We’ll continue to examine the expectations outlined in the “Action Plan” that was introduced last spring as part of SCASD’s “Complete Elementary Math Program” that includes TERC’s Investigations.  The Action Plan was intended to assuage the concerns of Board members and parents that Investigations is unchallenging and does not give enough attention to to computational fluency, knowledge of math facts, or algorithms.

In this post, we’ll look at when standard algorithms for whole number operations (like carrying, borrowing and long division) are expected to be mastered.  As with our previous comparison of fact fluency expectations, we’ll present expectations along with what is expected in California and Massachusetts (MA is two documents here and here).  California’s and Massachusetts’s are consistently rated as the best sets of math standards in the nation and the Action Plan was supposedly modeled on these standards.

1st Grade

SCASD: “1 digit +/- horizontal and vertical notation; 2 digit +/- (w/out regrouping)”

California: “Solve addition and subtraction problems with one- and two-digit numbers”

2nd Grade

SCASD: “2 digit +/- (regrouping); 3-4 [digit] +/- (w/out regrouping)”

California: “Find the sum or difference of two whole numbers up to three digits long.”

Massachusetts: “Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition (two 3-digit numbers and three 2-digit numbers) and subtraction (two 3-digit numbers).”

3rd Grade

SCASD: “3-4 digit +/- (regrouping); 2 by 1 digit x (with and w/out regrouping)”

California: “Find the sum or difference of two whole numbers between 0 and 10,000; Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 × 3 = __); Solve division problems in which a multidigit number is evenly divided by a one-digit number (135 ÷ 5 = __).”

Massachusetts: “Add and subtract (up to four-digit numbers) and multiply (up to two-digit numbers by a one-digit number) accurately and efficiently”

4th Grade

SCASD: “2-3-4 digit by 1 digit x (regrouping); 2 digit by 2 digit x (regrouping); 2-3 digit by 1 digit ÷”

California: “Solve problems involving multiplication of multidigit numbers by two-digit numbers; Solve problems involving division of multidigit numbers by one-digit numbers”

Massachusetts: “Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit numbers), and multiplication (up to three digits by two digits); Demonstrate in the classroom an understanding of and the ability to use the conventional algorithm for division of up to a three-digit whole number with a single-digit divisor (with or without remainders).”

5th Grade

SCASD: “2-3 digit by 2 digit x (whole numbers); 4-digit ÷ 1-digit Division; 4-digit ÷ 2-digit Division (multiples of 10)”

California: “Demonstrate proficiency with division, including … long division with multidigit divisors.”

Massachusetts: “Accurately and efficiently add and subtract whole numbers.  Multiply and divide (using double-digit divisors) whole numbers.”

In summary, it looks like SCASD’s expectations match those in MA in places, but more often is somewhat behind MA and a full year behind CA.  Note that, even by 5th grade, students in SCASD are not expected to know how to divide using a 2-digit divisor that is not a multiple of ten (e.g., 4,387 ÷ 37).  Note also that SCASD’s expectations seem to depend on whether “regrouping” (carrying and borrowing) is required.  SCASD 2nd graders will get practice with carrying only when adding 2-digit numbers, but not with 3- or 4-digit numbers.  What is the rationale for this?  It would seem that once the concepts behind carrying have been understood that those concepts would generalize readily to problems with more digits, as is done in CA and MA.  Finally, why does SCASD limit itself to 4-digit dividends and 3-digit factors, even by 5th grade?  If the division algorithm, for example, has been taught and understood, why can’t it  be applied to 84,387 ÷ 3 just as easily as it can to 4,387 ÷ 3?

In the final installment in this series, we’ll examine how fractions are covered in SCASD as well as in CA and MA.

Here are more content area standards from TIMMS 4th grade listings –

Geometric Shapes and Measures: Lines and Angles
1. Measure and estimate lengths.
2. Identify and draw parallel and perpendicular lines.
3. Compare angles by size and draw angles (e.g., a right angle, angles larger or smaller than a right angle).

Geometric Shapes and Measures: Two- and Three-dimensional Shapes
1. Identify common geometric shapes.
2. Know, describe, and use elementary properties of geometric figures.
3. Classify and compare geometric figures, (e.g., by shape, size or properties).
4. Recognize relationships between three-dimensional shapes and their two-dimensional representations.
5. Calculate areas and perimeters of squares and rectangles of given dimensions
6. Determine and estimate areas and volumes (e.g., by covering with a given shape or by recognizing that area is conserved).

Geometric Shapes and Measures: Location and Movement
1. Use informal coordinate systems to locate points in a plane.
2. Recognize and draw figures with line symmetry.
3. Recognize and draw reflections and rotations of figures.

Data Display: Reading and Interpreting
1. Read data from tables, pictographs, bar graphs, and pie charts.
2. Compare information from related data sets (e.g., given data or representations of data on the favorite flavors of ice cream
in four or more classes, identify the class with chocolate as the most popular flavor).
3. Use information from data displays to answer questions that go beyond directly reading the data displayed (e.g., combine data, perform computations based on the data, draw conclusions, and make predictions).

Data Display: Organizing and Representing
1. Compare and match different representations of the same data.
2. Organize and display data using tables, pictographs, and bar graphs.

Maybe we’d do better on some of these than the Number areas? Or, looking at the PSSA scores by area — maybe not. Again, I’d be curious to know when these are covered in Investigations – let me know if/when you come across these kinds of problems!

In a shocking development, a judge in Seattle has ruled in favor of parent, university professor, and a retired math teacher who sued the school district there to get them to reconsider their constructivist math curriculum choice, Discovery Math.   The district is indicating that it will appeal.  From the decision:

“The court finds, based upon a review of the entire administrative record, that there is insufficient evidence for any reasonable member to approve selection of the Discovering series.”

Here is the story in the Seattle Post-Intelligencer.

While parent groups all over the country have been struggling to change their districts’ math programs, I have not heard of a lawsuit being successful before now.  Even the plaintiffs’ own lawyer was initially pessimistic about their chances.

Despite this news I am personally opposed to a lawsuit here, and I say that knowing that some parents will disagree with me on this.  Lawsuits can be divisive and counterproductive, and I am confident that our Board is beginning to take steps on its own to address our math problems.  This may be naive, but I still think it’s early in the game for us in State College because successful efforts to change the curriculum in other districts across the country have generally taken longer than a few months.

Thanks to the reader who sent the link to this story about the sources of math anxiety in girls:

Little girls may learn to fear math from the women who are their earliest teachers. Despite gains in recent years, women still trail men in some areas of math achievement, and the question of why has provoked controversy. Now, a study of first- and second-graders suggests what may be part of the answer: Female elementary school teachers who are concerned about their own math skills could be passing that along to the little girls they teach.

Young students tend to model themselves after adults of the same sex, and having a female teacher who is anxious about math may reinforce the stereotype that boys are better at math than girls, explained Sian L. Beilock, an associate professor in psychology at the University of Chicago.

But boys have their own set of math problems, according to Richard Whitmire, author of the new book “Why Boys Fail: Saving Our Sons from an Educational System That’s Leaving Them Behind“.  From the book’s description on Amazon:

Even in their traditionally strong subjects of science and math, boys are hit at a young age with new educational approaches, stressing high-level reading and writing goals that they are developmentally unable to achieve. The gap between male and female achievement has reached the college level, where only 40 per cent of graduates next year will be male.

As I learn more about the math standards in the U.S. and elsewhere, it is much clearer just how for behind SCASD is using Investigations. It does a lousy job meeting PA standards — which don’t meet NAEP standards — which are rated more poorerly than those of  TIMSS. Sheesh!

Based on what was covered by Investigations in grade 4, at the end of last year I would bet that my kid wouldn’t have done well on an evaluation that covered these TIMMS standards in the area of Numerical Operations (bolded):

Number: Whole Numbers
1. Represent whole numbers using words, diagrams, or symbols.
2. Demonstrate knowledge of place value, including recognizing and writing numbers in expanded form.
3. Compare and order whole numbers.
4. Know the four operations ( +, −, ×, ÷) and compute with whole numbers.
5. Recognize multiples and factors of numbers; read weight and temperature scales marked in multiples.
6. Estimate computations by approximating the numbers involved.
7. Solve problems, including those set in real life contexts (for example, measurement and money problems).
8. Solve problems involving proportions.

Number: Fractions and Decimals
1. Recognize fractions as parts of unit wholes, parts of a collection, locations on number lines, and divisions of
whole numbers.
2. Represent fractions using words, numbers, or models.
3. Identify equivalent fractions; compare and order fractions.
4. Add and subtract simple fractions.
5. Show understanding of decimal place value including
recognizing and writing decimals using words and numbers.
6. Add and subtract decimals.
7. Solve problems involving simple fractions or decimals.
Note: Fourth-grade fractions items will involve denominators of 2, 3, 4, 5, 8, or 10.
Fourth-grade decimals items will involve decimals to tenths and/or hundredths.

Number: Number Sentences with Whole Numbers
1. Find the missing number or operation in a number sentence (e.g., if 17 + __ = 29, what number would go in the blank to
make the number sentence true?).

2. Model simple situations involving unknowns with expressions or number sentences.

Number: Patterns and Relationships
1. Extend patterns and find missing terms in them.
2. Describe relationships between adjacent terms in a sequence or between the sequence number of the term and the term.
3. Generate pairs of whole numbers following a given rule (e.g., multiply the first number by 3 and add 2 to get the second number).
4. Write or select a rule for a relationship given some pairs of whole numbers satisfying the relationship.

I’d be curious to know when these are covered in Investigations – let me know if/when you come across these kinds of problems in your 4th graders’ Investigations work!

The What Works Clearinghouse (WWC) of the U.S. Dept. of Education’s Institute of Education Sciences has issued a report on the Connected Mathematics Project (CMP), the constructivist mathematics curriculum currently in use in grades 6-8 in SCASD.  Seventy-nine studies of CMP were considered, but only one of these studies was found to meet the rigorous evidence standards of the WWC.  This was a 2000 study of 12,000 students in grades 6-8 in Texas.  The conclusion from the report was:

Based on the one study, the WWC found no discernible effects on math achievement.

The full report is available here and here.

Across the U.S., a Common Core of Standards is under development that may help address the marked variability across states.  Interestingly, the executive summary of Fordham Foundation’s Stars By Which to Navigate interim report (Oct 2009) indicated:

“Subject-matter experts reviewed the content, rigor, and clarity of the first public drafts of the “Common Core” standards released in September 2009 by the Common Core State Standards Initiative (CCSSI) of the National Governors Association and Council of Chief State School Officers. Using the same criteria, the same experts also reviewed the reading/writing and mathematics frameworks of the National Assessment of Educational Progress (NAEP); the Trends in International Mathematics and Science Study (TIMSS); and the Programme for International Student Assessment (PISA). Letter grades were awarded to each. The goal is to help U.S. educators and policymakers to judge the respective merits of these influential standards, de facto standards, and possible future standards.

In particular, how do the draft Common Core standards stack up alongside extant national and international benchmarks?

Here are the grades:
Common Core Reading/Writing/Speaking/Listening: B
Common Core Mathematics:  B
NAEP Reading:  B and NAEP Writing: B
NAEP Mathematics:  C
TIMSS Mathematics:  A
PISA Mathematics: D
PISA Reading: D” (p. 1)

As you may recall that PA Math Standards are weak and fall below the NAEP Math Standards (poorly rated here).  As Steve pointed out, SCASD’s fluency expectations are already two years behind CA & MA. Perhaps we should be pushing the SCASD to be truly visionary and aim to meet the top-rated TIMMS standards. Can you imagine what a powerhouse district we would have and how much better prepared our kids would be for their future schooling and employment? That’s certainly not going to happen with our kids receiving instruction using Investigations in the elementary grades.

This week SCASD sent home its “Elementary Fact Practice Parent Handbook”, which is a booklet containing advice for parents who want to help their kids with learning math “facts”, like “9 + 7 = 16″ and “8 x 4 = 32″.  It’s put together well, with descriptions of games that will be helpful for mastering facts and pointers to worksheets and websites.

The handbook also contains a listing of the “SCASD Fact Fluency Expectations” by grade level, which were revised as part of the “Action Plan”.  At Board meetings in the spring and then again in December of last year SCASD administrators told the Directors that the expectations for computation and fact fluency are “comparable” to those found in the Massachusetts and California state standards, but are they?

Here is a comparison of the SCASD fact fluency expectations along with what is expected in California and Massachusetts (MA is two documents here and here) for addition, subtraction, multiplication, and division of whole numbers:

Kindergarten

SCASD:  “Introduction of Addition and Subtraction Combinations through 10″

California: “Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10)”

Massachusetts: “Use objects and drawings to model and solve related addition and subtraction problems to ten”

1st Grade

SCASD:  “Mastery of Addition and Subtraction Combinations through 10; Introduction of Addition and Subtraction through 20″

California: “Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory”

2nd Grade

SCASD:  “Mastery of Addition and Subtraction Combinations through 20; Introduction of Multiplication Facts through 5 x 5″

California: “Know the multiplication tables of 2s, 5s, and 10s (to “times 10”) and commit them to memory”

Massachusetts: “Know addition facts (addends to ten) and related subtraction facts”

3rd Grade

SCASD:  “Mastery of Multiplication Facts through 5 x 9; Introduction of Multiplication Facts through 9 x 9″

California: “Memorize to automaticity the multiplication table for numbers between 1 and 10;  Use the inverse relationship of multiplication and division to compute and check results”

Massachusetts: “Know multiplication facts through 10 × 10 and related division facts”

4th Grade

SCASD:  “Mastery of Multiplication Facts through 12 x 12; Introduction of Division Facts through 81 ÷ 9″

5th Grade

SCASD:  “Mastery of Multiplication/Division Facts through 12 x 12″

We could split hairs about the meaning of “comparable” or that SCASD expects mastery of multiplication to 12 x 12 rather than 10 x 10, but the bottom line here is that students in SCASD are not expected to have mastered their whole number facts until 5th grade, two full years after kids in CA or MA.  But why even compare ourselves to CA or MA in the first place?  Because it is well known that Pennsylvania’s standards are vague and inadequate with respect to their attention to computation and math facts.   PA’s standards have received “D”  grades from the Fordham Foundation and the US Chamber of Commerce (and a negative review from the American Federation of Teachers), while the more rigorous and clear standards in CA and MA are rated “A” by those same organizations.

We’ll compare the computational standards (including use of the U.S. algorithms and fractions) in a future post.

Our community had a terrible loss this morning when SCASD Board President Rick Madore passed away suddenly after an apparent heart attack.  Here is the story in the CDT online.

I didn’t know Rick very well but one thing that stands out in my memory of him is his outstanding dedication as a Board member.  Everyone should know that service on the Board is not a paid position and that it requires a great deal of time outside of one’s regular job.  Last spring in the run-up to the math curriculum vote, he made a point of asking our group to send him studies and other information that would help him come to a decision.  He read everything we sent him (which was a lot) and when he followed up with questions for us I remember being very impressed with his unhesitating willingness to do this research so that he might make an informed decision that was the right one for the children of State College.

When I chatted with Rick after a recent Board meeting, he suggested to me that our group ought to recruit good candidates for the next Board election and I wondered if this was a hint that he wouldn’t be seeking re-election.  This was a discomforting thought because our town needs good people like Rick who are thoughtful and fair, and who are willing to give of themselves and their time as he did.  Our thoughts and prayers are with his family.

Have a look at this public letter to U.S. Secretary of Education Arne Duncan recommending serious math requirements and licensing for prospective K-8 teachers.  It is signed by numerous experts in mathematics who have been active in calling for change in the way math is taught is U.S. schools, including Penn State’s own George Andrews.  The letter begins:

If a first grade teacher read at the fifth grade level, we’d be outraged. But what if she had only third or fourth grade mathematics skills and lacked the conceptual understanding needed for teaching mathematics? Unfortunately, this is the reality for all too many licensed K – 8 teachers in this country. According to a recent report by the National Council on Teacher Quality, the current training that prospective K-8 teachers receive in the vast majority of this country’s education schools assures that this appalling situation will continue unchanged.