All Contents
Overview

Math matters. Elementary math skills are a strong predictor of who will graduate from high school.

When students struggle with the foundational math skills taught in the early elementary grades, they are likely to continue to struggle well into middle and high school.1 Mathematics knowledge is highly cumulative in nature, meaning that the ability to master each new concept is highly dependent upon having mastered the content that came before. Skill by skill. Grade by grade.

As early as in kindergarten, the role of teachers in building and cementing these math skills is all-important. Unfortunately, many elementary teachers do not themselves feel adequately confident of their own basic math skills.2 Potentially lacking confidence or sufficient content knowledge, they may dedicate less time to teaching math than students need, unsure of how to help their students avoid common misconceptions and errors. Compounding the issue, a basic problem for years has been the assumption that anyone who has graduated from high school has all the knowledge they need to teach elementary math. However, teaching elementary math requires a conceptual understanding of foundational mathematics and pedagogical knowledge, neither of which is addressed in a general mathematics course.3 Preparation programs must dedicate sufficient time to both.

While weak math instruction in elementary schools is not a new problem, learning loss as a result of the COVID-19 pandemic brings new urgency to this issue. Multiple studies find that elementary students, already well behind their peers in other nations, lost more learning in math than in reading during the pandemic.4

Amid these challenges, there's hopeful progress to report. In NCTQ's latest examination of the math preparation that educator preparation programs provide to aspiring teachers, we find that undergraduate programs across the nation are responding to the call to build in more time for math—not just for candidates to learn how to teach it, but also to develop the necessary conceptual understanding of the content. However, there is still room for improvement in how this additional instructional time is spent to ensure all essential mathematical content is adequately covered.

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Findings

Undergraduate programs are dedicating significantly more time to elementary school mathematics.

Since 2014, NCTQ has recommended that programs dedicate essentially four courses to mathematics (three courses in elementary math content and a single course in math pedagogy). Programs have responded accordingly, adding 1.5 credit hours of elementary mathematics to their graduation requirements. The addition means that programs have built-in 19% more time for math preparation, a significant increase.

Change in Undergraduate Elementary Mathematics Coursework Since 2014
0
1
2
3
4
5
6
7
8
9
10
2022 9.5
9.5

The average number of elementary mathematics course credits for undergraduate elementary programs in 2022.

738 total sample
2014 8
8.0

The average number of elementary mathematics course credits for undergraduate elementary programs in 2014.

738 total sample

Required elementary mathematics course credits

In spite of allocating more overall time to mathematics preparation, many undergraduate programs are not making optimal use of this instructional time.

Finding enough credit hours available to prepare teachers in both mathematics content and pedagogy may be the hardest step for programs to take—given the competition for credit hours in an institution—but it is a step many programs have accomplished. The next challenge for programs is to ensure the time is used optimally, ensuring that courses include all of the content topics and math pedagogy essential to fully preparing a new elementary teacher.

Only 15% of undergraduate programs earn an A by adequately covering all of the math topics and pedagogy that elementary teachers need (see Figure 3 below). However, most of the programs earning a B—38% of the sample—need to make only modest adjustments to existing coursework to qualify for an A.

Programs earning an A or a B constitute half the sample, another indication that programs are heading in the right direction.

Undergraduate program grades on the Elementary Mathematics standard in 2022
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As the chart below shows, programs on average exceed the time needed for math pedagogy, yet fall short of the instructional hours needed to develop a teacher candidate's deep understanding of math content.

Time dedicated to elementary math content versus pedagogy by undergraduate programs
0
15
30
45
60
75
90
105
Mathematics Content
85
85 hours

The average number of instructional hours in mathematics content provided by undergraduate programs.

105
105 hours

The minimum number of instructional hours to adequately address elementary mathematics content.

Mathematics Pedagogy
49
49 hours

The average number of instructional hours in mathematics pedagogy provided by undergraduate programs.

45
45 hours

The minimum number of instructional hours to adequately address elementary mathematics pedagogy.

Number of instructional hours

Program average
Recommended minimum Info icon
The minimum number of instructional hours required to address this topic area. See the full Methodology for more information.
0
15
30
45
60
75
90
105
120
135
150
Numbers & Operations + Algebraic Thinking (combined) Info icon
The number of hours estimated for Numbers & Operations and Algebraic Thinking are treated together because course descriptions often combine the two topics. This may occur because student standards (such as the Common Core State Mathematics Standards) combine Operations and Algebraic Thinking through the mid-elementary grades, with Algebra becoming more of a separate topic in later grades.
49
49 hours

The average number of instructional hours in Numbers & Operations + Algebraic Thinking (combined) provided by undergraduate programs.

65
65 hours

The minimum number of instructional hours to adequately address Numbers & Operations + Algebraic Thinking (combined).

Geometry & Measurement
24
24 hours

The average number of instructional hours in Geometry & Measurement provided by undergraduate programs.

25
25 hours

The minimum number of instructional hours to adequately address Geometry & Measurement.

Data Analysis & Probability
13
13 hours

The average number of instructional hours in Data Analysis & Probability provided by undergraduate programs.

15
15 hours

The minimum number of instructional hours to adequately address Data Analysis & Probability.

Mathematics Pedagogy
49
49 hours

The average number of instructional hours in mathematics pedagogy provided by undergraduate programs.

45
45 hours

The minimum number of instructional hours to adequately address elementary mathematics pedagogy.

Total
134
134 hours

The average number of instructional hours in mathematics content and pedagogy provided by undergraduate programs.

150
150 hours

The minimum number of instructional hours to adequately address elementary mathematics content and pedagogy.

Number of instructional hours

Program average
Recommended minimum Info icon
The minimum number of instructional hours required to address this topic area. See the full Methodology for more information.

Graduate programs have a very different story to tell. The average graduate program spends less than a single credit hour on math content.

The overwhelming majority of graduate programs preparing elementary teachers do not dedicate sufficient time to teaching mathematics content, explaining how 85% of graduate programs earn an F.

Graduate program grades on the Elementary Mathematics standard in 2022
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Where there is mathematics coursework, it is dedicated to mathematics pedagogy, largely ignoring elementary mathematics content. This distribution of instructional time appears to reflect measures taken at admission to ensure teacher candidates have demonstrated content knowledge mastery. However, only 16 programs (5% of the graduate sample) were found to require a passing score on a content knowledge licensing test that independently measures mathematics knowledge.

Time dedicated to elementary math content versus pedagogy by graduate programs
0
15
30
45
60
75
90
105
Mathematics Content
14
14 hours

The average number of instructional hours in mathematics content provided by graduate programs.

105
105 hours

The minimum number of instructional hours to adequately address elementary mathematics content.

Mathematics Pedagogy
38
38 hours

The average number of instructional hours in mathematics pedagogy provided by graduate programs.

45
45 hours

The minimum number of instructional hours to adequately address elementary mathematics pedagogy.

Number of instructional hours

Program average
Recommended minimum Info icon
The minimum number of instructional hours required to address this topic area. See the full Methodology for more information.
RECOMMENDATIONS

Accelerating Progress

The additional time that many undergraduate elementary programs have added in mathematics since 2014 represents real and significant progress. As many of these programs now only need to make moderate adjustments in how they use their time, we fully expect to be able to report in a few years that most new teachers coming through undergraduate programs are at long last receiving sufficient preparation in mathematics.

The status of graduate preparation is nowhere near as favorable, nor is their outlook as promising. With almost all graduate programs receiving a grade of F, both the institutions and the states that serve as their regulators must urgently identify viable ways to support teachers prepared at the post-baccalaureate level in mathematics, regardless of how someone enters the profession.

We offer eight ways teacher preparation programs and their state policymakers can make the progress that's needed:

Actions for teacher preparation programs

1

Dedicate ten credits to elementary mathematics and ensure the required coursework provides the necessary instructional time to address the essential knowledge elementary teachers need.

Lacking a system of strong diagnostic testing at the point of admissions, most candidates will need three content courses and one course in pedagogy. Specifically, elementary teachers need a strong conceptual understanding of four content topics (Numbers and Operations; Algebraic Thinking; Geometry and Measurement; and Data Analysis and Probability), in addition to Math Pedagogy. This knowledge is specialized, so should be aimed only at a teacher audience, not the broader campus population.

2

In the case of graduate programs, make much more extensive use of content knowledge tests during admissions, even if only for the purpose of diagnosing where candidates are going to need additional support and coursework rather than rejecting applicants.

Such testing does not mean that programs will need to turn down candidates if they fail. It does mean that they can accept candidates conditionally and that the candidates will need to take additional coursework to acquire the content knowledge needed of elementary teachers. Regardless, graduate programs also need to add more time to the requisite coursework dedicated to mathematics.

3

Build partnerships with nearby districts to create specific feedback loops related to elementary mathematics instruction.

Programs and districts could come together to review the course materials and content expectations for teacher candidates to determine if the program is meeting districts' needs. Consider the use of focus groups or surveys to understand specifically which key topic areas recent teacher candidates felt well-prepared to teach and which they did not.

4

Ensure student teaching placements occur with mentor teachers who have demonstrated knowledge of math content.

A large amount of recent research has demonstrated that student teachers who are paired with a more effective cooperating teacher are more effective in their first year of teaching. Programs should also consider how well-versed their program supervisors are in math content.

Actions for state policymakers

5

Make current standards for elementary mathematics preparation more explicit and assess programs on their alignment to the standards during the program approval process.

Currently, 11 states provide detailed math standards for elementary teacher preparation programs (see Appendix C). Arkansas, Massachusetts, and New Mexico stand out for their development of competencies for elementary teachers as a component of teacher preparation program requirements. Another 16 states adopted CAEP standards or require CAEP accreditation that also includes detailed math standards. Further, state education agencies should use these standards for elementary mathematics preparation in their review of programs.

6

Examine the state licensure tests for elementary licensure candidates to ensure alignment between what is required of elementary teachers and expectations for students.

Ensure that the licensure tests require candidates to demonstrate knowledge of the essential math topics found in the standard. Depending on the assessment, make revisions to strongly align content expectations for teachers with the math standard and standards for students (see above).

7

Hold programs accountable for fully preparing any candidate they have admitted by scrutinizing program pass rates on state licensing tests, particularly the first-time pass rates.

Only 21 states currently use pass rate data in their program approval systems, with none examining the first-time pass rates, the best indicator of program commitment to preparing all of their teacher candidates to meet state standards. To learn more about this issue and how many programs achieve high first-time pass rates regardless of the populations they serve, policymakers can review their state-specific dashboards here.

8

If shortages are a concern, consider creating a certification pathway in mathematics that would qualify a teacher to teach only the early elementary grades (K-2).

Early elementary grades require much less math knowledge and therefore less coursework would be required. This concentration would need to focus on fewer topics, but could ensure that all content-topics are covered with depth.

There has long been a broad consensus among mathematicians, math educators, the organizations which represent them, as well as independent organizations seeking to improve student outcomes in mathematics that elementary teachers need to acquire specialized content preparation in mathematics not found in the math coursework required of other college students. As the nation struggles to emerge from a global pandemic and our schools grapple with the consequence of so much instructional loss, this need takes on important new urgency. Fortunately, we can report evidence of progress, suggesting that the significant work that remains to be done is eminently achievable.

Promising Practices

Exemplar Resources

Varying designs of elementary mathematics coursework requirements for four programs are provided here as a resource. The University of Montana stands out for requiring identical coursework for both its undergraduate and graduate programs. All candidates pursuing initial elementary certification at the institution must complete the same five courses. This structure is unique not only in terms of the number of courses required at the graduate level but in that the institution did not create separate sets of courses for two programs preparing candidates for the same career.

CALIFORNIA
California State University - Northridge
Undergraduate
225 instructional hours
Download elementary mathematics coursework requirements
MONTANA
University of Montana
Undergraduate and Graduate
224 instructional hours
Download elementary mathematics coursework requirements
OKLAHOMA
University of Central Oklahoma
Undergraduate
279 instructional hours
Download elementary mathematics coursework requirements
WEST VIRGINIA
West Virginia State University
Undergraduate
180 instructional hours
Download elementary mathematics coursework requirements

A+ Programs

These 79 exemplary programs earn an A+ because they meet 100% of the target instructional hours under each of the five topics.

Indiana
Indiana University-Purdue University Indianapolis
A+ Undergraduate
Montana
Salish Kootenai College
A+ Undergraduate
Massachusetts
University of Massachusetts - Amherst
A+ Graduate
Michigan
University of Michigan - Flint
A+ Undergraduate
Oklahoma
University of Central Oklahoma
A+ Undergraduate
Connecticut
Western Connecticut State University
A+ Undergraduate
Virginia
Radford University
A+ Undergraduate
Louisiana
University of Louisiana at Lafayette
A+ Undergraduate
Show more See all program scores
How We Scored

Evaluation relies on two sources of data to produce a preliminary score for each program to review before the score is finalized:

  • Course syllabi

  • Course descriptions

To determine the validity of using course descriptions in lieu of syllabi, a pilot test evaluated 200 courses for which both course descriptions and syllabi were available to arrive at two independent assessments for how much time was dedicated to each of the five topics. The two forms of analysis produced similar results on average, within a range of 10 percentage points for each of the five topics. Further, the distribution of program grades under the two approaches was highly correlated, with no notable bias towards either approach. For more details from this pilot study, go here.

Methodology in brief

A team of analysts use course catalogs to determine the required coursework for each elementary program in the sample. Analysts then read course titles and descriptions to pinpoint all courses that are inclusive of mathematics. Teacher audience mathematics courses, inclusive of those that focus on both content and pedagogy are flagged for analysis. General audience mathematics courses (College Algebra, for example) are excluded from analysis.

A separate team of expert mathematics analysts evaluate syllabi and course descriptions using a detailed scoring protocol. Each course is analyzed for its coverage of five topics:

  • Numbers and Operations

  • Algebraic Thinking

  • Geometry and Measurement

  • Data Analysis and Probability

  • Mathematics Pedagogy

Two additional topic categories of "Other Content Topics" and "Other Pedagogy Topics" are also included in the coding process so that non-relevant material (science pedagogy, for example) can be accounted for while coding the totality of each course.

Course descriptions for every course are independently evaluated by two analysts. Twenty percent of syllabi are randomly selected to be scored by a second analyst. In both cases, disagreements are adjudicated by a third analyst.

Course coding

When coding a Course Description

When coding a Syllabus

Reference Count

Count of the total number of references to each identifiable topic (e.g., Numbers and Operations)

Unit Type

Identify the type of calendar (daily, weekly, etc.) used in the syllabus.

Unit Count

Identify the total number of units (days, weeks, etc.) that are defined in the course schedule.

Reference Count

For each unit, count of the total number of references to each identifiable topic.
 

Within the course description or details presented in the syllabus...

Within each unit specified in the syllabus...

1. Numbers and Operations

Count of the number of references to Numbers and Operations including any subtopics within that domain.

2. Algebraic Thinking

Count of the number of references to Algebraic Thinking, including any subtopics with that domain.

3. Geometry and Measurement

Count of the number of references to Geometry and Measurement including any subtopics with that domain.

4. Data Analysis and Probability

Count of the number of references to Data Analysis and Probability, including any subtopics with that domain.

5. Mathematics Pedagogy

Count of the number of references to mathematics instructional approaches.

6. Other Content Topics

Count of the number of references that address mathematics content topics outside any of the four content areas (e.g., references to trigonometry or calculus).

7. Other Pedagogy Topics

Count of the number of references that address general pedagogy or subject-specific pedagogy for subjects other than mathematics.

After coding all courses, the percentage of each course dedicated to each topic is calculated. The resulting percentages are multiplied by the total instructional hours for the course (1 semester credit hour is equal to 15 instructional hours). The instructional hour counts are summed across all courses within a program and those values are measured against the scoring rubric.

Info icon
Learn more about the development and scoring of the Elementary Mathematics standard.
See the full Technical Manual for Elementary Mathematics (2022) for details on the development of the standard, the sample of programs, scoring protocols, coding reliability between data sources, and supporting research.

A program's instructional hours under each topic are measured against individual targets. Instructional hours are counted up to the target, but not beyond, which means a program cannot make up for a lack of hours under one topic with excess in another. However, there is one exception.

While Numbers and Operations and Algebraic Thinking are separately coded and reported, for the purposes of scoring, the two topics are considered together. The result is that excess hours under either of those individual topic targets can be applied to the combined target.

Instructional hour targets

Numbers & Operations

Algebraic Thinking

Geometry & Measurement

Data Analysis & Probability

Math Pedagogy

Total

45 hours

20 hours

25 hours

15 hours

45 hours

150 hours

65 hours

Programs that require a passing score on a content knowledge licensing test as a condition of admissions into the preparation program automatically receive credit for 80% of the instructional hour targets for the four content topics (not Math Pedagogy), which is then added to the instructional hours these programs require through content coursework for the purpose of scoring this standard. The 80% credit is only given under this measure in instances where the content knowledge licensing test has an independent cut score for mathematics.

Grade rubric

Grade

Total Percentage of Instructional Hours Target

A+

At least 150 instructional hours across the five topics and 100% of the recommended target hours under each of the five topics.

A

At least 135 instructional hours across the five topics and at least 90% of the recommended target hours under each of the five topics

B

At least 120 instructional hours (80%) across the five topics

C

At least 105 instructional hours (70%) across the five topics

D

At least 90 instructional hours (60%) across the five topics

F

Fewer than 90 instructional hours (less than 60%) across the five topics

Note: In cases where a syllabus is not available and the course description does not provide enough detail to be coded, the course is designated as "couldn't be determined" (CBD). When that designation is applied to any course within a program, the program's grade is listed as "CBD."

Does mathematical content knowledge matter for elementary teachers?

In general, elementary students achieve more in math when taught by teachers with greater mathematics content knowledge.5

Unfortunately, completing a bachelor's degree or a teacher preparation program does not guarantee that teachers know the math they'll be expected to teach. One study found that many elementary teacher candidates had misconceptions about statistics and probability as they were about to enter student teaching (the culminating experience of most teacher preparation programs).6 A national survey found that few elementary teachers felt very well-prepared to teach specific elementary mathematics topics, and the proportion who felt very well-prepared declined between 2012 and 2018.7 This sense of inadequate preparation has persisted for decades; in 2002, surveys of elementary teachers in Michigan and Ohio8 indicate that they did not feel well prepared to teach the specific mathematics topics at the elementary level or slightly beyond.

Most research finds that teacher candidates' mathematics coursework seems to yield benefits for their students. Several studies demonstrated that teachers deliver stronger lessons on topics that they learned in their teacher preparation programs.9 One study of teacher preparation programs (both traditional and alternative) in New York City found that math courses correlated with increased student achievement in math during the second year of teaching,10 and another study found that not only the number of content courses but also the types of courses matters for building candidates' knowledge,11 although one study found no correlation between teachers' math education credits and student achievement in math.12

How much coursework do elementary teacher candidates need?

Prospective elementary teachers need mathematics courses which are designed specifically for teachers and which impart a deep understanding of elementary and middle school mathematics concepts.13 The Conference Board of the Mathematical Sciences (CBMS)14 recommends that aspiring elementary teachers take 12 semester-credit hours in "elementary mathematics content" covering numbers and operations, algebra, measurement and data, and geometry, while the National Council of Teachers of Mathematics (NCTM)15 recommends taking at least three college-level mathematics courses in the content essential to elementary grades, in addition to instruction on pedagogy.16 The Mathematical Education of Teachers II (MET II) study draws from the Common Core State Standards to recommend that elementary teachers be prepared in the domains of counting and cardinality, operations and algebraic thinking, numbers and operations, measurement and data, and geometry, as well as connections to mathematics topics typically addressed in the middle grades.17

Some research casts doubt on the extent to which current teacher preparation programs adequately meet the mathematics needs of aspiring elementary teachers. Several surveys of over 400 institutions, taken 6 years apart, found that most were not meeting the recommendation that elementary candidates take at least 12 semester credit-hours of mathematics content.18 Another study found that mathematics content courses were inconsistent in whether they engaged teacher candidates in the Common Core Standards for Mathematical Practice.19

What types of math courses should elementary teacher candidates take?

The preponderance of available research indicates that the mathematics content coursework needed by elementary teachers is neither pure mathematics nor pure methods but a combination of both.20 Teachers with more specialized content knowledge can better design lessons using math-science integration, use manipulatives in their lessons, and employ student-centered approaches to teaching mathematics.21

Experts suggest that educator preparation programs should structure requirements to address both subject matter knowledge (including common content knowledge and specialized content knowledge) and pedagogical content knowledge (including knowledge of content and students and knowledge of content and teaching).22 The approach described by Lee Shulman23—built off of early work begun by John Dewey and recently expanded by Deborah Ball—explains the complexity of teaching by delineating the domains of knowledge needed for teaching.

What should elementary teacher candidates learn about specialized mathematics content knowledge?

Elementary teachers need to grasp more than the mathematical knowledge and skills required in the curriculum. They need to master the mathematical knowledge that is unique to teaching.24 Examples of what teachers need to learn include being able to create and tailor representations of math problems to suit the "instructional purposes," being able to not only carry out but also explain algorithms for solving problems, and conducting error analysis.25 An effort to "unpack the mathematical work of teaching framework" to further explore what teachers should be able to do include activities such as, "Given conflicting explanations, determine which is valid and why," "Write a mathematically valid explanation for a process or concept," "Given a word problem, choose another word problem with the same structure," and "Given a set of representations, choose which does or does not show a particular idea."26

What should elementary teacher candidates learn about mathematics pedagogy?

Research on mathematics methods or pedagogy, although limited, also indicates the value of mathematics methods courses,27 including documented gains in mathematical knowledge for teaching.28 Research generally supports the importance of teachers' knowledge of fundamental math concepts as well as their ability to apply mathematics content in teaching (learned in mathematics methods courses), rather than their just knowing the mathematics content.29

Conclusion

In 2022, NCTQ revised its Elementary Mathematics Standard in keeping with the research that elementary teachers must be equipped with the mathematics content and pedagogical knowledge to effectively support learning by all students. NCTQ's examination of the opportunities a program provides to teacher candidates includes the domains of knowledge a teacher needs to bring to the classroom — the specialized mathematical content knowledge or knowledge of mathematics that is specific to teaching mathematics and pedagogical knowledge for teaching mathematics or the intersections between the mathematical content that comprises the curricula and knowledge of how student learn and effective methods to teach.30

Endnotes

  1. Claessens, A., & Engel, M. (2013). How important is where you start? Early mathematics knowledge and later school success. Teachers College Record, 115(6), 1-29; Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). The Groove of Growth: How Early Gains in Math Ability Influence Adolescent Achievement. Society for Research on Educational Effectiveness; Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691-697; Duncan, G. J., & Magnuson, K. (2011). The nature and impact of early achievement skills, attention skills, and behavior problems. Whither Opportunity, 47-70.

  2. Banilower, E. R., Smith, P. S., Malzahn, K. A., Plumley, C. L., Gordon, E. M., & Hayes, M. L. (2018). Report of the 2018 NSSME+. Horizon Research, Inc., Retrieved from http://www.horizon-research.com/report-of-the-2018-nssme; Banilower, E. R. et al. (2013). Report of the 2012 National Survey of Science and Mathematics Education. Horizon Research, Inc., Retrieved November 1, 2018, from http://www.horizon-research.com/2012nssme/wp-content/uploads/2013/02/2012-NSSME-FullReport1.pdf.

  3. National Council of Teachers of Mathematics. (2005, July). Highly qualified teachers: A position of the National Council of Teachers of Mathematics (Position Paper). Available at https://www.nctq.org/dmsView/NCTM_2005_HQ_teachers_statement. American Mathematical Society in Cooperation with the Mathematical Association of America. (2012). The Mathematical Education of Teachers II, 17. Available at https://www.nctq.org/dmsView/CBMS_Issues_in_Mathematics_Education.

  4. Halloran, C., Jack, R., Okun, J. C., & Oster, E. (2021). Pandemic schooling mode and student test scores: Evidence from US states. (No. w29497). National Bureau of Economic Research. Retrieved from https://www.nber.org/papers/w29497; Kuhfeld, M., Soland, J., & Lewis, K. (2022). Test Score Patterns Across Three COVID-19-impacted School Years. (EdWorkingPaper: 22-521). Annenberg Institute at Brown University. Retrieved from https://doi.org/10.26300/ga82-6v47.

  5. Blazar, D. (2015). Effective teaching in elementary mathematics: Identifying classroom practices that support student achievement. Economics of Education Review, 48, 16-29; Campbell, P. F., Nishio, M., Smith, T. M., Clark, L. M., Conant, D. L., Rust, A. H., DePiper, J. N., Frank, T. J., Griffin, M. J., & Choi, Y. (2014). The relationship between teachers' mathematical content and pedagogical knowledge, teachers' perceptions, and student achievement. Journal for Research in Mathematics Education, 45(4), 419-459; Hill, H., Charalambous, C. Y., & Chin, M. J. (2019). Teacher characteristics and student learning in mathematics: A comprehensive assessment. Educational Policy, 33(7), 1103-1134; Hill, H., Rowan, B., & Ball, D. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406; Kukla-Acevedo, S. (2009). Do teacher characteristics matter? New results on the effects of teacher preparation on student achievement. Economics of Education Review, 28, 49-57; Mapolelo, D. C., & Akinsola, M. K. (2015). Preparation of mathematics teachers: lessons from review of literature on teachers' knowledge, beliefs, and teacher education. International Journal of Educational Studies, 2(1), 01-12. Blazar (2015) found a relationship between teachers' mathematical content knowledge and two instructional characteristics: ambitiousness of instruction and frequency of errors and imprecision. While these characteristics also related to student achievement, the study did examine the direct relationship between teachers' math content knowledge and student achievement.

  6. Gorham Blanco, T., & Chamberlin, S. A. (2019). Pre-service teacher statistical misconceptions during teacher preparation program. The Mathematics Enthusiast, 16(1), 461-484.

  7. Banilower, E. R., Smith, P. S., Malzahn, K. A., Plumley, C. L., Gordon, E. M., & Hayes, M. L. (2018); Banilower, et al. (2013).

  8. Schmidt, W. H., & McKnight, C. (2002). Inequality for all: The challenge of unequal opportunity in American schools. New York: Teachers College Press, Columbia University.

  9. Hiebert, J., Berk, D., & Miller, E. (2017). Relationships between mathematics teacher preparation and graduates' analyses of classroom teaching. The Elementary School Journal, 117(4), 687-707; Hiebert, J., Berk, D., Miller, E., Gallivan, H., & Meikle, E. (2019). Relationships between opportunity to learn mathematics in teacher preparation and graduates' knowledge for teaching mathematics. Journal for Research in Mathematics Education, 50(1), 23-50; Morris, A. K., & Hiebert, J. (2017). Effects of teacher preparation courses: Do graduates use what they learned to plan mathematics lessons?. American Educational Research Journal, 54(3), 524-567; Suppa, S., DiNapoli, J., & Mixell, R. (2018). Teacher Preparation" Does" Matter: Relationships between Elementary Mathematics Content Courses and Graduates' Analyses of Teaching. Mathematics Teacher Education and Development, 20(2), 25-57.

  10. Boyd, D. J., Grossman, P. L., Lankford, H., Loeb, S., & Wyckoff, J. (2009). Teacher preparation and student achievement. Educational Evaluation and Policy Analysis, 31(4), 416-440. This study notes that its findings may differ from those in Harris & Sass (2011) because the Boyd study looked at "data on the characteristics of programs, courses, and field experiences," while the Harris study used course credit hours and hours of in-service training.

  11. Qian, H., & Youngs, P. (2016). The effect of teacher education programs on future elementary mathematics teachers' knowledge: a five-country analysis using TEDS-M data. Journal of Mathematics Teacher Education, 19(4), 371-396. This study found that (a) discrete structure and logic and (b) continuity and functions had the strongest effect on candidates' mathematics content knowledge.

  12. Harris, D. N., & Sass, T. R. (2011). Teacher training, teacher quality and student achievement. Journal of Public Economics, 95, 798-812. This study relates to several NCTQ standards. Although it meets the criteria for strong research, the study's findings run contrary to the conclusions of most strong research in the field.

  13. In this vein, a University of Virginia professor of psychology argued that elementary teachers need to be trained to understand and teach the "conceptual side of math," or else they cannot build a strong math foundation for their young students. Willingham, D. (2013). What the NY Times doesn't know about math instruction. Retrieved March 12, 2014 from http://www.danielwillingham.com/daniel-willingham-science-and-education-blog/what-the-ny-times-doesnt-know-about-math-instruction.

  14. Beckmann, S., Chazan, D., Cuoco, A., Fennell, F., & Findell, B. (2012). The Mathematical Education of Teachers II. In Issues in mathematics education/CBMS, Conference Board of the Mathematical Sciences (Vol. 17, pp. 1-86). Retrieved January 7, 2022 from https://www.cbmsweb.org/archive/MET2/met2.pdf.

  15. National Council of Teachers of Mathematics. (2005, July).

  16. National Council of Teachers of Mathematics. (No Date). Executive Summary: Principles and standards for school mathematics. Retrieved April 26, 2021 from https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf.

  17. Beckmann, S., Chazan, D., Cuoco, A., Fennell, F., & Findell, B. (2012).

  18. Masingila, J. O., Olanoff, D. E., & Kwaka, D. K. (2012). Who teaches mathematics content courses for prospective elementary teachers in the United States? Results of a national survey. Journal of Mathematics Teacher Education, 15(5), 347-358; Masingila, J. O., & Olanoff, D. (2021). Who teaches mathematics content courses for prospective elementary teachers in the USA? Results of a second national survey. Journal of Mathematics Teacher Education, 1-17.

  19. Max, B., & Welder, R. M. (2019). Engaging prospective elementary teachers in standards for mathematical practice within content courses for teachers. In Proceedings of the 41st Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1141-1145).

  20. Greenberg, J., & Walsh, K. (2008). No Common Denominator. Washington, D.C.: National Council on Teacher Quality. However, a recent international comparative study claims that while both content knowledge and pedagogical content knowledge are important, "…CK may not necessarily need to develop before PCK…future reforms should consider the best ways to foster content and pedagogical content knowledge as distinct constructs rather than working under the assumption that developing CK is a necessary prerequisite for developing PCK." (p 16). Murray, E., Durkin, K., Chao, T., Star, J. R., & Vig, R. (2018). Exploring Connections between Content Knowledge, Pedagogical Content Knowledge, and the Opportunities to Learn Mathematics: Findings from the TEDS-M Dataset. Mathematics Teacher Education and Development, 20(1), 4-22.

  21. An, S. A. (2017). Preservice teachers' knowledge of interdisciplinary pedagogy: the case of elementary mathematics—science integrated lessons. ZDM, 49(2), 237-248; Greenstein, S., & Seventko, J. (2017). Mathematical Making in Teacher Preparation: What Knowledge Is Brought to Bear?. North American Chapter of the International Group for the Psychology of Mathematics Education; Son, J. W. (2016). Preservice teachers' response and feedback type to correct and incorrect student-invented strategies for subtracting whole numbers. The Journal of Mathematical Behavior, 42, 49-68.

  22. Loewenberg Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407.

  23. Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-23.

  24. Loewenberg Ball, D., Thames, M. H., & Phelps, G. (2008). p. 400.

  25. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?. American Educator.

  26. Selling, S. K., Garcia, N., & Ball, D. L. (2016). What does it take to develop assessments of mathematical knowledge for teaching?: Unpacking the mathematical work of teaching. The Mathematics Enthusiast, 13(1), 35-51.

  27. Promoting Rigorous Outcomes in Mathematics and Science Education (2006, December). Knowing mathematics: What we can learn from teachers (Research Report, Vol. 2). East Lansing, MI: Michigan State University; Cavanna, J. M., Drake, C., & Pak, B. (2017). Exploring Elementary Mathematics Teachers' Opportunities to Learn to Teach. North American Chapter of the International Group for the Psychology of Mathematics Education; Santagata, R., Yeh, C., & Mercado, J. (2018). Preparing elementary school teachers to learn from teaching: A comparison of two approaches to mathematics methods instruction. Journal of the Learning Sciences, 27(3), 474-516; Giles, R. M., Byrd, K. O., & Bendolph, A. (2016). An investigation of elementary preservice teachers' self-efficacy for teaching mathematics. Cogent Education, 3(1), 1160523.

  28. Laursen, S. L., Hassi, M. L., & Hough, S. (2016). Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers. International Journal of Mathematical Education in Science and Technology, 47(2), 256-275.

  29. Ball, D., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook on Research on Teaching (4th ed.). Washington, D.C.: American Educational Research Association; Guyton, E., & Farokhi, E. (1987). Relationships among academic performance, basic skills, subject matter knowledge, and teaching skills of teacher education graduates. Journal of Teacher Education, 38, N5.

  30. Loewenberg Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

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