Common Core and Constructivist Math: Khan Academy Math Will Save the Day

Link to Mobil-Friendly Version:
Links Below to Sections of This Essay

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]

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.
 CFA 2: Automatic execution of the standard algorithms for addition, subtraction, multiplication, and division.  
CFA 3: Proficiency With Fractions (including Decimals, Percents, and Negative Fractions
CFA 4: Analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area.

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