Tuesday, September 16, 2014

It's not about politics!

On progressive ideals and reform math...

Alice Crary and W. Stephen Wilson. The Faulty Logic of the ‘Math Wars’. New York Times, New York, NY, June 16 2013.

"it would be naïve to assume that we can somehow promote original thinking in specific areas simply by calling for subject-related creative reasoning"

"It is easy to see why the mantle of progressivism is often taken to belong to advocates of reform math. But it doesn’t follow that this take on the math wars is correct. We could make a powerful case for putting the progressivist shoe on the other foot if we could show that reformists are wrong to deny that algorithm-based calculation involves an important kind of thinking."


W. S. Wilson. SBAC Math Specifications Don’t Add Up. 
Flypaper, Thomas B. Fordham Institute, September 19, 2011.
It's [23] on the Professor's website here.

The conceptualization of mathematical understanding on which SBAC will base its assessments is deeply flawed. The consortium focuses on the Mathematical Practices of the Common Core State Standards for Mathematics (CCSS-M) at the expense of content, and they outline plans to assess communication skills that have nothing to do with mathematical understanding.

Sunday, September 14, 2014

"Adaptive Reasoning" in Mathematics Education

Adding It Up page 129-131

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions. In mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning. One uses it to navigate through the many facts, procedures, concepts, and solution methods and to see that they all fit together in some way, that they make sense. In mathematics, deductive reasoning is used to settle disputes and disagreements. Answers are right because they follow from some agreed-upon assumptions through series of logical steps. Students who disagree about a mathematical answer need not rely on checking with the teacher, collecting opinions from their classmates, or gathering data from outside the classroom. In principle, they need only check that their reasoning is valid.

Research suggests that students are able to display reasoning ability when three conditions are met:  They have a sufficient knowledge base, the task is understandable and motivating, and the context is familiar and comfortable.37

Principles of Effective Instruction and the 3rd Common Core Standard for Mathematical Practice

This list is adapted from Principles of Instruction: 
Researched-Based Strategies That All Teachers Should Know

1. Begin a lesson with a short review of previous learning:  Daily review can strengthen previous learning and can lead to fluent recall. 

The most effective teachers ensured that students efficiently acquired, rehearsed, and connected knowledge.  Many went on to hands-on activities, but always after, not before, the basic material was learned.

2.  Limit the amount of material students receive at one time.  Present new material in small steps with student practice after each step, and assist students as they practice this material.

3.  Give clear and detailed instructions and explanations.  Provide many examples.

4.  Think aloud and model steps.  Providing students with models and worked examples can help them learn to solve problems faster.

Many of the skills taught in classrooms can be conveyed by providing prompts, modeling use of the prompt, and then guiding students as they develop independence.

5.  Use more time to provide explanations.

6.  Ask a large number of questions and check the responses of all students:  Questions help students practice new information and connect new material to their prior learning.

7.  Provide a high level of active practice for all students.

8.  Guide students as they begin practice.  Successful teachers spend more time guiding students’ practice of new material.

9.  Check for student understanding at each point to help students learn the material with fewer errors.

The most successful teachers spent more time in guided practice, more time asking questions, more time checking for understanding, and more time correcting errors.

10.  Ask students to explain what they have learned.

11.  Obtain a high success rate:  It is important for students to achieve a high success rate during classroom instruction.

12.  Require and monitor independent practice:  Students need extensive, successful, independent practice in order to develop well-connected and automatic knowledge and skills.

13.  Provide systematic feedback and corrections, reteach material when necessary.
Construction viable arguments and critique the reasoning of others. 
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.  They make conjectures and build a logical progression of statements to explore the truth of their conjectures.  They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.  They justify their conclusions, communicate them to others, and respond to the arguments of others.  They reason inductively about data, making plausible arguments that take into account the context from which the data arose.  Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is.  Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.  Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.  Later, students learn to determine domains to which an argument applies.  Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.