Ze’ev Wurman was a U.S. Department of Education official under George W. Bush, is currently an executive with MonolithIC 3D Inc. In 2010 Wurman served on the California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California, and is coauthor with Sandra Stotsky of “Common Core’s Standards Still Don’t Make the Grade” (Pioneer Institute, 2010) Here is his review of the new SC Math Standards... In K-5 and 6-8, the SC draft is, indeed, very much based on the Common Core. Sometimes the language is changed, rarely improving the original, and frequently actually even perverting the CC original and creating less-clear and sometimes completely incomprehensible standards. As part of the changes, some important things have been dropped, such as requiring mastery with the standard algorithm for addition and subtraction in grade 4, or multiplication of whole number using the standard algorithms in grade 5. In grade six the Common Core capstone standards for decimals and for dividing integers are present, yet the original clear demand for using the standard algorithm has been changed to using a standard algorithmic approach, which means any odd algorithm, be it the awkward partial sums or the Everyday Maths lattice method will do. Not to mention the unnecessary awkward wording. Further, little has been added in K-8, while essentially most of Common Core has been retained. So there is an early and unnecessary focus on 3D shapes, there is the delayed learning about area of triangles (grade 6) or the sum of their angles (grade 8) or circles (grade 7). And everything, overall, is as fast as common core, which is to say not very fast -- this draft will not lead to many kids taking early Algebra 1. In the high school the situation is a bit better. While many CC idiosyncrasies remain -- the excessive focus in Algebra 1 on functions is the most prominent, but there are a few others as well -- most of the content is in place. Still, topic like sum of arithmetic (finite) and geometric (finite and infinite) sequences, and quadratic inequalities seems to have been forgotten, mathematical induction is absent, Conic Sections that typically belong in Algebra 2 were pushed into pre-Calc, and the Calculus standards are rather skimpy. Still, more trigonometry and geometry content than in Common Core is in place, which is a good thing even as the language is often poor. Finally, I will point out to two generic issues. (a) First are the process standards -- if they could be pushed to the BACK of the standards it would help. Common Core did not do that, and the result is that most teachers (and test makers) focus on them rather than focus on the content itself. Pushing them to the back would help a bit. (b) Second, these are supposed to be CONTENT standards, while instructional practices are supposed to be left to local schools and teachers. Yet these standards sometime use the word DISCOVER which is defined as (p. 24) The word discover in a standard indicates that students will be given the opportunity to determine a formula through the use of manipulatives or inquiry-based activities. In other words, the standards directly and explicitly dictate pedagogy -- not a good idea, and perhaps even unconstitutional in South Carolina if LEAs are supposed to be in charge of curriculum. This is in addition to a lot of Common Core verbiage dictating various strategies or using visual fraction models that the draft carried-in unchanged. Bottom line, K-8 is essentially badly re-written Common Core with a few minor and inconsequential changes. 9-12 is augmented and almost-decent set of standards that is in a need of serious language cleanup for clarity. Now to some specific examples of issues. This is NOT an exhaustive review, but rather examples of problems and issues. Process Standards (p. 6-7) 4.b Interpret mathematical models in the context of the situation. Presumably this means in the context of the original problem rather than some undefined situation. This ill-defined use of situation is quite prevalent throughout these standards. 6.a Determine when an approximation, an estimation, or an exact answer is most appropriate. I cannot see the difference between approximation and estimation, yet the standard implies there is a difference. 6.b Specify units of measure according to the context of the situation. Again, the pernicious and undefined situation. How about problem instead? Further, I would nor use according but rather appropriate. 7.a Recognize complex objects as being composed of more than one simple object. This is a rather poor choice of words. Complex objects may be composed of simpler objects, yet those simpler objects do not need to be simple themselves. Further, one or more is confusing as it is unclear whether it refers to more than one type of object, or to more than one instance of possibly the same type, or both. Why not simply something like Recognize when complex objects are composed of simpler ones? In general, the process standards are poorly written and are sometimes duplicative such as 2.b, 2.c, 2.d, 4.a and 4.b -- all of them essentially say the exact same thing. 2.b Describe a given situation using mathematical representations. 2.c Translate between mathematical representations and their meanings. 2.d Connect the meaning of mathematical operations to the context of a given situation. 4.a Identify relevant quantities and develop a model to describe their relationships. 4.b Interpret mathematical models in the context of the situation. South Carolina Portrait of a College- and Career-Ready Mathematics Student (p.8) This section is probably not very important, yet one wonders what exactly is interdependent thinking or whether that truly is what South California students should exhibit, or what is its connection to mathematics. I assume South Carolina doesnt want to raise students that will join the Borg. Interdependent Thinking and Collaborative Spirit: Student collaborates effectively with others and respectfully critiques varied perspectives. Similarly, one wonders what Self-Reliance and Autonomy or Effective Communication have to do with a student who is mathematically ready for college. This whole page doesnt really belong to mathematics standards. K-5 Standards K.NS.1 Count forward by ones to 100. 1.NSBT.1a. count to 120, starting at any number within 120 2.NSBT.2 Count within 1000 by 2s, 5s, 10s, and 100s beginning with 0. These three minor standards are a hallmark of trying to blindly follow in Common Core footsteps. Most state standards expected students to count to 20-30 in Kindergarten, 100 in the first grade, and 1000 in the second grade. Common Core, for some strange reason, decided to push Kinders to count to 100. Yet it didnt truly expect to accelerate facility with large number so it left grade 2 at 1000, like state standards always did. Then it found itself in a bind -- what would first graders be expected to do? So Common Core pulled 120 out of the blue, and stuck it there. Why not 127? Why not 144? Only God and CC writers know. Yet this baseless choice of 100/120/1000 became the hallmark of blind following of Common Core rather than the more natural (and more reasonable, truly -- no other standard in K deals with number over 20, and no other standard in grade 1 deals with numbers over 100) 20/100/1000. South Carolina seems proudly to follow in Common Cores mindless steps. K.ATO.3 Compose and decompose numbers up to 10 using objects, drawings, and equations. Equations (symbolic sentences) should not be expected in Kindergrten. K.G.5 Model two-dimensional shapes using multiple representations. The intent of this standard is unclear. What multiple representations? Presumably neither equations nor coordinate space. If a standard is not obvious to me, it certainly wont be obvious to parents or teachers. 1.NSBT.7 Decompose two-digit numbers in multiple ways and record the decomposition in expanded form and as an equation. Which multiple ways? Isnt the expanded form and equation one and the same? Why not simply say Decompose two-digit numbers into expanded form or even simpler, write two-digit numbers in expanded form? Everyone would understand that. 2.NSBT.1 Understand place value within 1,000 by demonstrating that: c. three-digit numbers can be decomposed in multiple ways. Like in the previous case, it is unclear what the multiple ways means here, particularly that grade 2 already has a standard (2.NSBT.3) that says Read, write and represent numbers to 1000 using ... expanded form. 1.MDA.2 Use nonstandard physical models to show the length of an object as the number of same size units of length. nonstandard physical models is meaningless gibberish. 2.NSBT.5 Add and subtract fluently within 100. 2.ATO.2 Demonstrate fluency with basic addition facts and related subtraction facts within 20. The first standard already expects fluent addition and subtraction up to 100. What is the meaning of the second standard expecting fluent addition and subtraction to 20? 2.G.3 Identify two-dimensional regular and irregular shapes as polygons and non-polygons. It is unclear what precisely this standard has in mind. Perhaps some examples might illustrate it, but examples are notably missing from all these standards. Are non-convex polygons included? Are self-crossing polygons expected? 2.MDA.4 Measure to determine how much longer one object is than another, using standard length units. Presumably what was meant is standard units of length (e.g., cm., m.) 3.NF.1 Develop an understanding of fractions as numbers. b. A fraction a/b is the quantity formed by a parts of size 1/b; c. Represent a fraction on a number line based on counts of a unit fraction. (c) seems correct yet is almost incomprehensible. Why not simply Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0? This is lifted directly from Common Core, but at least it is well written. If one steals, at least one should steal intelligently. 3.NF.2 Explain fraction equivalence by demonstrating an understanding that: a. two fractions are equal if they are the same size, based on the same whole, or at the same point on a number line; Based on (a), this standard implies that 7/1 equals 11/1, or 3/1 equals 27/1. All of them are based on the same whole, the unit 1. 3.NF.2 Explain fraction equivalence by demonstrating an understanding that: b. fraction equivalence can be represented using set, area, and linear models; Perhaps what was meant was area models and number line. Perhaps not. I am not clairvoyant. I am not even sure what was meant by set. 3.G.1 Understand that shapes in different categories may share attributes but the shared attributes can define a larger category. Reading this on its own is so abstruse as to be meaningless. At least Common Core (3.G.1) offered examples clarifying what is meant. 3.G.4 Identify a 3-dimensional shape based on a given 2-dimensional net and explain the reasoning. Probably overly demanding for 3rd grade. 4.NSBT.4 Add and subtract multi-digit whole numbers. As compared even to the Common Core, it misses Fluently in the beginning and using the standard algorithm at the end. 4.NF.2 Compare two given fractions with different numerators and different denominators using a variety of methods, and represent the comparison using the symbols ,=. The lack of specificity as to the variety of methods makes this standard unhelpful. Even Common Core (4.NF.2) is much clearer. 4.NSBT.1 Understand that, in a multi-digit whole number, a digit represents ten times what it would represent in the place to its right. 5.NSBT.1 Understand in a multi-digit whole number, a digit in one place represents 10 times what it represents in the place to its right, and represents 1/10 times what it represents in the place to its left. 5.NSBT.7 Understand in a multi-digit whole number, a digit in one place represents 10 times what it represents in the place to its right, and represents 1/10 times what it represents in the place to its left. The idea that a student will learn in one year that the digit to the left is x10, while he will have to wait a year too learn that the digit to the right is 1/10 seems somewhat undemanding, to put it mildly. Repeating it twice a year later seems Freudian. 5.NSBT.x 5. Fluently multiply multi-digit whole numbers using the standard algorithm. This is a missing standard, clearly forgotten when carrying from Common Core. 5.MDA.3 Understand the concept of volume measurement. a. Recognize volume as an attribute of right rectangular prisms; Grade 3 (3.MDA.2) and grade 4 (4.MDA.2) already dealt with volume, so clearly the concept of volume is now familiar. The way the standard is written implies that volume being an attribute of right rectangular prism is, in some sense, unique. This is confusing and misleading. Ideally, simply delete this redundant standard. Alternately, at least use something like recognize volume as an attribute of any three-dimensional shape. 6.NS.2 Fluently compute the division of multi-digit whole numbers using a standard algorithmic approach. 6.NS.3 Fluently compute the addition, subtraction, multiplication, and division of multi-digit decimal numbers using a standard algorithmic approach. Change to the standard algorithm(s). As it is now, it allows for any and every invented algorithm to be used, whether efficient or not. 6.GM.1 Solve real-world and mathematical problems involving area of polygons. a. Compute the area of right triangles by composing two triangles into a rectangle. b. Compute the area of other triangles by composing two triangles into a parallelogram. c. Compute the area of special quadrilaterals and polygons by decomposing these figures into triangles and rectangles. Regarding (b) and (c), is there anywhere in these standards where students learn to calculate the area of an arbitrary triangle? This is typically done in grade 5, and I couldnt find it anywhere. Without this knowledge, this standard cant be taught. Regarding (b), these standards do not expect students to learn how to calculate an area of parallelogram until high school geometry. 7.NS.e Apply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to add and subtract rational numbers. Makes little mathematical sense. How can some of these properties (e.g., distributive or inverse) assist in adding or subtracting rational numbers? Sloppy. 7.NS.3 Apply the concepts of all four operations with rational numbers to solve real-world and mathematical problems. Awkward language. The concepts of all four operations? Really? At least the Common Core languages is relatively clean: Solve real-world and mathematical problems involving the four operations with rational numbers. 7.GM.2 Construct triangles and other geometric figures. a. Construct triangles given all measurements of either angles or sides. b. Decide if the measurements determine a unique triangle or no triangle. c. Construct other geometric figures given specific parameters about angles or sides. Awful language. Regarding (b), it doesnt leave an option for multiple triangles. Regarding (c) what exactly are parameters about angles or sides? Gibberish. 8.GM1-8.GM.4 Introduce rigid transformations as the basis for recognizing and proving triangle congruence and similarity. This is one of the most prominent experimental features of Common Core that has a no track record of success, and it has been inserted wholesale into this draft. =================================== The examples above are just that -- examples of bad language and bad mathematics. There are many more examples of this in the K-8 section of the draft. High School Standards The high school standards are -- in general -- significantly better than the K-8 standards. Most of the required mathematics content seems to be there, with the notable exception of rather poorly defined Calculus standards. In amny cases the language needs some clean up too. Regarding Algebra 1, the content seems relatively heavy and would benefit from trimming some of the function-theory and exponential functions standards (move to Alg. 2) Algebra 2 misses conic sections (they are placed in pre-calc instead) and sums of series, both arithmetic and geometric. Quadratic inequalities are missing from either Algebra 1 or Algebra 2. Algebra 2 or pre-calc also miss mathematical induction.
Posted on: Tue, 18 Nov 2014 20:48:42 +0000
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