# Introduction

There are four modules in the *Introduction to the Revised Mathematics TEKS* professional development series. Each module is approximately three hours. Each module gives you the opportunity to focus on the TEKS for one of the following high school mathematics courses: Algebra I, Geometry, or Algebra II.

Be sure to download the journal for this module.

**Note:** Please download the latest version of Adobe Reader, which enables you to type directly into the PDF and save your work.

This is the last of four modules to introduce the revised mathematics TEKS. The four modules are

- Revised Math TEKS (9–12) with Supporting Documents,
- Applying the Mathematical Process Standards,
- Completing the Gap Analysis, and
**Achieving Fluency and Proficiency**.

# Definitions

Watch the video and read the Merriam-Webster definitions for the terms below. Use the dictionary definitions to create your own definitions and record them in your journal.

# Research Activity

Download the *Research Reading* document. Read the research information, and consider how it relates to your understanding of computational fluency, mathematical proficiency, and automaticity.

# Conceptual Understanding

Watch the following video, and write your definition of conceptual understanding in your journal.

Watch the following video, and review the National Research Council definition of conceptual understanding below.

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.

A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes.

**Reflection Activity**

- How is the National Research Council definition of conceptual fluency similar or different from your own?
- How is conceptual understanding different from mathematical proficiency?

Watch the following video for a possible response.

# Vertical Learning Progression Activity

Watch the video overview of the vertical learning progression activity.

Watch the next two videos and use the *Vertical Alignment *documents for grades 5–8, Algebra I, and Algebra II, or for grades 5-8 and Geometry, to complete the** Vertical Learning Progressions Activity** in your journal.

After you have completed the activity in your journal, watch the video below for possible responses.

# Developing Mathematical Proficiency

Watch the video, and respond to the reflection question on the **Developing Mathematical Proficiency** page in your journal.

How does pairing a content standard with a process standard allow students to become mathematically proficient?

Watch the next video, and refer to the vertical progression pages you completed as you consider the following question:

Why is it important that the student expectations in the mathematical proficiency column be coupled with the process standards?

After you have finished recording your thoughts in your journal, watch the video below for possible responses.

# Student Activities

After watching the video, explore the student activities (**Franchesca's Fractions**, **Inigo's Integers**, and **Ra'Neisha's Rationals**) in your journal or download the PDF.

These three activities reflect the application of grades 5, 6, and 7 TEKS related to properties of operations and rational number calculations. Students are expected to be fluent with rational numbers by the end of grade 7.

Consider the following questions:

- How do these activities collectively help build computational fluency?
- How might strategies based on properties support students struggling with computation as they use a calculator on the STAAR Algebra I end-of-course assessment?
- How might these strategies support discussions about reasonableness of solutions?
- How do the properties of operations affect simplifying expressions?

# Open Array Method

How might building on an open array representation support students as they work to develop fluency with multiplying binomials?

# Developing Fluency

Studying whole number operations by examining and using relationships among representations and algorithms allows for "algebraic reasoning" including

- taking apart, working with pieces, and putting back together;
- thinking about the meaning of operations; and
- developing number sense.

How could we take the strategies from the open array methods and extend them to algebra and beyond?

Review the examples in your journal or download the PDF and record your reflections in your journal.

# Drill or Practice?

How do the activities that we just looked at compare with some of the more traditional paper and pencil activities? Look at the example below.

Watch the videos, and consider how the provided definitions of drill and practice are reflected in the examples. Record your thoughts in your journal.

Source: Van De Walle, J. (2004). Elementary and Middle School Mathematics Boston: Pearson

# Case Studies

Watch the video, and review the *Case Study* student work samples for Student A and Student B in your journal. Record your observations, and respond to the following questions on the **Case Study Recording Sheet** in your journal:

- What evidence indicates the student has computational fluency as related to polynomial operations?
- What evidence indicates the student has mathematical proficiency as related to polynomial operations?
- What evidence is missing?

When you have finished, watch the video below for possible responses.

What steps might the teacher take to address computational fluency and mathematical proficiency?

# Reflection

How are computational fluency and mathematical proficiency similar? How are they different?

Summarize your observations by completing the Venn diagram in your journal.

# Conclusion

Thank you for participating in this module. Please consider continuing your professional development by accessing the other modules in the* Introduction to the Revised Mathematics TEKS* series.

- Revised Math TEKS (9–12) with Supporting Documents
- Applying the Mathematical Process Standards
- Completing the Gap Analysis