Dear Superintendent Bergeson:
At the suggestion of my UW colleagues Viginia Warfield and Cliff Mass, I am writing to convey my thoughts/comments on the current revision of the K-12 math standards in our state. I have been following the debate between the "reform math" and the "other" side for the last few years, as this is an important issue that impacts the future of math education in our state. FYI, a summary of my educational background is appended below.
First, I appreciate the passion and the efforts on all sides to improve the standards. Clearly we share a common goal that the revised standards lead to math curricula that promote interest and understanding, while preparing our K-12 students well for the future.
Which set of standards will best achieve that goal?
Upon comparing the OSPI standards draft (January 31, 2008 version) and the Where's The Math (WTM) standards draft (February 2008 version), I find the two to have much in common, though also with some key differences on calculator use, explanation, standard algorithms, and the level of arithmetic/algebra expected. This reflects differences in philosophies on what math K-12 students need to learn and can understand.
Another consideration is the availability of resources (textbooks, curricula, teacher training) to help students to meet the standards.
I agree with the Strategic Teaching review that the OSPI standards represent a significant improvement over the previous standards. I also agree with its critiques and recommendations that the OSPI draft has various deficiencies that need to be addressed. In fact, it will be worthwhile to have a side-by-side comparison of the OSPI standards with the WTM standards, conducted by Strategic Teaching or another independent source.
Comments on the two standards from the Gates Foundation and from leaders in the state's high-tech industry will also be insightful.
My preference would be for the two sides to work together to create a joint draft acceptable to both. Failing that, I take some encouragement that the two drafts are much closer to each other than I initially expected.
If I have to choose between the two current drafts, I would choose the WTM standards. Overall, I find the WTM standards to be clearer, more concise, more rigorous, and more measurable. Some example comparisons:
* Multiplication/ division:
The OSPI standards do not introduce multiplication until Grade 3 and ask students in Grade 4 to "demonstrate mastery of multiplication ... facts through 10 by 10". In contrast, the WTM standards introduce multiplication in Grade 2 and ask students in Grade 3 to "recall from memory multiplication facts for numbers between 1 and 10."
(For further contrast, my school in Taiwan required us to memorize the multiplication table for numbers 1 to 10 in Grade 2. This is still true today, as is the case in Japan . Here we debate whether children can/should memorize the multiplication table in Grade 2. In Japan this is given.)
Knowing the multiplication table by heart is essential for understanding division and multi-digit multiplication, which in turn are needed for understanding fractions, which in turn is needed to understand algebra and trignometry, so postponing the former to Grade 4 would force the other topics to either be postponed or be taught before the students are sufficiently ready.
* Explanation: The OSPI draft has more emphasis on explanation, such as the following requirements in Grade 2 (pages 16, 17):
"Explain and use strategies for remembering addition and subtraction facts to 20."
"Compute two-digit sums and differences efficiently and accurately using a method that can be generalized, including the standard algorithms, and explain why the procedure works."
"Add and subtract two-digit numbers mentally and explain the strategies used."
I appreciate the OSPI authors' desire to promote understanding by students, but simply asking students to explain may not be the best way to achieve that. It would be difficult to measure which explanation is valid and which is not, and students can just as easily end up memorizing one particular set of explanations instead of focusing on actual understanding.
It might be more effective to ask students to "show intermediate steps" in the case of an algorithm being used. If a student can do mental arithmetic correctly and quickly, there seems no need to ask for an explanation (just as there is no need for a students who has learned to play the notes on a piano correctly and quickly to be required to explain how he/she does it). Explanation may also disadvantage children from immigrant families for whom English is a second language.
* Core Processes:
The OSPI standards has a Core Processes section for each grade, with the aim of extending and deepening students' understanding of core mathematical concepts. However, the performance expectations look quite similar from grade to grade. And the given examples seem to fall well short of the stated aim.
For example, Grade 5 has the core concepts of "division of multi-digit numbers, surface area and volume of rectangles, ..., fractions and decimals, ..." (page 43).
But the given example of Armida buying 50 boxes of water bottle involves only simple 2-digit multiplications (e.g., 44 times 50) and a simple 4-digit subtraction (4500 subtract 2200).
Grade 7 has the core contents of negative numbers, rational numbers, linear equations, surface area and volume, proportionality.
But the given example involves only calculating the average and the median of the daily high and low temperatures in Spokane .
* Technology:
Our department has at one time banned all calculators from our first-year calculus classes due to concerns with students relying on calculator to multiply 7 by 9, say. We now allow calculators, but only those that have no graphing nor symbolic capabilities. This contrasts with the OSPI standards which write (in the section on Technology):
"At the high school level, graphing calculators become essential tools as all students tackle the challenges of algebra and geometry to prepare for a range of postsecondary options in a technological world."
I use technology all the time in my work, writing computer programs in different languages to implement mathematical algorithms for solving various optimization problems in real life. (One of my current programming project involves the Rosetta C++ computer package from David Baker's group at UW for predicting the 3D structure of proteins from their amino acid sequence. This is crucial for predicting the biochemical functions of new proteins.) However, for learning and understanding math, calculators should be used sparingly and only when the calculation is too tedious to do by hand (e.g., computing the square root of 7 to 3 decimal places). Computer programming is a great way for students to learn about rigor and algorithms, but it is different from using technology as a black box.
I was brought up on what one might label as "traditional" math, and I feel it prepared me well both in profiency and understanding, as well as finding possibly multiple solutions in multiple ways. To me, math is a language for quantifying and relating patterns through rigorous reasoning.
As in learning any new language (including music), memorizing and practicing in varied contexts using varied strategies are needed to build proficiency, working knowledge, and understanding. Profiency and understanding at one level is essential before proceeding to the next level. A student that cannot mentally calculate the fraction (1-8)/(8-1) will invariably struggle to simplify the algebraic expression (1-x)/(x-1), which I have seen in students of my first-year calculus class at the UW. Coherent and well motivated lesson plans can make all this fun and connect math to the real world, instead of becoming mindless drill-and-kill and teaching-to- the-test.
Regardless of which set of standards is adopted, I hope the OSPI will remain open to further improvements in the standards, and view this as an evolving process (albeit there are resource constraints) . It will be helpful to periodically seek the responses of parents, teachers, and K-12 graduates.
A number of my UW colleagues in math and stats (Ioana Dumitriu, Jack Lee, Boris Solomyak, Tatiana Toro, Marina Meila) have expressed their disappointments at various aspects of the curricula for their children based on Connected Math Project, Everyday Math, TERC, or Integrated Math.
Another UW math colleague William McGovern is happy with Integrated Math, though he also indicates that his son prefers Integrated Math III, which is written in a more "traditional" style, than I and II. Hence the choice of textbook, curricula, as well as teacher preparation, will be key.
Whatever the curricula, they should promote understanding as well as profiency, for example, by showing students more than one way to find the solution(s).
Thank you for patiently reading this far. Please feel free to contact me if you have any questions.
Sincerely,
Paul Tseng
Professor, Mathematics, UW
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A summary of my educational background: I lived in Taiwan until age 11 (Grade 6) when our family moved to Vancouver , Canada where I completed my middle and high school and undergraduate study (in engineering math at Queen's Univ.).
Thus I experienced math education in two different cultures.
In 1986 I received my PhD from MIT in Operations Research, which involves research in math with applications to decision/operation problems in the real world. The Operations Research program at MIT is interdepartmental, so my research background is mixed, ranging from Management Science to Electrical Engineering and Computer Science. After PhD I spent 1 year as a research fellow at the Management Science department in Univ. British Columbia and then 3 years at the Laboratory for Information & Decision Systems, which is a part of the Electrical Enginnering & Computer Science Department, at MIT, before coming to the math department at UW in 1990, where I have stayed since. My research has both theoretical and applied aspects.
Some of it, in the form of computer programs that implement mathematical algorithms for solving large structured optimization problems, are used by companies like Boeing, UPS, AT&T to improve their operation efficiency.
I collaborate widely with colleagues in engineering, computer science, management science, math, biochemistry, statistics, economics from UW and from all over the world.