Why Sally Can’t Teach Math:
What Needs To Be Re-Invented In Elementary Teacher Education

Printed from: https://newbostonpost.com/2021/08/19/why-sally-cant-teach-mathwhat-needs-to-be-re-invented-in-elementary-teacher-education/

Bloomsbury Publishing Company is developing a series of books that presents research studies about re-inventing teacher education. Everybody know that teachers need better training than they’re getting now. But for some reason many people don’t seem to know what sort of training that should be.

Most state legislators probably don’t know what they are already funding with their appropriations to state colleges and universities for teacher training. They know that the purpose for public education has changed:  schools no longer seek to find and challenge ambitious or capable students but, instead, try to improve the academic achievement of their lowest achievers.  Puzzlingly, state legislators have not tried to find out what their departments of education, usually under the governor’s office, are telling their teacher preparation programs to do in the form of licensure regulations and tests.

Required coursework in the arts and sciences is what needs to be re-invented, in Massachusetts, and possibly in other states, if aspiring elementary-school teachers are to be able to teach the content knowledge that low-achievers and other struggling students need to learn.

The purpose of this essay is to spell out the arts and sciences coursework (not concepts, skills, or standards) that prospective elementary teachers in kindergarten through eighth grade should study or address in mathematics, in addition to whatever coursework they are required to take in their teacher preparation programs. The academic coursework I recommend is based in part on what the Bay State required in 2000 after revising its program approval and licensing regulations to address the Massachusetts Education Reform Act of 1993. This revision was done at the request of the commissioner of education and approved by the state’s board of education. 

The recommended coursework in this essay is also based on guidelines in 2008 spelling out what the state’s training programs should offer aspiring elementary teachers as preparation for a stand-alone licensure test in elementary-school mathematics. (While I helped put together such preparation standards in the late 1990s and early 2000s, I retired from what was then called the Massachusetts Department of Education in the fall of 2003 and had no professional involvement with the new mathematics test.)

With the approval of new state boards of education appointed after gubernatorial elections in Massachusetts in 2006 and 2014, many of the academic courses required of prospective elementary teachers in the 2000 regulations are no longer in the regulations. They have been completely replaced by ominous language, which I will discuss below. Nevertheless, it should be stressed that students in the Bay State from 2005 to 2017 have had the highest state averages on National Assessment of Educational Progress tests in both Reading and Mathematics and in both Grade 4 and Grade 8.

The standards developed 20 years ago were based on the assumption that getting kids to do as well as possible in math was a good thing. Perhaps that kind of achievement is no longer desirable. But it was considered so at the time. The approach reflected not only the K-12 standards my staff and I developed or revised, but also the licensing regulations and tests that were approved by the commissioner of education and the State Board of Education at the time.

For the sake of argument, let’s pretend that excelling at math is a good thing. Let’s also assume that the standards that preceded high math test scores in Massachusetts helped bring them about.

How did we get here?

Mathematics is probably the most difficult subject for elementary-school teachers to teach successfully if they expect students to be able to take advanced coursework in high school in mathematics and science or major in mathematics-intensive subjects when they attend college. Moreover, teaching and learning mathematics has received much attention only in recent decades, from both researchers and the federal policy makers. (On the other hand, reading has received constant attention for over a century from local, state, and federal officials — and the teaching of beginning reading is still controversial.)

The United States first became aware of the need to improve mathematics education nationally during World War II, a time when the small number of high school graduates with adequate knowledge of mathematics became woefully evident. In 1950, Congress created the National Science Foundation — which includes mathematics in its scope because so much of science depends on math. The Russians’ launch of Sputnik into space in 1957 added a sense of urgency to improving mathematics education in the United States. The National Defense Education Act in 1958 provided funds for institutes to develop new curricular materials for grades K-12 and for qualified students to pursue advanced education in the sciences and engineering. In fact, more students majored in mathematics in the 1960s and early 1970s than at any other time in the nation’s history. Yet, the mathematics education most students experienced in K-12 during the last half of the 20th century did not lead to a strong mathematically literate high school population overall in the 21st century. 

The final report of the National Mathematics Advisory Panel, established in 2006 during the presidency of George W. Bush, came out in March 2008, under the direction of Larry Faulkner, former president of the University of Texas at Austin. The charge to the Panel was to recommend “how students can be best prepared for entry into Algebra.” Algebra has long been considered the gateway to advanced science and mathematics coursework in high school and beyond. The Panel noted that success in Algebra I rests on proficiency with whole numbers, fractions, and certain aspects of geometry and measurement. These are the critical foundations for the study of algebra.    The Panel’s report further stressed the teaching of fractions (on pages 28 and 29), suggesting that knowledge of fractions is the most important foundational skill that is not developed effectively in our students.

The Panel’s task force on teacher education, headed by Deborah Loewenberg Ball, recommended “direct assessments of teachers’ actual mathematical knowledge.” 

“Research on the relationship between teachers’ mathematical knowledge and students’ achievement confirms the importance of teachers’ content knowledge. It is self-evident that teachers cannot teach what they do not know. However, because most studies have relied on proxies for teachers’ mathematical knowledge (such as teacher certification or courses taken), existing research does not reveal the specific mathematical knowledge and instructional skill needed for effective teaching, especially at the elementary and middle school level. Direct assessments of teachers’ actual mathematical knowledge provide the strongest indication of a relation between teachers’ content knowledge and their students’ achievement …” (page xxi).

This task force also recommended that “teachers must know in detail the mathematical content they are responsible for teaching …both prior to and beyond the level they are assigned to teach.”

To find out what problems young students have in mathematics, the Panel surveyed 743 Algebra I teachers in 2008. These teachers rated their students’ background preparation for Algebra I as weak.

The task force said:  “The three areas in which teachers reported their students to have the poorest preparation were rational numbers, word problems, and study habits. For changes in the curriculum leading up to Algebra I, teachers most often cited the need for a greater focus at the elementary school level on proficiency with basic mathematical concepts and skills.”

Survey results supported the Panel’s recommendations for teacher preparation and professional development. The Panel recommended that a sharp focus be placed on systematically strengthening teacher preparation, early career mentoring and support, and ongoing professional development for teachers of mathematics at every level, with special emphasis on ways to ensure appropriate content knowledge for teaching.

That is:  Not only should future math teachers learn the material while they’re in school; they should continue to re-learn it after they start teaching school.

Now, that’s the sort of approach Massachusetts used around 20 years ago. And it seemed to be working – the state started finishing number one in the country in grade 4 and grade 8 national mathematics tests. So what was the problem?

The fly in the pie chart was the dreaded “gap.” Despite all the gains among students in mathematics overall, there remained a continuing gap between the scores of low-income students and other students, on National Assessment of Educational Progress tests as well as on tests in the Bay State.

The question was:  What should be done about it?

In 2008 and 2009, governors, commissioners of education, and members of state boards of education were asked to agree to adopt Common Core’s College and Career-Ready Standards — sets of K-12 standards for mathematics and English language arts that would, it was claimed, ensure all students’ readiness for college-level coursework. Top elected and appointed officials in most states voted in 2010 and 2011 to adopt Common Core’s standards, although their legislatures were not asked to discuss or approve the decision. They also voted to join a testing consortium that would help to ensure via statewide tests that each state would meet Common Core’s goal of college readiness for all students.

How’s it been going?

Ten years later, the goal has not been met. In fact, on the latest National Assessment of Educational Progress tests, the gap between high-achieving and low-achieving students increased.  Worried NAEP officials have had no explanation for this phenomenon. (For evidence of their puzzlement, see this NAEP statement from October 2020; for some of the ugly data, see this April 2020 Pioneer Institute report.)

Nor do they know how to address the growing gaps between blacks and Asian-Americans in school achievement. Was teaching part of the problem? Not so far as we can tell.

Massachusetts is not at the top of the rank-ordered list of states needing to close gaps – and for good reason. When the mathematics scores of the state’s low-income students were compared with the scores of low-income students in the other states (when relevant data were available in 2007), it turns out that the low-income students in the Bay State were tied for first place in Grades 4 and 8. Their gains also showed up on the state’s tests – that is, Massachusetts Comprehensive Assessment System tests (known as MCAS). For example, in 2001, only about 15 percent of black and Latino 10th-graders scored at the proficient and advanced levels on the MCAS mathematics test. The percentages rose to about 45 percent in 2007 — a three-fold increase in the percentage of those who are proficient or advanced. Interestingly, the percentage of black and Latino 10th-graders who were proficient or advanced in 2007 (45 percent) was only slightly below the percentage of white students who were proficient or advanced in 2001 (50 percent).

Thus, the figures tell students and teachers a very different story from the usual analysis. As others’ scores have risen, so have the Bay State’s low-income students’ scores. The gap is large not because the performance of the state’s low-income students is worse than similar students in other states or because they haven’t shown much improvement, but because the performance of the state’s other students is so much better than those in other states.

In other words, the rising tide lifted all boats – but it lifted the boats of non-black and non-Latino students higher than those of low-income students.

While we don’t have definitive explanations of why this is the case, the performance of the state’s higher-income students may help to explain the new arts and sciences requirements in the Bay State, to be discussed below.

The Panel’s report noted that current integrated approaches at the high school level (a national phenomenon) may make it more difficult for students to take advanced mathematics course work in their senior year than a single-subject approach, beginning with Algebra I in Grade 8, that enables students to take an Algebra II course by their sophomore year. This possibility, which was based on an analysis of one state’s standards, was supported by a report to the Massachusetts Board of Education in 2000 on the sequence of mathematics courses needed for taking calculus in grade 12. This report was based on responses from mathematics department chairmen in 17 school districts in Massachusetts; almost all said, in 2000, that in order to take calculus in grade 12, most students would need to take what they called an honors-level Algebra I course in Grade 8. (The survey results were in a March 2000 internal report of what was then called the Massachusetts Department of Education; the report was called “Course Progression and Placement Leading to Enrollment in Advanced Placement Calculus in the Twelfth Grade.”)

In Massachusetts, the slowly increasing percentage from 2001 to 2007 of Grade 8 students who reported on MCAS (state test) surveys that they were enrolled in an Algebra I course or in Geometry (suggesting that they probably took Algebra I in Grade 7) may have been a major factor accounting for the state’s number-one finishes on the mathematics test given by NAEP in Grade 8.

In the meantime, it seemed to state-based educators that in order to strengthen the performance of its low-income elementary-school or middle-grade students in mathematics, the state would need to strengthen the capacity of its elementary-school teachers to teach mathematics — and to require a stand-alone licensure test in elementary mathematics, in addition to the stand-alone reading test it already required as of 2000.

The new stand-alone reading test for all prospective elementary teachers in the 2000 revision of the state’s regulations for teacher training did not entail new coursework in the arts and sciences. It meant changes in the reading-methods courses already required in teacher preparation programs. But this was not the case with the new subject test in elementary-school mathematics. Changes in approach to teaching mathematics were not the central issue driving the new mathematics test as they had been in reading.

So, how many mathematics courses in the arts and sciences should aspiring elementary teachers now take? According to the commissioner of education in the Bay State:  three or four.


As the authors of these guidelines to approved program providers wrote:

“Most approved programs for teaching licenses at the elementary level will need to expand the number and depth of mathematics courses that are available to their candidates. As in every subject area, candidates will have developed different levels of competence in mathematics prior to enrolling in the program. However, the research is clear that competence across the population in general, including candidates for licenses at the elementary level, is lower in mathematics than in reading, writing, and language arts. For those candidates enrolling with typical knowledge and fluency in mathematics, attaining the necessary level of content knowledge will normally require at least three to four college-level, subject-matter courses, i.e., 9–12 semester-hours, taught by mathematics faculty, potentially in partnership with education faculty. These should be taken after any necessary remedial courses and either integrated with or taken prior to math methods courses.”

This recommended guideline was written chiefly by in-state mathematics educators, mathematicians (Richard Bisk and Solomon Friedberg), and a scientist (Andrew Chen). It accompanied the State Board of Education’s approval of a stand-alone mathematics licensure test for prospective elementary teachers.

And it caused a flap at the college level. Whose credits would these courses reflect — arts and sciences credits for a college diploma or education credits for a teacher preparation program?

(How that came out is not clear to me. The Board of Higher Education in the Bay State might have available statistics on what the state’s preparation programs across the state now offer or require.)

Because colleges and universities in and outside the state have different course structures, schedules, general education requirements, and other constraints, and because high school graduates enrolling in these colleges may have varying degrees of mathematical preparation despite the common adoption across states of Common Core’s mathematics standards, the Massachusetts Department of Elementary and Secondary Education decided to recommend relative weightings for the four strands in its K-12 mathematics curriculum framework rather than attempt to define the specific courses that would prepare prospective teachers for the new stand-alone elementary mathematics test.  These weightings are as follows:


1.  Number & Operations               45%

2.  Functions & Algebra                 25%

3.  Geometry & Measurement       20%

4.  Statistics & Probability             10% 


By 2009, the Bay State required prospective elementary-school teachers to pass both a stand-alone reading test and a stand-alone elementary mathematics test, in addition to a General Curriculum test for the other major subjects taught in the elementary grades. The pass rate for the stand-alone mathematics test in recent years has been about 60 percent, usually a little less than the pass rate for those taking the reading test.

Whether or not aspiring elementary teachers take three or four courses in the arts and sciences, the major change that may have occurred in their academic requirements is in the academic coursework they take. Aspiring elementary teachers may experience methods courses in their preparation programs addressing the standards their future elementary students must address. But it is not as clear as it used to be what specific courses they will take in the arts and sciences to support their students. 

The first edition of the revised program approval regulations (in 2000) required thirty-six hours of arts and sciences coursework in the major subjects they teach in the elementary schools. (Those subects are composition; American, British, and other literature; mathematics; science; U.S., European, and world history; geography; economics; and U.S. government.)

The standards in 2000 led to a complaint by Margaret McKenna, then-president of Lesley University (a teachers college in Cambridge), that 36 hours of arts and sciences coursework for prospective elementary teachers would damage the enrollment of prospective middle school teachers in Lesley’s preparation program for middle school teachers. She recommended that the requirement be abolished. In response, the state education commissioner and I agreed to reduce the number of arts and sciences credit hours to 18.

But even that 18-credit compromise requirement, intended to ensure that prospective elementary teachers would have some academic background for teaching literature, U.S. history, mathematics, and science in K-8, is not in the current regulations.

Instead, current regulations require only that prospective teacher candidates for grades 1-6 “must demonstrate the necessary depth and breadth of content knowledge” needed to support all students in mastering expectations outlined in the following Massachusetts Curriculum Frameworks:


1.  2017 English Language Arts (ELA)/Literacy Framework: Grades Pre-K-8

2.  2017 Mathematics Curriculum Framework: Grades Pre-K-8

3.  2016 Science and Technology/Engineering (STE) Curriculum Framework: Grades Pre-K-8

4.  2018 History and Social Science Framework: Grades Pre-K-8


The source is Subject Matter Knowledge (SMK) Guidelines, updated September 2019, Massachusetts Department of Elementary and Secondary Education, page 13.

It is quite clear from it that teaching method (known as “pedagogy” in the education business) takes priority over academic content. The entire section under Context spells that out. Here’s a portion from page 3:

“As you can see from the continuum of content knowledge for educators above, after the provisional licensure stage of an educator’s career, all assessments associated with SMKs begin to assess content knowledge through pedagogy. Whether this is through the content-specific performance assessment of the pre-practicum gateways or during employment through the Educator Evaluation system, eventually it is inappropriate to separate out content knowledge from pedagogical skill.”

In other words:  It’s more important how you teach something than what you know.

We do not know if this philosophy of teaching, learning, and assessment was developed by teachers using classroom teaching experience to support prioritizing pedagogical skill. We do not know what the “necessary depth and breadth of content knowledge” is for a potential grade 5 or grade 6 classroom teacher teaching a Common Core-aligned curriculum. We do not know who would decide what content knowledge is necessary for that college student, nor how it would be decided.

This philosophy requires some empirical support before it is implemented in practice. But no references are provided for research, philosophy, or curriculum assessment.

In other words:  It’s a new theory with shaky theoretical underpinnings with no research to back it up and no method for obtaining research about it in the future.

Just how shaky is it? Instead of math, let’s consider reading or English language arts as an example.

The literature professor or instructor, for example, would be unlikely to want to teach academic courses in literature in classes designed only for prospective elementary-school teachers, since these future elementary-school teachers would be told not to first master the material but rather to master how to present material.

Why would a mathematics professor or instructor react differently? A college physics, science, or mathematics course – teaching actual content to students – may be disallowed for those planning to teach math or a mathematics-intensive subject or area.

Using this pedagogy-prioritized approach to curriculum development, a college instructor of literature, faced with a group of college students intending to teach literature to K-8 students, will likely choose to have them read middle school literary and non-literary texts that are suitable for them to use themselves as teachers in their own upper elementary grades and to show them how to address the standards for students in those grades.

On the other hand, college students not intending to become teachers may be able to take college literature classes and read literary or non-literary works written for adults.

Which group of students are likely to come out better educated? The ones reading texts meant for sixth-graders? Or the ones reading texts meant for adults?

Despite the suggestion in this state education document that teachers should have content knowledge two grade spans above and below the level of their license, there is no incentive for a college instructor to try to educate teacher-intending college students beyond a grade 8 reading level. This content-limiting perspective on academic coursework for college students intending on a teaching career will no doubt take care of the problem perceived by many education reformers today of well-educated teachers gravitating to jobs in high schools with large numbers of motivated or ambitious secondary students.

Dumb down their coursework and you dumb down the teachers.

If pre-kindergarten-to-grade-8 teachers are taught in their arts and sciences coursework just enough subject matter to enable them to address the Pre-K-to-grade-8 standards in their state, it’s true that their academic backgrounds will be equalized, since all states use Common Core-aligned standards. That is one way to re-invent teacher education.

But it is unlikely to address the learning needs of many if not most minority students in our schools, because it won’t cover up the failure of Common Core’s standards to live up to their promises.


Sandra Stotsky was Senior Associate Commissioner at what is now called the Massachusetts Department of Elementary and Secondary Education from 1999 to 2003. She was in charge of revising or developing all K-12 standards in major subjects as well as all teacher and administrator licensing requirements and professional development criteria. A version of this article has appeared in Heartland Daily News.


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