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