Sequencing

=Sequencing in Undergraduate Mathematics Courses=

Mathematics Topics
Functions/rate of change **[277]** Trigonometry **[308]** Conic Sections **[308]** Transformational Geometry **[308]** Proportional Reasoning **[177]** Fractions (division) **[177]** Integers **[177]** Probability **[301]** Statistics **[301]** Triangle Congruence Criteria **[362/308]** Linear/Slope Variable **[177]** Quadratic **[277,308]** Exponential **[277]** Logarithms **[277]** Inverse Functions **[277]** 2D and 3D Geometry **[308,362]** Proof / Reasoning **[362,308]** Rules for Exponents **[276/277]** Matricies Making Sense of Symbolic Manipulation Perspective/patterns

Systems of equations **[276]** Radicals and irrational numbers **[276]** Ratio and Rate **[177]** Quantitative Reasoning **[276]** Connection sequences to Functions **[277]** Inverse Functions **[277]** Function covariation **[277]** Algorithms **[308]** Number Theory **[308]** Modeling **[308]** Multiple Representations **[308]**

Literacies
Definition: A literacy is fluency in creating and interpreting a particular kind of text in a way that is acceptable in a community of practice.

Conceptual Mathematical Explanations **[177, 276, 308] ** Fundamental Mathematical Concept Description **[276, 277, 377-8, 476/496] ** Learning Goals **[276?, 277, 377-8, 476/496] ** Task Design **[277, 377-8, 476/496] ** Lesson Plan **[377-378, 476/496 [[file:Lesson Plan Template October 2014.doc|Lesson Plan Template]]**
 * Understanding/Critiquing tasks;
 * Modifying/Creating a task

Using Student Thinking
Recognizing their own thinking and that of their peers **[177]** Analyzing and understanding student thinking (making sense of student thinking) **[177, 277, 377]** Sequencing student responses **[377]**

Developing PSTs' Identities as Teachers
Illuminating and examining beliefs about schooling, students, mathematics, and mathematics learning and teaching **[276]** Knowledge of //Principles and Standards **[276]**// Knowledge of //Principles to Action **[276]**// Knowledge of //CCSSM **[276]**// Articulation of a mathematics teaching philosophy **[276]**

Creating and Using Tasks
Identifying what FMCs are addressed by a task **[277]** Creating/Modifying worthwhile tasks **[277]** Creating tasks that develop a particular FMC **[277]** Sequencing tasks to develop understanding and fluency **[377]**

**Pedagogy (some skills)**
Anticipating student thinking **[277, 377]** Orchestrating Class Discussions **[377]** Lesson Sequencing **[377]**

Preparation for Inclusive Mathematics Instruction
Considering positioning and its effect on identity Considering issues of race, class and gender Accommodating students with special needs Creating multiple ways to be good at math Teaching mathematics for social justice

=Math Ed 177=

Mathematics Topics

 * Proportional Reasoning
 * Fractions (division)
 * Ratio and rate
 * Integers
 * Linear/Slope Variable


 * Comments: ** These four topics are more than enough for one semester's work in 177. Faculty who have taught it usually cover 3 of the 4 carefully, and parts of the fourth as time permits.

Literacies

 * [[file:Conceptual Math Explanations final.docx|Conceptual Mathematical Explanations]] __[This literacy should be done regularly as part of this course]__


 * Comments: ** We can begin working on this literacy in 177, and it will continue into the next courses. Dan's work goes here.

Using Student Thinking

 * Recognizing their own thinking and that of their peers


 * Comments: **
 * Group work and discussion is a natural way to bring this out.
 * The course should be taught so that it is based on student thinking, but there should also be explicit discussions about the modeling we do.
 * Students should also learn how to listen to peers and learn to understand how they are thinking.
 * Show kids who use more sophisticated thinking (e.g. about proportional reasoning) than our students do -- help them see that the systems has failed them.
 * (points to the anticipation we want to see in 277/377)

Preparation for Inclusive Mathematics Instruction

 * === A focus on how traditional math instruction has positioned students. ===
 * Disrupt our students' acceptance of current/traditional ways of teaching mathematics (Benny, disaster studies, problematic student thinking)


 * Comment: T**he course learning outcomes state that we want MthEd 177 students to realize "that mathematics instruction that does not build relational understanding has failed a majority of adolescents, even themselves, **//and that they have a moral obligation to seek the necessary knowledge and expertise to teach mathematics in ways that enable all adolescents to develop conceptual understanding, procedural fluency, and facility with authentic mathematical practices" (emphasis added)//**


 * Modeling the disruption of traditional positioning (e.g. who's good at math).


 * Comment: ** One pathway into these ideas is the following activity as students get accustomed to working in a group (Thanks, Kate):
 * Ask students to write three sentences on an index card about people they have class with (but using pseudonyms). Because classes are small here, I normally have them write one sentence about someone in their class, one sentence about someone from other class, and one sentence they think someone would write about them. I try not to give them too much advice here but I say things like, "Think about what you think about other people in the class. What roles do each of you play in class?" Students write things like, "Yoda is opinionated but in a respectful way, he likes to share but is willing to consider the opinions of others.", "Jordan is really good at coming up with unique (and brilliant) ways of looking at and describing problems", "Frank is always commenting, he likes being smarter than the rest of the class", "Sally Jane is slow which holds us back." I collect the index cards and read a selection of them to the class and ask them to consider questions such as: How did these come to be our impressions of others? Do you have the same positioning in each of your classes? Is positioning intentional or unintentional? What are the consequences to your position in any of your classes (both positive and negative)?
 * These kinds of questions can be used to disrupt and/or highlight current positionings that are happening in class.
 * What are you smart in/about?
 * What are others in the group smart in/about?
 * How can you tell?

Questions, comments, and suggestions:
Kate: I think 177 might be too early for a reading that specifically addresses positioning and I am not sure to what extent we want use the language of "positioning" to talk about how mathematics is positioned in this class either, but I am going to drop this reading in that I've used with the 377 students so we might consider its use and placement in the sequence of courses. Link to document: The document ends with this set of questions that can be used to highlight a range of ideas related to positioning:
 * 1) Who is considered ‘smart’ in my classroom? About what (e.g., procedures? concepts? representations?)? Who is considered a ‘struggling’ learner? By whom?
 * 2) Who is talking (the teacher, which students specifically)? What do they talk about? What kind of mathematical language do they use?
 * 3) What does it mean to ‘know’ mathematics in my classroom? Is mathematics about procedures, concepts, and/or something else?
 * 4) What kind of mathematical practices (e.g., argumentation, explanation, just answers) do we engage in? What is emphasized, thinking processes or doing processes?
 * 5) In what ways do I use my authority in the classroom?

=MthEd 276=

**Texts**

 * Professional Standards for Teaching Mathematics (electronic - NCTM membership)
 * Principles and Standards for School Mathematics (electronic - NCTM membership)
 * Principles to Actions
 * Common Core State Standards for Mathematics

Pedagogical Topics

 * Worthwhile Mathematical Tasks (identifying and implementing)
 * Mathematical Discourse (roles and tools)
 * Learning Environment (e.g., norms, positioning)
 * Content Standards
 * Process Standards

Mathematics Topics
These mathematical topics along with others are used as a context to teach about the standards and illustrate pedagogy
 * Systems of Equations
 * Radicals and irrational numbers
 * Quantitative reasoning: units of, operations on, and relationships between quantities

Comments:

Literacies

 * [[file:Conceptual Math Explanations final.docx|Conceptual Mathematical Explanations]]
 * Mathematical Analysis Paper -
 * Problem Statement - clear statement of the task and expected mathematics
 * Process Description - thoughtful description of the use of the process standards
 * Solution - clear explanation with convincing argument
 * Reflection - How have I grown as a mathematician?


 * Fundamental Mathematics Concept Description - includes: title, description, example [[file:Concept description of an FMC 9.11.14.doc|FMC definition/description]]


 * Learning Goals - Measurable actions that either a) help students develop an understanding of the FMC or b) demonstrate an understanding of the FMC. [[file:Goals for Mathematics Lessons 9.11.14.docx|Mathematical Learning Goals definition/description]]


 * Observing, Analyzing and Reporting on student thinking (The students match readings and class discussions to observations through identifying missed opportunities as well as citing episodes of the standard being met. They support claims with evidence - student work, student discourse, specific questions, etc)
 * Philosophy and Beliefs
 * Worthwhile Mathematical Task
 * Teachers role in discourse
 * Students role in discourse
 * Tools to enhance discourse
 * Learning Environment
 * Student Interview
 * Student engagement in authentic mathematical practice
 * Reflection on teaching experience


 * Comments:** Students continue to develop their fluency in creating and evaluating conceptual mathematical explanations, but emphasis should be given to learning how to write learning goals and fundamental mathematics concept descriptions. In addition students learn to observe, analyze and report on student thinking.

Using Mathematical Thinking

 * Recognizing student thinking in the classroom
 * Beginning to identify how teachers use or don't use student thinking
 * Introduction to identifying student thinking in terms of conceptual understanding and procedural fluency
 * Introduction to anticipating student thinking by working through tasks in multiple ways
 * Introduction to selecting, sequencing and connecting student thinking through use of student work

Comments:

Preparation for Inclusive Mathematics Instruction

 * Observe, analyze and report on positioning of students in the classroom environment

Comments:

Creating and Using Tasks

 * Identify learning goals addressed by the task
 * Create a worthwhile task
 * Implement a task with students

Comments:

Mathematical Discourse
> tools and representations for enhancing discourse.
 * Observe, analyze and report on discoure in terms of teachers role, students role, tasks that elicit discourse and

Comments:

Developing PSTs' Identities as Teachers

 * Illuminating and examining beliefs about schooling, students, mathematics, and mathematics learning and teaching
 * Knowledge of // Principles and Standards //
 * Knowledge of // Teaching Standards //
 * Knowledge of // CCSSM //
 * Articulation of a mathematics teaching philosophy

Comments:

= = =MthEd 277=

Mathematics Topics

 * Functions from a Rate of Change (covariation) perspective. Specifically exponential as compared to quadratic and linear
 * Inverse functions (exponential and log)
 * Connecting sequences to functions (arithmetics to linear and geometric to exponential).

Comments:

Literacies

 * Fundamental Mathematics Concept Description - includes: title, description, example [[file:Concept description of an FMC 9.11.14.doc|FMC definition/description]] Because this is a critical part of critiquing, writing or modifying tasks, students should __write many FMC descriptions__ during this course.
 * Learning Goals [[file:Goals for Mathematics Lessons 9.11.14.docx|Mathematical Learning Goals definition/description]]
 * Task Description - critiquing, writing or modifying lesson plans should be a driving thread of this course so students should have multiple experiences doing each of these activities during this course. [[file:Task Analysis v3 2016.02.02.docx|Mathematics Task Analysis]]


 * Comments:** Task descriptions incorporate the previously learned literacies of conceptual mathematical explanations and fundamental mathematics concept descriptions. By now, students should be able to consistently provide acceptable conceptual mathematical explanations and evaluate others' explanations. In contrast, it is likely that students will not have mastered the ability to write fundamental mathematics concept descriptions, and will need additional support in writing these descriptions.

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments:

= = =MthEd 377-378=

Mathematics Topics

 * Professors Choice with the following parameter: It should be related to the Common Core (grades 7-12)

Comments:

Literacies

 * [[file:Concept description of an FMC 9.11.14.doc|Fundamental Mathematics Concept Descriptions]] - includes: title description, example
 * [[file:Goals for Mathematics Lessons 9.11.14.docx|Learning Goals]]
 * Task Description
 * Lesson Plan (Students need experience writing many lesson plans - __at least 6__. This may be done in groups or individually). [[file:Lesson Plan Template October 2014.doc|Lesson Plan Template]]


 * Comments:** Lesson plans will include fundamental mathematics concept descriptions and task descriptions.

Pedagogies

 * Lesson sequencing - students need experience thinking about how one lesson builds on another. They can either plan sequential lessons or peers can plan sequential lessons thus forcing them to think about how a lesson taught one day builds on or toward a lesson taught on another day.
 * Anticipating student thinking - a critical part of the lesson plan literacy is anticipating student thinking. By writing at least 6 lesson plans they should have repeated experience doing this kind of anticipation.
 * Orchestrating Discussions - Students need to have repeated experiences thinking about orchestrating discussions. Once again, this is part of the lesson plan literacy so writing multiple lesson plans will help them think about this practice in depth even if they don't actually enact the discussion for all of the lesson plans.

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments:

= = =MthEd 300=

Mathematics Topics

 * First item

Comments:

Literacies

 * First Item

Comments:

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments:

= = =MthEd 301=

Mathematics Topics

 * First item

Comments:

Literacies

 * Conceptual mathematics explanations

Comments:

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments:

= = =MthEd 308=

Mathematics Topics

 * Algorithms (e.g. square root, nth root, Euclidean)
 * Number Theory (e.g. factorizations, Fibonacci sequence)
 * Functions (quadratic from a geometric perspective, trigonometric)
 * Geometric constructions and transformations (Specifically be able to construct the transformation that will map congruent or similar figures to each other.)

Cutting across all of these Mathematical topics, students should have experiences with the following processes and ideas:
 * Modeling(e.g. analyzing real-life problems with CAS systems, fitting models to data [perhaps through CBLs or other experiments], analyzing functional form and coefficients and how they relate to the situation being modeled, extraneous solutions in real-life problems, modeling phenomena with matrices or differential equations, etc.)
 * Multiple Representations (e.g. graphical, algebraic, tabular, numeric, verbal, dynamic vs static, interactive vs non-interactive, etc.)
 * Proof (e.g. Issues of what constitutes a mathematical proof when engaging in mathematics through technology, using technology to make conjectures which can then be proven.)

Comments:

Literacies

 * [[file:Conceptual Math Explanations final.docx|Conceptual Mathematics Explanations]]
 * Geometry sketches using sketchpad

Comments:

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments:

= = =MthEd 362=

Mathematics Topics

 * First item

Comments:

Literacies

 * [[file:Conceptual Math Explanations final.docx|Conceptual Mathematics Explanations]]
 * Mathematical Proofs
 * Creation of axiomatic systems


 * Comments:** Students who take this course may still not have mastered the ability to write convincing proofs. They will need explicit instruction on proof writing, particularly in identifying and correcting mistakes in proofs. At the same time, geometry provides a unique opportunity to build an axiomatic system starting at the ground floor. Students need to learn how to identify and avoid relying on assumptions about or objects in the axiomatic system that they have yet to define or prove.

Using Mathematical Thinking

 * First item

Comments:

Preparation for Inclusive Mathematics Instruction

 * First Item

Comments: