Sections of This Essay
- The National Math Panel (NMP) Identified the Critical Foundations of Algebra (CFA)
- Common Core Math "Standards" do not Adequately Cover the NMP's Critical Foundations of Algebra (CFA)
- Common Core Math Standards are not Clearly Written
- How Did This Happen? Constructivist Math Educators Were in Control.
- Khan Academy Math Will Save the Day
- Are National Math Standards a Good Idea?
Alarmed by the constructivist approach to K-8 math education, the National Mathematics Panel (NMP) explained how students need to be prepared for algebra, the gateway to higher math and positions in STEM fields. Here are quotes (with page numbers) from the 2008 NMP Final Report: [Bold and underline emphasis added]
- Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement should be understood as the Critical Foundations of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels. [Page 46]
- Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division. Additionally it requires a solid understanding of core concepts, such as the commutative, distributive, and associative properties. [Page 19]
- By the term proficiency, the Panel means that students should understand key concepts, achieve automaticity as appropriate (e.g., with addition and related subtraction facts), develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems. [Pages 17 and 50]
- The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected. [Pages 24 and 78]
- Proficiency with whole numbers is a necessary precursor for the study of fractions. [Page 17]
- Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra. [Page 19]
- The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. [Page 46]
- Students should be able to analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area. [Page 46]
Common Core Math "Standards" Do Not Adequately Cover the National Math Panel's Critical Foundations of Algebra (CFA)CFA 1: Automatic recall of addition and related subtraction facts, and of multiplication and related division facts.
- "By end of Grade 2, know from memory all sums of two one-digit numbers." CCM Section 2.OA Page 19
- "By the end of Grade 3, know from memory all products of two one-digit numbers." CCM Section 3.OA Page 23
- Comment: The CCM document only offers these two short sentences. There is no mention of the subtraction facts that are related to the single-digit number facts for addition, and there is no mention of division facts that are related to the single digit number facts for multiplication. This is a significant CCM omission.
- Comment: Why memorize the single digit number facts? Because later mastery (and automaticity) for the standard algorithms depends on the (prerequisite) ability to automatically recall the single digit number facts. By ingenious design, the standard algorithms reduce multi-digit computations to (multiple) single digit number facts. But this is not mentioned in CCM. This is a significant CCM omission.
- "Fluently add and subtract multi-digit whole numbers using the standard algorithm." CCM Section 4.NBT Page 29
- "Fluently multiply multi-digit whole numbers using the standard algorithm." CCM Section 5.NBT Page 35
- "Fluently divide multi-digit whole numbers using the standard algorithm." CCM Section 6.NS Page 42
- "Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation." CCM Section 6.NS Page 42
- Comment: We have only the preceding four short sentences and there's no definition or example of a standard algorithm in the CCM document. Also, there's no CCM mention of the place value concepts of carrying and borrowing. These are significant CCM omissions.
- Comprehensive and excellent CCM coverage for this CFA category is found in CCM sections 3.NF, 4.NF, 5.NF, 6.NS, and 7.NS.
- But the CCM failure to adequately cover CFA 1 and CFA 2 (above) means that CCM does not clearly identify the prerequisite knowledge needed for student mastery of proficiency with fractions.
- Note: Excellent CCM Support for CFA Content Category 3 must be credited to Berkeley Mathematics Professor Emeritus, Hung-Hsi Wu. Professor Wu further contributed an important paper, Teaching Fractions According to the Common Core Standards. Professor Wu was a member of the National Mathematics Advisory Panel that identified "The Critical Foundations of Algebra" (CFA).
- Apply the area and perimeter formulas for rectangles in real world and mathematical problems. CCM Section 4.MD Page 31
- Comment: These rectangle-related formulas are not given anywhere in the CCM document.
- Know the formulas for the area and circumference of a circle and use them to solve problems. CCM Section 7.G Page Page 50
- Comment: The circle-related formulas are not given and pi is not mentioned anywhere in the CCM document.
- Find the area of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms. CCM Section 6.G Page 44.
- Comment: Formulas for area are not given in the CCM document. Students are expected to work with other students to discover formulas.
- Know the formulas for the volumes of cones, cylinders, and spheres. CCM Section 8.G Page 56
- These formulas for volume are not given in the CCM document.
- Summary Comment: Generally speaking, the CCM document does not provides formulas for perimeter, area, surface area, and volume. These are significant CCM omissions.
- Basic: Each standard deals with a core knowledge K-8 math topic that all K-8 math students should learn..
- Focused: Each standard covers exactly one math topic, where a math topic is a small closely related set of math facts and math skills
- Specific: Each standard should be stated in the most explicit possible way.
- Teachable: It must be possible to teach the topic in a step-by-step manner.
- Measurable: Student mastery can be easily evaluated by an objective test.
- Concise: Standards should be stated using the minimal number of words needed for clarity.
- Examples are provided for each standard to provide maximum clarity.
Keeping the preceding list of characteristics in mind, consider the following computation-related CCM "standards:"
- Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. CCM Section 2.NBT Page 19
- Fluently add and subtract within 1000, using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. CCM Section 3.NBT Page 24
- See later example of how the Khan Academy satisfies this CCM "standard."
- Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. CCM Section 4.NBT Page 29
- Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. CCM Section 4.NBT Page 30
- Find whole-number quotients with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. CCM Section 5.NBT Page 35
- Note: CCM mentions finding remainders in grade 4, but not in grade 5.
- Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. CCM Section 5.NBT Page 35
- They specify constructivist teaching methods and are not limited to basic computation-related content. For computation, the basic content should focus on the facts and skills associated with the standard algorithms. Key subtopics here are the concepts of carrying and borrowing relative to place value. Carrying and borrowing are not mentioned in the CCM document.
- The term "strategies" is regularly used in the CCM document, but examples of computation-related strategies are not found in the CCM document. More generally, examples related to these computation-related "standards" are not found in the CCM document. Without specific examples, constructivist math educators can claim that their K-8 programs are CCM-compliant.
- To learn about the constructivist claim that learning the standard algorithms is actually harmful for students, see The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms Papers
- See Standard Algorithm for Multiplication vs. Constructivist Partial Products for an example of a constructivist "CCM-compliant" alternative to the standard algorithm for multiplication.
- See Standard Algorithms in the Common Core State Standards to see how the CCM failure to clearly define the standard algorithms is being exploited by those who don't want to acknowledge the connection between the (true) standard algorithms and later success in algebra.
Khan Academy. For orientation to the features of Khan Academy Math, click on Math Education Game-Changer: Khan Academy Math. Next click on Khan Academy Support for Common Core Math . This gift from the Khan Academy will transform math education. Faced with Khan's free and detailed support for Common Core Math, constructivist math educators will find it difficult to be vague about the meaning of "standard algorithm" and the meaning of any specific Common Core Math standard.
Here's one example of Khan support for Common Core Math: Consider the Common Core Math Standard: Fluently multiply multi-digit whole numbers using the standard algorithm (listed above under CFA 2). The Khan Academy lists this standard under Grade 5: Number and Operations in Base Ten. Scroll down and find this standard identified as 5.NBT.B.5. Note that Khan offers 200 questions (exercises) associated with just this one standard. Click on multi-digit multiplication to get started. Note that each exercise is accompanied by multiple hints that gradually reveal the steps of the solution. At any time, the student can interrupt the process of doing exercises and watch a video for this standard (multi-digit multiplication).
Here's an example showing that you won't find constructivist math distractions in Khan Academy math. As mentioned in a comment under CFA2 above, for grade 3 CCM, we have the vague statement: "Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction." A footnote says "a range of algorithms may be used." The CCM document doesn't identify possible "algorithms" that may be in this "range of algorithms," and CCM does not offer any examples related to this "standard" This vagueness opens the door for constructivist alternatives to the standard algorithms. But you won't find constructivist alternative "algorithms" in Khan Academy math. To verify this, find the videos and exercises for 3.NBT.A.2, under Grade 3: Numbers and Operations in Base 10. If an algorithm is used in an exercise, it will be the standard algorithm.
As explained in Math Education Game-Changer: Khan Academy Math, the Khan Academy approach (short video lessons linked to exercises with solution hints) is superior to traditional classroom instruction. For example, with the Khan approach, learning is not limited to the content traditionally covered in the "grade" associated with the student's age. Say the Khan student is attempting to learn the standard algorithm for multi-digit multiplication. That algorithm reduces the multiplication problem to a problem that is solved using the standard algorithm for multi-digit addition. The Khan user interface makes it easy for the student to review the standard algorithm for multi-digit addition. On the other hand, prerequisite knowledge covered in an earlier grade is not easily reviewed in the classroom model.
- Local school boards do not have the resources to develop a set of math standards. They have no choice but to use standards developed elsewhere.
- The K-12 math that students need to learn does not differ by locality. So choosing math standards developed elsewhere is the smart thing to do.
- Many students move. Sometimes they move multiple times.
Who should write national math standards? Considering the excellent California math standards, writing another set of math standards is not necessary. The authors in California had excellent writing skills and extensive experience teaching math beyond the K-12 level. They had advanced degrees in mathematics, not math education. They understood in detail what math should be mastered at the K-12 level to be well prepared for learning more advanced math.
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