Current through Register Vol. 42, No. 11, August 30, 2024
The Mathematics Coaching Endorsement (grades K-5) is designed
for teachers who hold a valid Alabama Professional Educator Certificate in
early childhood education, elementary education, or collaborative special
education teacher (K-6) and have at least three years of teaching experience in
grades K-5. The mathematics coaching endorsement shall be offered only at the
post baccalaureate level and may not be included within an initial educator
preparation program.
1.
Content
and Pedagogical Knowledge.
a.
Mathematics Coaching Courses. The mathematics coaching endorsement
shall be comprised composed of four courses with embedded field experiences.
The courses are:
1. K-2 Content and
Pedagogical Knowledge
2. 3-5
Content and Pedagogical Knowledge
3. Coaching Principles in the Law
4. Literacy in Mathematics Education
b.
Definitions
1.
Numeracy. Numeracy is defined herein as the ability to understand
and work with numbers. Numeracy is the knowledge, skills, behaviors, and
dispositions that students need to use mathematics in the world and having the
dispositions and capacities to use mathematical knowledge and skills purposely.
The candidates are exposed to numeracy standards at the initial certification
level. The individuals completing this endorsement are expected to demonstrate
advanced knowledge and abilities within this content domain.
2.
Dyscalculia. A term used to
refer to a pattern of learning difficulties characterized by problems
processing numerical information, learning a arithmetic facts, performing
accurate of fluent calculations, difficulties with mathematical reasoning, and
difficulties with word reasoning accuracy.
c.
Analyze, apply, and
synthesize are the keywords used to describe the level of intensity and
performance that individuals who complete this endorsement are expected to
display. This skillset includes maintaining objectivity and clarity in the best
interest of all learners, including those struggling with number sense, and
maintaining public trust using current scientifically supported best
practices.
d.
Curriculum. The curriculum is reflective of the recommendations of
the National Council of Teachers of Mathematics (NCTM), the Conference Board of
the Mathematics Sciences (CBMS), the United States Department of Education (US
DoE), the Council for the Accreditation of Educator Preparation (CAEP) the
Mathematics Sciences Research Institute (MSRI), and the Alabama Coaching
Framework. These standards have been aligned with the Alabama Course of Study
(ACOS) to ensure that these individuals display critical thinking abilities to
coach novice teachers in the development of the mathematical practices that
students in the K-5 grade ban should develop.
e.
Pedagogical and Andragogical
Framework. This endorsement provides the necessary coaching framework to
assist these coaching candidates in heightening the mathematical knowledge base
of novice teachers to assist K-5 learners. The andragogy factors in the learned
experiences of adult teachers and potential fears and misconceptions to assist
in the development of more confidence and greater teaching principals. The
mathematics coach will utilize these teaching practices to ensure that
mathematics teachers optimize opportunities to enhance student understanding
and application. Coaching candidates learn how to guide novice teachers through
assisting students in working through productive struggle as they garner
greater mathematical abilities.
f.
Assessing, Planning and Designing Contexts for Learning. Coaching
candidates learn how to guide and facilitate new teachers in assessing,
planning, and designing contexts to support the development of a coherent
curriculum and an understanding of how mathematical topics and expectations are
connected to each other throughout the elementary grades. This connection
throughout the academic curriculum requires coaching candidates to exhibit
patience as they model strategies to demonstrate understanding related to
student learning, curricular practices and standards, academic language, and
assessments within and across K-5 grade levels. This involves each of the
following:
1.
Analyze, apply, and
synthesize data to plan sequences of instruction that include goals,
appropriate materials, activities, and assessments, and that support engagement
in learning through evidence-based practices.
2.
Analyze, apply, and
synthesize data from formative and summative assessments to determine
student competencies and learning needs, and use this assessment data to
provide feedback, improve instruction, and monitor learning.
3.
Analyze, apply, and
synthesize data to differentiate instructional plans to meet the needs
of diverse students in the classroom.
4.
Analyze, apply, and
synthesize data to develop accommodations for students with dyscalculia
or a math learning disability and provide specific strategies to assist them
such as:
i. Early warning signs, screenings,
and recommendations for intervention,
ii. Use of visual representations,
iii. Use of instructional examples and
concrete objects,
iv. Student
verbalization,
v. Use of
heuristic/multiple strategies,
vi.
Provide ongoing feedback, and
vii
Review strategies and connect to previous learning.
1.
K-2 Content and
Pedagogical Knowledge Course.
a.
K-2 Content and Pedagogy Knowledge. Effective mathematics coaches
analyze, apply, and synthesize major mathematical concepts, algorithms,
procedures, connections, and applications in varied contexts, within and among
mathematical domains.
b.
Numerical Practices. Numerical Practices consist of concepts
within number and operations base ten, and operations and algebraic
thinking.
c.
Foundations of
Counting. Analyze, apply, and synthesize the intricacy of counting,
including the distinction between counting as a list of numbers in order and
counting to determine a number of objects. (ACOS K.1, K.2, K.3, K.4, K.
5, 1.10)
d.
Operations with Numbers: Base Ten. Upon endorsement completion
mathematics coaches shall be able to do the following:
1.
Analyze, apply, and
synthesize how the base-ten place value system relies on repeated
bundling in groups of ten and how to use varied representations including
objects, drawings, layered place value cards, and numerical expressions to help
reveal the base-ten structure. (ACOS K.14, 1.11, 1.12, 2.6, 2.7, 2.8,
2.9, 4.6, 4.7, 4.8, 4.9, 5.3, 5.4, 5.5)
2.
Analyze, apply, and
synthesize how efficient base-ten computation methods for addition,
subtraction, multiplication, and division rely on decomposing numbers
represented in base ten according to the base-ten units represented by their
digits and applying (often informally) properties of operations, including the
commutative and associative properties of addition and multiplication and the
distributive property, to decompose a calculation into parts. (ACOS
K.10, K.11, K. 12, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.13, 1.14, 1.15,
2.1, 2.2, 2.10, 2.11, 2.12, 2.13, 2.14, 3.10, 3.11, 3.12, 4.10, 4.11, 4.12,
5.6, 5.7, 5.8)
3.
Analyze, apply, and synthesize how to use drawings or manipulative
materials to reveal, discuss, and explain the rationale behind computation
methods. (ACOS K.13, K. 15,1.13, 2.1, 2.2, 2.3, 2.4, 2.10, 2.11, 2.12,
2.13, 2.14, 2.21, 2.22, 2.2.2, 2.3, 2.4, 2.10, 2.11, 2.12, 2.13,2.14, 2,21,
2,22, 24c, 3.1, 3.2, 3.3, 3.5, 3.6, 3.8, 3.9, 3.11, 3.12, 4.2, 4.3b, 4.10,
4.11, 4.12, 5.7)
e.
Operations and Algebraic
Thinking. Upon endorsement completion mathematics coaches shall be able
to do the following:
1.
Analyze, apply,
and synthesize the different types of problems solved by addition,
subtraction, multiplication, and division, and meanings of the operations
illustrated by these problem types. (ACOS K.9, 1.1, 1.2, 2.1, 3.3, 3.8,
4.1, 4.2, 4.3, 5.1)
2.
Analyze, apply, and synthesize teaching/learning paths for
single-digit addition and associated subtraction and single-digit
multiplication and associated division, including the use of properties of
operations. (ACOS K.8, K.12, 1.3, 1.4, 1.5, 1.6, 2.2, 3.1, 3.2, 3.5,
3.6, 3.7)
3.
Analyze, apply, and synthesize foundations of algebra within
elementary mathematics, including understanding the equal sign as meaning "the
same amount as" rather than a "calculate the answer" symbol. (ACOS 1.7,
3.4)
4.
Analyze,
apply, and synthesize numerical and algebraic expressions by describing
them in words, parsing them into their component parts, and interpreting the
components in terms of a context. (ACOS K.10, K.11, 1.8, 2.3, 2.4, 3.8,
4.3, 5.1)
5.
Analyze, apply, and synthesize lines of reasoning used to solve
equations and systems of equations. (ACOS K.13, 1.9, 2.5, 3.9, 4.4,
4.5, 5.2)
f.
Measurement, Data Analysis and Geometry. Measurement is the
process of finding a number that shows the amount of something. It is a system
to measure the height, weight, capacity or even number of certain objects. It
is the process of quantifying something and then possibly making comparisons
between two or more objects or concepts. Typically, measurements involve 2
parts-a numeric value and the specific unit. Data Analysis is the ability to
formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them. Geometry is the study of different types
of shapes, figures, and sizes in real life. Coaches will develop greater
knowledge and abilities within geometry to assist novice teachers with the
ability to instruct K-5 learners.
1.
Measurement. Upon program
completion mathematics coaches shall be able to do the following:
i.
Analyze, apply, and
synthesize the general principles of measurement, the process of
iterations, and the central role of units: that measurement requires a choice
of measurable attribute, that measurement is comparison with a unit and how the
size of a unit affects measurements, and the iteration, additivity, and
invariance used in determining measurements. (ACOS K.16, K.17, 1.17,
1.18, 1.19, 1.20, 2.17, 2.18, 2.19, 2.20, 2.23, 2.24, 4.21,
5.17)
ii.
Analyze,
apply, and synthesize how the number line connects measurement with
number through length. (ACOS 2.21, 2.22, 4.22)
2.
Data Analysis (Statistics
and Probability). Upon program completion mathematics coaches shall be
able to do the following:
i.
Analyze,
apply, and synthesize appropriate graphs and numerical summaries to
describe the distribution of categorical and numerical data. (ACOS
K.15, 1.16, 2.15, 3.16, 3.17, 5.16)
ii.
Analyze, apply, and
synthesize the understanding that responses to statistical questions
should consider variability.
(ACOS 2.16, 4.20, 5.16, 6.22)
3.
Geometry. Upon program
completion mathematics coaches shall be able to do the following:
i.
Analyze, apply, and
synthesize geometric concepts of angle, parallel, and perpendicular; use
them in describing and defining shapes; and describing and reasoning about
spatial locations (including the coordinate plane). (ACOS K.18, K.19,
K.20, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 5.20, 6.25)
ii.
Analyze, apply, and
synthesize how shapes are classified into categories, and how reasoning
is used to explain the relationships among the categories. (ACOS K.21,
K.22, K.23, 1.21, 1.22, 2.25, 2.26, 3.26, 5.21, 5.22, 5.23)
3.
3-5
Content and Pedagogy Knowledge Course.
(a)
3-5 Content and Pedagogy
Knowledge. Effective mathematics coaches analyze, apply, and
synthesize major mathematical concepts, algorithms, procedures,
connections, and applications in varied contexts,
within and among mathematical domains.
(b)
Numerical Practices.
Numerical Practices consist of concepts within number and operations base ten,
and operations and algebraic thinking.
(c)
Operations with Numbers: Base
Ten. Upon program completion mathematics coaches shall be able to do the
following:
1.
Analyze, apply, and
synthesize how the base-ten place value system relies on repeated
bundling in groups of ten and how to use varied representations including
objects, drawings, layered place value cards, and numerical expressions to help
reveal the base-ten structure. (ACOS K.14, 1.11, 1.12, 2.6, 2.7, 2.8,
2.9, 4.6, 4.7, 4.8, 4.9, 5.3, 5.4, 5.5)
2.
Analyze, apply, and
synthesize how efficient base-ten computation methods for addition,
subtraction, multiplication, and division rely on decomposing numbers
represented in base ten according to the base-ten units represented by their
digits and applying (often informally) properties of operations, including the
commutative and associative properties of addition and multiplication and the
distributive property, to decompose a calculation into parts. (ACOS
K.10, K.11, K.12, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.13, 1.14, 1.15,
2.1, 2.2, 2.10, 2.11, 2.12, 2.13, 2.14, 3.10, 3.11, 3.12, 4.10, 4.11, 4.12,
5.6, 5.7, 5.8)
3.
Analyze, apply, and synthesize how to use drawings or manipulative
materials to reveal, discuss, and explain the rationale behind computation
methods. (ACOS K.13, K.15,1.13, 2.1,2.2, 2.3, 2.4, 2.10, 2.11, 2.12,
2.13, 2.14, 2.21, 2.22, 2. 2.14, 2,21, 2,22, 24c, 3.1, 3.2,
3.3, 3.5, 3.6, 3.8, 3.9, 3.11, 3.12, 4.2, 4.3b, 4.10, 4.11, 4.12,
5.7)
4.
Analyze,
apply, and synthesize how to extend the base-ten system to decimals and
use number lines to represent decimals. Explain the rationale for decimal
computation methods. (ACOS 5.3, 5.4a, 5.5, 5.8)
(d)
Operations and
Algebraic Thinking. Upon program completion mathematics coaches shall be
able to do the following:
1.
Analyze,
apply, and synthesize the different types of problems solved by
addition, subtraction, multiplication, and division, and meanings of the
operations illustrated by these problem types. (ACOS K.9, 1.1, 1.2,
2.1, 3.3, 3.8, 4.1, 4.2, 4.3, 5.1)
2.
Analyze, apply, and
synthesize teaching/learning paths for single-digit addition and
associated subtraction and single-digit multiplication and associated division,
including the use of properties of operations. (ACOS K.8, K.12, 1.3,
1.4, 1.5, 1.6, 2.2, 3.1, 3.2, 3.5, 3.6, 3.7)
3.
Analyze, apply, and
synthesize foundations of algebra within elementary mathematics,
including understanding the equal sign as meaning "the same amount as" rather
than a "calculate the answer" symbol. (ACOS 1.7,
3.4)
4.
Analyze,
apply, and synthesize numerical and algebraic expressions by describing
them in words, parsing them into their component parts, and interpreting the
components in terms of a context. (ACOS K.10, K.11, 1.8, 2.3, 2.4, 3.8,
4.3, 5.1)
5.
Analyze, apply, and synthesize lines of reasoning used to solve
equations and systems of equations. (ACOS K.13, 1.9, 2.5, 3.9, 4.4,
4.5, 5.2)
(e)
Operations with Numbers: Fractions. Upon program completion
mathematics coaches shall be able to do the following:
1.
Analyze, apply, and
synthesize fractions as numbers, which can be represented by area and
set models and by lengths on a number line. Define
a/b
fractions as
a part, each of size
1/ b.
Attend closely to the whole (referent unit) while solving problems and
explaining solutions.
(ACOS 1.23, 2.27, 3.13, 3.14)
2.
Analyze, apply, and
synthesize addition, subtraction, multiplication, and division problem
types and associated meanings for the operations extended from whole numbers to
fractions. (ACOS 4.15, 4.16, 5.11, 5.14, 5.15)
3.
Analyze, apply, and
synthesize the rationale for defining and representing equivalent
fractions and procedures for adding, subtracting, multiplying, and dividing
fractions. (ACOS 3.15, 4.13, 4.14, 4.17, 4.18, 4.19, 5.9, 5.10,
5.12
4.
Analyze,
apply, and synthesize the connection between fractions and division, a/b
= a÷b, and how fractions, ratios, and rates are connected via
unit rates. (ACOS 5.11)
5.
Analyze, apply, and
synthesize proportional relationships from other relationships, such as
additive relationships and inversely proportional relationships. (ACOS
5.13, 7.2)
6.
Analyze, apply, and synthesize unit rates to solve problems and to
formulate equations for proportional relationships. (ACOS 5.13,
7.1,7.2)
(f)
Measurement, Data Analysis and Geometry. Measurement is the
process of finding a number that shows the amount of something. It is a system
to measure the height, weight, capacity or even number of certain objects. It
is the process of quantifying something and then possibly making comparisons
between two or more objects or concepts. Typically, measurements involve 2
parts-a numeric value and the specific unit.
Data Analysis is the
ability to formulate questions that can be addressed with data and collect,
organize, and display relevant data to answer them.
Geometry is
the study of different types of shapes, figures, and sizes in real life.
Coaches will develop greater knowledge and abilities within geometry to assist
novice teachers with the ability to instruct K-5 learners.
1.
Measurement. Upon program
completion mathematics coaches shall be able to do the following:
i.
Analyze, apply, and
synthesize the general principles of measurement, the process of
iterations, and the central role of units: that measurement requires a choice
of measurable attribute, that measurement is comparison with a unit and how the
size of a unit affects measurements, and the iteration, additivity, and
invariance used in determining measurements. (ACOS K.16, K. 17, 1.17,
1.18, 1.19, 1.20, 2.17, 2.18, 2.19, 2.20, 2.23, 2.24, 4.21,
5.17)
ii.
Analyze,
apply, and synthesize how the number line connects measurement with
number through length. (ACOS 2.21, 2.22, 4.22)
iii.
Analyze, apply, and
synthesize what area and volume are and give rationales for area and
volume formulas that can be obtained by finitely many compositions and
decompositions of unit squares or unit cubes, including formulas for the areas
of rectangles, triangles, and parallelograms, and volumes of rectangular
prisms. (ACOS 3.18, 3.19, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 4.23,
5.18, 5.19, 6.26, 6.27, 6.28)
2.
Data Analysis (Statistics and
Probability). Upon program completion mathematics coaches shall be able
to do the following:
i.
Analyze, apply,
and synthesize appropriate graphs and numerical summaries to describe
the distribution of categorical and numerical data.
(ACOS K.15, 1.16,
2.15, 3.16, 3.17, 5.16)(ii)
Analyze, apply, and synthesize that responses to statistical
questions should consider variability. (ACOS 2.16, 4.20, 5.16,
6.22)
3.
Geometry. Upon program
completion mathematics coaches shall be able to do the following:
i.
Analyze, apply, and
synthesize geometric concepts of angle, parallel, and perpendicular; use
them in describing and defining shapes; and describing and reasoning about
spatial locations (including the coordinate plane).
(ACOS K.18, K. 19,
K.20, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 5.20, 6.25)
(ii)
Analyze, apply, and
synthesize how shapes are classified into categories, and how reasoning
is used to explain the relationships among the categories. (ACOS K.21,
K.22, K.23, 1.21, 1.22, 2.25, 2.26, 3.26, 5.21, 5.22, 5.23)
4.
Coaching Principles in the Law Course. The Alabama State
Department of Education (ALSDE) defines coaching as a supportive, job-embedded,
ongoing, and differentiated professional learning practice focusing on growth
and achievement for all. In accordance with The Alabama Numeracy Act, the K-5
mathematics coaching endorsement shall prepare individuals to demonstrate
conceptual understanding and procedural fluency regarding major concepts of
mathematics appropriate for grades K-5. The Coaching Principles in the Law
course may only be taken after successful completion of the K-2 and 3-5 content
and pedagogical knowledge courses.
a.
Professional Dispositions and Practices. Demonstrate the pillars
of effective coaching according to the Alabama Coaching Framework:
1. Leads by example and influence,
2. Builds a relationship-oriented
collaborative approach,
3. Applies
knowledge and experience of adult and student learning,
4. Utilizes effective communication to
promote growth, and
5. Incorporates
evidence and data to support instructional improvement.
b.
Framework: The Alabama
Coaching Framework document, developed by the Alabama State Department of
Education (2020), was designed to improve outcomes for equitable teaching and
learning.
c.
Course
Content. Candidates shall:
1.
Demonstrate coaching strategies including goals, principles, and approaches in
the Alabama Coaching Framework.
2.
Understand adult learning principles that support collaboration with the
ultimate goal of improved student performance.
3. Demonstrate leadership skills.
4. Understand the roles of school-based
mathematics coaches.
5. Understand
research on the science of learning.
6. Translate research findings to effective
instruction.
7. Conduct coaching
cycles.
8. Demonstrate ability to
work with school administrators in disaggregating data and developing
strategies.
9. Demonstrate ability
to effectively present complex information to and engage with various
stakeholders.
10. Participate
actively and co-facilitate the professional learning community of mathematics
educators.
11. Analyze and organize
data for interpretation and application.
5.
Literacy in Mathematics Education
Course. An excellent mathematics program in Alabama requires educators
to hold themselves and their colleagues accountable for seeking and engaging in
professional growth to improve their practice as lifelong learners in order to
promote student understanding of mathematics as a meaningful endeavor
applicable to everyday life. Professionals are dedicated to learning and
improving their craft, which ultimately benefits students. Designing and
enacting effective lessons and valid assessments requires teachers to increase
their knowledge and skill throughout their careers. Teaching in ways that
promote student collaboration in learning mathematics from and with each other
requires adults to model effective collaboration in their own learning and
progress. The Literacy in Mathematics Education course may only be taken after
successful completion of the K-2 and 3-5 content and pedagogical knowledge
courses.
a.
Professional
Dispositions. Demonstrates habits of an effective teacher according to
the Interstate Teacher Assessment and Support Consortium (InTASC) standards and
the Alabama Mathematics Course of Study (2019).
b.
Framework: InTASC Standards.
Pursuant to the mission of improving the academic achievement of all students
in the public schools of Alabama, mathematic coaches will guide K-5 teachers in
aligning their instruction with the Interstate Teacher Assessment and Support
Consortium (InTASC) standards.
1.
Learner Development. The coach models how learners grow and
develop, recognizing that patterns of learning and development vary
individually within and across the cognitive, linguistic, social, emotional,
and physical areas, and designs and implements developmentally appropriate and
challenging learning experiences.
2.
Learning Differences. The
coach models understanding of individual differences and diverse cultures and
communities to ensure inclusive learning environments that enable each learner
to meet high standards.
3.
Learning Environments. The coach models how to create environments
that support individual and collaborative learning, and that encourage positive
social interaction, active engagement in learning, and
self-motivation.
4.
Content
Knowledge. The coach models how to utilize the central concepts, tools
of inquiry, and structures of the discipline(s) he or she teaches and creates
learning experiences that make these aspects of the discipline accessible and
meaningful for learners to assure mastery of the content.
5.
Application of Content. The
coach connects concepts and uses differing perspectives to engage learners in
critical thinking, creativity, and collaborative problem solving related to
authentic local and global issues.
6.
Assessment. The coach models
how to use multiple methods of assessment to engage learners in their own
growth, to monitor learner progress, and to guide the candidate's and learner's
decision making.
7.
Planning
for Instruction. Based on the appropriate Alabama Course(s) of Study,
the coach models how to plan instruction that supports every student in meeting
rigorous learning goals by drawing upon knowledge of content areas, curriculum,
crossdisciplinary skills, and pedagogy, as well as knowledge of learners and
the community context.
8.
Instructional Strategies. The coach models how to use a variety of
instructional strategies to encourage learners to develop deep understanding of
content areas and their connections, and to build skills to apply knowledge in
meaningful ways.
9.
Professional Learning and Ethical Practice. The coach emphasizes
engagement in ongoing professional learning and the use of evidence to
continually evaluate his/her practice, particularly the effects of his/her
choices and actions on others (learners, families, other professionals, and the
community) and illustrates how to adapt practice to meet the needs of each
learner.
10.
Leadership and
Collaboration. The coach suggests that novice teachers seek appropriate
leadership roles and opportunities to take responsibility for student learning,
to collaborate with learners, families, colleagues, other school professionals,
and community members to ensure learner growth, and to advance the
profession.
c.
Course Content. Candidates shall:
1. Have knowledge of historical developments
in mathematics, including the contributions of underrepresented groups and
diverse cultures.
2. Demonstrate
knowledge of the basic structures and problem types of word problems for all
operations and proper sequencing to support student understanding of the
meaning of the operations.
3.
Understand the developmental nature of mathematics and the interconnections
among mathematical concepts.
4.
Demonstrate knowledge of common errors and misconceptions about the operations
and how to help students learn.
5.
Demonstrate knowledge of the phases students move through in developing
fluency.
6. Use their knowledge of
student diversity to affirm and support full participation and continued study
of mathematics by all students. Student diversity includes gender, ethnicity,
socioeconomic background, language, special needs, and mathematical learning
styles.
7. Use appropriate
technology to support the learning of mathematics.
8. Use appropriate formative and summative
assessment methods to assess student learning and program
effectiveness.
9. Use formative
assessments to monitor student learning and to adjust instructional strategies
and activities.
10. Use summative
assessments to determine student achievement and to evaluate the mathematics
program.
11. Know when and how to
use student groupings such as collaborative groups, cooperative learning, and
peer teaching.
6.
Unique Field Experience
Requirements: At a minimum, field experiences shall include placements
where candidates:
a. Observe building-based
coaches provide assistance and scaffold support on a daily basis in grade
levels K-2 and 3-5.
b. Practice a
mini-coaching cycle, according to prescribed expectations, with a teacher in
his/her school under the guidance of the building-based math coach in grade
levels K-2 and 3-5.