From the agenda for the Monday, May 9, Board meeting:

“Pamela Francis, Marybeth Irvin and Deirdre Bauer review information and present the recommended Math Program and Implementation Plan for Board consideration.”

What will be announced on Monday is anyone’s guess.  The previous updates to the Board on the pilot process have not contained any indication of the reception received by any of the programs under consideration.  Just to recap, the three programs under consideration are:

1. Investigations (2nd Ed.): Yes, the current program, controversial though it may be, is being pilot tested alongside the two other programs identified last spring by the Math Review Committee.  It’s possible that the Math Leadership Team will simply recommend that no change in the math program be made.

2. enVision Math: This program has the same publisher (Pearson) as Investigations and it is possible to integrate Investigations units into the enVision program.  This is the manner in which enVision was pilot tested in SCASD.

3. Math Expressions: Released in 2005, this program (according to its publisher, Houghton Mifflin Harcourt) “combin[es] aspects of traditional approaches with the most powerful elements of reform teaching”.

Both enVision and Expressions were authored by math educators active in developing and implementing the NCTM standards, and both programs are touted as “standards-based”, as is Investigations.  The most substantial differences between  enVision or Expressions and what we have now in SCASD is that these two programs make use of a textbook and do not avoid the use of worked examples and other forms of direct instruction.

Of course all the programs claim to meet the Common Core Standards (also now the PA state standards), but the authors of strict constructivist programs like Investigations and Everyday Math have published objections to the new standards, perhaps because their programs do not align well with Common Core.

UPDATE: It now appears that this item has disappeared from the May 9 agenda and will be discussed on May 23.

The National Math and Science Initiative has put out a Parent’s Guide with ideas about how to help our kids be successful. Among the recommendations:

Grades 1 through 6 are about creating interest and mastering basic skills. Think and speak positively about math and science. Pay attention to math and science teaching in your child’s elementary school. Review your child’s science and math homework. Don’t perpetuate the outdated cultural bias that girls aren’t as good at math. By grade 5 make sure your child has mastered the necessary skills.

In grade 7 – Talk to teachers about whether or not your child is performing at grade level. This is important, because if your child is not performing “at level” in math, science, and English, now’s the time to get some extra help. Likewise, if your child shows interest in math and science, now’s the time to ask about special opportunities for summer or advanced work. Register your child to take Algebra I in the eighth grade. It’s essential preparation for courses to follow.

For grade 8 – If your child has a specific career in mind, make sure he or she is taking the necessary math and science courses make that choice possible. Encourage your child to take Pre-AP or “honors” courses. Stress the importance of taking as much math and science as possible in high school. Your child should take Algebra I in the eighth grade to prepare for math courses to follow. Algebra is the gateway course for math and science literacy for all students, not just the academically gifted.

In grade 9 – Students should take Geometry, part of the essential foundation to move forward in today’s technologically competitive world. Your child also should take Pre-AP Biology to put them on a path to take both Chemistry and Physics before graduation.

For grade 10 – This is the time to take Algebra II, another keystone for success in today’s world. Your child should also take Pre-AP Chemistry, which is the prerequisite to all of the AP science courses.

In grade 11 – This is the time to take Algebra II, Pre-Calculus, or AP Statistics, to acquire an even stronger math and science foundation. Usually students take Pre-AP or honor physics at this point. Students can also take a second science course such as AP biology or AP chemistry. This is the year most students start taking AP classes like AP English language or AP science.

Let’s compare and contrast the following first-grade assignments from about a third of the way through the year:

From Investigations:

From Singapore Math:

So, in which assignment is something about math actually being taught and learned?

If this weren’t so stupid, it would almost be funny. Almost.

I recently came across a reference (NY Times, behind paywall) to a report in the journal Science describing a study suggesting that testing can be used effectively to reinforce learning. The study, by psychologists at Purdue University,

… found that students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods.

One of those methods — repeatedly studying the material — is familiar to legions of students who cram before exams. The other — having students draw detailed diagrams documenting what they are learning — is prized by many teachers because it forces students to make connections among facts.

I’m mentioning this study not in support of the PSSAs or other testing, but because of this quote in the Times article:

Howard Gardner, an education professor at Harvard who advocates constructivism — the idea that children should discover their own approach to learning, emphasizing reasoning over memorization — said in an e-mail that the results “throw down the gauntlet to those progressive educators, myself included.”

“Educators who embrace seemingly more active approaches, like concept mapping,” he continued, “are challenged to devise outcome measures that can demonstrate the superiority of such constructivist approaches.”

This a priori assumption of which approach is best reminds me of true-believer syndrome among people who believe in psychic powers.  If a test shows that your favorite approach to learning is not working, this ought to lead researchers (and administrators and school boards) to question their initial assumption rather than send them looking for a new test.

From Katharine Beals’ Out in Left Field blog, a lovely comparison of 5th grade assignments:

I. The third problem set in “Unit 4: What’s that Potion?” in the 5th grade (TERC) Investigations Math Student Activity Book, Session 1.1:

Everyday Uses of Fractions, Decimals, and Percents

List in the spaces below the everyday uses you find for fractions, decimals, and percents. Cut out your examples from used magazines and newspapers, and attach them to this sheet.

Everyday Uses of Fractions

[space to attach]

Everyday Uses of Decimals

[space to attach]

Everyday Uses of Percents

[space to attach]

II. The third problem set in “Unit 3: Fractions” in the 5th grade Singapore Math Primary Mathematics Workbook 5A, pp. 53-54:

Add. Give each answer in its simplest form.

1.

(a) 7/8 + 3/4 = 7/8 + ?/8 =

(b) 2/3 + 4/9 = ?/9 + 4/9 =

(c) 4/5 + 3/10 =

(d) 3/4 + 7/12 =

(e) 5/6 + 2/3 =

(f) 1/2 + 9/ 10 =

2.

(a) 1/6 + 3/4 = ?/12 + ?/12 =

(b) 5/9 + 1/2 = ?/18 + ?/18 =

(c) 1/2 + 3/5 =

(d) 2/5 + 3/4 =

(e) 9/10 + 1/6 =

(f) 3/10 + 5/6 =

There was an article about the Pennsylvania System of School Assessment tests (PSSAs) in the CDT yesterday, and CNN even came to SCASD last week to do a story on parents who pull their kids out of standardized tests.  With important implications for standardized testing dictated by No Child Left Behind (NCLB), it’s not surprising that this has become a hot political issue.  It’s become critically important to schools and districts that a high proportion of their students register as “proficient” or “advanced” on these tests.  In SCASD, depending on the grade level and the year, about 70% – 90% of students are “proficient” or “advanced” in math, but what does this mean in terms of students’ specific math skills?

To answer this question, one has to understand that NCLB requires states to label performance on their tests as “advanced”, “proficient”, “basic”, and “below basic”, but how those labels are applied is up to each state and can vary from year to year.  State standards can be compared, however, by examining student performance on a national test called the NAEP.  The U.S. Department of Education has done this analysis, and Pennsylvania proficiency standards are near the bottom of the pack.  In other words, the threshold for proficient in Pennsylvania is a lot closer to Alabama’s threshold than it is to Massachusetts’s.

But what about the actual test questions?  What kinds of questions are SCASD students answering correctly when they are rated proficient or advanced?  Example questions and scoring guides for past years’ PSSAs are available here, and it is worth taking a look.  Here is a question from the 2009-10 fifth-grade math test:

It is important to know how to measure using a ruler, but to me this question seems too easy to be on the 5th grade test and, in fact, measuring in fractions of an inch with a ruler is something expected to be mastered in 3rd grade under the Common Core Standards, which recently became Pennsylvania’s.  Only 46% of Pennsylvania fifth-graders were able to measure the paper clip correctly on last year’s test.

Another question from the 5th grade test:

17. A. Mr. Stevens bought 15 books for his store. He paid $11 for each book. He then sold all 15 books for $25 each.  What is the difference between the total amount of money Mr. Stevens sold the books for and the total amount of money he paid for the books? Show or explain all your work.

17.B. Mr. Stevens has $220 to purchase books at $8 each (tax included).  How many books can Mr. Stevens buy? Show or explain all your work.

Again, this is something 5th graders ought to be able to do, but it matches the 4th grade, not the 5th grade, standard under Common Core.

We all want our schools to be making “adequate yearly progress” (to use the NCLB term), but what’s much more important to parents is whether their own kids are progressing at the rate they ought to be.  Up until now, Pennsylvania and SCASD standards have not been very helpful in making that determination.  Maybe this will change once the new Common Core Standards are implemented across Pennsylvania, but even those standards are lacking in key respects.

One good way to assess your child’s math progress at home is with the free placement tests available at SingaporeMath.com.  For those not familiar with Singapore Math,

In the United States the Singapore Math Method of teaching mathematics is based on the primary textbooks and syllabus from the national curriculum of Singapore. These textbooks have a consistent and strong emphasis on problem solving and model drawing, with a focus on in-depth understanding of the essential math skills recommended in the NCTM Curriculum Focal Points (National Council of Teachers of Mathematics), the National Mathematics Advisory Panel,  and the proposed Common Core State Standards.

The St. Vrain School District, located just north of Denver, has replaced TERC’s Investigations math curriculum with Math Expressions, one of the programs currently undergoing pilot testing for potential adoption in SCASD.  Alignment with the new Common Core Standards was a key factor driving the change:

One reason the district’s textbook adoption committee chose Math Expressions, George said, is that it aligns well with the new state standards.

Some math concepts will be taught in different grades than they have been to meet those standards.

Fractions that Wilson previously taught in fifth grade are taught in fourth grade in Math Expressions, Wilson said.

It remains to be seen whether cost will enter into the Board’s decision in SCASD, where cutbacks are being planned to deal with a reduced state appropriation.  In the Colorado district, which has about four times as many students as SCASD, it wasn’t cheap:

[Math Expressions] will cost $925,000 to purchase, based on this fall’s expected enrollment, and the district will pay as much as $200,000 per year for workbooks for students in kindergarten through second grade, according to the school district. The other programs the district considered had similar or higher costs, George said.

As the Colorado administrator points out, the costs on all these programs are similar, or higher.  It’s also worth noting that programs without textbooks that rely exclusively on “consumable” workbooks may be more expensive on an ongoing basis.

Marybeth Irvin, Deirdre Bauer, and Pam Francis provided another update on the math curriculum piloting for the SCASD Board of Directors this evening.  The pilot is going according to schedule and the plan is for the administration to submit its recommendation to the Board by May 9th.

The discussion following the presentation was brief and focused on implementation.  Director Dorothea Stahl asked if a plan was being developed for staging the implementation of whatever curriculum is chosen, so that it wouldn’t be implemented “across the board” immediately.  Marybeth Irvin said that such considerations were being made, along with the costs of the curriculum materials and staff development.  Director Penni Fishbaine said that she thought that many parents and teachers in the District would be up to the challenge of an implementation that was not necessarily phased in gradually.

On March 5’s Weekend Edition Saturday, NPR’s Scott Simon interviewed Stanford mathematician Keith Devlin on what is going on lately with elementary mathematics.  Devlin is very friendly to this new way of teaching math, with it’s emphasis on number sense and conceptual understanding.  He’s right – attention to number sense and conceptual understanding is important and these are things that are emphasized in TERC’s Investigations.   Devlin is also right, however, when he says that:

“You cannot become good at algebra without a mastery of arithmetic, but arithmetic itself is no longer the ultimate goal.”

No one these days sees mastery of arithmetic as an end in itself.  What’s much more important is that being successful in algebra requires facility with arithmetic and math facts and this is where TERC’s Investigations has been failing kids in SCASD.  TERC sees memorization of math facts as being in competition with true understanding, and sees the process of memorizing these facts as making kids “miserable“.  Devlin doesn’t see it that way:

“Sooner or later, and I would always recommend soon as possible, you actually do have to sit down and learn the multiplication tables.  There is no alternative to learning them.  From what we know about the way the brain works, we have a strong reason to believe that the only way to actually master the multiplication tables is to learn them linguistically, almost like the chanting that you and I probably had to do, but if all you do is master them as a chant you will never be able to make sophisticated uses of numbers.”

A parent and math remediation specialist has made a video showing a third grader attempting to add some large numbers using techniques she learned in TERC’s “Investigations” and using the “standard algorithm” of stacking the numbers and carrying:

The comments at the end of the video are especially good.


via kitchentablemath, the sequel