discussion 3

1  Effective Practices for Learning Mathmatics

Effective Practices for

Learning MathEmatics

2  Effective Practices for Learning Mathmatics

One thing I think all of us can agree on, and which has been noted by Bennett and Rol- heiser (2001), is that teaching is a complex and creative undertaking. I’m also reasonably sure that all of us would agree that effective teachers are those who sometimes seem to have superhuman powers when it comes to teaching and helping students learn.

Research shows that teacher knowledge about teaching and the learning process is the most powerful predictor of student success (Marzano, 1998; Greenwald, Hedges, Laine, 1996). But just what is it that effective teachers do that makes such a difference in student learning?

Effective Practices for Learning Mathematics

3  Effective Practices for Learning Mathmatics

Linda Darling-Hammond (1998) de- scribed expert teachers as those who use their knowledge about children and their learning to fashion lessons that connect ideas to students’ experiences. They’re the teachers who create a wide variety of learning opportunities that make subject matter come alive for those who learn in different ways.

Teaching Works, directed by Dr. Deb- orah Ball, cites effective mathematics teaching to include high-leverage practic- es as activities or tasks central to teaching that powerfully promote student learning

and are fundamental to competent teachers. These practices cover a wide range of tasks and activities, from recognizing and identifying common pat- terns of student thinking in a content domain, to using specific methods to assess students’ learning on an ongoing basis.

As you can see, there are some common traits in both of these descriptions related to ef- fective teaching. First of all, effective teachers must know their content and have the abil- ity to translate complex and difficult content ideas into learnable topics. Secondly, they know and understand important pedagogy about classrooms, how they work, and how they promote learning. But unless teachers have the teaching skills to use their content and pedagogical knowledge to bring about student learning, then effective teaching does not occur (Figure 1).

Figure 1

Bennett and Rolheiser (2001) said, “The real world does not work in rows nor do people interact by simply recalling facts in the absences of discussion…unless they are playing Trivial Pursuit.” So if students are able to work in groups to inquire, to process informa- tion, to confront and resolve authentic issues, problems, and task demands, then increased learning opportunities for learning mathematics occur. But just what practices do effec- tive teachers employ at any grade level to provide these opportunities for learning?

4  Effective Practices for Learning Mathmatics

Creating Connections Learner Connections

Effective teachers know from research that the most successful strategies for aid- ing students in reaching specific cognitive, social, and critical thinking goals are strat- egies that place learning in the hands and minds of the learners. They also know and understand that students learn and process in different ways.

They are keenly aware that students use different sensory channels/pathways or modalities (Eiszler, 1983) through which to give, receive, and store information and that some prefer one modality – visual, auditory, or kinesthetic – over another. Therefore, they realize the importance of including opportunities for using all mo-

dalities in learning. As a result, effective teachers include a variety of research-based strategies, activities, and structures such as those identified by Marzano, Wolfe, and Kagan which appeal to different learning modalities and intelligences for both inde- pendent and collaborative student engage-

ment for teaching and learning important mathematics concepts and skills.

Content Connections

Brain research has shown that the best way to make information meaningful is to associate or compare a new concept with an unknown concept, or to hook some- thing unfamiliar with something familiar (Wolfe). There are a couple of different ways to approach this in mathematics.

One way is the constructive use of stu- dents’ prior mathematics knowledge. The Common Core State Standards for Math- ematics Progressions documents, informed by research on children’s cognitive devel- opment and the logical structure of math- ematics, provide an excellent resource for defining and illustrating the progression of a mathematics topic across a number of grades. The example problems can be pre- sented to students to solve and/or teach- ers can use them as a resource in planning instruction and presenting problems to students that provide connections to prior knowledge and concepts.

A second way to connect content is by presenting students with problems that are realistic and within a context they can identify with. In doing so, they are more apt to “mathematize” the problem or relate the real-world context of a problem with the mathematical task required to solve it. Barwell (2005) identifies three dimen- sions of problems that students pay atten- tion to including genre (language), mathe- matical structure, and personal experience

5  Effective Practices for Learning Mathmatics

(life). He recommends specific activities to help students better understand the structure of problems which in turn aids them in mathematizing the problem. For additional information and activities that Barwell recommends for helping students understand the structure of problems in order to mathematize them, read our Solving Real World Problems = Language + Mathematics + Life blog.

Encouraging Engagement Mathematical Tasks

There is no decision a teacher makes that has a greater impact on students’ oppor- tunities to learn and on their perceptions about what mathematics is than the selec- tion or creation of the tasks with which they engage students in studying math- ematics (Lappan and Briars, 1995). Ef- fective teachers know that students need to be able to work independently and collaboratively in order to make sense of ideas, and that when groups of students of varying mathematical achievements work together insights come at different levels, and these insights tend to enhance over- all understanding. (Hunter, 2005; Sfard & Kieran, 2001; Woo, 2002.)

But as you and I both know, not all tasks are created equal, and different tasks pro- mote different levels and kinds of think- ing and reasoning. Cognitive demand rep- resents the type of thinking and the level/ depth of complexity of thought you expect students to engage in when completing a task. Stein, Smith, Henningsen, & Silver

(2000) identify four levels of cognitive de- mands of mathematical tasks, as shown in Figure 2 below.

Figure 2

High-level cognitive demand tasks allow students to engage in active inquiry and exploration, or encourage students to use procedures in ways that are meaningfully connected with concepts or understand- ing. Low-level tasks, on the other hand, are those that involve simple recall of facts, formulas, and rules or using proce- dures that are focused merely on produc- ing a correct answer rather than develop- ing mathematical understanding. So when choosing tasks, effective teachers choose mathematically productive tasks that:

• Work with a wide range of students

• Have multiple entry points

• Have different methods/pathways for solving, and sometimes have multiple solutions

Effective teachers also know the relevance,

6  Effective Practices for Learning Mathmatics

importance, and value of interacting with students while they engage in learning ex- periences and solving problems. They not only encourage students to interact with each other through – asking and answer- ing questions, but they also assist those who need support in their problem solving ef- forts.

Tools

When a teacher chooses a range of math- ematically productive tasks, they are also providing opportunities for students to use a wide range of tools. Effective teachers know that using a variety of mathemati- cal tools shapes the way that students think and helps build conceptual understanding of the hows and whys of mathematics (Fu-

son et al. 1992). So they provide a variety of tools and resources ranging from – pen- cil and paper, graphs, number lines, rul- ers, diagrams, graphing calculators, digital tools, and other appropriate manipulatives for students to use as they hone their ability to select and use appropriate tools to solve problems.

Virtual versions of concrete manipulatives typically used in mathematics education are available at no cost online. Many avail- able virtual manipulatives are paired with structured activities or suggestions to aid implementation in the classroom. Sites that include popular interactive online ma- terials for math education include the likes of http://nlvm.usu.edu/ and http://www. shodor.org/.

7  Effective Practices for Learning Mathmatics

Perfecting Practice The art and science of teaching go hand-in-hand with the art of being informed by both the science of teaching and an individual’s personal experiences over time. The art of teaching cannot be perfected without considering the science of teaching, and pressing the science of teaching without considering the art of teaching is unacceptable in the teaching and learning process.

There are a number of readily available free online resources and tools that are based upon both the art and science of teaching that effective teachers have and use within their teaching reper- toire to provide learning opportuni- ties for students. The following list in- cludes a range of lessons, videos, and interactive tools that provide multiple opportunities for students to learn mathematics and develop their prob- lem solving skills.

http://betterlesson.com—Over 10,000 complete Common Core–aligned lessons from 130 Master Teachers.

http://nrich.maths.org/—FREE enrichment materials (Problems, Articles and Games) for all ages that really can help to inspire and engage learners and embed RICH tasks into everyday practice.

http://www.insidemathematics.org/tools-for-educators—A professional resource for educators passionate about improving students’ mathematics learning and performance.

http://illuminations.nctm.org/—Quality standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers.

http://achievethecore.org/—Common Core-aligned exemplar lessons and units with supporting resources.

We hope you find this guide a useful resource for identifying and implementing effective mathematical practices that lead to increased opportunities for students to learn math- ematics. Your feedback is valuable to us, so we invite you to share your questions, com- ments, and suggestions here.  Thanks!

8  Effective Practices for Learning Mathmatics

Have you heard about LearnBop’s new Paid Pilot Program? Chances are we have one in your area.

Click here to learn more!