Thursday, September 25, 2008
Wednesday, June 11, 2008
The data don't distinguish between which students are sophomores, juniors or seniors when they take the ACT, which students may have repeated courses or what year they started the pathway (7th, 8th or 9th grade). But it does give some idea of how much math "preparation" each course pathway provides at least for the years for which data is available.
The average ACT score is highest for students in the honors algebra pathway. Note that the average ACT score of students in the regular (non-honors) algebra pathway is higher than the average ACT of honors integrated math (IM) students, with regular (non-honors) IM students having the lowest scores. But the year-to-year increase in ACT scores is similar for the algebra, honors algebra and honors integrated pathways. The increase for the regular IM pathway is less from year 2 to 3, but similar for year 3 to 4.
Looking at how much ACT scores change year-to-year in the course sequence suggests where students might have a difficult time moving up to higher level math. For example, regular IM students that try to take either of the AP Calculus courses may struggle since the jump in ACT scores is quite big. (As a matter of fact, very few regular IM students enroll in AP math courses.) Honors algebra pathway students may not be adequately challenged by AP calculus AB (one semester of college calculus in a year-long high school course) since the average ACT score actually drops from precalc to AP calc AB; they should likely be encouraged to enroll in AP calculus BC (one year of college calculus in a year-long high school course).
Another way of looking at the data shows that after four years in regular IM, ACT scores are about the same as students with just two years of algebra pathway coursework. Although students in 2nd year IM have higher ACT scores than 1st year algebra students, after four total years of IM coursework, their ACT scores are lower than algebra students with just three years of coursework.
Comparing ACT scores of groups of students that start algebra and integrated pathways in the same year and with similar Terra Nova scores in 8th grade would likely make clear whether one pathway or the other provides better preparation for the ACT and potentially give some guidance for math placement.
Tuesday, June 10, 2008
In Fall 2006, the National Council of Teachers of Mathematics (NCTM) released a Curriculum Focal Points document. Pearson Scott Foresman, publishers of the elementary mathematics textbooks Investigations in Number, Data, and Space 2nd edition subsequently released a document claiming Investigations is aligned with the NCTM Focal Points.
First, a word about the Investigations 2nd Edition materials. Each teacher receives a box measuring 12" x 14" x 14" containing nine Teacher's Unit Guides, a three-ring "Resources Binder" containing student worksheets for photocopying, and an "Implementation Guide." Students receive a workbook "Student Activity Book" and the "Student Math Handbook." The nine units have catchy names that obfuscate the mathematical content they purport to contain. For example, the unit on addition and subtraction is called "Thousands of Miles, Thousands of Seats;" the unit on multiplication and division is called "How Many People? How Many Teams?"
Since reliable research indicates that algebra is key to post-secondary academic success (Answers in the Toolbox, 1999) and fractions are key to algebra (National Math Advisory Panel report, 2007), I will focus on the NCTM Focal Points that directly relate to fractions. The Investigations unit on fractions is called "What's That Portion?"
The first NCTM focal point regarding fractions is:
"Understand fractions and fraction models to represent addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators."
Contrast the language of the NCTM document with the Grade 5 mathematics benchmarks published by Achieve, Inc., a national consortium of state governors and business leaders:
"Understand how to add and subtract fractions.
Specifically: Add fractions with unequal denominators by rewriting them as equivalent fractions with equal denominators.
Understand and use the general formula
a/b + c/d = (ad + bc) / bd
Note: There is no need to find a least common denominator. The easiest common denominator of ab and cd is most often bd."
NCTM focal points do not provide examples of the type of problems students would be expected to solve. But disregarding the problems with the NCTM Focal Points, Investigations fails to meet even these nebulous standards. The publisher claims that
NCTM focal points do not provide examples of the type of problems students would be expected to solve. But disregarding the problems with the NCTM Focal Points, Investigations fails to meet even these nebulous standards. The publisher claims thatInvestigations meets this focal point with several activities in Unit 4: What’s that Portion? specifically:
pp. 95-96 “Introducing Adding Fractions”
pp. 96-98 “Clock Fractions”
pp. 98-100 “Adding Clock Fractions”
In these activities, students learn to mentally visualize “familiar” fractions on a clock face to add them. If sufficient drill is provided in class, this technique might work for problems with special denominators easily represented on a clock face (fractions with denominators that are factors of 60, although this particular fact is never pointed out in the Investigations lessons), including:
12 (5 min), 10 (6 min), 6 (10 min), 5 (12 min), 4 (15 min), 3 (20 min), and 2 (30 min)A typical student reaction is presented in the teacher's guide: “I’m starting to remember halves, thirds and fourths. Like I know 1/2 is 6, 1/3 is 4, ..." However, this technique would be useless for adding other simple fractions with denominators that are not factors of 60 like 7, 8, or 9 for example.
The concept of common denominators is specifically delayed for “later grades.” On page 100 of Unit 4: What's that Portion? a note to the teacher states: "One useful strategy students encounter in later grades for adding and subtracting fractions is finding equivalent fractions with a common denominator.”
The "clock face" activity and fraction tracks are not just examples of activities used to teach adding and subtracting fractions; these are the only procedures presented for adding and subtracting fractions in 5th grade Investigations materials.No universally applicable, accurate and efficient procedure for adding and subtracting fractions is ever presented in the Investigations lessons. Instead a hodge-podge of short-cut methods and estimating techniques displace fast accurate calculations.
NCTM Focal Points also state:
"Apply understanding of decimal models, place value, and properties to add and subtract decimals."
Investigations claims to meet this standard with several activities in Unit 6 entitled "Jeweler's Gold" (p. 93-97). During the “Jeweler’s Gold” activity in one full hour of class time, students solve exactly 1 problem:
.3 + 1.14 + .085
The majority of class time is spent making a poster to explain their solutions. If students finish quickly, another problem is suggested for “an additional challenge”:
2.05 + 0.76 + 1.3
The next section covers "Strategies for Adding Decimals" p. 98-104. During one more hour of class time, students order the decimals .625, .025, .3, .8, .75 and then add together the three largest:
.8 + .75 + .625
using a “hundredths grid” if necessary. Nowhere do the Investigations materials discuss alignment of the decimal point for stack-and-carry addition.
A "Teacher Note" on page 133 of Unit 6 states: "Often students are taught 'rules' or 'tricks' to add decimals, including lining up the decimal point." The professional development notes go on to likewise dismiss the value of filling in blank spaces with zeros. Students are never let in on these "tricks."
Failing to provide adequate practice with calculations is as bad as failing to cover essential concepts; lack of repetition masquerades as “deep thinking.”
Standard procedures for adding and subtracting fractions and decimals
NCTM Focal Points also states that students should:
"Develop fluency with standard procedures for adding and subtracting fractions and decimals."
Investigations claims to meet standard 3 with:
• Unit 4: What’s that Portion?
pp. 102-103 “Roll around the Clock” game
pp. 105-106 “Writing Fractions Problems”
p. 108 “Subtracting Fractions”
pp. 122-123 Activity 1
pp. 129-131 Discussion 1
pp. 132-134 Math Workshop 2
• Unit 6: Decimals on Grids and Number Lines
pp. 105-106 Activity 3
pp. 108-109 Activity 1
Investigations never presents "standard" methods for adding and subtracting fractions; therefore, it is baseless to claim that Investigations develops "fluency" with such methods.
Summarizing Investigations 5th grade lessons on subtracting fractions, we have:
- “Subtraction Fraction Equations” chart (A list of subtraction facts like: 8/10-3/10=4/8 presumably to be memorized since no explanation of how they are calculated is given.)
- Using a clock face (discussed above)
- Playing the "Fraction Track" game (Students move between zero and one by fractional amounts; they use premarked number lines with denominators up to tenths but have no way of determining if one fraction is greater or less than another except by using the track as a measuring stick.)
These are not standard procedures nor are they universally applicable. Without the concept of common denominators, students can not even compare simple fractions. For example, on page 51 of the "Student Math Handbook" (textbook), the fractions 7/12 and 4/10 are placed in order on a number line by noting that 7/12 is a "little more than 1/2" and 4/10 is a "little less than 1/2" so therefore 7/12 must be greater than 4/10. However, could students order 8/9 and 9/11 using any method presented in the Investigations materials? No; the "fraction tracks" provided don't have denominators greater than 10, and the denominator 11 is not easily represented on a clock face. This is just one example of how problems that are not cherry-picked to be solvable by Investigations' methods are ignored by the curriculum, yet this is supposedly the way to make children think critically about math.
Investigations vs. Singapore Math
Investigations students will be drawing pictures of clocks in 5th grade while Singapore students will be solving problems like these presented on the Singapore Math web site 5th grade placement tests:
•Find 32% of $96.
•Express 8 5/8 as a decimal correct to 2 decimal places.
•Express each as a percentage:
(c) 215 out of 500
Investigations' non-standard techniques, inadequate practice and low expectations lead to students having a poor conceptual understanding of mathematics and poor problem-solving skills as reflected in dropping test scores in 4th and 7th grade at Columbia Public Schools.
Monday, November 12, 2007
According to Resendez and Manley (2005) cited on the U.S. DOE website the TerraNova CTBS is a reliable and valid standardized test that offers broad coverage of the mathematics content in most textbooks and reflects the National Council of Teachers of Mathematics (NCTM) standards.
According to Assessment Standards For Missouri Public Schools on the Missouri DESE web site "the advantages of these items are: 1) they are effective in measuring students’ breadth of content knowledge; and 2) a large number of these items can be administered and scored in a short amount of time."
A major criticism of TERC materials is their inadequate coverage of standard mathematical content like definitions and terminology, a criticism which is clearly supported by CPS test results.
Thursday, September 20, 2007
COMMENTS addressed to the National Mathematics Advisory Panel, September 6, 2007:
I represent a parent group in Columbia, Missouri. Our community is a microcosm of the national math debate, albeit perhaps a late blooming one. All the players are assembled for yet another season of "mathematics on the verge of a nervous breakdown."
The math education department at the state university located in Columbia is heavily funded by the NSF to promote teacher development using particular math curricula. Many of their graduate students earn master's degrees by participating in the implementation of these curricula in the public schools. The local public school implemented these curricula in 2001, in part to gain access to MU graduate students for CPS classrooms.
But who evaluates the effectiveness of these curricula? It goes without saying that public school administrators like to present student achievement in the best possible light. Students and faculty at MU's math education department have published numerous papers, not surprisingly, supporting the effectiveness of their own efforts; however many of these same papers have been found to lack sound research by the What Works Clearinghouse.
At the same time, nationally-normed, standard assessments of student achievement are being ignored. An eight-year record of CPS student scores on the Iowa Algebra Aptitude Test spanning the period of implementation of CMP seems to indicate a significant drop in algebra readiness but has not been carefully examined by the school district or university researchers. ACT test scores have dropped and remedial math rates of students attending state universities have escalated since adopting these math curricula.
Parents are justifiably concerned. Many parents who work at the University of Missouri in the math, engineering, food science, economics, psychology, and other departments have signed a petition opposing the current math curricula used in the public schools. These scientists, engineers, mathematicians, technicians and physicians know intimately the demands of a career requiring mastery of mathematics and they are speaking up to say the local public schools are failing students who have aspirations to follow a STEM career path.
Likely, this script is is all too familiar to the panel. You have been tasked with advising the President and Secretary Spellings "on the best use of scientifically based research on the teaching and learning of mathematics."
I conclude with one important point, the cliffhanger for the season: how best can evaluations of effectiveness and assessment of student performance be separated from and independent of development and implementation of curricula? The basic tenet of slightly adversarial, peer-reviewed research is lost when researchers are paid by textbook publishers and administrators play dual roles implementing curricula and assessing their impact.
I thank you for the crucial and urgent work you are doing on behalf of our students, families and nation.
Columbia Parents for Real Math
Tuesday, September 11, 2007
I am from the Village of Ridgewood in New Jersey. So you may say I am a villager with a pitchfork. That American Gothic image of the farm couple, they would not send their children to your schools. They would demand a practical math education for farmers - needing to learn how to multiple and divide dozens of eggs. But lest I be called a "shopkeeper," let me also invoke the image of our professors of mathematics who will refuse to let their children learn reform math because they know as teachers of mathematics that reform math does not provide the proper foundation for higher level mathematics, the sciences or engineering.
My friends you may not know this, but you are witnessing the first shot of the revolution. With this speech, I begin the revolt of the parents of the children in your schools against all forms of reform math.
I am here not to petition you or to plea with you, but to tell you , I/We will refuse to allow our children to be taught in this ridiculous way. I/We will not be intimidated by your PhDs in Education, I represent the thousands of parents who have achieved great success with traditional math education and are the generation of proof that it works. We will begin to send back your silly TERC books, filled with only simple answers derived using standard algorithms. We will instruct our children to refuse to draw pictures, not to write math stories and to call it an "equation" with symbols rather than the silly "math sentence." If you have the audacity to chide them for using REAL math, we will come down to the school en masse and picket with large signs that say NO MORE BAD MATH or JUST SAY NO TO TERC, MATH IS A TERRIBLE THING TO WASTE.
You can decide to allow reform math to continue, but you will find us, your customers and clients, no longer cooperative. We will throw out the TERC2 and CMP2 workbooks and send our children in with traditional texts. Our children will become so smart, they will teach the teachers.
When in the course of human events it becomes necessary for parents to dissolve the ties which have connected them with their child's public school a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
We will boycott Pearson Publishing and its subsidiaries until its salespeople stop pushing reform math programs on our schools based on weak and laughable research. We will boycott the schools that use reform math and we will inform every minority group that they are getting substandard mathematics education compare to their contemporaries.
Welcome to history, welcome to the beginning of the parental revolution, welcome to the beginning of the day when the public was put back in public education. Let this be our declaration of independence and the start of our revolutionary war.
We hold these truths to be self-evident, that all children are created equal; that they are endowed by their Creator with certain unalienable Rights, that primary among our children's rights is the right to an adequate and true education. Within that right, there shall be included a strong mathematical education.
A math education defined by mathematicians rather than educators and includes the following tenets. 1) Our schools should focus on math programs on the basis of their content and from hereon pedagogy will driven by the clear detailed well documented mathematical content. 2) A math program should include a logical sequencing of topics, honoring the scholarly subject entitled mathematics. 3) A quality math program will not include for any grade other than Kindergarten the use of scissors, glue, paperclips, M&M or any other object that is now defined as a manipulative and acceptable for exceeding assessment benchmarks. No, our children will have the high and honorable goal of a math program desiring them to use the abstract symbols and language of mathematics. 4) A quality math program will emphasize the learning of necessary math facts & standard algorithms. 5) The math program should use the proper language of mathematics and not invent new unnecessary or watered down terms.
We the parents of the children in the public education system are not happy. These are our children whose educational fate you decide. Shame on those of you for not including the educated parents of this country in this debate; shame on those of you for having 48 speakers and only three of them parents; and shame on those of you for ignoring parents concerns, for it is OUR children who will be known as the LOST Mathematical Generation, OUR children who will not be able to make change without a calculator, OUR CHILDREN who in their elementary years are being limited in their future by reform maths limitation of its teachings. It is our children you doom and it is done without even giving a PARENT THE CHOICE FOR THE CHILD.
Thomas Jefferson's vision of public education would NOT have included drawing circles to add and subtract. Jefferson would be angered when he saw mathematics taught with scissors and glue. Jefferson would be irate when he saw that educators dismiss the outcries of parents. Jefferson would weep at the thought that his dear United States of America would lose an entire generation of its educated society because a British publisher wanted to make more money selling manipulatives with programs like TERC and CMP than selling real textbooks. Just as patriots broke open tea chests and heaved them into Boston harbor, with other patriots at other seaports following that example and staging similar acts of resistance, so too should parents be be throwing TERC2 and CMP2 workbooks into a harbor or river or recycling bin - our very own Boston tea party.
We, the parents, will ultimately triumph because it is Our children, not children of the state or education system. And for OUR children, their education is more important and held more dearly than any social, political, economical, or ideological agenda.
It is on the shoulders of parents across this nation, that a generation of children will not be lost in their math education. And those that recognize this and stand in recognition will provide to the future of this great nation, mathematically capable citizens to lead us throughout the 21st century. And that success will be none for reform math.
Written by Joan O'Keefe of the Village of Ridgewood, New Jersey and published here with her permission. She is a member of a parent group opposing TERC and CMP in her child's school district. Their web site is http://www.vormath.info/. Her opinions are her own and we welcome discussion of the issues she raises.
Tuesday, September 4, 2007
The quality of the secondary math education in the United States has long become a standard joke among mathematicians around the world. It is very painful to see how the most powerful country trails behind European, Asian and third world countries in the level of mathematical
skills of high school graduates.
To my opinion, the major reason is that, unlike many countries, algebra is not included in the middle school curriculum in the United States. Algebra was invented a few thousand years ago as the universal language of mathematics, allowing to avoid lengthy word explanations of mathematical procedures and making mathematical studies logical and connected. It is
essential that the students start learning algebra as early as in the 5th grade, which brings mathematical formulas to them naturally later in their lives. It is still possible for the best students to start algebra in the 8th grade and be successful, but in general American students continue looking at mathematics as a foreign language. I think the switch to starting algebra in the 5th grade must be made as soon as possible, by adopting a system used in one of the countries like Russia, France or Germany. American students are as smart as anybody, and they and their teachers will quickly adjust to this system.
The so-called Connected Mathematics Project, Integrated Math and other recent innovations are even worse than the "traditional" American system. They further water down the curriculum and leave the students largely unprepared for college mathematics. Instead of bringing our children back to the level of "ancient Greeks playing with stones on the beach,"
American math education must quickly switch to the most advanced methods of mathematical learning.
Alexander Koldobsky is a math professor with fifteen years of experience in teaching mathematics at the college level in the US and a parent of two children who went all the way through the American educational system. His opinions are his own, and we welcome discussion of the issues he raises.