teacher explain it in field tested verbage.

Math-U-See's approach to fractions is also interesting, I'm not sure what I think about it. The rectangles do make more sense to me than the traditional circle/pie method. From their downloads page, in sample lesson pages, click on Epsilon.

For an even more interesting and thought-provoking review, read Wild About Math!'s Calculus in 4th grade? I'm really not sure what I think about that one. Luckily, we're still doing kindergarten math so I have time to figure it all out.