Developing Teacher Content Knowledge

The first step in developing our content knowledge as teachers is have a sense of "how much math we need to know".

Compare the following problems:

from Liping Ma's "Knowing and Teaching Elementary Mathematics"

• What do you need to know to be able to answer the problem on the left?
• What do you need to know to answer the problem on the right?
• Which problem was harder for you? What made it harder?

Many teachers would agree with the statement, "I understand fractions" based on their comfort answering procedural problems like the one on the left. A deeper understanding of fractions is required for the problem on the right.

In the Common Core era, there is a much larger emphasis on developing a greater depth of knowledge in our students (See Develop Teacher Understanding and Comfort Using Mathematical Tasks with High Cognitive Demands).

If we are going to help our students develop the level of conceptual understanding they need, we have to deepen our sense of what it means to "understand" different mathematical concepts and ideas.

One final recommendation: Before delving into advanced math topics like polynomials or trigonometry, first focus on developing a profound understanding of fundamental mathematics. The print resources below are highly recommended in helping you do that. When possible, discuss them and do the activities described within each with colleagues.  

Recommended Print Resources for Developing Mathematical Content Knowledge

• Recommended Articles for Deepening Math Content Knowledge

• Math Matters: Understanding the Math You Teach by Suzanne Chapin and Art Johnson  

This is a great teaching resource and very helpful when it comes to deepening your understanding of the mathematical concepts you teach. Math Matters provides an in-depth study with 14 chapters covering number sense, computation, addition, subtraction, multiplication, division, fractions, decimals, percents, ratios, algebra, geometry, spatial sense, measurement, statistics, and probability. Over 100 activities give readers an opportunity to connect ideas, compare and contrast concepts, and consider how students understand the mathematics presented.

• Knowing and Teaching Elementary Mathematics by Liping Ma
• Knowing and Teaching Elementary Mathematics - a brief article exploring some of the findings from Liping Ma's book of the same name.

• The "Developing Essential Understanding" series from NCTM (the National Council for Teachers of Mathematics) - The Essential Understanding series addresses topics crucial to student development but often difficult to teach. Each book in the series gives an overview of the topic, highlights the differences between what teachers and students need to know, examines the big ideas and related essential understandings, reconsiders the ideas presented in light of connections with other mathematical ideas, and includes questions for readers’ reflection.

• The Fostering Geometric Thinking Toolkit: A Guide for Staff Development by Mark Driscoll

These professional development materials are a series of group study guides focused on geometric thinking, intended for use by grades 5–10 mathematics teachers in a professional development setting. Twenty two-hour sessions combine to produce forty hours of meeting time and offer the following:

• a conceptual framework to help teachers understand middle school students’ thinking in geometry and measurement and to guide them in engaging their students’ thinking more productively

• hands-on investigation of rich mathematical problems in geometry and measurement and tools for discussion and reflection aimed at deepening teachers’ understanding of geometric thinking

• structured approaches to gathering and analyzing data about how students’ thinking about geometry and measurement develops

• structured approaches to discussion among teachers about mathematics, curriculum, student thinking, and other issues related to teachers’ practice

• The Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10   by Mark Driscoll

Progressions Documents for the Common Core Math Standards

The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels. They explain why standards are sequenced the way they are, point out common student misconceptions and offer pedagogical strategies for dealing with those misconceptions.   

Resources for Developing Mathematical Content Knowledge  

• Why does this work? Where does this procedure come from? - The link above will take you to a series of questions that ask you to figure out the math behind the procedures and formulas you may have memorized. One way to begin to get deeper into mathematics is to ask yourself why things work and try to build your answer using the math that you know (as opposed to just reading or hearing a quick explanation). 

• Teachers Investigating Adult Numeracy (TIAN) Bundles - The TIAN Bundles are resources for ABE math teachers to explore math together with their peers, have examples of activities that can be used with students, and read and reflect on research on numeracy instruction. There are 5 "bundles": (1) Number Sense: Flexibility & Fluency, (2) Operation & Symbol Sense, (3) Number Sense: Integers, (4) Algebraic Thinking and (5) Geometric Thinking.

• *Specialized Teacher Knowledge - This recommended online resource offers a concise review of relevant content for teaching mathematics. It is from the "Powerful Tools Teachers Use to Help With Sense-making in Mathematics". It was developed by SERP (Strategic Education Research Partnership).

• *LearnZillion - Over 2,000 lessons built by teachers, connected to Common Core Content Standards for Mathematics. Videos break down content and talk about common student misconceptions.  Please note, these videos are intended to be used as lessons for students - but they work well for teachers to deepen their conceptual understanding. We recommend these videos over the Khan Academy videos because (a) LearnZillion is specific to the Common Core, (b) LearnZillion videos are created by teachers and reflect an understanding of both content as well as pedagogical content knowledge and (c) LearnZillion is structured in such a way that users can understand the coherence of how the lessons fit together. 

• *Illustrative Mathematics Fractions Progression - A series of brief videos that give an overview of how the concepts of fractions build off of each other and off of operation work with whole numbers.

•  Challenging Problems and Tasks - Ratio and Proportional Relationships - Developed for EngageNY by EduTtron, these 12 problems and tasks are not for teachers to just turn around and give to students. They were created for teachers to dissect them with their colleagues to improve their own knowledge. They are provided as examples and models for teachers/teacher teams to create problems of equal depth.

Online Courses for Developing Mathematical Content Knowledge

• Visualizing Algebra - A free online course from Udacity. "Learn the basics of algebra through intuition and problem-solving! From fractions to factors to functions, we'll cover a breadth of topics."
  

• Introduction to Statistics: Making Decisions Based on Data - A free online course from Udacity. "In this class you'll learn to visualize, and interpret data. Thought-provoking examples and opportunities to combine statistics and programming will keep you engaged and challenged."

• Statistics: The Science of Decisions - A free online course from Udacity. "We live in a time of unprecedented access to information. You'll learn how to use statistics to interpret that information and make decisions."

• The Equipartitioning Learning Trajectory: a K-5 Foundation for Rational Number Reasoning - The Mathematics Learning Trajectories MOOC-Ed series introduces you to Learning Trajectories (LT) as frameworks for interpreting and implementing the Common Core State Standards. Our first course introduces Equipartitioning. Grounded in fair sharing activities that create equal-sized groups (from collections) or parts (from single wholes), Equipartitioning provides a robust foundation for students’ conceptual understanding of division, multiplication, ratio, and fractions. We recommend this course for elementary and middle grades educators seeking a deeper understanding of students’ learning of fundamental mathematics for their own instructional practice. This course is on-line and free, though you do need to register for it with a gmail email address.