Mathematics research and teacher understanding

This website is a great place to get started to understand  the research regarding mathematics education and the teacher. Deborah Ball, who began her career as an elementary teacher, is now the Dean of Education at the University of Michigan.  Her work has been instrumental in demonstrating that a teacher's deep understanding of the math they teach plays a critical role in student achievement.  You might start here by going to publications and reading some of Ball's earlier work (Magical Hopes and The mathematical understandings that prospective teachers bring to teacher education are a good start).

Article Questions:
Magical Hopes--
1) What is the author saying in her title?  In what do teachers have "magical hopes" and why does Ball see this as problematic?
2) In your own words, describe a strong response to the student misunderstanding that 1/6 + 1/6 = 2/12.  Use information from the article to frame your response.

Pre-Service Teachers--
1) As you read this article, stop to answer each math question on your own as they come up in the article. Gauge your own understanding and try to understand areas where you may need to strengthen your content knowledge. Try not to be hard on yourself or to disengage because of frustration.  This difficulty with content knowledge is, unfortunately, widespread. By reading this article, and following this trail, you are beginning the journey towards improvement!

Dr. Griffin is a developmental psychologist who has worked extensively to document how young students learn math. Her work led to the development of a program now called Number Worlds (see site on this trail). On this page, start with the short article Building number sense with number worlds and then consider reading the longer version--Fostering the development of whole-number sense: teaching mathematics in the primary grades.
Article Questions:
1) How does Dr. Griffin describe student acquisition of mathematics knowledge?
2) What area of mathematics development and understanding is critical to student success, particularly for those students who have fallen behind early on?
3) How does the program Number Worlds address the students' underlying understanding of mathematics?
4) What are the "worlds" of number worlds? Describe each one.

This site gives you detailed information regarding the program Number Worlds developed by Dr. Griffin. Originally titled "Right Start"  Dr. Griffin published the program on her own for years as she did research on the effects of the program. This is one of the few mathematics programs that is research tested and has consistently shown results.  Designed for students who are already behind as they enter kindergarten the program develops a students understanding of the number line in its different forms.

Activities:
1) Click on the icons on the left of the site to understand the different forms of number and the number line.  Check to see how well you answered the question posed earlier in this trail about the different "worlds" in number worlds. Consider whether you have been teaching these worlds explicitly in your classroom.

2) Click on the video examples to see how teachers engage students in deep thinking about number.  This is critical to student success. Choose one video and describe the type of questions the teacher asks the students.  a) How is the teacher's questioning technique the same or different from what you are used to?  b) What seems to be the purpose behind her questions?

This assessment has been cited (Geary, Jordan and Flojo, 2005) as the best robust probe of young students' developmental level in mathematics. It is relatively quick to use (5 minutes for young kids (age 4-5) and up to 15 minutes for later elementary students.  This assessment can be used to understand student level in math and also to use as a pre and post test for teachers to understand if their interventions are having an impact over the course of a few months or the school year.

Activity--
Administer the Number Knowledge Test to an elementary school student who is understood to be below grade level in math (or a middle school student who is well below grade level).
Were you surprised by the results in any way?
What could the student do, or not do that may inhibit his or her ability in mathematics?
How might this probe be useful to you and your instruction?

This book is essential. Liping Ma came to the United States after being a teacher in China and became a student of Deborah Ball (see webpage on this trail). The responses of teachers in the U.S. and China are compared for four different elementary mathematics topics. 
This book preview has many pages in pdf form to give you a good feel for the dialogue and discussion that these questions generate. Truly amazing. Read this book and begin to consider both how you were taught, how you now teach and how you can improve.

Activity--
As you did when reading the Deborah Ball piece on pre-service teachers, stop to answer each mathematical question as it comes up first. Then go on to read how other teachers answered the same question. Consider what you can learn about your own instruction by considering your content knowledge.

John Woodward created the program Transitional Math and focuses on mathematics, special education and the middle school-aged student. In particular, his work attempts to combine the strong points of a constructivist approach with the structured instruction needed for the struggling learner.
This site was created to give a comprehensive one-stop place for teachers to go for activities, ideas and research.

Activity:
1) Read the section on Journaling.  This is an important consideration regarding mathematics instruction that is under-appreciated.  Students benefit from making meaning of the mathematics they are learning and writing supports that meaning-making process.
How can you bring writing into your math instruction? Try some of the prompts provided in this section to get you started!

Here in John Woodward's homepage you will find some excellent articles. In particular, Dr. Woodward has done several articles looking into how students with special needs fare in the constructivist classroom (the first few articles).  He also has many suggestions regarding fluency (see articles and link on this trail).  Finally, you may want to consider reading the article "Mathematics reform in the U.S.: Past to present." This article provides an overview of the back and forth nature of how the education community has approached mathematics over the past 50 years.

Article Questions:
1) In the article "Effects of reform-based mathematics instruction in five third-grade classrooms" what do the researchers find?  Do struggling students perform well in these classrooms?  What conditions help to support the struggling students within these classrooms?

2)  In the article  "Developing automaticity in multiplication facts..." What does the author recommend as key instructional strategies for developing this automaticity?

Dr. Woodward has written a book on teaching math facts.  He argues his position that math facts are an important element to future student success in the Overview for Teaching Facts.  Many of his strategies are very useful for teaching facts in a way that is not just rote drill. While it is not available on this webpage, Dr. Woodward suggests the strategy for multiplication of thinking about the "extended facts" and supporting students in understanding powers of ten (40x3 is an 'extended fact for 4x3). This is nice way to demonstrate the power of base ten while working on facts.

The Problem solving strategies are not, in my opinion, as strong.  For instance, "guess and check" is a consistent 'strategy' utilized.  Guess and check is not so much a strategy as it is a back-up plan.  Consider relying on the other strategies more heavily.

Also of note here is the link to the National Library of Virtual Manipulatives.  This website has a LARGE variety of applets designed to help students work through problems visually and to develop meaning as they develop their fluency.  There are several applets that support math fact fluency and meaning.  Look particularly at the 1-2 and 3-5 grade sections.

Question and Activity:
1) Read the Subtraction Facts link.  How does Dr. Woodward's suggested strategies tie-in with other readings you have done on this trail?  Note particularly any convergence with the work of Sharon Griffin.

2) Go to the Virtual Manipulatives link, click onto virtual library and then click onto the 3-5 grade set of manipulatives for number and operation.  Investigate the applets and consider how they might support "math fact" instruction.  In particular, try Number Line Arithmetic, Number Line Bounce, Rectangle Multiplication and Rectangle Division.  Consider how these might support student learning and try them with a student.  Report on student engagement and the amount of "repetitions" the student(s) were able to get into a given period of time.  Do you think these applets would support student fluency with math facts over time?  Why or why not?

This article reviews how a school system attempts to implement Singapore Math. Singapore students are consistently at the top of the international comparisons for math achievement. 
As you read this article, look particularly at the discussion on "Bar Modeling" and consider this strategy as compared to "guess and check" (see site on this trail Teaching Math facts and problem solving).

Question--
Consider the Bar Modeling presented in this article.  How is this method different (for student learning) from Key Word Strategies?  Bar modeling is supported by research but Key Word Strategies are not.  Why do you think this is so?

Here is some further discussion about the Miracle Math article. It is down in the middle of this page.
This back and forth begins to give you an idea about the discourse taking place in the United States regarding math instruction.  Where do we begin?  Who do we look towards?  How do we implement change once we've decided upon what change we want?

Discussion Questions--
Consider your own classroom and your own school. 
1) After following this trail, what changes do you think could and should be implemented?
2) What information will you seek, what will you work on and what changes will you make in your own classroom?
3) If you were in charge, how would you go about ensuring that these changes took place for the whole school?

Finally, let's consider an instructional tool.  How do we  analyze different program that are out there considering the research? Take a look at the information in this website and then complete the activity below.

Activity:
Watch the demo video of Hands-on-Equations and consider the readings you have done on this trail. 
1) Do you think the authors you have read through this webtrail would consider this a sound instructional technique?  Why or why not?
2) Would this program help support your own understanding of algebra? If so, how? If not, what do you think is missing?