Creating a Mathematics Environment

Over at Math Hub, Jennifer Chintala, in Top 10 Ways to Strengthen Classroom Math Instruction,offers some strong tips for math educators.   Building on Chintala’s piece, I would go a step farther and add a short addendum to her first suggestion: Create a mathematics environment

An important strategy in strengthening mathematics instruction is establishing an environment where students are comfortable asking questions, undaunted by problem-solving activities, and secure in the belief that even some mistakes may have some redeeming instructional value.  A mathematics classroom should be non-threatening to all students. Their interaction and discussion about problem solving methods may often reveal alternative creative processes, processes and methods that may yield correct conclusions.  Alternative methods of solving problems offer opportunities for large-group or small-group discussions as to why certain methods work and others don’t.  Such dialogue is important for students to develop their understanding of numeracy, patterns, logic, and spatial reasoning.

In the same way that  adults learn through mistakes, students discover important facts through trial and error and discussion.  A mathematics educator that provides students with time for reflection on arithmetic processes and patterns in logic and geometry will create a mathematics classroom environment that improves mathematics vocabulary and stimulates interest in real problem-solving activities.  That’s what makes learning mathematics fun!

From Novice to Expert

Tony Schwartz over at Harvard Business Review gets it exactly right: the key to excellence is practice.  Specifically, deliberate practice.  Building on the work of Anders Ericsson, Schwartz argues that whatever role our genetic inheritance plays, it is the type of effort we put into an endeavor that determines how good we become:

Like everyone who studies performance, I’m indebted to the extraordinary Anders Ericsson, arguably the world’s leading researcher into high performance. For more than two decades, Ericsson has been making the case that it’s not inherited talent which determines how good we become at something, but rather how hard we’re willing to work — something he calls “deliberate practice.” Numerous researchers now agree that 10,000 hours of such practice as the minimum necessary to achieve expertise in any complex domain.

Ericsson’s research on human performance and what it takes to move from novice to expert has informed our own research here at MetaMetrics and has recently been popularized by writers like Geoffrey Colvin and Malcom Gladwell.  As Malbert Smith has written in ‘Education Reform: Making this the ‘Best of Times’: (more…)

Targeting for Success in Math

Over at Scholastic’s Math Hub, Carolyn Kaemmer has an interesting interview with Harvard psychologist,  Dr. Jon Star on conceptual understanding in math.  In the interview, Dr. Star tackles the controversial distinction between ‘knowing versus doing’ in mathematics, e.g. does the math student really understand what they’re doing or just following a process.  Star argues that a student’s math performance can, in fact, be determined through various modes of assessment, including multiple choice tests, though he qualifies his argument with the requirement that the assessment prompts need to be carefully designed.

The Quantile Framework is directly relevant here.  Unfortunately, assessments often fail to inform classroom teachers of their students needs or of their progress in mathematical development.  When summative assessments offer a Quantile measure for the test-taker, however, the teacher has information that is instructionally actionable.  The Quantile Framework for Mathematics helps the educator to determine the material that various students in the classroom are ready to learn when the Quantile measure of the student closely matches the Quantile measure of the skill or concept. When such a match occurs, students will perform more successfully and develop more confidence in their mathematics ability. (more…)

Differentiating Math Instruction

Scholastic’s Math Hub has a great post on using the Quantile Framework to differentiate math instruction.  Utilizing the Quantile Framework allows teachers to identify prerequisite math skills that students may need in order to be successful with a new math skill or concept and to then target students at the appropriate level:

The Quantile Framework measures student mathematical ability, the curriculum and teaching materials on the same developmental scale. Quantile measures help teachers determine which skills and concepts a student is ready to learn and those that will require more instruction. Educators can then use this information to better focus instruction to incorporate the necessary prerequisite skills that may be missing and accurately forecast understanding.

Math Hub does a nice job of summarizing the purpose of the benefits of the Quantile Framework.  Take a look.

Work and Wrestle: Targeting Students with the Quantile Framework for Mathematics

Hat tip to Marshall Memo for pointing to the latest edition of Better for James Hiebert and Douglas Grouws’ article, ‘Which Instructional Methods Are Most Effective For Math?’ (subscription required).  Hiebert and Grouws argue  that when it comes to teaching math skills and concepts, there are a number of essential elements in ensuring students gain conceptual understanding of the skills and concepts being taught.

First, teachers should continually draw relationships between what is being taught and past material covered.  Here’s the Marshall Memo summarizing what Hiebert and Grouws label the ‘Work and talk’ approach:

  • Examining relationships among facts, procedures and ideas within a lesson and across lessons.
  • Exploring reasons why procedures work as they do.
  • Solving problems using different procedures and then looking at similarities and differences between them. (more…)
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