Overview
The Content Expertise Key Assessment is a project intended
to allow the teacher candidate to demonstrate content knowledge for mathematics
teaching. In this assignment,
mathematics teacher candidates will review the Amsco Intergrated Algebra 1 textbook using the Common Core State Standards (CCSS) Mathematics Curriculum Materials
Analysis Project toolkits from the National Council of Supervisors of
Mathematics (NCSM). The NCSM toolkit consists of three tools: (1) Mathematics
Content Alignment; (2) Mathematics Practices Alignment; (3) Overarching
Considerations—Equity, Assessment, and
Technology (http://www.mathedleadership.org/ccss/materials.html) .
The integrated algebra 1 textbook includes 16
chapters.
|
1. Number Systems |
9. Graphing Linear Functions and
Relations |
|
2. Operations and Properties |
10. Writing and Solving Systems of Linear
Functions |
|
3. Algebraic Expressions and Open
Sentences |
11. Special Products and Factors |
|
4. First Degree Equations and Inequalities
in One Variable |
12. Operations with Radicals |
|
5. Operations with Algebraic Expressions |
13. Quadratic Relations and Functions |
|
6. Ratio and Proportion |
14. Algebraic Fractions, and Equations
and Inequalities Involving Fractions |
|
7. Geometric Figures, Areas, and Volumes |
15. Probability |
|
8. Trigonometry of the Right Triangle |
16. Statistics |
Purpose
Being an effective teacher requires a strong command of your
disciplinary knowledge and practices.
Deep content understanding is needed to make connections between
concepts as well as between concepts and procedures. It is also necessary for
making connections with real world phenomena. All
teachers certified in secondary mathematics should know, understand, teach, and
be able to communicate their mathematical knowledge. Mathematics teachers often
exercise this knowledge through analyses of textbook materials during adoption
cycles. In between adoption cycles, teachers regularly make decisions about the
appropriateness of textbook materials for their short- and long-term
instructional goals and supplement the textbook materials accordingly.
The content expertise project
provides teacher candidates with an opportunity to develop and demonstrate
their content understanding through a three-step textbook analysis, reflection,
and plans to supplement the textbook materials where needed to support the
goals of the Common Core Standards for Mathematical Content and Practice.
Teacher candidates will use the peer-reviewed and field-tested toolkit
developed by Dr. William S. Bush, the National Council of Supervisors of
Mathematics and supported by the Council of Chief State School Officers,
Brookhill Foundation, and Texas Instruments; explicit connections will be made
to help teacher candidates recognize the importance of connecting
research-based products to their teaching.
|
Connections to NCTM (2012) Standards for Teacher Preparation |
|
Content
Understanding |
|
A.1. Number and Quantity To be
prepared to develop student mathematical proficiency, all secondary
mathematics teachers should know the following topics related to number and
quantity with their content understanding and mathematical practices
supported by appropriate technology and varied representational tools,
including concrete models: |
|
A.1.1 Structure, properties,
relationships, operations, and representations including standard and
non-standard algorithms, of numbers and number systems including integer,
rational, irrational, real, and complex numbers |
|
A.1.2 Fundamental ideas of
number theory (divisors, factors and factorization, primes, composite
numbers, greatest common factor, least common multiple, and modular
arithmetic) |
|
A.1.3 Quantitative reasoning and
relationships that include ratio, rate, and proportion and the use of units in
problem situations |
|
A.2. Algebra To be
prepared to develop student mathematical proficiency, all secondary
mathematics teachers should know the following topics related to algebra with
their content understanding and mathematical practices supported by
appropriate technology and varied representational tools, including concrete
models: |
|
A.2.1 Algebraic notation,
symbols, expressions, equations, inequalities, and proportional
relationships, and their use in describing, interpreting, modeling,
generalizing, and justifying relationships and operations |
|
A.2.3 Functional representations
(tables, graphs, equations, descriptions, recursive definitions, and finite
differences), characteristics (e.g., zeros, intervals of increase or
decrease, extrema, average rates of change, domain and range, and end behavior),
and notations as a means to describe, reason, interpret, and analyze
relationships and to build new functions |
|
A.2.4 Patterns of change in
linear, quadratic, polynomial, and exponential functions and in proportional
and inversely proportional relationships and types of real-world
relationships these functions can model |
|
A.3. Geometry and Trigonometry To be
prepared to develop student mathematical proficiency, all secondary
mathematics teachers should know the following topics related to geometry and
trigonometry with their content understanding and mathematical practices
supported by appropriate technology and varied representational tools,
including concrete models: |
|
A.3.1 Core concepts and
principles of Euclidean geometry in two and three dimensions and
two-dimensional non-Euclidean geometries |
|
A.3.2 Transformations including
dilations, translations, rotations, reflections, glide reflections;
compositions of transformations; and the expression of symmetry in terms of
transformations |
|
A.3.3 Congruence, similarity and
scaling, and their development and expression in terms of transformations |
|
A.3.4 Right triangles and
trigonometry |
|
A.3.5 Application of periodic
phenomena and trigonometric identities |
|
A.3.6 Identification,
classification into categories, visualization, and representation of two- and
three-dimensional objects (triangles, quadrilaterals, regular polygons,
prisms, pyramids, cones, cylinders, and spheres) |
|
A.3.7 Formula rationale and
derivation (perimeter, area, surface area, and volume) of two- and
three-dimensional objects (triangles, quadrilaterals, regular polygons,
rectangular prisms, pyramids, cones, cylinders, and spheres), with attention
to units, unit comparison, and the iteration, additivity, and invariance
related to measurements |
|
A.3.8 Geometric constructions,
axiomatic reasoning, and proof |
|
A.3.9 Analytic and coordinate
geometry including algebraic proofs (e.g., the Pythagorean Theorem and its
converse) and equations of lines and planes, and expressing geometric
properties of conic sections with equations |
|
A.4. Statistics and Probability To be
prepared to develop student mathematical proficiency, all secondary
mathematics teachers should know the following topics related to statistics
and probability with their content understanding and mathematical practices
supported by appropriate technology and varied representational tools,
including concrete models: |
|
A.4.1 Statistical variability
and its sources and the role of randomness in statistical inference |
|
A.4.2 Creation and
implementation of surveys and investigations using sampling methods and
statistical designs, statistical inference (estimation of population
parameters and hypotheses testing), justification of conclusions, and
generalization of results |
|
A.4.3 Univariate and bivariate
data distributions for categorical data and for discrete and continuous
random variables, including representations, construction and interpretation
of graphical displays (e.g., box plots, histograms, cumulative frequency
plots, scatter plots), summary measures, and comparisons of distributions |
|
A.4.4 Empirical and theoretical
probability (discrete, continuous, and conditional) for both simple and
compound events |
|
A.4.5 Random (chance) phenomena,
simulations, and probability distributions and their application as models of
real phenomena and to decision making |
|
Pedagogical and Professional Connections
to Content Understanding |
|
2a) Use problem solving to develop conceptual understanding, make sense
of a wide variety of problems and persevere in solving them, apply and adapt
a variety of strategies in solving problems confronted within the field of
mathematics and other contexts, and formulate and test conjectures in order
to frame generalizations. |
|
2b) Reason abstractly, reflectively, and quantitatively with attention
to units, constructing viable arguments and proofs, and critiquing the
reasoning of others; represent and model generalizations using mathematics;
recognize structure and express regularity in patterns of mathematical
reasoning; use multiple representations to model and describe mathematics;
and utilize appropriate mathematical vocabulary and symbols to communicate
mathematical ideas to others. |
|
2c) Formulate, represent, analyze, and interpret mathematical models
derived from real-world contexts or mathematical problems. |
|
2d) Organize mathematical thinking and use the language of mathematics
to express ideas precisely, both orally and in writing to multiple audiences. |
|
2e) Demonstrate the interconnectedness of mathematical ideas and how
they build on one another and recognize and apply mathematical connections
among mathematical ideas and across various content areas and real-world
contexts. |
|
2f) Model how the development of mathematical understanding within and
among mathematical domains intersects with the mathematical practices of
problem solving, reasoning, communicating, connecting, and representing. |
|
3a) Apply knowledge of curriculum standards for secondary mathematics
and their relationship to student learning within and across mathematical
domains. |
|
7c) Develop knowledge, skills, and professional behaviors across both
middle and high school settings; examine the nature of mathematics, how
mathematics should be taught, and how students learn mathematics; and observe
and analyze a range of approaches to mathematics teaching and learning,
focusing on tasks, discourse, environment, and assessment. |
Requirements
1. Teacher candidates must participate in all training sessions.
2. Teacher candidates must submit all training material products during EDUC 426/628 on BlackBoard.
3. Teacher candidates must
a.
Code all 16 chapters of Amsco Integrated Algebra 1 (https://umbc.box.com/s/a0dsasqaoi5bt1k7v719)
using Toolkit 1.
b.
Code all 16 chapters of Amsco Integrated Algebra 1 using Toolkit 2.
c.
Code all 16 chapters of Amsco Integrated Algebra 1 along with the Teacher Materials using Toolkit 3.
d.
Submit coding results, reflection, questions,
and syntheses for all three tools to TK-20.
i. Toolkit
1 coding results along with a reflection and synthesis that addresses the
following essential outcome questions.
1. Which CCSSM content was addressed by Amsco Integrated Algebra 1?
2. Which grade level(s) are most appropriate for this content?
3. Was there relevant content missing?
4. What was the quality of balance between conceptual understanding and procedural skills? How well were connections between the two developed?
5. What alterations, additions, or deletions would you make based on your coding results?
ii. Toolkit
2 coding results along with a reflection and synthesis that addresses the
following essential outcome questions.
6.
(Mathematical Practices ˆ Content) To what
extent do the materials demand that students engage in the Standards for
Mathematical Practice as the primary vehicle for learning the Content
Standards?
7.
(Content ˆ Mathematical
Practices) To what extent do the materials provide opportunities for students
to develop the Standards for Mathematical Practice as Òhabits of mindÓ (ways of
thinking about mathematics that are rich, challenging, and useful) throughout
the development of the Content Standards?
8.
To what extent do accompanying
assessments of student learning (such as homework, observation checklists,
portfolio recommendations, extended tasks, tests, and quizzes) provide evidence
regarding studentsÕ proficiency with respect to the Standards for Mathematical
Practice?
9. What
is the quality of the instructional support for studentsÕ development of the
Standards for Mathematical Practice as habits of mind?
iii. Toolkit
3 coding results along with a reflection and synthesis that addresses the
following essential outcome questions.
10. Equity:
To what extent do the materials contain embedded support for elements of
equity consistently within and across grades?
11. Assessment:
To what extent do the materials contain embedded support for elements of
assessment consistently within and across grades?
12. Technology:
To what extent do the materials contain embedded support for elements of
technology consistently within and across grades?
13. Overall:
To what extent do the materials incorporate the Overarching
Consideration elements to advance studentsÕ learning of mathematical content
and engagement in the mathematical practices?
iv. Overall recommendations, strategies, additions and/or deletions, and supplemental materials needed to successfully use Amsco Integrated Algebra 1 to teach Common Core Mathematics.
Process
1. Introduction to the NCSM toolkit.
2. Receive training on the use of Toolkit 1 (Content Coverage and Balance).
3. Code the content coverage and balance of mathematical understanding and procedural skills (i.e., apply Toolkit 1) for a textbook chapter, Amsco Geometry Chapter 4 (https://umbc.box.com/s/w8j1m04o1x9lmhfdzmon) as a training exercise. The rubrics for Toolkit 1 (p. 11) are:
Content Coverage Rubric:
Not Found (N)—The
mathematics content was not found.
Low (L)—Major
gaps in the mathematics content were found.
Marginal (M)—Gaps
in the mathematics content, as described in the Standards, were found and these
gaps may not be easily filled.
Acceptable (A)—Few
gaps in the mathematics content, as described in the Standards, were found and
these gaps may be easily filled.
High (H)—The mathematics content was fully formed
as described in the Standards.
Balance
of Mathematical Understanding and Procedural Skills Rubric:
Not Found (N)
- The content was not found.
Low (L) – The content
was not developed or developed superficially.
Marginal (M)
- The content was found and focused primarily on procedural skills and
minimally on mathematical understanding, or ignored procedural skills.
Acceptable (A)
-The content was developed with a balance of mathematical understanding and
procedural skills consistent with the Standards, but the connections between
the two were not developed.
High (H) – The content was developed with a balance of
mathematical understanding and procedural skills consistent with the Standards,
and the connections between the two were developed.
4. Discuss Toolkit 1 coding results in large and small groups. Essential outcome questions:
a.
Which CCSSM
content was addressed by Amsco Geometry
Chapter 4?
b.
Which grade
level(s) are most appropriate for this content?
c.
Was there
relevant content missing from the chapter?
d.
What was the
quality of balance between conceptual understanding and procedural skills? How
well were connections between the two developed?
e.
What alterations,
additions, or deletions would you make based on your coding results?
5. Receive training on the use of Toolkit 2 (Standards of Mathematical Practice).
6. Code the textbook chapter, Amsco Geometry Chapter 4 (https://umbc.box.com/s/w8j1m04o1x9lmhfdzmon) using Toolkit 2 as a training exercise.
7. Discuss Toolkit 2 coding results in large and small groups. Essential outcome questions (from p. 79 of Toolkit):
a.
(Mathematical Practices ˆ
Content) To what extent do the materials demand that students engage in the
Standards for Mathematical Practice as the primary vehicle for learning the
Content Standards?
b.
(Content ˆ
Mathematical Practices) To what extent do the materials provide opportunities
for students to develop the Standards for Mathematical Practice as Òhabits of
mindÓ (ways of thinking about mathematics that are rich, challenging, and
useful) throughout the development of the Content Standards?
c.
To what extent do accompanying
assessments of student learning (such as homework, observation checklists,
portfolio recommendations, extended tasks, tests, and quizzes) provide evidence
regarding studentsÕ proficiency with respect to the Standards for Mathematical
Practice?
d. What
is the quality of the instructional support for studentsÕ development of the
Standards for Mathematical Practice as habits of mind?
8. Discuss Toolkit 2 summative coding decisions with justification:
Summative
Assessment Rubric (from p. 79 of
Toolkit)
á
(Low) – The Standards for Mathematical Practice
are not addressed or are addressed superficially.
á
(Marginal) The Standards for Mathematical Practice are
addressed, but not consistently in a way that is embedded in the development of
the Content Standards.
á
(Acceptable) – Attention to the Standards for
Mathematical Practice is embedded throughout the curriculum materials in ways
that may help students to develop them as habits of mind.
9. Receive training on the use of Toolkit 3. From p. 84 of the Toolkit, the three overarching considerations are described as:
a.
Equity: Allowing for the widest
possible range of students to participate fully from the outset, along with
appropriate accommodations to ensure maximum participation of students with
special education needs.Ó
b.
Assessment is a critical part of
classroom instruction, and curriculum materials can provide a variety of levels
of support with regard to information to teachers about student learning.
c.
The increasing
availability of technology offers
opportunities to use technology mindfully in ways that enable students to
explore and deepen their understanding of mathematical concepts.
10. Code the textbook chapter, Amsco Geometry Chapter 4 (https://umbc.box.com/s/w8j1m04o1x9lmhfdzmon) using Toolkit 3 as a training exercise.
11. Discuss
Toolkit 3 coding results in large and small groups. The rubric for
answering questions about Overarching
Considerations (from p. 14 of Toolkit):
Not Found (NF): The curriculum materials do not support
this element.
Low (L): The curriculum materials contain limited
support for this element, but the support is not embedded or consistently
present within and across grades.
Medium (M): The curriculum materials contain support
for this element, but it is not always embedded or consistently present within
and across grades.
High (H): The curriculum materials contain
embedded support for this element so that it is consistently present within and
across grades.
The
rubric describes the extent to which the materials provide teachers support in
these three critical overarching considerations. In
contrast to the previous tools, we suggest here that reviewers consider
supporting materials in addition to the teacher and student materials.
At
the end of Tool 3, reviewers are asked to summarize their responses through
questions about the three overarching considerations. These questions were
designed to provide guidance and stimulate discussion to determine the degree
to which these issues were addressed in the curriculum materials.
12. Code
all 16 chapters of Amsco Integrated
Algebra 1 (https://umbc.box.com/s/a0dsasqaoi5bt1k7v719)
using Toolkit 1.
13. Code
all 16 chapters of Amsco Integrated
Algebra 1 using Toolkit 2.
14. Code
all 16 chapters of Amsco Integrated
Algebra 1 along with the Teacher Materials using Toolkit 3.
15. Submit
final products.
Rubric
|
Criterion |
Limited (1) |
Developing (2) |
Proficient (3) |
Exemplary (4) |
|
A.1.1 Structure, properties,
relationships, operations, and representations including standard and
non-standard algorithms, of numbers and number systems including integer,
rational, irrational, real, and complex numbers |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and Technology
considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.1.2 Fundamental ideas of
number theory (divisors, factors and factorization, primes, composite
numbers, greatest common factor, least common multiple, and modular
arithmetic) |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly connected
to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.1.3 Quantitative reasoning and
relationships that include ratio, rate, and proportion and the use of units
in problem situations |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.2.1 Algebraic notation,
symbols, expressions, equations, inequalities, and proportional
relationships, and their use in describing, interpreting, modeling,
generalizing, and justifying relationships and operations |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and Technology
considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.2.3 Functional representations
(tables, graphs, equations, descriptions, recursive definitions, and finite
differences), characteristics (e.g., zeros, intervals of increase or
decrease, extrema, average rates of change, domain and range, and end
behavior), and notations as a means to describe, reason, interpret, and
analyze relationships and to build new functions |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.2.4 Patterns of change in
linear, quadratic, polynomial, and exponential functions and in proportional
and inversely proportional relationships and types of real-world
relationships these functions can model |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.1 Core concepts and
principles of Euclidean geometry in two and three dimensions and
two-dimensional non-Euclidean geometries |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.2 Transformations including
dilations, translations, rotations, reflections, glide reflections;
compositions of transformations; and the expression of symmetry in terms of
transformations |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.3 Congruence, similarity and
scaling, and their development and expression in terms of transformations |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.4 Right triangles and
trigonometry |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.5 Application of periodic
phenomena and trigonometric identities |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.6 Identification,
classification into categories, visualization, and representation of two- and
three-dimensional objects (triangles, quadrilaterals, regular polygons,
prisms, pyramids, cones, cylinders, and spheres) |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and Technology
considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.7 Formula rationale and
derivation (perimeter, area, surface area, and volume) of two- and
three-dimensional objects (triangles, quadrilaterals, regular polygons,
rectangular prisms, pyramids, cones, cylinders, and spheres), with attention
to units, unit comparison, and the iteration, additivity, and invariance
related to measurements |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.8 Geometric constructions,
axiomatic reasoning, and proof |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.3.9 Analytic and coordinate
geometry including algebraic proofs (e.g., the Pythagorean Theorem and its
converse) and equations of lines and planes, and expressing geometric
properties of conic sections with equations |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.4.1 Statistical variability
and its sources and the role of randomness in statistical inference |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.4.2 Creation and
implementation of surveys and investigations using sampling methods and
statistical designs, statistical inference (estimation of population
parameters and hypotheses testing), justification of conclusions, and
generalization of results |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.4.3 Univariate and bivariate
data distributions for categorical data and for discrete and continuous
random variables, including representations, construction and interpretation
of graphical displays (e.g., box plots, histograms, cumulative frequency
plots, scatter plots), summary measures, and comparisons of distributions |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.4.4 Empirical and theoretical
probability (discrete, continuous, and conditional) for both simple and
compound events |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly
connected to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
A.4.5 Random (chance) phenomena,
simulations, and probability distributions and their application as models of
real phenomena and to decision making |
Content Understanding does not
clearly balance conceptual and procedural understanding. |
Content Understanding includes
conceptual understanding, procedural understanding, but they are unbalanced,
or the connections between concepts and procedures are unclear. Equity, Assessment, and
Technology considerations are addressed. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise, but may not be explicitly connected
to the content. |
Content Understanding includes
conceptual understanding, procedural understanding, and connections between
concepts and procedures. Equity, Assessment, and
Technology considerations are clear and concise in relation to the content. |
|
Pedagogical
and Professional Connections to Content Understanding |
||||
|
2a) Use problem solving to
develop conceptual understanding, make sense of a wide variety of problems
and persevere in solving them, apply and adapt a variety of strategies in
solving problems confronted within the field of mathematics and other
contexts, and formulate and test conjectures in order to frame
generalizations. |
Analysis
of curriculum material support of learner engagement in problem solving for
develop conceptual understanding, for making sense of a wide variety of
problems and persevering in solving them, for applying and adapting a variety
of strategies in solving problems confronted within the field of mathematics
and other contexts, and for formulating and testing conjectures to frame
generalizations is minimally or not evident. |
Analysis
of curriculum material support of learner engagement in problem solving for
develop conceptual understanding, for making sense of a wide variety of
problems and persevering in solving them, for applying and adapting a variety
of strategies in solving problems confronted within the field of mathematics
and other contexts, and for formulating and testing conjectures to frame generalizations
is vague, implicit, or imprecise. |
Analysis
of curriculum material support of learner engagement in problem solving for
develop conceptual understanding, for making sense of a wide variety of
problems and persevering in solving them, for applying and adapting a variety
of strategies in solving problems confronted within the field of mathematics
and other contexts, and for formulating and testing conjectures to frame
generalizations is explicit, but connections to Toolkit
2 analysis are vague or implicit. |
Analysis
of curriculum material support of learner engagement in problem solving for
develop conceptual understanding, for making sense of a wide variety of
problems and persevering in solving them, for applying and adapting a variety
of strategies in solving problems confronted within the field of mathematics
and other contexts, and for formulating and testing conjectures to frame
generalizations is clear and concise with supporting
evidence from Toolkit 2. |
|
2b.1)
The teacher engages learners in abstract, quantitative, and reflective
reasoning with attention to units. |
Analysis
of curriculum material support of learner engagement in abstract,
quantitative, and reflective reasoning with attention to units is minimally or not evident. |
Analysis
of curriculum material support of learner engagement in abstract,
quantitative, and reflective reasoning with attention to units is vague, implicit, or imprecise. |
Analysis
of curriculum material support of learner engagement in abstract,
quantitative, and reflective reasoning with attention to units is explicit,
but connections to Toolkit 2 analysis are vague or implicit. |
Analysis
of curriculum material support of learner engagement in abstract,
quantitative, and reflective reasoning with attention to units is clear and concise with supporting evidence
from Toolkit 2. |
|
2b.2)
The teacher facilitates learnersÕ ability to construct viable arguments and
proofs and critique the reasoning of others. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to construct viable arguments and proofs and critique the
reasoning of others is minimally or not evident. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to construct viable arguments and proofs and critique the
reasoning of others is vague, implicit, or
imprecise. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to construct viable arguments and proofs and critique the
reasoning of others is explicit, but
connections to Toolkit 2 analysis are vague or implicit. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to construct viable arguments and proofs and critique the
reasoning of others is clear and concise with
supporting evidence from Toolkit 2. |
|
2b.3)
The teacher facilitates learnersÕ ability to represent and model
generalizations using mathematics, to recognize structure, and to express
regularity in patterns of mathematical reasoning. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to represent and model generalizations using mathematics,
to recognize structure, and to express regularity in patterns of mathematical
reasoning is minimally or not evident. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to represent and model generalizations using mathematics,
to recognize structure, and to express regularity in patterns of mathematical
reasoning is vague, implicit, or
imprecise. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to represent and model generalizations using mathematics,
to recognize structure, and to express regularity in patterns of mathematical
reasoning is explicit, but connections
to Toolkit 2 analysis are vague or implicit. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to represent and model generalizations using mathematics,
to recognize structure, and to express regularity in patterns of mathematical
reasoning is clear and concise with
supporting evidence from Toolkit 2. |
|
2b.4)
The teacher facilitates learnersÕ ability to use multiple representations to
model and describe mathematics. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to use multiple representations to model and describe
mathematics is minimally or not evident. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to use multiple representations to model and describe
mathematics is vague, implicit, or
imprecise. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to use multiple representations to model and describe
mathematics is explicit, but connections
to Toolkit 2 analysis are vague or implicit. |
Analysis
of curriculum material to help facilitate
learnersÕ ability to use multiple representations to model and describe
mathematics is clear and concise with
supporting evidence from Toolkit 2. |
|
2d) Organize mathematical
thinking and use the language of mathematics to express ideas precisely, both
orally and in writing to multiple audiences. |
Analysis
of curriculum materials to organize mathematical thinking and to
guide students in the use of the language of mathematics to express ideas
precisely, both orally and in writing to multiple audiences is minimally or not evident. |
Analysis
of curriculum materials to organize mathematical thinking and to
guide students in the use of the language of mathematics to express ideas
precisely, both orally and in writing to multiple audiences is vague, implicit, or
imprecise. |
Analysis
of curriculum materials to organize mathematical thinking and to
guide students in the use of the language of mathematics to express ideas
precisely, both orally and in writing to multiple audiences is explicit, but connections
to the toolkit analyses are vague or implicit. |
Analysis
of curriculum materials to organize mathematical thinking and to
guide students in the use of the language of mathematics to express ideas
precisely, both orally and in writing to multiple audiences is clear and concise with
supporting evidence. |
|
2e) Demonstrate the
interconnectedness of mathematical ideas and how they build on one another
and recognize and apply mathematical connections among mathematical ideas and
across various content areas and real-world contexts. |
Analysis
of curriculum material to help students
develop the ability to demonstrate the interconnectedness of mathematical ideas and how they
build on one another and recognize and apply mathematical connections among mathematical ideas
and across various content areas and real-world contexts is minimally or not evident. |
Analysis
of curriculum material to help students
develop the ability to demonstrate the interconnectedness of mathematical ideas and how they
build on one another and recognize and apply mathematical connections among mathematical ideas
and across various content areas and real-world contexts is vague, implicit, or imprecise. |
Analysis
of curriculum material to help students
develop the ability to demonstrate the interconnectedness of mathematical ideas and how they
build on one another and recognize and apply mathematical connections among mathematical ideas
and across various content areas and real-world contexts is explicit, but connections to the toolkit analyses are vague or
implicit. |
Analysis
of curriculum material to help students
develop the ability to demonstrate the interconnectedness of mathematical ideas and how they
build on one another and recognize and apply mathematical connections among mathematical ideas
and across various content areas and real-world contexts is clear and concise with supporting evidence. |
|
2f) Model how the development of
mathematical understanding within and among mathematical domains intersects
with the mathematical practices of problem solving, reasoning, communicating,
connecting, and representing. |
Analysis of
curriculum material to model how the development of mathematical
understanding within and among mathematical domains intersects with the
mathematical practices of problem solving, reasoning, communicating,
connecting, and representing is minimally or not evident. |
Analysis of
curriculum material to model how the development of mathematical
understanding within and among mathematical domains intersects with the
mathematical practices of problem solving, reasoning, communicating,
connecting, and representing is vague, implicit, or imprecise. |
Analysis of
curriculum material to model how the development of mathematical
understanding within and among mathematical domains intersects with the
mathematical practices of problem solving, reasoning, communicating,
connecting, and representing is explicit, but connections
to Toolkit 2 analysis are vague or implicit. |
Analysis of
curriculum material to model how the development of mathematical
understanding within and among mathematical domains intersects with the
mathematical practices of problem solving, reasoning, communicating,
connecting, and representing is clear and concise with
supporting evidence from Toolkit 2. |
|
3a) Apply knowledge of
curriculum standards for secondary mathematics and their relationship to
student learning within and across mathematical domains. |
The application of knowledge of curriculum
standards for secondary mathematics and their relationship to student
learning within and across mathematical domains is minimally or not evident. |
The application of knowledge of curriculum
standards for secondary mathematics and their relationship to student
learning within and across mathematical domains is vague, implicit, or imprecise. |
The application of knowledge of curriculum
standards for secondary mathematics and their relationship to student
learning within and across mathematical domains is explicit. |
The application of knowledge of curriculum
standards for secondary mathematics and their relationship to student
learning within and across mathematical domains is clear and concise with supporting evidence. |
|
7c.1) Demonstrate knowledge, skills, and professional
behaviors at both middle and high school settings. |
The demonstration of knowledge, skills, and
professional behaviors is minimally or not evident or is minimally evident. |
The demonstration of knowledge, skills, and
professional behaviors is vague, implicit, or imprecise. |
The demonstration of knowledge, skills, and
professional behaviors is explicit. |
The demonstration of knowledge, skills, and
professional behaviors is clear and concise with supporting evidence. |
|
7c.2) Examine the nature of mathematics,
how mathematics should be taught, and how students learn mathematics. |
Examination of the nature of mathematics, how
mathematics should be taught, and how students learn mathematics is minimally or not
evident or is minimally evident. |
Examination of the nature of mathematics, how
mathematics should be taught, and how students learn mathematics is vague, implicit, or
imprecise. |
Examination of the nature of mathematics, how
mathematics should be taught, and how students learn mathematics is explicit. |
Examination of the nature of mathematics, how
mathematics should be taught, and how students learn mathematics is clear and concise
with supporting evidence. |
|
7c.3) Observe and analyze a range of approaches to
mathematics teaching and learning, focusing on tasks, discourse, environment,
and assessment. |
The observation and analysis of a range of
approaches to mathematics teaching and learning, focusing on tasks,
discourse, environment, and assessment is minimally or not evident. |
The observation and analysis of a range of
approaches to mathematics teaching and learning, focusing on tasks,
discourse, environment, and assessment is vague, implicit, or imprecise. |
The observation and analysis of a range of
approaches to mathematics teaching and learning, focusing on tasks,
discourse, environment, and assessment is explicit. |
The observation and analysis of a range of
approaches to mathematics teaching and learning, focusing on tasks,
discourse, environment, and assessment is clear and concise with supporting
evidence. |
|
7c.4)
Participate in innovative or transformative initiatives, partnerships, or
research projects related to the teaching of secondary |
The
innovation/transformative nature of the SLO research project for enhancing
the teaching of secondary mathematics is
minimally or not evident. |
The
innovation/transformative nature of the SLO research project for enhancing
the teaching of secondary mathematics is
vague, implicit, or imprecise. |
The
innovation/transformative nature of the SLO research project for enhancing
the teaching of secondary mathematics is
explicit. |
The
innovation/transformative nature of the SLO research project for enhancing
the teaching of secondary mathematics is
clear and concise with supporting evidence. |
Connections to other assessments
1. EDUC 412: Praxis II Content Knowledge Test Analysis and Reflection
2. EDUC 426 and Internship: Curriculum Unit Plan
3. Internship: Praxis II Content Knowledge Test for Professional Licensure
Graders
EDUC 426/628 Instructor
Additional Resources
1. Practice Materials for Praxis II Content Knowledge Licensure Exam
|
Praxis II Content Test |
Test Code |
Passing
Score |
|
5161 |
160 |
2. The National Council of Supervisors of Mathematics (NCSM) website: http://www.mathedleadership.org/
3. National Council of Teachers of Mathematics (NCTM) research briefs: http://www.nctm.org/news/content.aspx?id=8468. Students may find the following research briefs particularly helpful:
a. Algebraic Thinking in Arithmetic
b. Algebraic Reasoning in School Algebra
c. Selecting the Right Curriculum
d. Producing Gains
e. Formative Assessment