The Truth About The New Jersey
 Math Standards

• The NJMS Are Based On the NCTM Standards
• The NJMS Are Not Genuine Math Standards
• The NJMS Offer A Program For Replacing Traditional K-12 Math With Calculators, Math Appreciation, and Content-Independent "Process Skills"

The New Jersey Standards (NJMS) are a "child" of The NCTM Standards, from The National Council of Teachers of Mathematics (NCTM). Similar to the parent, they fail to identify the specific, grade-by-grade math content that should be taught and learned during the K-12 years. Instead, they promote the NCTM's "progressive" program for replacing traditional K-12 math with "manipulatives", calculators, "guess and check", estimation, and content-independent "process skills". These NJMS substitution strategies are disguised as 16 so-called "content standards", each with multiple "cumulative process indicators" (CPIs) listed for the ends of grades 4, 8, and 12.

Yes, New Jersey high school graduates will be able to guess the solution to 3x + 2 = 17, and they will be able to use a calculator to estimate the solution to 6x +2 = 22, but don't be surprised if they can't find the exact answer in the second case. Expect to hear that "exact answers" aren't important and that modern math is all about calculators. Don't be surprised when they reach for a calculator to find 10% of 300, or to find the new selling price of an $8 item, now offered at a 25% discount.

The NJMS  achieves its lean 26-page appearance by speaking in generalities and by suppressing the underlying "progressive" philosophy found in the NCTM Standards. That's all found in the 670-page New Jersey Mathematics Curriculum Framework (NJMCF), a 1500 "activities" cookbook for math self-discovery. Following the lead of the NCTM, the following "progressive" axioms have been assumed by the writers of the NJMCF:

1. Belief that children must be allowed to follow their own interests to personally discover the math knowledge that they find interesting and relevant to their own lives.
• Rejection of the concept of a common core of basic math knowledge that all children should learn during the K-12 years
2. Belief that children must "construct" mathematical knowledge for themselves.
• Rejection of teacher-directed knowledge transmission.
3. Belief that all knowledge must be acquired as a byproduct of social interaction in real-world settings.
• Rejection of classroom learning.
4. Belief in the primary importance of general, content-independent "process" skills.
• Rejection of the need to remember any specific math content.
5. Belief that calculators have fundamentally changed the nature of math.
• Rejection of the need to acquire traditional paper-and-pencil math skills.
6. Belief that learning must always be an enjoyable, happy experience, with knowledge emerging naturally from games and group activities.
• Rejection of any attempt to challenge a child to work harder.

New Jersey math educators want you to believe that their philosophy is backed up by recent research in cognitive psychology. Don't buy it. They've been trying to sell this philosophy for 80 years. If they talk excitedly, using high-sounding terms like "constructivism", "situated learning", "holistic learning", "developmentally appropriate", and "research-based", you'll know you're experiencing a strange encounter with a member of the progressive educationist cult. Good luck!

The good news is that genuine researchers are also appalled by the claims of today's progressive educationists. This truth was recently explained by E. D. Hirsch, Jr. in his April 10, 1997 Address to The California State Board of Education. In his presentation Professor Hirsch discussed the difficulty of publishing an important research article, Applications and Misapplications of Cognitive Psychology to Mathematics Education,  even though one of the co-authors, Herbert A. Simon, is a winner of the Nobel Prize, and the other co-authors, John R. Anderson and Lynne M. Reder, have the highest respect of those who do genuine research in cognitive psychology. Most progressive educationists don't know about these scholars and their research. Those who do know don't want you to know.

Rather than wading through the 670-page NJMCF, you can find what you need to know by reading this document, together with the linked sections from The Truth About the NCTM Standards. This will be more than enough for you to fully appreciate how New Jersey educationists want to emphasize "progressive" social goals and the fundamental progressive gospel of "discovering math through problem solving". Note: At any time you can use the NJMCF links to check the context of NJMCF quotes. Page references are given.

The NJMS Are Not Genuine Math Standards

• For math standards which do qualify, see the Mathematics Content Standards for California Public Schools, the Mathematics Standards of Learning for Virginia Public Schools, and the Mathematics Standards for Kindergarten to Geometry found at 2 plus 2: The Home of Mathematically Correct .

• Not focused
• The 16 NJMS "content standards" are very general learning objectives, not focused math topics. Each NJMS "standard" has an associated list of "Cumulative Progress Indicators" (CPIs) which still fail to identify focused math topics. But this is the lowest level of detail found in the NJMS! Consider the very general nature of the following CPIs, listed for the end of grade 4:

(Note: The NJMS "content standards" are numbered from 4.1 to 4.16 in the NJMS, and 1 to 16 in the NJMCF. We will use the NJMCF 1 to 16 approach. Also, the notation N.K indicates CPI  K for standard N.)
1. CPI 1.1: Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematical content appropriate to early elementary grades.
• CPI 1.10 substitutes "content appropriate to the middle grades", and CPI 1.15 substitutes "content appropriate to the high school grades".
• The Truth: The detailed identification of "mathematical content appropriate" for each grade is exactly the role of math standards!
2. CPI 2.2: Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.
3. CPI 3.1: View mathematics as an integrated whole rather than as a series of disconnected topics and rules.
4. CPI 4.1: Make educated guesses and test them for correctness.
5. CPI 5.2: Select and use calculators, software, manipulatives, and other tools based on their utility and limitations and on the problem situation.
6. CPI 6.1 Use real-life experiences, physical materials, and technology to construct meaning for whole numbers, commonly used fractions, and decimals.
7. CPI 7.5 Investigate and predict the results of combining, subdividing, and changing shapes.
8. CPI 8.5:  Use a variety of mental computation and estimation techniques
9. CPI 10.4: Explore, construct, and use a variety of estimation strategies.
10. CPI 11.5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts.
11. CPI 12.7: Make predictions based on intuitive, experimental, and theoretical probabilities.
• The NJMCF goes one level below the CPI, associating multiple activities and games with each CPI. But there's still an overall failure to identify focused math topics, and there's frequently no evident reason as to why a particular activity or game is associated with a particular CPI. Somehow, the NJMCF "activities" are supposed to help New Jersey kids discover some math. What math? That's usually not clear.
• Not specific
• New Jersey educationists conspicuously avoid being specific about math content for the K-12 years. They believe that each kid should be allowed to explore broadly.
• Not basic
• New Jersey math educators don't believe in identifying and teaching core math knowledge. Instead, they endorse broad, discovery learning (Link to Chapter 4 of The Truth About The NCTM Standards)
• Not teachable
• The NJMS and NJMCF are too broad and unfocused to be teachable. But "teachable" isn't a term that has meaning for progressives. Their endorsement of discovery learning implies a corresponding rejection of teacher-directed knowledge transmission. "Progressive" educators believe it is wrong for teachers, or anyone else, to decide what kids should learn. They say the main role of the teacher is to create rich enabling environments (Link to Chapter 4 of The Truth About The NCTM Standards)
• Not measurable
• Because of their fundamental belief in "broad content" and "discovery learning", traditional objective testing is ruled out. All "teachers" can do is to attempt to "discover" what each kid has personally discovered. That is, all they can do is "test to find success". (Link to Chapter 4 of The Truth About The NCTM Standards)
• Not linked to grade
• The NJMS and NJMCF are not specific about what should happen in each grade. They discuss general learning goals for the ends of grade 4, 8, and 12. There is no possibility for either student or teacher accountability.
• Concise, to a fault
• CPI's are concisely stated. That is, they do "express much in few words". The problem is that they express far too much, using general words.
• Redundant
• Consider the redundancy in the following examples:
1. CPI 8.5: Use a variety of mental computation and estimation techniques.
2. CPI 8.6: Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods and check for reasonableness of results.
3. CPI 8.10: Develop, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers.
4. CPI 10.4: Explore, construct, and use a variety of estimation strategies.
5. CPI 10.6: Determine the reasonableness of an answer by estimating the result of operations.
6. CPI 10.7: Apply estimation in working with quantities, measurement, time, computation, and problem solving.
7. CPI 10.8: Develop, apply, and explain a variety of different estimation strategies involving quantities and measurement.
• Not genuine math
• This is the most alarming problem and the topic of the next section.
• Not brief
• Relative to the 217 page NCTM parent, the 26 page NJMS is brief. But the fair comparison is to include the 670-page NJMCF, for a total 696 pages.
• Not selective
• Today's progressive educationists are repelled by the very thought of a narrow selection of content. They preach broad, open-ended exploration, not carefully selected math topics.
• Not pedagogically neutral
• The NJMS don't explicitly preach "progressive" teaching methods. But they assume them! All the key words are there. The detailed progressive rational and "teaching" methods are found in the 670-page NJMCF

The NJMS Offer A Program For Replacing Traditional K-12 Math With Calculators, Math Appreciation, and Content-Independent "Process Skills"

Note: Links below are to related sections of The Truth About The NCTM Standards.

1. Problem Solving (The first "content standard" and the first "process skill")
• The NJMCF say that the problem-solving "process skills" to be learned are "making a list, making a chart or a table, drawing a diagram, making a model, simplifying the problem, looking for a pattern, using manipulatives, working backwards, eliminating possibilities, using a formula or equation, acting out the problem, using logic, using guess and check, using a spreadsheet, using a computer sketching program like Geometer's Sketchpad, The Geometry SuperSupposer, or Cabri, writing a computer program, or using a graphing calculator." (Page 71, NJMCF)
• The Truth:  Problem solving in the context of math refers to the application of remembered math knowledge, with the step-by-step use of context-specific math facts tightly linked to context-specific math skills. It may involve remembering and applying a whole series of linked facts and skills. If there are gaps in knowledge, the whole process breaks down. But New Jersey math educators think this is too difficult for today's kids. They would rather distract New Jersey students with thousands of hours of open-ended games and "activities", all with the goal of "discovering" general "problem-solving skills". They say that such "skills" are all that one needs to be a "mathematical problem solver". It's a cult-like belief in magic! They would have us believe that it's possible to "do math" without remembering any specific math.
2. Communication (The second "content standard" and the second "process skill")
• "Most experiences relating to mathematical communication will involve the use of natural language". (Page 75, NJMCF)
• The Truth: Basic communication skills are more important than math skills, but they're not math. Learning about "communication" in the context of math refers to learning the precise language and symbols of math. It's not about "natural language".
3. Connections (The third "content standard" and the third "process skill")
• "Mathematics must be approached as a whole" (Page 76, NJMCF)
• In a very real sense, the whole is greater than the sum of its parts." (Page 76, NJMCF)
• The Truth: Similar to "whole language", progressive educationists preach "whole math". They would have us believe that kids are ready for complex math in the first grade. But genuine math knowledge must be acquired through a step-by-step buildup process, from simple to complex, with the initial hiding of complexities.
4. Reasoning (The fourth "content standard" and the fourth "process skill")
• "Reasoning" CPIs for "the end of Grade 4" say students:
1. CPI 4.1: Make educated guesses and test them for correctness.
2. CPI 4.2: Draw logical conclusions and make generalizations.
3. CPI 4.3: Use models, known facts, properties, and relationships to explain their own thinking.
4. CPI 4.4: Justify answers and solution processes in a variety of problems.
• The Truth: It's all unbelievable, but sadly true. Kids just starting out in math are to learn that "guess and check" is the first principle of reasoning. Next comes a weak form of inductive reasoning whereby kids "generalize" based on the experience of a few tests with a calculator. As for "explain their own thinking" and "justify answers", progressives just want them to be happy with "their own thinking". New Jersey kids won't learn about the importance of accuracy and correct answers. They will learn about "socially correct thinking", where every kid's ideas are to be seen as valuable, not subject to criticism. They won't learn about mathematically correct thinking, with the quick identification and correction of errors in understanding.
5. Tools and Technology (The fifth "content standard")
• "Problem Solving", "Communication", "Connections", and "Reasoning", are also the first four NCTM "standards". But now we have something different! The NJMS go beyond the NCTM Standards by elevating calculators, computers, and "manipulatives" to a full-fledged "standard". First they say "These tools should be used, not to replace mental math and paper-and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics". (Page 137, NJMCF) But later they hedge, with "Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, and other kinds of numbers. With calculators that perform these operations quickly and accurately, however, the instructional emphasis now should be on understanding the meaning and uses of operations, and on estimation and mental skills, rather than solely on developing paper-and-pencil skills." (Page 251, NJMCF)
• The Truth: Don't be fooled by the hedging language. Calculator "replacement" will occur because New Jersey math educators believe it should. They insist that every child must have a calculator, beginning in kindergarten, and they leave it up to the kids to "choose" the "appropriate" computational method. Of course, any child will opt for the simplicity of calculator button-pushing, when faced with the more difficult alternative of memorizing basic math facts and acquiring paper-and-pencil skills. But progressive educators aren't bothered by this. They say that traditional computational skills are now obsolete.
• The result? If easily-acquired calculator skills are substituted for the more-difficult, step-by-step buildup of remembered number manipulation skills, New Jersey students will never be able to build more sophisticated math knowledge in the brain. They will be missing the key, first step towards growing intellectually in the ways uniquely conveyed by learning genuine math.
6. Number Sense (The sixth "content standard")
• "A "sense-building mode" is best established when students are provided with opportunities to explore number relationships, are encouraged to question and challenge, and are allowed to discover strategies and techniques of their own that ease the path to the solution of mathematical problems." (Page 173, NJMCF)
• The Truth:  Math isn't the place for self-discovered, personal methods. Each child needs to learn the standard methods of traditional K-12 math. Otherwise, Sarah may "discover" that 2 x 2 = 22, using "her own techniques", and progressive teachers won't want to tell her she's wrong.
7. Geometry and Spatial Sense (The seventh "content standard")
• "Traditionally, elementary school geometry instruction has focused on the characterization of shapes; at the secondary level, it has been taught as the prime example of a formal deductive system. While these perspectives of the content are important, they are also limiting. In order to develop spatial sense, students should be exposed to a broader range of geometric activities at all grade levels". (Page 209, NJMCF)
• The Truth:  Perhaps you interpreted the preceding statement as supporting both the "traditional" geometry topics and the new "broader range"? With all the extra time made available by the use of calculators, why not? But no! Progressive educators want to replace traditional math content with vague, open-ended discovery learning. In particular, they reject plane geometry, because it requires careful, disciplined thinking, and it can't be done by a calculator. They think it's too hard for New Jersey kids.
8. Numerical Operations (The eighth "content standard")
• Initially, the NJMCF sounds encouragingly traditional, with "Fourth-graders must know the basic facts of the multiplication table" (Page 251, NJMCF) and ""Most paper-and-pencil procedures should still be taught and one-digit basic facts should still be committed to memory" (Page 252, NJMCF) (emphasis in the original). But then the progressive writers take over. They say "Students should be able to quickly and easily recall one-digit sums and differences. The most effective way to accomplish this has been shown to be the focused and explicit use of basic fact strategies". The "strategies"? They're called "counting on", "counting back", "make ten", "doubles", and "near doubles". (Page 254, NJMCF)
• The Truth: Today's progressive educators detest the idea of memorizing basic number facts. They would prefer to have kids learn an array of "techniques" for "constructing" the sum or difference each and every time. What's the point? Why not memorize the addition and multiplication tables, freeing the mind to learn more advanced math? The result of these "figure it out each time" methods will be kids who need a calculator to multiply 7 times 9.
9. Measurement (The ninth "content standard")
• "Estimation of measures should be the focus of any work that students do with measurement. Indeed, the very concept that any continuous measurement is inexact -- that it is at best an "estimate" -- is a concept that must be developed throughout the grades." (Page, 281, NJMCF)
• The Truth:  The historical development of integral calculus was primarily motivated by the search for a general method for measuring the exact area of any region in the plane which is bounded by a continuous perimeter. Perhaps New Jersey math educators haven't heard. The search was successful. Integral calculus is routinely applied, thousands of times daily, to exactly measure areas and volumes. Will New Jersey high school graduates go away thinking that all "measurement is inexact"?
10. Estimation (The tenth "content standard")
• "One of the estimation emphases for very young children is the development of the idea that guessing is an important and exciting part of mathematics." (Page 311, NJMCF)
• "People who use mathematics in their lives and careers find estimation to be preferable to the use of exact numbers in many circumstances." (Page 309, NJMCF)
• The Truth:  Since progressives reject the idea of learning the traditional procedures of exact math, they are left with no alternative but to elevate guessing and estimating, and trash exactness. This program will result in kids who can't find exact answers. More alarmingly, New Jersey students won't see the importance of exact answers. Do you want them building your bridges?
11. Patterns, Relationships, and Functions (The eleventh "content standard")
• "From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and by doing so, make mathematical discoveries." (Page 335, NJMCF)
• The Truth: The pattern-recognition learning activities found in the NJMCF amount to aimless playing, not the carefully focused, step-by-step buildup of math knowledge.
• "Students need to have experience with situations involving linear, quadratic, polynomial, trigonometric, exponential, and rational functions" (Page 365, NJMCF)
• The Truth: The "experience" they are talking about consists of "activities" using graphing calculators. With or without one, New Jersey high school graduates will be encouraged to use "guess and check" and estimation for solving any equation.
12. Probability and Statistics (The twelfth "content standard")
• "From weather reports to sophisticated studies of genetics, from election results to product preference surveys, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Students need this mathematics to help them judge the correctness of an argument supported by seemingly persuasive data." (Page 371, NJMCF)
• "They should not believe an argument merely because various statistics are offered. Rather, they should be able to judge whether the statistics are meaningful and are being used appropriately." (Page 371, NJMCF)
• The Truth:  The preceding two statements are typical of the totally unrealistic expectations being sold by progressive educators. They would have kids believe that they are savvy experts because they have discussed "the theoretical probabilities of different events such as the possible sums of a pair of dice." (Page 371, NJMCF)
13. Algebra (The thirteenth "content standard")
• "Modern technology provides tools for supplementing the traditional focus on algebraic techniques, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen." (Page 405, NJMCF)
• The Truth:  This isn't about "supplementing" traditional algebra. It's about replacing it with calculators. Consider CPI 13.11: "Explore linear equations through the use of calculators, computers, and other technology." Yes, without a graphing calculator, New Jersey kids won't be able to find the equation of a straight line.
14. Discrete Mathematics (The fourteenth "content standard")
• "The material in this chapter is drawn from activities that have been reviewed and classroom-tested by the K-12 teachers in the Rutgers University Leadership Program in Discrete Mathematics over the past nine years; this program is funded by the National Science Foundation." (Page 442, NJMCF)
• The Truth: Sounds impressive, but, sad to say, it was all a waste of time and money. All these NJMCF "activities" are based on the terribly misguided belief that each kid can personally discover the major facts of K-12 math, without direct instruction from teachers or books. Somehow the key ideas will all emerge from hundreds of activities. For example, let's take the key idea of counting formulas for permutations and combinations, and consider how they're handled in the 46 pages of Discrete Math "activities" in the NJMCF. Yes, the words "permutation" and "combination" are mentioned, and there is an "activity" (Page 481, NJMCF) which suggests that kids will discover these formulas based on generalizing from one case. But if you're looking for any suggestion of unusual importance, or expect to see the derivation, explanation, or application of these formulas, don't bother! It's expected that each kid will somehow recognize the importance as they personally rediscover key ideas that countless mathematicians have carefully refined over hundreds of years. If this underlined clause sound like an exaggeration, consider the following quote from page 28 of the 1992 "research" paper, A Constructivist Alternative to the Representational View of Mind in Mathematical Education, (Journal for Research in Mathematics Education, 23, 2-33). These progressive gurus actually said "it is possible for students to construct for themselves the mathematical practices that, historically, took several thousands of years to evolve". Simply amazing!
15. Conceptual Building Blocks of Calculus (The fifteenth "content standard")
• "An intuitive feel for the mathematics of infinity, limit, and change are accessible and necessary for all students. Although some students will go on to study these concepts in a formal calculus class, this standard does not advocate the formal study of calculus in high school for all students or even for all college-intending students. Rather, it calls for providing opportunities for all students to informally investigate the central ideas of calculus." (Page 487, NJMCF)
• The Truth: Do you sense the fear? Apparently genuine calculus is too difficult for most New Jersey high school students, so New Jersey educationists have decided to substitute their own "informal" calculus. Parents of New Jersey, don't subject your children to this nonsense. Let them wait for genuine calculus. Otherwise, they will have to eventually unlearn the "intuitive feel" proposed here.
• The Truth: With the proper, grade-by-grade buildup of math knowledge during the K-10 years, the average student could begin to learn genuine calculus, beginning in the third year of the high school.
16. Excellence and Equity for All Students (The sixteenth and last "content standard")
• "The first fifteen standards set out high expectations for all students, and this final standard insists that all students need to, can, and will meet those standards." (Emphasis in the original) (Page 517, NJMCF).
• The Truth: The good news is that New Jersey kids won't have any difficulty meeting these so-called "standards", because there are no measurable expectations. The bad news is that New Jersey kids will "graduate" from high school with the math education that they will pick up from their parents, supplemented by the bits-and-pieces they manage to "discover" under the heading of "math" during the K-12 years.

• The Anti-Content Mindset: The Root Cause of the "Math Wars"
• Memorize Multiplication Facts?  Cheney, Yes.  Romberg, Abstain.
• Understanding NCTM Math "Standards" and NCTM "Math Reform"  [Home Page]