Monday, April 30, 2012

Common Core or Common Excuse?

I spoke at the Board of Education meeting on 25 April 2012.    The remarks (below) are primarily in response to the comments made by the district’s Math Coordinator at the Board of Education Work Session on 4 April 2012. 

Following are my prepared remarks, which I altered slightly when I read them at the meeting.

During the work session on April fourth, the administration presented three alternatives for transitioning to a mathematics curriculum based on Common Core standards. Dr. Lulow indicated that the administration was outlining options in response to “a public concern raised about the quality of the math curriculum and our willingness to make adjustments in the schedule” of a math curriculum review. 

The first two of these options are based on using the current or a Common Core edition of Everyday Math.  The third option allows for the adoption of a different curriculum to begin in fall 2012.  The points in favor of the first two options revolve around the lower cost for professional development and material, and the minimal change required.  I use minimal change to mean, in Mr. Southworth’s words, that “there is a very close, very close alignment” and that there was “not dramatic change coming up.”

Yet elsewhere in the presentation, Mr. Southworth said the Standards are “organized into domains and…. that’s a significant difference in how we have approached mathematics in the past.”  Later he indicated, in regard to which grades were impacted the most, that “the most significant shifts between what we are currently doing and Common Core are occurring at the elementary school.”  He used fractions and numeracy as examples of the extent and nature of the changes required. 

Peter Sherr asked what major things needed to be taught earlier or later, to which Mr. Southworth replied that that was a difficult question and that “we’re really well aligned.”  Yet, from the very same document that the administration used to claim a 92% alignment, the percentage of standards being taught earlier or later in K-5, is 35%.  For the test years (grades 3-5), the percentage is 44%.

The point here is not to highlight inconsistencies, but to show that any way you slice the pie (that’s a reference to fractions), there is a significant change required. 

Still unaddressed, however, is the question of whether Everyday Math is the proper foundation for this transition.  The lack of progress in raising CMT scores, particularly at the Goal and Advanced levels, the widening of the gap relative to other districts, and the slow progress made in closing subgroup gaps in our district, indicate to me that the curriculum is failing our students.  Do we really want to continue with this program under options one and two, in whatever mutated form it might take?

I proposed a fourth alternative.  Begin the curriculum review now.  Begin implementation in fall 2013 for grades two through five, and in fall 2014 for Kindergarten and first grade.  That will ensure that all grades will have almost two full years of an aligned curriculum before the SBAC tests.  You will have the proposed twelve to eighteen months to do the review and preparation.  You will save the expense for the option one or option two changes, which would likely have been discarded when the scheduled 2014 review was concluded.  And, most importantly, you will have taken the first step toward addressing the impact of this failing program on the lost generation of our students.

Thank you.

________________________________________________________________

Some additional comments and thoughts:

In the presentation made by the administration on the fourth, they promised to post a summary of the options and costs outlined by the Math Coordinator.  This was posted, but then removed (I printed a copy).  The interesting thing is that during the meeting (and you can watch the tape), the third option was to “set aside the current set of curriculum and to adopt a new Common Core edition.”   This was later clarified that the option was to “adopt a K-5 Common Core Edition of a math program” (about 25 minutes into the tape).  This appears to open it up to all curricula, not just Everyday Math.  Yet the summary that was posted (and removed) contained the words “All options assume K-5 Core Math package is Everyday Math.”

The Math Coordinator also made the comment that he thought the administration could switch to a new edition of Everyday Math (i.e., the Common Core edition) without BoE approval, albeit with some additional funding.  Yet if you listen carefully to his rationalization for not switching to another curriculum (about 30 minutes into the tape), the same logic (?) should apply to adopting (actually not adopting) the Common Core edition of Everyday Math.  Talk about doubling down on a losing hand!

During the work session, Board member Peter von Braun asked the Math Coordinator “what are the problems that reduce student achievement in mathematics, what causes those problems, and why does this program (Everyday Math) solve those problems.”  The Math Coordinator’s response was to highlight new students coming into the district with different math backgrounds, students with “other handicaps” such as language difficulties, and the fact that Greenwich does not group students by ability, but differentiates in the classroom.  Don’t other districts also have these factors with which to contend?  So why are we falling behind? 

If you listen to the math Common Core discussion during the work session, and I have several times, the essence is that the administration (while not recommending any particular path) is moving toward making a little bit of change to the current curriculum (don’t forget, it is already close) to get us through to 2014, then do a full scale review.  This would be acceptable IF the current program was not failing, and IF that failure was not jeopardizing the future of our children, and IF the alignment was as good as suggested.  Chairman Moriarty said at the conclusion of the conversation that next steps need to be determined, but no plans were made or information requested (as far as I heard).  We cannot afford to wait for 2014. 

I am betting that at some point a group of parents will realize the impact Everyday Math had on their children (when they drop out of college), and will institute a law suit against the Board of Education and the Greenwich Public Schools for educational malpractice.  This has not worked in the past, but with the focus on accountability, who knows what legal avenue some clever lawyer will find to push the issue.

Monday, April 23, 2012

Another Dimension of Everyday Math

“In Unit 9, children will develop a variety of strategies for multiplying whole numbers.  They will begin by using mental math (computation done by counting fingers, drawing pictures, making diagrams, and computing in one’s head).  Later in this unit, children will be introduced to two specific algorithms, or methods, for multiplication: the partial products algorithm and the lattice method.”
A friend of ours, who has a child in the third grade, passed along this page from the third grade homework book for Everyday Math.  This “Family Letter” is used to introduce parents to what is coming up in the next chapter.  Our friend’s note included the words “troubling” and “irritating” to describe the contents of the Family Letter.

To some extent, I feel like this is an episode out of The Twilight Zone, where I can almost hear Rod Serling say “Imagine if you will, a school, like many schools, except that the third grade children count on their fingers because they don’t know their addition facts yet.  You have just entered (pause), the EDM Zone.”  Key weird music. 

Do we really feel comfortable with a program that encourages third graders to count on their fingers and draw pictures to figure out addition problems?  Shouldn’t they have mastered their basic addition facts in first grade?  Not according to the Teacher’s Reference Manual for Grades 1-3.  On page 196, the table indicates that the “Hard facts” (like 8+7) should be mastered by the end of third grade.  At least Common Core is forcing mastery to an earlier grade.

And is this the best way to teach multiplication?  Are our children really going to learn multiplication by counting on their fingers and drawing pictures?  Now if “computing in one’s head” means recalling number facts automatically, then they are partly right.  But I would interpret that phrase as meaning the students are actually figuring out a multiplication problem like 3x8 by thinking something like, “I know 3x4=12, and twice 4 is 8, so 3x8 must equal 12 +12 or 24.” 

Herein lays the problem with Everyday Math, and its proliferation around the US.  It says all the right things, and any teacher or administrator can point to these statements to defend the curriculum.  For example, who can argue with “Automatically knowing basic number facts is as important to learning mathematics as knowing words by sight is to reading.”  In execution, however, it fails miserably.  And when it fails, the publishers can always claim that the implementation by the school was flawed.

Our friend goes on in the e-mail, “So then I read more of the letter, which notes that soon the kids will learn the lattice method of multiplication.  I only glanced at what that method entails but it looks pretty ridiculous. But what's worse is that the letter tells parents why they are teaching this method and the reasons are like "because it is historically interesting"!  Not because it's a method that leads to fewer errors, or one that is faster or some other sensible reason.” 

The exact quote on the Lattice Method is “Third Grade Everyday Mathematics introduces the lattice method of multiplication for several reasons: This algorithm is historically interesting; it provides practice with multiplication facts and addition of 1-digit numbers; and it is fun.  Also, some children find it easier to use than other methods of multiplication.

Let’s take each one of these reasons. 

1.    Historically interesting: this is another meaningless statement.  I am sure there are very few if any third grade students who care about the history of the method dating back to “Hindu origins in India before A.D. 1100.”

2.    Provides practice: EDM is so desperate to convince parents that it provides practice for multiplication and addition facts that it has to use multiplication problems to support its claim.  Isn’t that a bit like saying we practice spelling by spelling words?  And couldn’t the same thing be said about the traditional algorithm?

3.    It is fun: in the Teacher’s Reference Manual, the authors state that “To our surprise, lattice multiplication has become a favorite of many children.”  Why?  Because they are spending time drawing rather than practicing math.  Fun or results?  No wonder they call this Math Lite.

The Teacher’s Manual also claims that the methodology is very efficient.  I have already dealt with that fallacy in a previous post. 


The Manual goes on to say you can use lattice multiplication for “problems that are too large for long multiplication (i.e., the traditional algorithm) or for most calculators…”  Since the traditional algorithm is more efficient, the first half of this statement is also fallacious.  And isn’t it interesting that the justification for not teaching the traditional algorithm is that we can always use a calculator, but we should learn to use lattice multiplication because it can handle problems calculators can’t?  My solution: get rid of the calculators and get rid of lattice multiplication (i.e., stop wasting time), and focus on the traditional algorithm.  Better yet, get rid of Everyday Math!  Send it to (pause) The Twilight Zone.

Sunday, April 22, 2012

Test Questions to Question

You may have seen articles in the NYT and WSJ (and elsewhere) about a question given on the NYS eighth grade English Language Arts standardized test. 




You may have also seen a blog posting from The Happy Scientist, a Florida blogger who was preparing questions to aid students taking the science Florida Comprehensive Assessment Test, the FL equivalent of Connecticut’s CMTs.  He presented some interesting findings, and even more interesting feedback from state education officials.


My comment was:
My brother is a former Florida public school teacher, who went on to private, parochial, prison and military schools (the prisoners were the most-well behaved). He had similar concerns about standardized tests.
Two comments:
1. while it is a possible defense to say that the students should not know this or that at the fifth grade level (see comment 2), thereby eliminating a correct response, why not simply come up with three responses that are clearly not (scientifically) correct? Yes, there is always going to be one or two that slip through, or there will be a scientist or mathematician who can posit some reasonable test or answer (as some of your knowledgeable readers have shown), but any reasonable review process by real scientists or good science teachers would at least indicate which answers/definitions need to be examined in more depth before inclusion.
2. the state administrator's response that students should not know X, since it is not in the standards for fifth grade, is reflective of what I call the RTTM (Race to the Middle). The thinking is that we only need to get all of our students to the NCLB "Proficient" level (as it is called here in Connecticut), so any effort beyond that (Gifted and Talented, Advanced Learning, self study, good teaching, etc.) is a waste of time and money. Yet everyone is focused on college readiness and on preparing students for the STEM subjects. Proficient does not begin to cut it if one is interested in engineering, especially when many of the state standardized tests actually award Proficient level status for what the NAEP Math and ELA would consider Basic or Below Basic.
Check out the Math and Reading NAEP vs. state test comparisons:
One question: has anyone actually had experience in using FOIA to get access to the test questions?
So, what’s your point, Brian?  Well, this certainly couldn’t happen here in Connecticut, could it?  Our daughter brought home a practice test for the third grade CMT Mathematics test.  Of the 56 problems on the test, six were estimation problems, three relating to arithmetic operations and three relating to measurement.  Why pick on estimation problems?  This is one of the strands, as mentioned before, with which our students have a difficult time and (coincidentally?) on which Everyday Math places much emphasis.  Not wanting to blame everything on Everyday Math (although my office is still a mess), I wanted to see if there was another explanation for the relatively low scores on this strand.

In examining the three arithmetic operations questions, one was ambiguous (i.e., two of the answers could be correct, depending on how you interpreted the question), one was unambiguous only because the two interpretations yielded the same answer, and one was clearly written.  See if you agree with my thinking.

Question 1: What is the closest estimate of $54 - $38?  Round to the nearest ten.
A.   $10
B.   $20
C.   $30
D.   $90
The answer is A, because you round the 54 to 50 and the 38 to 40, and then subtract.  My daughter put down B, and I agreed with her, since the actual difference of $16 rounds to $20.  I can understand both answers, which is actually the problem.  The wording is poor, as either answer would be defendable. 

Question 2: What is the best estimate of 64 + 77?
A.   130
B.   140
C.   169
D.   180
Only 140 would fit both possible interpretations (round first then add, or add first then round).  Interestingly, no instructions are given for rounding, as in the first problem above.  So my vote is for ambiguous wording.

Question 3: There are 48 pencils and 53 pens in the school supply closet.  Round the numbers to the nearest 10 to estimate the total number of pencils and pens in the closet.
A.   80
B.   90
C.   100
D.   110
Much better wording here, including the clarifying word “numbers” indicating that rounding comes first.  If the first problem had said “Round the numbers to the nearest ten.” most of the ambiguity would have been removed.

I may be off-base on the first problem, as our children are taught (I hope) to round, then subtract or add (that is the useful skill).  This review raises three questions in my mind: First, why make it confusing by having two possible correct answers, and providing insufficient wording to draw a single conclusion?  Second, why is there different wording between problem 1 and problem 2?  And third, if the first problem confuses a Math ALP student, what percent of other students are going to be confused (especially since they are learning using Everyday Math)?

The point is that with only a quick look at a practice test, presumably based on actual test guidelines, I was able to find some ambiguous questions (or answers).  Maybe we have an issue, like Florida and New York.  To be fair, the example tests problems on the CSDE site appear well written.  So this might be a one-off issue with a bad practice test, and the real tests are okay.  But doesn’t that mean that the low strand scores are due to something else, like Everyday Math?

And don’t forget that one of the reasons Everyday Math focuses on estimation is so that the students can check the results of their calculations when they use a calculator.  If we eliminate calculators from K-5 (like Texas wants to: http://www.chron.com/news/houston-texas/article/Early-math-lesson-Put-calculators-aside-3496172.php), not only would we have students who could do their basic calculations better, but we wouldn’t have to spend so much time on estimation.  Sounds like a win-win situation to me.

Saturday, April 21, 2012

Congratulations GHS Math Team

Congratulations to the Greenwich High School Math Team on winning the Connecticut State Association of Mathematics Leagues annual competition.  This group of seven students, guided by their coach Gordon Jones, beat the competition from across the state, giving GHS its fourth straight title. 

I had a chance to meet one of the senior co-captains.  He was a very engaging, well-spoken young man, who obviously is also very good at mathematics.  He appears headed for an engineering degree at his chosen university.  Best of luck!


This was a great performance by all of these young people.  Greenwich should be proud.  And thanks to the team advisor and coach.


Saturday, April 14, 2012

A Place to Go for Math Fun – Review of MathAlive! at the Smithsonian

We just returned from our vacation, which included a trip to the MathAlive! exhibit at the Smithsonian Institution in Washington, D.C.  The exhibit is in the Smithsonian International Gallery in the S. Dillon Ripley Center (a little difficult to find, but it is near the Castle). 

Our daughter (eight years old) had a good time, as there were many interactive displays.  Most of the math was slightly more advanced than what she can do, and was probably appropriate for sixth to eighth graders.  There wasn’t a lot of explanation of the details of the math, which would have made it interesting for high school grades. 

My favorite was the snow board, which had three individuals racing down a slope.  Two controlled snow boards and one had a joy stick to control the angle of the board versus the snow (hence the connection with math).  You really don’t think too much about math as (at least in my case) you try to survive the course, but you do live to snow board another day. 

There was also a music related exhibit which showed how fractional notes/beats from a variety of instruments make up a tune.  Many of the interactive games will be easy for any student with video game experience (hence my need to do the cat rescue game about 20 times before I got to level two). 

The exhibit was not crowded when we went, and we spent about two hours.  I would not rate it as a must see, but an interesting use of time for the right age group.  The exhibit did do a good job of showing the variety of applications of math to our world.  And you can pick up a brochure about where math appears in other Smithsonian museums and research centers.  Look for the Pocket Guide to Math at the Smithsonian. 

The exhibit runs through 3 June 2012, and you can get more information at:
http://www.mathalive.com/about-the-exhibit.html

Saturday, April 7, 2012

Board of Education Work Session

I attended most of the Board of Education work session held 4 April 2012.  I missed the first portion (baseball coach’s meeting), which dealt with the Common Core State Standards and the impact on Greenwich Public Schools.  I will be reviewing the discussion once it is posted on the web, but if the presentation attached to the agenda is any indication, the administration is still denying that the changes in standards between the old CT standards and the Common Core are significant:


” In 2011, the State provided districts with Crosswalk documents that demonstrate a correlation between the new CCSS, [and the] Connecticut State Standards for Mathematics and Reading and Language Arts. These documents demonstrated a 90-92% alignment among the CCSS, former State Standards and current assessments.”


In reality, the Reading and Language Arts match is only 80%, and that is before you subtract the weak matches, which push the total of changes required to 32% (i.e., only a 68% match where the match is considered Excellent or Good).  The Math match is 92%, but that is before subtracting another 24% for weak matches, bringing the more realistic match to only 68% for Math also. 


Now it is possible that the Greenwich standards exceed the old state standards, so that our specific match rate would be higher.  I have seen no evidence of this coming from the administration, and the fact that they continue to quote the source document match rates for Math makes me believe that the Greenwich standards are not different. 


Based on what was posted on the internet, the district Math Coordinator presented three alternatives to transition to the Common Core for math.  All of these alternatives involved keeping Everyday Math, or changing (notice I didn’t use the word “upgrade”) to the CCSS edition of Everyday Math. 
http://www.greenwichschools.org/uploaded/district/Board_of_Education/meeting_materials/2011-12_meetings/4-4-12_meeting/Addendum_to_4-4-12_Transition_to_Math_CCSS.pdf


Missing as an alternative is to initiate a Math Curriculum review now so that we can transition to a program that works as soon as possible.  Any of the suggested pathways would only delay the termination of the failing Everyday Math program, extending the lost generation of our children another few years.  Even under the new Board Policy for curriculum reviews the schools would have 12-18 months to complete such a review, likely putting any implementation into 2015-16 school year.


I also noticed that this slide only addresses K-5.  Are there no changes to be made for the rest of the grades? 
I am sure I will have more comments on this when I get to review the video.



Sunday, April 1, 2012

Another Thing to Blame on Everyday Math???

While I was cleaning up my office yesterday morning, I uncovered (literally) the Teacher’s Reference Manual for Everyday Mathematics ®, Grade 1-3, 2007 (the edition currently used in our schools).  Over the last few months I have marked various pages for ease of reference to material which I find particularly interesting, where you can define interesting any way you want.  I flipped to the page (page 96) which discusses Algorithms and Procedures.  As I read, I “reacquainted” myself with all of the concern and angst associated with this failed program. 

The authors include the following paragraph at the start of the section on Computational Algorithms (section 11.1.1, page 97): “Several teachers have asked the Everyday Mathematics authors about the role of computational algorithms in elementary school mathematics.  Should traditional paper-and-pencil algorithms be taught? Should children be expected to use these algorithms to solve complex computational problems? Should calculators be used in the classroom?  If so, in which circumstances and under what conditions?”

The authors then spend the next one and one-half pages discussing what, to me, are obvious answers (yes, yes, no, with no answer needed for the fourth question). 

Their answer, however, included two stories, “the need to rethink the school mathematics curriculum in light of the widespread availability of calculators and computers outside of school,” a discussion of estimation and approximation, a discussion of “buggy algorithms”, and a rationale for introducing different algorithms.  When they finally got to the end of the section, which I took to be their position, which they call a “moderate” position, they conclude that :

“(D)uring the early phases of learning an operation, Everyday Mathematics encourages children to invent their own procedures.  Children are asked to solve arithmetic problems from first principles about situations in which operations are used, before they develop or learn systematic procedures for solving such problems.  This helps them to understand the operations better and also gives them valuable experience solving nonroutine problems.

Later, when children thoroughly understand the concept of the operation, several alternative algorithms are introduced.”

As we have come to expect with Everyday Math, there are plenty of cautions and caveats which make it appear that they have considered all of the angles.  For example: “Children certainly still need:

·         To know the meanings and uses of all the arithmetic operations in order to function in the practical world and to succeed in mathematics in high school and beyond;

·         To know the basic addition and multiplication facts automatically, especially to solve mental-arithmetic problems in our technological society [I guess subtraction and division facts don’t count anymore];

·         To understand and be able to apply paper-and-pencil algorithms for addition, subtraction, multiplication, and division of whole numbers, decimals, fractions, especially in an environment of standardized testing [heaven forbid that you would actually have to know this outside of a test].”

If you are looking to find the core of what is wrong with Everyday Math, look no further.  There are so many issues with this philosophy (a philosophy reflected throughout the program), it is hard to know where to start.  Let me begin with:

1.    Inventing incorrect algorithms, which stick with the child

2.    Inventing algorithms which do not apply to the generalized case.

3.    Putting the burden on teachers to correct all of the errors, when a teacher may not have the background to realize all of the flaws (see number two).

4.    The introduction of calculators.

5.    The introduction of estimation and approximation, with a justification that they need to be able to judge the output of calculators and computer spreadsheets.

6.    The concept of solving from first principles for a grade school aged student.

Regarding the first three, several teachers with whom I spoke said that this was not a problem for them, since they don’t teach it that way to begin with.  My guess is that these are the more astute teachers, who recognize the deficiencies of the Everyday Math program.  This does raises two questions, however.  First, why not use a curriculum which allows proper teaching of these critical elements of arithmetic?  Second, if the teacher is not astute enough to recognize the deficiencies, are they really equipped to fix the errors?

And speaking of fixing incorrect invented algorithms, the authors of Everyday Math wrote: “Research carried out in the past 30 years by Kurt Van Lehn and others has shown that many children develop “buggy” algorithms that resemble standard procedures but do not work properly.  In subtraction, for example, some children always subtract the smaller digit from the larger digit.  Van Lehn has shown that bugs such as this develop because children are trying to carry out procedures they don’t understand and can’t remember well enough to reproduce accurately.  Procedures that are well understood, on the other hand, are more easily recalled, are more easily “repaired” when they are not recalled accurately, and are more easily modified to fit new situations.”

So EDM says invent first, and solve problems from first principles, “before they develop or learn systematic procedures….  This helps them to understand the operations better…”  A little later they say, “prematurely teaching traditional paper-and-paper algorithms can foster persistent errors and “buggy” algorithms…”  But the research they quote says by inventing first, they introduce bugs because they don’t understand.  This is a perfect example of the authors of Everyday Math getting, as an old boss of mine use to say, “tangled in their own underwear.” 

Correct me if I’m wrong, but isn’t it the teacher’s role to teach a topic so that the child understands?  But that is not how Everyday Math works.  The student needs to learn for himself. 

Interestingly, the authors also highlight research done by Van Lehn, who “said this about using the traditional subtraction algorithm in some of his research: ‘(O)rdinary multidigit subtraction….is a virtually meaningless procedure [for] most elementary school children… When compared to procedures they use to operate vending machines or play games, subtraction is as dry, formal and as disconnected from everyday interests as the nonsense syllables used in early psychological investigations were different from real words.  This isolation is the bane of teachers…’ ” 

So the researcher doesn’t like the traditional subtraction algorithm, because it is meaningless and dry (i.e., from the Everyday Math school of fun math, no doubt).  The authors of Everyday Math make the researcher sound as confused as they are. And you can provide your own joke about what vending machines have to do with this topic.

But it gets more confusing.  The Everyday Math authors go on to say “(T)he program aims to make children active participants in the development of algorithms.  Such participation requires a good background in the following three areas:

·         The decimal system for number writing In particular, children need to understand place value.

·         Basic facts To be successful at carrying out multistep computational procedures, children need to know basic facts automatically.

·         The meaning of operations and the relationship among operations To solve 37 – 25, for example, a student might reason What number must I add to 25 to get 37?”

Since Everyday Math fails miserably at getting anywhere near mastery of basic facts, the whole premise of this invented algorithm route falls apart.  And doesn’t having “a good background” mean that you understand?  And if you understand what addition and subtraction mean, should you really have difficulty with learning a method (the efficient traditional algorithms, developed over centuries by intelligent people) for adding or subtracting multi-digit numbers, instead of spending (wasting?) time inventing a method that may not be correct, and may not generalize?
But back to our list.  Calculators!   The authors of Everyday Math say “In the modern world, most adults reach for a calculator when faced with any moderately complicated arithmetic computation.  This behavior is sensible and should be an option for children, too.”  (Wait for it)  They then backpedal and say: “Nevertheless, children benefit in the following ways from developing their own noncalculator procedures:

·         Children are more motivated when they do not have to memorize traditional paper-and-pencil algorithms without understanding why they work [stop and think about this last sentence and see if it means anything to you]…(C)hildren generally understand their own methods, as obscure as they may be to others.”

Notice, “their own” method, not traditional methods.  So calculators are okay, but children benefit from using invented algorithms.

So we have calculators and computers, and we use apps and spreadsheets.  Then, in order to check that these tools are giving us the right answer, we need to be able to estimate and approximate.  However, if we just spent the time wasted learning estimation and approximation (which should be learned later, but not in grades 1-4) actually practicing our number facts and algorithms, we not only would learn the facts and procedures to mastery, but we would be developing number sense.  This would make learning estimation and approximation very easy, at the appropriate time.  And it should be noted that Everyday Math fails as a curriculum for estimation and approximation.  How do I know?  Check the “estimating solutions to problems” strand results on the grades 3-5 CMT’s.  The only scores that come close to being as low as this strand are the “approximating measures” and “mathematical applications” strands.  The mathematical applications strand is education-speak for solving word problems.  Isn’t it strange how the lowest scores are in the areas that Everyday Math considers important?

I must admit to a small moment of panic when I saw the words “first principles.”  I flashed back to several engineering and graduate school classes, where we had exam questions asking us to begin with first principles and prove this, or solve for that, or show how to do something.  Example: Imagine you are stranded on a deserted, sandy island.  Describe step-by-step what you would do to manufacture a certain silicon-based polymer (polydimethysiloxane).  Be sure to include all chemical reactions necessary, and indicate where you would source the materials required for both the synthesis of the polymer and the production machinery.  You have three blue books (exam notebooks) and three hours.  Go. 

Now I recognize that this may not be exactly the type of learning experience the authors of Everyday Math were getting at when they used the phrase in the manual (or maybe they had a similar experience in education school, and are now getting back at the world), but solving arithmetic problems and inventing algorithms from true first principles for first through third graders is an interesting, albeit comical, notion.  This is like asking a student to act like an expert, without the appropriate knowledge and experience.

In summary, I am not sure if the authors ever clearly answered the questions asked, and if they did, I am sure the qualifications and provisions they attached to those answers made any conclusion meaningless. 

Oh, and my office is still not clean.  Another thing to blame on Everyday Math???