The Case for National Math Standards

Are National Math Standards a Good Idea?

Yes, national math standards make sense because:

• Local school boards do not have the resources to develop a set of K-12 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 and it remains very stable over time.  So choosing math standards developed elsewhere is the smart thing to do.
• Many students move.  Sometimes they move multiple times.  The math content they are expected to learn in a particular grade in their new school should be the same math content they were expected to learn in that same grade in their prior school.  

What to Choose for National Math Standards?

As discussed in another essay, the Common Core Math Standards don't offer a good choice. They are not clearly written and their failure to provide examples opens the door for any math program to claim "alignment" with the Common Core Math Standards.  On the other hand, the Khan Academy implementation of the Common Core Math Standards offers a good choice, and it's the easiest thing to do. But Chapter 2 of the 2005 Mathematics Framework for California Public Schools offers the best choice.  This 86 page K-12 math standards document achieves maximum clarity through the use of excellent examples. Additionally, Chapter 3 of the California Framework offers 85 pages of excellent grade-by-grade teaching guidelines. 
 
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.  

Will National Math Standards be Controlled by the Federal Government?

Simply choose the 2005 California Math Standards [Chapter 2 of the 2005 Mathematics Framework for California Public Schools] and don't allow any changes for at least 5 years.  The K-12 math that should be learned today has not changed since 2005.  All members of the Expert Panel [page 127 in The State of State Math Standards, 2005] agreed that California offered the best math standards in the country.  This was true in 2005, and [as one member of that Expert Panel] I know it remains true today and it will remain true for many years.

Copyright 2015  William G. Quirk, Ph.D.

Common Core and Constructivist Math: Khan Academy Math Saves the Day

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?

The National Math Panel Identified the Critical Foundations of Algebra (CFA)

The National Council of Teachers of Mathematics (NCTM) released the NCTM Standards in1989. In spite of “standards” in the title, the NCTM Standards did not contain specific math learning standards. Instead, the NCTM emphasized that traditional K-8 math content should receive "decreased attention," with "increased attention" for math appreciation, calculator skills, hands-on activities, and content-independent reasoning skills.  Prior to the release of the NCTM Standards, most American children memorized the single digit number facts and learned how to carry and borrow as necessary steps in mastering the standard algorithms for addition, subtraction, multiplication, and division. They next learned about equivalent fractions and common denominators as necessary steps in mastering the standard procedures for adding, subtracting, multiplying, and dividing fractions. They also learned standard formulas and standard math terminology.  But these traditional K-8 math topics are no longer being taught in most American public schools. The anti-traditional NCTM Standards approach is known as constructivist math.  The "math wars" are about traditional math education vs. constructivist math education  Constructivist math programs, such as Everyday Math, TERC's Investigations, and Connected Math, now dominate K-8 math education in American public schools.

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.

 CFA 2: Automatic execution of the standard algorithms for addition, subtraction, multiplication, and division.  

• "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.

CFA 3: Proficiency With Fractions (including Decimals, Percents, and Negative Fractions

• 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). 

CFA 4: Analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area.

• 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.

Common Core Math Standards are Not Clearly Written

K-8 math standards should be limited to content. They should not specify teaching methods. Clearly written math standards have the following characteristics:

• 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.

See Chapter 2 of the 2005 Mathematics Framework for California Public Schools for math standards that satisfy the preceding list of characteristics.

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

The preceding CCM computation-related "standards" are not clearly written. They fail to provide specific, teachable, and measurable learning expectations for K-8 computation.

• 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.

How Did This Happen?  Constructivist Math Educators Were in Control.Constructivist math educators substitute "math appreciation" content for traditional K-12 math content.  They emphasize mental math for simple computations and promote the use of calculators for more difficult computations.  They claim that there's no longer a need to master the standard algorithms, due to the power of calculators.  They fail to recognize that mastery of standard arithmetic is necessay foundational knowledge for later master of algebra, the gateway to higher math.

• 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.

Constructivists math educators openly oppose memorization.  [See The Parrot Attack on Memorization.]  This explains the failure to provide formulas in CCM, and it also explains why CCM offers just two short sentences for memorization of single digit number facts.

Khan Academy Math Will Save the Day

Common Core Math is taught in a very student-friendly and mathematically correct way at the 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.

Are National Math Standards a Good Idea?

Yes, national math standards make sense because:

• 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.

But what to choose for national math standards? As discussed above, Common Core Math is not a good choice. If a school is already in the process of attempting to implement Common Core Math, then the Khan Academy  Implementation of Common Core Math offers a very good choice, and it's the easiest thing to do.  Chapter 2 of the 2005 Mathematics Framework for California Public Schools offers an excellent choice.  This 86 page K-12 math standards document achieves maximum clarity through the use of excellent examples. Additionally, chapter 3 of the California Framework offers 85 pages of excellent grade-by-grade teaching guidelines.

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.

Home Page for Bill Quirk

Copyright 2015 William G. Quirk, Ph.D.

Math Education Game-Changer: Khan Academy Math

Features of Khan Academy Math       

• It's Free!   [Financially supported by Bill Gates, Google, and others who care about genuine math education.]
• It's available 24 hours a day, every day.
• It's exactly the math that students should learn. There are no constructivist math distractions.
• It completely covers all important math topics:  elementary school math, middle school math, high school math, and college math.
• Every important math topic is covered with one or more internet videos.
• A typical video is 10 to 15 minutes long.  This fits the average child's attention span.
• The student hears the instructor (usually Salman Khan), who is not seen, but appears to be sitting to the left of the student.
• The unseen instructor speaks and writes as he explains the current math topic.
• Salman Khan believes that viewing the instructor's face is an unnecessary distraction for the student.
• Each video is linked to exercises that are designed to test the student's understanding of the video.  
• Each exercise comes with multiple hints.  After each hint, the student can request another hint, until the problem is completely solved, typically after 3 or 4 hints.  
• The student can request exercise after exercise, without ever seeing an exercise repeated.
• The student can easily move back and forth from the video to related exercises.
• There's a "knowledge map" and other orientation features that: 
• Show how the current video relates to other videos (including prerequisites for the current video).
• Shows the student what is possible next, based on what the student has already done.  What is possible next is not limited by the student's age.
• The Khan system "remembers" what videos the student has viewed and how the student has performed on the exercises.  
• Multiple features encourage the student to move beyond initial understanding to full mastery (long-term remembering concepts and methods).  
• You don't know what you don't remember.  
• There are user interfaces for students, parents, coaches, and teachers.   
• It's far superior to classroom learning. 
• Each student works at his or hers own pace, at any time and in any place.  
• Each student can take all the time needed to master each topic.  Unlike classroom learning, there's no need to move on because the teacher is ready to move on.
• Learning is not limited to the content traditionally covered in the "grade" associated with the student's age.  
• There's no fear of making a mistake in front of a teacher or peers.
• There's no multiple hours delay between a lesson and "homework." The student can easily switch back and forth from a video to exercises related to the math content covered in the video 
• The student can repeat a video, rewind to any earlier point in the video, or fast forward over points already understood.
• The student can easily fill in gaps by using videos that cover content from earlier grades.
• In some ways, it's superior to tutorial learning.  
• The student doesn't have to remember what the tutor said. 
• The student isn't pressured by a tutor's questions.
• The student isn't distracted by the tutor's face or behavior.
• And it's free!

Copyright 2015 William G. Quirk, Ph.D.