<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school student, being exposed to formal mathematical proof for the first time, to something like that?

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

BarryGarelick 7/6/05

A coda from Catherine and Barry to kick off comments:

Catherine: Barry, I've forgotten theorems & postulates.

Do you want to add a quick definition?

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

BG: (Offstage voice in the funhouse) Yes, that's a good way of putting it. It is a proposition that is accepted without proof. What is logically deduced from postulates and definitions are theorems, which because they can be deduced, can be proven.

Catherine (in front of crazy mirrors): I think I've got the two mixed up.....

Also, do you know how popular Core Plus is?

BG: (enters, walking on ceiling): Fairly prevalent throughout Michigan and Minnesota. Used in other states too, but those are the main ones. Google on "Bachelis; Core Plus" You'll find a paper he did on it. He did a survey of students in two high schools outside of Detroit; one used Core Plus, the other a normal program. Students using Core Plus did poorly in math in the university. Chris Hirsch, the PI for Core Plus threatened Bachelis with legal action. Tom Parker of MSU did a paper on Core Plus as well using statistical data showing performance in freshman year mathematics; similar to what Bachelis did. Also criticized by Hirsch.

(pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k


I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

---

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson


Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)



All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page


Thanks, Jo Anne, for taking the time to do this!



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

---

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

<!--

• • Set ALLOWTOPICCHANGE = TWikiAdminGroup
  • Set ALLOWTOPICRENAME = TWikiAdminGroup

$pattern(.*?\%STARTINCLUDE\%\n*(.*?)\%STOPINCLUDE\%\n.*) -->

<!-- Do not change this setting!

<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school student, being exposed to formal mathematical proof for the first time, to something like that?

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

BarryGarelick 7/6/05

A coda from Catherine and Barry to kick off comments:

Catherine: Barry, I've forgotten theorems & postulates.

Do you want to add a quick definition?

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

BG: (Offstage voice in the funhouse) Yes, that's a good way of putting it. It is a proposition that is accepted without proof. What is logically deduced from postulates and definitions are theorems, which because they can be deduced, can be proven.

Catherine (in front of crazy mirrors): I think I've got the two mixed up.....

Also, do you know how popular Core Plus is?

BG: (enters, walking on ceiling): Fairly prevalent throughout Michigan and Minnesota. Used in other states too, but those are the main ones. Google on "Bachelis; Core Plus" You'll find a paper he did on it. He did a survey of students in two high schools outside of Detroit; one used Core Plus, the other a normal program. Students using Core Plus did poorly in math in the university. Chris Hirsch, the PI for Core Plus threatened Bachelis with legal action. Tom Parker of MSU did a paper on Core Plus as well using statistical data showing performance in freshman year mathematics; similar to what Bachelis did. Also criticized by Hirsch.

(pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k


I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

---

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson


Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)



All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page


Thanks, Jo Anne, for taking the time to do this!



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

---

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

<!--

• • Set ALLOWTOPICCHANGE = TWikiAdminGroup
  • Set ALLOWTOPICRENAME = TWikiAdminGroup

$pattern(.*?\%STARTINCLUDE\%\n*(.*?)\%STOPINCLUDE\%\n.*) -->

CarolynJohnston - 06 Jul 2005

<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school student, being exposed to formal mathematical proof for the first time, to something like that?

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

BarryGarelick 7/6/05

A coda from Catherine and Barry to kick off comments:

Catherine: Barry, I've forgotten theorems & postulates.

Do you want to add a quick definition?

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

BG: (Offstage voice in the funhouse) Yes, that's a good way of putting it. It is a proposition that is accepted without proof. What is logically deduced from postulates and definitions are theorems, which because they can be deduced, can be proven.

Catherine (in front of crazy mirrors): I think I've got the two mixed up.....

Also, do you know how popular Core Plus is?

BG: (enters, walking on ceiling): Fairly prevalent throughout Michigan and Minnesota. Used in other states too, but those are the main ones. Google on "Bachelis; Core Plus" You'll find a paper he did on it. He did a survey of students in two high schools outside of Detroit; one used Core Plus, the other a normal program. Students using Core Plus did poorly in math in the university. Chris Hirsch, the PI for Core Plus threatened Bachelis with legal action. Tom Parker of MSU did a paper on Core Plus as well using statistical data showing performance in freshman year mathematics; similar to what Bachelis did. Also criticized by Hirsch.

(pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k


I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

---

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson


Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)



All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page


Thanks, Jo Anne, for taking the time to do this!



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

---

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

<!--

• • Set ALLOWTOPICCHANGE = TWikiAdminGroup
  • Set ALLOWTOPICRENAME = TWikiAdminGroup

$pattern(.*?\%STARTINCLUDE\%\n*(.*?)\%STOPINCLUDE\%\n.*) -->

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

Also, do you know how popular Core Plus is?

This page is NOT actually intended to be edited! It's the page that you would see if you pulled up the Core Plus archives!!!

<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school student, being exposed to formal mathematical proof for the first time, to something like that?

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

BarryGarelick 7/6/05

A coda from Catherine and Barry to kick off comments:

Catherine: Barry, I've forgotten theorems & postulates.

Do you want to add a quick definition?

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

BG: (Offstage voice in the funhouse) Yes, that's a good way of putting it. It is a proposition that is accepted without proof. What is logically deduced from postulates and definitions are theorems, which because they can be deduced, can be proven.

Catherine (in front of crazy mirrors): I think I've got the two mixed up.....

Also, do you know how popular Core Plus is?

BG: (enters, walking on ceiling): Fairly prevalent throughout Michigan and Minnesota. Used in other states too, but those are the main ones. Google on "Bachelis; Core Plus" You'll find a paper he did on it. He did a survey of students in two high schools outside of Detroit; one used Core Plus, the other a normal program. Students using Core Plus did poorly in math in the university. Chris Hirsch, the PI for Core Plus threatened Bachelis with legal action. Tom Parker of MSU did a paper on Core Plus as well using statistical data showing performance in freshman year mathematics; similar to what Bachelis did. Also criticized by Hirsch.

(pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k


I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

---

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson


Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)



All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page


Thanks, Jo Anne, for taking the time to do this!



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

---

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

<!--

• • Set ALLOWTOPICCHANGE = TWikiAdminGroup
  • Set ALLOWTOPICRENAME = TWikiAdminGroup

$pattern(.*?\%STARTINCLUDE\%\n*(.*?)\%STOPINCLUDE\%\n.*) -->

Also, do you know how popular Core Plus is?

<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school student, being exposed to formal mathematical proof for the first time, to something like that?

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

BarryGarelick 7/6/05

A coda from Catherine and Barry to kick off comments:

Catherine: Barry, I've forgotten theorems & postulates.

Do you want to add a quick definition?

(Isn't one of them supposed to be a kind of 'given,' and the other the logical deduction from the given?)

BG: (Offstage voice in the funhouse) Yes, that's a good way of putting it. It is a proposition that is accepted without proof. What is logically deduced from postulates and definitions are theorems, which because they can be deduced, can be proven.

Catherine (in front of crazy mirrors): I think I've got the two mixed up.....

Also, do you know how popular Core Plus is?

BG: (enters, walking on ceiling): Fairly prevalent throughout Michigan and Minnesota. Used in other states too, but those are the main ones. Google on "Bachelis; Core Plus" You'll find a paper he did on it. He did a survey of students in two high schools outside of Detroit; one used Core Plus, the other a normal program. Students using Core Plus did poorly in math in the university. Chris Hirsch, the PI for Core Plus threatened Bachelis with legal action. Tom Parker of MSU did a paper on Core Plus as well using statistical data showing performance in freshman year mathematics; similar to what Bachelis did. Also criticized by Hirsch.

(pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k


I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

---

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson


Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)



All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page


Thanks, Jo Anne, for taking the time to do this!



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

---

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

<!--

• • Set ALLOWTOPICCHANGE = TWikiAdminGroup
  • Set ALLOWTOPICRENAME = TWikiAdminGroup

$pattern(.*?\%STARTINCLUDE\%\n*(.*?)\%STOPINCLUDE\%\n.*) -->

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

It would be like teaching second graders that it doesn't matter whether the earth goes around the sun or vice versa, because all motion is relative per Einstein's theory of relativity. In early grades, it makes sense to teach kids that the planets revolve around the sun. Later, maybe high school but usually college, discussion of relative motion is introduced and students understand that viewing the sun as center of the solar system is for utilitarian reasons but that all reference frames hold. Core Plus' approach of proving SAS for high school students is inappropriate. And the way they have done it is incorrect, to boot.

<!-- this section searches for all subject areas to show them in the drop down menu --> select another subject area

Entries from CorePlus

One thing we tried to do was to create topics for all the major contenders in the curriculum game, constructivist and not, so that people searching for information about some new curriculum they've been handed could find information about it easily (hat tip to David Klein for the suggestion!). As a result, I've created topics for curricula that I personally know nothing about, and CorePlus is one of those.

But BarryGarelick is very familiar with CorePlus, and here is his input on it. Thanks again, Barry -- and you'll see this post back on the page that I took it from! -- Main.CarolynJohnston - 07 Jul 2005

The CorePlus program

Core Plus is a so-called "integrated math" program. It has undergone one set of revisions so far, and I believe is undergoing another one. So far, Western Michigan University which develops the program has received $11 million in grant money from NSF-EHR to do this.

At last glance, Core Plus doesn't introduce the quadratic equation until the 11th grade, thereby rendering many problems difficult or unsolveable until then. (It generally is presented in a first year algebra course).

Also, their treatment of geometry is a bit unusual. In most texts, the congruence relationship between triangles that depends on SIDE ANGLE SIDE (SAS) is stated as a postulate. Core Plus states it as a theorem, and proves it using the law of cosines.

Since the law of cosines is dependent on similar triangles and the SAS congruence theorem itself, some might say this is circular.

I wrote Dr. Hirsch (the PI for Core Plus at Western Michigan University) about this, and he responded as follows:

"With respect to our approach to sufficient conditions for similarity and congruence, it would be helpful for you to carefully examine the development in our texts. See Course 1 Unit 5 for initial work with the Pythagorean Theorem; Course 2 Unit 2 for initial work with similarity via size transformations; Course 2 Unit 6 for development of the trigonometric ratios; Course 3 Units 1 and 3 for development of the Law of Sines and the Law of Cosines; and Course 3 for the proofs of sufficient conditions for similarity and congruence.

"The geometry work has been reviewed by two research geometers: James King and Doris Schattschneider. Professor Schattschneider is working closely with us on the revision of the geometry units.

"Hope this helps. There is really no substitute for a careful examination of the texts themselves."

Disregarding the haughtiness of his answer, I asked for the opinions of three mathematicians on his approach: Hung-Hsi Wu of Berkeley, Jim Milgram of Stanford and Richard Askey of U of Wisconsin.

Jim Milgram pointed out that postulates are not God given. One can assume any number of propositions to be a postulate and then the theorems and corollaries follow logically from it. There is a way in which the SAS congruence relationship between triangles can be proven, but it is an advanced approach to geometry, and one which Core Plus does not rely on in its proof. Dr. Milgram could not be sure that Core Plus was mistaken in its approach without a thorough examination of the text, but said that in any event, such an approach for a high school course was not advisable. (Actually his words were a bit stronger than that).

Dr. Wu is himself a geometer who teaches at U.C. Berkeley. He stated outright that Core Plus' approach to "proving" the SAS congruence relationship using the law of cosines was circular:

"You cannot define sine and cosine, in the usual sense of leg of right triangle to hypotenuse, WITHOUT knowing similarity of triangles. Otherwise the sine and cosine functions would be function of angles OF A PARTICULAR RIGHT TRIANGLE rather than a function of the angle itself. This being the case, using sine and cosine to prove SAS is circular reasoning. So CORE-PLUS teaches INCORRECT mathematics, but what else is new?"

Dick Askey from U. of Wisconsin concurred. (Also Larry Gray of U of Minnesota in his own comments about Core Plus on his website; he is head of undergraduate dept of Mathematics).

Whether Core Plus corrects this in their next version will be "interesting". In any event, even if they succeeded in a proof of a proposition that is normally presented as a postulate, this raises the question of why on earth you would subject a high school