## Thursday, November 18, 2010

### Ignite event at CMC-S Math Conference

So what could I do to "ignite" thinking about mathematics education given the constraints of using 20 slides and just 5 minutes (with the slides auto-advancing every 15 seconds)? This was the challenge I faced as November 6 approached and I prepared for a so-called Ignite event. The result? See for yourself at Key Curriclum's Sine of the Times blog. Be sure to check out the other presenter's ideas, too, on You Tube. I am sure you'll find something to ignite your thinking about mathematics teaching and learning!

## Sunday, October 17, 2010

### Teaching and Learning Fraction Concepts and Operations

The Institute of Education Studies (IES), part of the U.S. Department of Education, has recently released a research-based 90-page report titled, "Developing Effective Fractions Instruction for Kindergarten through 8th Grade." The report begins by detailing the poor level of understanding of fraction concepts and skills among U.S. students. For example data from international and national exams have shown that "50% of 8th-graders could not order three fractions from least to greatest...[and] fewer than 30% of 17-year-olds correctly translated 0.029 as 29/1000" (p. 6).

A panel of experts reviewed dozens of studies and identified five (5) recommendations for improving fraction instruction. From page 1 of the document:

- Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.
- Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.
- Help students understand why procedures for computations with fractions make sense.
- Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems.
- Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.

This list fits well with the approaches I've found successful in my own classroom (previously) and in the experiences of my credential students in their classrooms. Learning to make sense of fraction concepts requires deliberate activities that allow for sense-making to occur.

Some resources to help you design such learning activities are:

Have fun with these! It is always great to see students "get it" when working with fractions.

## Sunday, September 5, 2010

### The Importance of Questioning

For the past few years I've taught methods courses for both beginning and experienced teachers of mathematics. One element of these courses has been the use of videos of teachers in the classroom to prompt discussion of specific aspects of practice. The lesson captured in a set of video clips here, "Looking for Squares" by Lisa Brown, is one of the best examples I've found of a teacher's use of questioning to guide students' exploration and eventual sense-making of a mathematical concept, in this case square numbers and square root. It might be helpful to read "Questioning our Patterns of Questioning" by Herbel-Eisenmann and Brefogel, to provide a context for thinking about the use of questioning in a mathematics classroom. What specific strategies and teaching skills does Ms. Brown use to promote, sustain, and engage student thinking? How does this support their sense-making?

## Sunday, May 9, 2010

### Content Knowlege vs. Pedagogical Content Knowledge

Since its introduction by educational psycholigist Lee Shulman in the late 1980s, the term "pedagogical content knowledge" (PCK) has been used often to refer to the particular knowledge needed by teachers to effectively create learning environments that support student sense-making in mathematics (and other disciplines). However, until recently there has been no good way to measure a teacher's PCK.

This is no longer the case and we now need to learn from the work that has led to greater insights into PCK among teachers of mathematics. With the recent work of U. S. mathematics education researchers Heather Hill and Deborah Ball some progress has been made with this at the elementary and middle school levels and researchers led by Jurgen Baumert in Germany have helped to capture PCK among secondary teachers of matheamtics. The bottom line is that while content knowledge is clearly necessary for teachers of mathematics, it is not sufficient. The most effective teachers also have insights into the mathematics content their students will learn that allows them to identify misconceptions, set up lessons in which students make connections, and provide students with questions and prompts that help them access important concepts.

This is no longer the case and we now need to learn from the work that has led to greater insights into PCK among teachers of mathematics. With the recent work of U. S. mathematics education researchers Heather Hill and Deborah Ball some progress has been made with this at the elementary and middle school levels and researchers led by Jurgen Baumert in Germany have helped to capture PCK among secondary teachers of matheamtics. The bottom line is that while content knowledge is clearly necessary for teachers of mathematics, it is not sufficient. The most effective teachers also have insights into the mathematics content their students will learn that allows them to identify misconceptions, set up lessons in which students make connections, and provide students with questions and prompts that help them access important concepts.

## Monday, April 19, 2010

### Thinking about Ability - Fixed vs Growth Mindset

In the U.S. many folks perceive there to be a "math ability" that one either has or does not have (see Uttal's research). This actually does much harm to how we structure mathematics learning and limits achievement among students. As a teacher I constantly sought ways to engage students in making sense of mathematical concepts and relationships and believe that with few exceptions the majority of people can learn the mathematics in the K-12 curriculum. This was shown to be true among my students, many of whom went from believing mathematics to be a black hole of nonsensical symbols and rules to realizing the logical structure of mathematics and developing the power to reason mathematically to solve problems. This was possible in large part because I felt they were capable of making significant improvements to their knowledge of mathematics.

Important research from Carol Dweck and her colleagues at Stanford has shed light on the relationship between one's "mindset" toward ability (mathematical or otherwise) and the actions to which this leads in terms of opportunities to learn and feedback to learners. Ultimately, these have serious consequences on learning outcomes. The idea is this: if one believes ability to be fixed - a fixed mindset - this will lead to actions that a) serve to identify who is high-ability and not and b) provide feedback that reinforces ability status, directly and indirectly. The result - a few students are "smart" in mathematics while many others are not. The smart ones must strive to retain that labeling through "looking smart" (often at any cost) while the others see no reason to work at learning what they are being told is beyond their ability. The result is that actual learning among all students suffers.

What Dweck and colleagues found is that changing this script can have pronounced effects. Taking a "growth mindset" that views ability as derived more from effort than some innate quality leads to very difference choices about learning environment and feedback to learners. If one believes most all students can learn mathematics, those who are struggling to do so are in need of support. Feedback such as, "if you work hard at this you will improve" leads to greater motivation and, as their research has shown, greater success. For students who do well with mathematics, the feedback in the growth mindset tells them, "you made a good effort at this and have done well." This is critical because when the mathematics does get challenging (and it will!), these students will persist, putting in more effort, rather than look for ways to simply maintain their "looking smart" status (e.g., shortcuts, cheating, or bowing out).

See here for more about this work and suggestions for teachers and parents.

Important research from Carol Dweck and her colleagues at Stanford has shed light on the relationship between one's "mindset" toward ability (mathematical or otherwise) and the actions to which this leads in terms of opportunities to learn and feedback to learners. Ultimately, these have serious consequences on learning outcomes. The idea is this: if one believes ability to be fixed - a fixed mindset - this will lead to actions that a) serve to identify who is high-ability and not and b) provide feedback that reinforces ability status, directly and indirectly. The result - a few students are "smart" in mathematics while many others are not. The smart ones must strive to retain that labeling through "looking smart" (often at any cost) while the others see no reason to work at learning what they are being told is beyond their ability. The result is that actual learning among all students suffers.

What Dweck and colleagues found is that changing this script can have pronounced effects. Taking a "growth mindset" that views ability as derived more from effort than some innate quality leads to very difference choices about learning environment and feedback to learners. If one believes most all students can learn mathematics, those who are struggling to do so are in need of support. Feedback such as, "if you work hard at this you will improve" leads to greater motivation and, as their research has shown, greater success. For students who do well with mathematics, the feedback in the growth mindset tells them, "you made a good effort at this and have done well." This is critical because when the mathematics does get challenging (and it will!), these students will persist, putting in more effort, rather than look for ways to simply maintain their "looking smart" status (e.g., shortcuts, cheating, or bowing out).

See here for more about this work and suggestions for teachers and parents.

## Sunday, April 11, 2010

### Getting Ready for NCTM

Why would 10,000 teachers of mathematics descend on San Diego for a week? When the annual NCTM Conference is being held there! Check out the amazing list of sessions here.

As a member of the planning committee, I will be in charge of making sure things run smoothly with sessions in the Marriott, so be sure to stop by and say, "Hi!" if you're there.

As a member of the planning committee, I will be in charge of making sure things run smoothly with sessions in the Marriott, so be sure to stop by and say, "Hi!" if you're there.

## Monday, April 5, 2010

### Welcome!

Welcome to my new foundational-level mathematics blog! I plan to share resources and perspectives on the teaching of mathematics at the middle school and early high school level from my experience as a teacher and math educator. My philosophy of teaching is to create a learning environment in which students are encouraged and supported in making sense of mathematics. My work in public Title I schools in California with learners of all backgrounds convinced me that students have the potential to understand mathematics and the our work as teachers is to uncover and nurture this (as well as prodding it along when needed!). As a professor my scholarship looks at democratic practices in mathematics teaching and learning that work toward creating more equitable learning outcomes. See here for some of my academic musings: http://faculty.fullerton.edu/mellis/Articles%20about%20Mathematics%20Education.htm

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