Research and professional standards provide ample guidance about the traits of sound assessments. In the case of assessments that will be generative for mathematics teacher preparation, literature and standards focused on teacher education and mathematics education are certainly useful. One source that was particularly pivotal in our consideration of the recommendations below was the National Council of Teachers of Mathematics Assessment Standards for School Mathematics (NCTM, 1995). Although the standards presented in that document were focused on assessment in PreK–12 mathematics 20 years ago, the ways in which they identify goals and traits of highquality approaches to assessment make them both timely and applicable to assessment in mathematics teacher preparation.
The recommendations on the features of assessments provide a set of criteria by which individual assessments and systems of assessments can be judged. More generally, they relate to the content and forms of assessment as well as to routines that support the implementation of assessments and uses of assessment information. The recommendations can be used proactively as goals for assessment selection and development and can also shape dispositions guiding implementation of assessments.
Recommendation AF.1. Focus on Mathematics Teaching in Assessments
Effective assessments are tailored to provide information relevant to the development of mathematics teacher candidates.
Effective assessments in mathematics teacher preparation address key knowledge, skills, and dispositions that teacher candidates need to support mathematics learning. Effective assessments address candidates' understanding of mathematical concepts and their skills in using mathematics to solve problems. At the program level, effective assessments facilitate the capture and aggregation of information about mathematics teaching. Two components of assessment practice are entailed in the focus on mathematics teaching. First, assessments must be sensitive to the mathematical aspects of teaching, as opposed to relying on generic or subjectmatterneutral approaches. Second, although programs can assess aspects relevant to mathematics teaching outside of practice, assessments must appraise candidates’ engagement in mathematics teaching, including supporting candidates’ selfassessment of their own mathematics teaching. Given the complexity of mathematics teaching, those preparing teachers of mathematics often engage candidates in approximations of teaching in which some aspects of teaching are authentic but other aspects are temporarily suspended. Assessments can and should be used in these contexts to provide information on candidates’ growth that can be used to shape candidates’ subsequent opportunities to learn about mathematics teaching.
Recommendation AF.2. Promote Equity and Access in Assessments
Effective assessments are designed, implemented, interpreted, and used to advance equity and access.
Effective assessments in mathematics teacher preparation allow all candidates equitable opportunities to demonstrate their knowledge, skills, and dispositions. This process includes access for each and every candidate to assessments through which they can demonstrate their abilities to teach mathematics and to receive useful, mathematicsteachingfocused feedback. Additionally, criteria defining mathematics teaching proficiency honor diverse approaches in the service of students’ meaningful and productive engagement with mathematics. All assessments are routinely analyzed for signs of bias or patterns of outcomes that indicate that the assessment is functioning inequitably. Results of assessments should support candidates' agency, leading to improvement of their opportunities to learn about mathematics teaching, including about differentiation of supports and resources to enhance the performance of each and every candidate. The tools used should produce reliable results and be valid in what they measure.
Recommendation AF.3. Embody Openness in Assessments
Effective assessments are part of a transparent process in which those assessing, those being assessed, and other stakeholders know the assessment focus, the nature of assessment methods and criteria, and the uses of the information gathered.
Transparency in assessment means that all involved in assessments and those stakeholders who use information from assessments are fully informed and have suitable roles in assessment selection, implementation, and use of assessment results. The goals for assessments must be clear to all involved, and those involved must validate that assessments provide useful and actionable information. That the assessment process be open to mathematics teacher candidates is particularly important. This openness can be attained in many ways, for example, by including selfassessment components in assessments and involving candidates in the development of assessment criteria. Candidates and other users of assessment results receive performance information in timely ways and in forms that are easily interpreted and meaningful. Although at times some aspects of assessments need to be secure, effective mathematics teacher preparation programs maximize opportunities to enhance the openness of the process and content of assessments.
Recommendation AF.4. Support Valid Inferences and Action Based on Assessments
Effective assessments appropriately include multiple measures that are well suited to gather the types and amounts of information needed to make valid inferences and take action.
Assessments of mathematics candidate and program quality should be comprehensive, utilizing multiple measures to address the range of relevant standards. Relying on a single data point in required assessments (such as a score on a test required for teacher certification) will not provide the breadth of information that is needed. Effective assessments must specifically address mathematics teaching beyond general measures that provide inadequate information about candidates' readiness to teach mathematics. Effective assessments extend beyond a sole focus on candidates’ mathematical knowledge to include attention to their instructional practices and dispositions related to mathematics, as outlined in Chapter 2. The set of assessments chosen could include those that are locally designed and others that are selected from assessments that have been developed and used by the wider educational community. Users of assessments must be aware of the limitations of each assessment that is chosen and account for those limitations through warranted modifications or specific approaches to implementation, analysis, and interpretation of assessment information.
Assessment approaches should be chosen for their capacities to capture information most relevant to the knowledge, skills, and dispositions targeted in the assessment. Ease of administration is not the leading consideration in choosing assessment methods, just as assessment targets should be chosen by their importance rather than the ease of assessing them. Further, many valued outcomes are not easily or well conveyed quantitatively. Questionnaires, interviews, focus groups, observational notes, video recordings of teaching, and other records of practice provide critical complementary data. Although the learning of mathematics achieved by their students may be the ultimate assessment of a candidate’s quality, measures of student success that focus on a single highstakes assessment are unlikely to be valid or to provide information that can guide improvement.
Table 8.5 provides an overview of potential attributes and measures of interest to mathematics teacher educators. A collection of measures that provide a program the most meaningful data regarding their candidates' mathematical knowledge, instructional practice, and dispositions should be selected, judged on the basis of the values and goals of the program.
Table 8.5. Mathematics Teacher Preparation Quality: Attributes and Evidence
Attributes 
Possible Evidence 
Program admissions and recruitment criteria (Potentially relevant for AC.1, AC.3, and AP.4) 

Quality and substance of instruction (Potentially relevant for AC.1 and AP.2) 

Quality of clinical experiences (Potentially relevant for AC.1, AC.2, AC.3, AP.2, and AP.3) 

Faculty/instructor quality (Potentially relevant for AP.2 and AP.3) 

Effectiveness in preparing new teachers who are employable and stay in the field (Potentially relevant for AC.1, AC.2, and AP.4) 

Note. Adapted from Evaluation of Teacher Preparation Programs: Purposes, Methods, and Policy Options (p. 27), by M. J. Feuer, R. E. Floden, N. Chudowsky, and J. Ahn, 2013, Washington, DC: National Academy of Education. Copyright 2013 by National Academy of Education.
These examples are not comprehensive. Moreover, do not conclude from this list that the focus should be on the number of assessments used. Rather, the examples convey an array of assessment data that could be relevant, contexts in which assessments may occur, and people who could beneficially be involved in such assessments.
Finally, although many assessments related to mathematics teacher preparation may be mandated by university or state policy, significant benefits may ensue from including measures used across many universities. First, such common measures can be designed to align to the standards in this document, utilizing expertise across universities to provide economy of scale in their development, validation, and perhaps even scoring. Second, having common measures allows for comparisons across programs to better assess the progress being made in particular areas. For example, the Mathematics Teacher Education Partnership, a collaboration of secondary mathematics teacher preparation programs, has adopted several measures that include a common classroomobservation protocol (Gleason, Livers, & Zelkowski, 2015), a Partnershipdeveloped survey in which candidates selfassess their readiness to teach, a Partnershipdeveloped program selfassessment of progress along various dimensions, and a survey of the number of candidates produced by programs. These common measures are incorporated into each program’s existing assessment system to provide information about the progress of the Partnership as a whole as well as the progress of individual programs in comparison to progress of the Partnership as a whole (Martin, W. G., & Gobstein, 2015). Such work also provides a foundation for collaboration on ways of using information provided by assessments and approaches to enhance teacher preparation in areas for which assessments across institutions show similar performance (joint work to design new efforts) or difference in performance (one institution with an area of strength could help partner institutions at which performance is less robust).
Recommendation AF.5. Embody Coherence and Sustainability in Assessments
Effective assessments are aligned with teacher preparation goals, opportunities to learn, and intended uses of information they generate. They are designed and enacted in ways that can be sustained over time.
In effective mathematics teacher preparation programs, assessments are logical outgrowths of program goals and learning experiences. This coherence makes the content within assessments and approaches to assessment seem natural and expected—instead of surprising, unfair, or alarming. Further, the strong connection also positions users of assessment data to see the ways in which insights relate to learning opportunities and project into actions.
Coherence across assessments includes coherence of the content assessed and the implementation and interpretation of assessments. Those implementing assessments do so in accordance with the specific requirements specified for each assessment. In addition, they have routines and norms that can be used across assessments to ensure appropriate and purposeful engagement as well as to mitigate sources of bias and misuse of assessment information. Formative and summative assessments complement each other and collectively provide multiple points of data on candidate and program outcomes.
Assessments in mathematics teacher preparation need to be chosen, enacted, and used in ways that can be replicated over time, not simply at a particular point in time. Routinely, efforts are made to enhance sustainability of the assessment system. However these efforts do not lead to assessments within the system being lightly or frequently changed. Attending to sustainability in these ways supports the collection of data necessary to answer important longitudinal questions about course and program effectiveness; the nature and degree of consensus among and between candidates, mentor teachers, and program instructors with respect to mathematics teaching quality; and the correlation, or at least correspondence, between particular assessments used by programs.