Jun 9, 2015 by


PODCAST:  Senior engineer, Henry W. Burke, gives the Nebraska State Board of Education the “nuts and bolts” about good math standards. Every state should follow Burke’s expert advice so that America’s STEM students will be able to compete in the world’s economy.


Mr. Burke’s 6.5.15 presentation starts at marker 3:15 and ends at 13:10: 


LINK TO VIDEO: http://www.education.ne.gov/Movies/StateBoard/June_2015_Board_Meeting.mp4




Analysis of Draft Nebraska Mathematics Standards

By Henry W. Burke



“Students whose last high school math course was Algebra II or lower have less than a 40 percent chance of earning any kind of four-year college degree.”


  1. Math Teacher Nakonia Hayes


In preparation for this report, I asked Nakonia Hayes for advice on Math Standards.  I met Nakonia (Niki) Hayes last June when Nakonia and I were speakers at the #CANiSEE Solutions Conference in Austin, Texas. 


Notable speakers at the Conference included: Dr. Sandra Stotsky, Dr. James Milgram, Dr. Peg Luksik, Jane Robbins, Sarah Perry, Lori Mashburn, Jenni White, Mary Bowen, Jeanine MacGregor, Alice Linahan, Rebecca Forest, Frank Gaffney, Glyn Wright, Nakonia Hayes, Henry Burke, and others.

Nakonia (Niki) Hayes was a K-12 math teacher, counselor, and principal. She also worked outside of teaching, mainly in journalism. She now operates a tutoring academy in Waco, Texas, providing help in math, reading, and writing.

In 2008 Niki started work on the biography of one of America’s great teachers, John Saxon.  Publishers said no one wanted to read a story about a math teacher, so she published it herself.  The book is entitled “John Saxon’s Story: A Genius of Common Sense in Math Education,” by Nakonia (Niki) Hayes. 


Her mission is to have John Saxon honored for his superior teaching methods and his continued record of success with students today. Everyone who cares about American K-12 education should join her mission.  When free to choose, people choose Saxon.  More than one million homeschoolers use Saxon textbooks!

Nakonia Hayes was a member of the Texas math curriculum standards writing team that developed the strong Type #1 2012 Texas Math TEKS (Texas Essential Knowledge and Skills). 

In response to my request for assistance, Nakonia Hayes provided numerous suggestions on creating good math standards; and I incorporated most of her suggestions and comments into this report.  When a state is embarking on developing good education standards, the state agency should enlist experienced classroom teachers to write the standards.  In a similar way, I heeded this advice by obtaining the assistance of experienced math teacher Nakonia Hayes for this report.

  1. Texas Math TEKS

Because the Texas Math TEKS (Texas Essential Knowledge and Skills) are the best (and only) Type #1 Standards in the country, I will spend some time on these Standards. 

Nakonia Hayes described the Texas Math TEKS development process this way:

            First, our TEKS document is a brand name product that was developed by 80 citizens who put in 12-hour days during three separate meetings over four months. We were charged with developing quality standards that would benefit our children and Texas citizens. We built our TEKS starting with a draft first created by a panel of mathematics experts that was commissioned by the Texas Education Agency (TEA); then we researched specific states with outstanding math standards at the time (such as Minnesota, Massachusetts, and Indiana).

            Most importantly, we brought to the table professional knowledge and experiences as educators in Texas classrooms. We knew our state’s children and their needs. The TEKS were personal to us.


Nakonia Hayes wanted to restrict the use of calculators for daily problems in the elementary grades.  Reformers on the writing team were pushing for technology in K-12 rather than traditional methods (paper and pencil) of student learning.  Ms. Hayes explained:

            Even though I vociferously advocated for standard algorithms and the restriction against calculator use among elementary students in Grades K-5, I was losing the debate. Therefore, I contacted Dr. James Milgram, one of the panel experts hired by TEA, and asked for his help.  He stepped forward, and a higher-up official at the TEA also got involved.  References to the Common Core by the TEA staff ceased.  The required teaching of standard algorithms and the restricted use of calculators in Grades K-5 were adopted in the final Math TEKS document.   

Nakonia Hayes related how the writing team wanted to create standards that were explicit, direct, and clear:

            Despite some philosophical differences on what we should include in the Math TEKS, our group did agree that the standards had to be explicit, direct, and clear. They had to be understandable not only for elementary teachers (many of whom fear mathematics and want clarity and brevity in instructions) but also for parents as well.

At the outset, the TEKS writing team made it imperative that the math standards must be traditional Type #1 standards.  Hayes explained:

            Our TEKS writing team agreed that the new TEKS standards had to be measurable with objective criteria and that each element had to be testable through objective measurements.  Our team knew that the new TEKS would not be perfect but that they needed to be traditionally oriented standards (a.k.a., Type #1) as compared with the 1997 TEKS which were “fuzzy” standards (a.k.a., Type #2).



The Texas Math TEKS are definitive and clear.  In contrast, the Common Core Standards are wordy and include complex explanations.  Ms. Hayes offered this comparison between the Texas TEKS and the Common Core:


TEKS, Grade 5, Number and Operations 3.H:

            “Represent and solve addition and subtraction of fractions with unequal denominators, referring to the same whole using objects and pictorial models and properties of operation.”


Common Core, (same standard but labeled NF1 and NF2):

             “Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)  Solve word problems involving addition and subtraction of fractions referring to the same whole including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2.”


In past testimonies before the Nebraska State Board of Education, I have suggested that the NDE should base the Nebraska Standards on the Texas TEKS.  Nebraska officials might try to argue that Texas students do not score very well on the national tests.  Why should we base our standards on the Texas Standards? 


In addition to the obvious problems associated with comparing states (e.g., differences in demographics, much higher minority percentages in Texas, language barriers, etc.), the Nebraska Department of Education (NDE) must understand several issues.


First of all, the new Texas Type #1 TEKS have not been fully implemented; and the former Standards were Type #2 standards.  In 2012 the Texas Math TEKS were adopted by the elected members of the Texas State Board of Education.  However, the K-8 Math TEKS were not required to be fully implemented into the schools until 2014 (when the textbooks and instructional materials, IMs, were available for purchase).  The High School (Grades 9-12) Math TEKS must be implemented by the 2015-2016 School Year (when the Math IMs are available to the school districts).


Another major problem is CSCOPE.  Billed as a “curriculum management system,” CSCOPE is essentially Common Core by another name.  Because it was developed by Texas Education Service Centers (similar to Nebraska’s Education Service Units) and widely marketed, many school districts bought into the CSCOPE system.  Roughly 80 % of the Texas public schools purchased CSCOPE with taxpayer funds


When Donna Garner (and other researchers) compared the Texas test scores of various school districts using CSCOPE with non-CSCOPE districts, most of the non-CSCOPE school districts invariably produced higher test scores.  Because CSCOPE is based on the failed Type #2 Common Core philosophy, this should come as no surprise.


Finally, CSCOPE and the Common Core Standards are illegal in Texas.  On 6.17.14, Texas Attorney General Greg Abbott (now Governor of Texas) issued his opinion about the illegality of the Common Core Standards in Texas.  The Texas Attorney General (TAG) ruled:

            “Texas school districts are required to provide instruction in the essential knowledge and skills…they may not use the Common Core State Standards Initiative to comply with this requirement.”


In a weak attempt to fool parents and teachers, CSCOPE has been renamed the “TEKS Resource System.”  Like Common Core, alert parents can readily see through this rebranding guise.


The NDE should not compare Nebraska’s student test scores with Texas’ student test scores for the following reasons: Texas has much higher minority percentages, the new Type #1 Texas TEKS have not been fully implemented, and 80 % of the Texas schools have been using CSCOPE (a sister to Common Core).


  1. Realities of Today’s School Children


Good teachers understand that school children today have short attention spans.  The children have limited attention spans for listening, remembering, and performing tasks; yet these qualities are needed today in school and in the workplace. 


Teachers should know that they must develop children’s attention span and thus their memory development.  As borne out in different studies, we recognize that the “multi-task” worker is not normally excellent in his performance.  In fact, he is usually mediocre on one or more levels. 


When children come to school with low attention span, why would we have them work on many tasks at the same time?  If the student is not allowed to build short memory into long memory, the student will continue the trait of random, disconnected learning.  This hampers math education, which depends upon linear, explicit understanding.


Word problems present another layer in teaching.  If the students are not reading and writing at grade level, their math skills will suffer in word problems.  Before math teachers can insist that young students write about mathematics, the students must be competent in the writing conventions (usage, spelling, punctuation, and grammar). 


When the teacher forces the students to explain endlessly how they worked a problem, this will drive the students away from word problems.  It is often a boring chore and will be seen as “busy work” and of no value.  It does not show “critical thinking.”  Getting the right answer makes children eager to learn more mathematics; positive reinforcement definitely works!


Students today have very poor deductive reasoning skills.  Unless the concept is very clear, the students will not be able to deduce the meaning.  It is important for educators not to assume students understand a concept.  Because students cannot deduce concepts that seem rather obvious to us as adults, we must “spell out” concepts and ideas to students in very clear terms.


In other reports, I have described the Common Core English Standards.  Those standards impose an arbitrary requirement of 50 % informational texts and 50 % literature.  In high school, this requirement jumps to 70 % informational texts and 30% literature.  Why is this important?


By reducing the study of complex literary texts and by requiring teachers to place a heavy emphasis instead on informational text, Common Core decreases the students’ opportunities to develop analytical thinking skills.  When students have poor critical thinking skills, this has a dramatic impact on mathematics and the other sciences, especially on engineering.


Dr. Sandra Stotsky sat on the Common Core Validation Committee.  Because the Standards were so deficient, she refused to validate the Common Core Standards for English.  Dr. Stotsky stated the following about the informational text requirement:

            This theory – that exposure to technical manuals rather than great stories will make students better readers, and ultimately better employees — is not only preposterous on its face, but refuted by all available research.


Even in non-Common Core states like Nebraska, the influence of Common Core is experienced through Common Core-aligned textbooks and instructional materials (IM’s).  The weak Nebraska Education Standards have opened the door for Common Core-aligned instructional materials in the state.  Also, many of the new teachers have been influenced by their college professors to include Common Core concepts in their classrooms.


  1. Memorizing the Multiplication Tables Improves Reasoning


Is there a connection between memorizing the multiplication tables (times tables) and developing reasoning skills?  Can that memory activity strengthen a student’s later ability to perform complex calculations?


The title for a National Post article says it all — “Math wars: Rote memorization plays crucial role in teaching students how to solve complex calculations, study says.”

          In a finding sure to inflame the math wars, a team of neuroscientists has revealed the crucial role played by rote memorization in the growing brains of young math students.


The new research, reported by American and South Korean scientists in the journal Nature Neuroscience, illustrates the “overlapping waves” theory of childhood cognitive development, in which advances are not abrupt but rather gradual shifts between strategies.  The study, involving children aged 7 to 9, required them to solve single-digit addition problems.  The article explains:

          Memorizing the answers to simple math problems, such as basic addition or the multiplication tables, marks a key shift in a child’s cognitive development, because it helps bridge the gap from counting on fingers to complex calculation, according to the new brain scanning research.

            The progression from counting on fingers to simply remembering that, for example, six plus three equals nine, parallels physical changes in a child’s brain, in which the hippocampus, a key brain structure for memory, gradually takes over from the pre-frontal parietal cortex, an area of higher order reasoning.

It is a gradual process, like “overlapping waves;” but it clearly shows that, for the growing child’s brain, rote memorization is a key step along the way to efficient mathematical reasoning.  Repeated problem solving during the early stages of arithmetic skill development contributes to memory “consolidation” and the ability to recall basic arithmetic facts.  As these problem-solving skills mature, the student depends less on inefficient strategies (like counting fingers) and more on memory-based procedures.  Repetition and memory serve as stepping stones for mature calculation. 

The tried and true methods of addition, subtraction, multiplication, and division still work very well.  The studies’ findings make a mockery of so-called “discovery-based learning.”

            One critic of the government’s adoption of “discovery-based learning,” Ken Porteous, a retired engineering professor, put it bluntly: “There is nothing to discover. The tried and true methods of addition, subtraction, multiplication and division work just fine as they have for centuries.

There is no benefit, and in fact a huge downside, to students being asked to discover other methods of performing these operations and picking the one which they like.  This just leads to confusion and frustration, and a strong dislike for mathematics; these students will invariably drop out of any form of mathematics course at the earliest opportunity.

The article explains the studies’ findings as follows:

            At the start, the greater role of the prefrontal cortex “likely reflects high levels of working memory and executive processing needed for implementing counting strategies, especially at a stage when children are still learning to solve arithmetic problems,” according to lead author Shaozheng Qin of Stanford University School of Medicine.

            The increasing activity of the hippocampus as time went on, however, “is consistent with its known role in learning and memory for encoding and retrieval of facts and events, and matches our observation of greater reliance on memory-based retrieval of addition facts” over the course of the year-long experiment.

            “In particular, the hippocampal system appears to be critical for children’s learning of mathematics in ways that are not evident in adults who have mastered basic skills,” the authors writeIt appears to play a “critical, time-limited, role” in fostering “the gradual establishment of long-lasting knowledge represented in the neocortex,” a brain area of higher order functions.

The process is “time-limited” because the role of the hippocampus seems to taper off once this knowledge has been consolidated elsewhere in the brain.  This emphasizes the importance of teaching the multiplication tables to children while they are young.  Timing is important!


  1. Math Content and Algorithms for K-12 Grades


Most adults (including adults on school boards) do not know what an “algorithm” is.  When we use the term in a sentence about mathematics, their brains freeze; and they think they are just stupid about math.  They become cranky and stop listening.  To overcome this “brain freeze,” adults should explain algorithms in easy-to-understand terms; a clear picture of an algorithm is needed. 


For example, adults should show the traditional algorithms for adding or subtracting, multiplying or dividing, including the formula to find area or perimeter of a simple figure or shape.  When students see that the formula for the area of a rectangle is: Area = Length x Width, they can apply this “algorithm” to their classroom at school or living room at home.  The more often this procedure is applied, the more likely that it will become a “principle” to get the correct answer.


Nakonia Hayes identified the following traditional content items (standards) for inclusion in K-12 Education Standards:  


K-3 Grades

Focus on teaching students to add and subtract two-digit numbers, with carrying and borrowing, accurately up through 1,000.  They need to do this without counting on their fingers, and they must have automatic recall in the process.  The standard should state the expectation of “automatic recall.”


Students must know their multiplication tables through the tens by the end of third grade.  This means immediate mental recall, not having to stop and to try to remember.  The students must have these math areas firmly implanted in their knowledge base before they begin 4th grade.


Grade 4

Focus on mastering one and two-digit multiplication and division, while continuing to review adding and subtracting.


Grade 5

Focus on mastering word problems with the four basic operations – adding, subtracting, multiplying, and dividing.  Also include square roots and exponents.


Grade 6

Focus on mastering fractions and decimals in computation and word problems, plus adding and subtracting integers.


Grade 7

Focus on mastering fractions, decimals, and percents in computation and word problems, plus multiplying and dividing integers.


Hayes suggests teachers can include some geometry, financial literacy, and data analysis based on real world problems, avoiding the faux stuff often found in textbooks that promotes a social agenda.  No calculators should be used until the 8th grade.  The standards should not tell the teachers how to accomplish these goals in the standards but should leave the “how” up to the teacher.


Grade 8

Focus on real algebra because students are competent in the four basic operations that cover whole numbers, negative numbers, fractions, decimals, and percents.  (If a person can’t multiply, he can’t divide.  If he can’t divide, he can’t succeed with algebra’s rational expressions.)


Courses for Grades 8-12

The math courses should be taught as follows:

Grade 8 – Algebra I

Grade 9 – Algebra II

Grade 10 – Geometry

Grade 11 – Trigonometry

Grade 12 – Calculus (or at least Pre-Calculus)


If the students do not have Calculus in High School, they will have to take remedial classes in college for STEM careers (Science, Technology, Engineering, and Math).  Students going to community colleges need at least Algebra II to avoid taking remedial courses at the community college level.  Do not insert Geometry between Algebra I and Algebra II; it is a great wedge that causes students to do poorly in Algebra II, thus turning them off to math and science.


  1. Critique of Nebraska Draft Math Standards


The Draft Nebraska Mathematics Standards were released to the public by the Nebraska Department of Education (NDE) on 4.3.15.  These are the links for the 2015 Draft Math Standards:   






Do the Nebraska Draft Mathematics Standards satisfy the criteria established by math teacher Nakonia Hayes? 


Compliance and Deficiencies:

  1. Grades K-3 — “Add and subtract two-digit numbers.” – The Nebraska Standards satisfy this criteria.
  2. Grades K-3 — “Know multiplication tables through the tens.” – The Nebraska Standards do not require students to memorize the multiplication tables.  (This is a major failure.)
  3. Grade 4 – “Master one-digit and two-digit multiplication and division.” –The Nebraska Standards do not require two-digit division in the 4th Grade.  This criteria is satisfied in Grade 5.
  4. Grade 5 – “Master word problems with the four basic operations – adding, subtracting, multiplying, and dividing.” – The Nebraska Standards do not include word problems for multiplying and dividing in Grade 5.
  5. Grade 5 – “Master square roots and exponents.” – The Nebraska Standards do not satisfy this requirement in Grade 5, but the Standards address this in Grade 6.
  6. Grade 6 – “Master fractions and decimals in computation and word problems. – The Nebraska Standards satisfy this criteria. 
  7. Grade 6 – “Add and subtract integers.” – The Nebraska Standards include this criteria.
  8. Grade 7 – “Master fractions, decimals, and percents in computation and word problems.” – The Nebraska Standards satisfy this criteria in Grade 6.
  9. Grade 7 – “Multiply and divide integers.” – This criteria is satisfied.
  10. Grade 8 – “Focus on real Algebra, Algebra I.” – The Nebraska Standards include Algebra I in Grade 8.
  11. Grade 9 – “Algebra II” – The Nebraska Standards end in the middle of Algebra II (like Common Core).  Because the Nebraska Standards group Grades 9-11      together, the standards cannot be identified with a certain grade level.
  12. Grade 10 – “Geometry” – The Nebraska Standards cover Geometry reasonably well in various grade levels.
  13. Grade 11 – “Trigonometry” – The Nebraska Standards do not cover Trigonometry.
  14. Grade 12 – “Calculus” – The Nebraska Standards do not cover Calculus.

The Nebraska Draft Mathematics Standards do a mediocre job of satisfying the Hayes Math Criteria in Grades K-8.  A major deficiency is the failure to require students to memorize the multiplication tables.  Without this requirement, students will suffer in their ability to handle complex calculations (as described in an earlier Section).  Such students will become overly dependent on calculators and computers; and not gain a sense for reasonable answers.


The Nebraska Math Standards have major problems at the High School level!


The Draft 2015 Math Standards include standards for each grade level K-8; however, the writers of the 2015 Draft Mathematics Standards for high school decided to group Grades 9-11 together and wrote a separate set of standards for Grade 12 (Advanced Topics).  In high school, the standards need to be explicit for each course (e.g., Algebra I, Algebra II, Geometry, etc.)


Teachers must know exactly what should be covered for every single grade/course with no exceptions.  When Grades 9-11 are lumped together, the teachers and students are unsure what falls into each grade level/course.


Without specific standards for each grade/course, teachers will have to guess what should be covered in a particular grade/course; and there will be no real accountability at each grade level or course for teachers nor for their students.


I am guessing that the NDE took its lead from the highly questionable and non-piloted Common Core Standards.  For Math, Common Core includes standards for Grades K-8, and High School.  The NDE mimics the Common Core approach and groups Grades 9-11 together, and adds Grade 12 (Advanced Topics).  Since the Common Core authors grouped several grades together, the NDE evidently decided to follow the same wrong pathway.


As covered in previous reports, the Nebraska Math Standards will not prepare students for careers in STEM (Science, Technology, Engineering, and Math).  Likewise, Nebraska High School graduates (under the Nebraska Standards) will be poorly prepared for college and the workplace.  If a Nebraska student wants to pursue college and certain careers, the student must take math courses beyond the requirements of the Nebraska Math Standards.


The government data for STEM are quite compelling.  It is extremely rare for students who begin their undergraduate years with coursework in pre-calculus (or an even lower level of mathematical knowledge) to achieve a bachelor’s degree in a STEM area.

            U.S. government data show that only one out of every 50 prospective STEM majors who begin their undergraduate math coursework at the precalculus level or lower will earn a bachelor’s degree in a STEM area.

            Moreover, students whose last high school math course was Algebra II or lower have less than a 40 percent chance of earning any kind of four-year college degree.

            In addition, the National Center for Education Statistics (NCES) publication STEM in Postsecondary Education shows that only 2.1 percent of STEM-intending students who had to take pre-college mathematics coursework in their freshman year graduated with a STEM degree.




Several startling conclusions are revealed in the NCES study, “STEM in Postsecondary Education: Entrance, Attrition, and Coursetaking Among 2003−04 Beginning Postsecondary Students.”  These include:


  1. If the incoming STEM college students have taken Calculus in High School, 69 % of them will complete a STEM degree.
  2. If the incoming STEM students must take introductory college-level math, only 15 % of the students will earn a STEM degree.
  3. If the incoming non-STEM students must take introductory college-level math, 26 % of the students will leave college without a degree.




[Note: Introductory college-level math courses are initial or entry-level math courses that represent prerequisites for other courses/degrees; these courses are commonly referred to as “gatekeeper” or “gateway” courses.]





The first mistake that the Nebraska Department of Education (NDE) made was basing the 2015 Nebraska Mathematics Standards on the poor, Type #2 Nebraska 2009 Mathematics Standards.  A poor foundation will never yield good standards.


In numerous reports and testimonies before the State Board of Education (SBOE), I have suggested that Nebraska should utilize the Texas Math TEKS as the basic foundation for our Standards.  Simply modify and adjust the TEKS to suit Nebraska’s needs.


Also I have pleaded with the SBOE to include Saxon Math in the Nebraska Math Standards.  Saxon Math produces proven results and higher test scores.


Reliable scientific research proves that memorizing the multiplication tables helps to develop reasoning skills.  Because this neurological growth is time-sensitive, the memorization must take place at an early age.


For this report, I obtained the input from an excellent math teacher, Nakonia Hayes.  In addition to her teaching expertise, Ms. Hayes served on the 2012 Texas Math TEKS writing team.


The Nebraska Draft Mathematics Standards do a mediocre job of satisfying the Hayes Math Criteria in Grades K-8.  A major deficiency is the failure to require students to memorize the multiplication tables. 


The Nebraska Draft Standards are especially deficient at the High School level.


The Nebraska Department of Education (NDE) made a major blunder when it grouped Grades 9-11 together.  The NDE needs to separate each grade level and each course.  Unless this is done, teachers will not know what should be covered in each grade level/course.


The NDE should list the Math Standards as follows:

          Grade 8 – Algebra I

          Grade 9 – Algebra II

          Grade 10 – Geometry

          Grade 11 – Trigonometry

          Grade 12 – Calculus


An NCES study reveals that Calculus in High School is a prime determinant for success with STEM courses in college.


Also non-STEM college students can benefit from an adequate high school math background. 


“Students whose last high school math course was Algebra II or lower have less than a 40 percent chance of earning any kind of four-year college degree.”



Bio for Henry W. Burke

Henry Burke is a Civil Engineer  with a B.S.C.E. and M.S.C.E.  He has been a Registered Professional Engineer (P.E.) for 37 years and has worked as a Civil Engineer in construction for over 40 years. 

Mr. Burke had a successful 27-year career with a large construction company. 

Henry Burke serves as a full-time volunteer to oversee various construction projects. He has written numerous articles on education, engineering, construction, politics, taxes, and the economy.

Henry W. Burke

E-mail:  hwburke@cox.net

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