Tag: NCTM

Posted in Algebra, Mathematics education

Develop your ‘Variable Sense’

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Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) x ? a = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

[iframe https://math4teaching.com/wp-content/uploads/2012/10/meaning_of_variable.html 600 360]

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.

Posted in Mathematics education

NCTM Process Standards vs CCSS Mathematical Practices

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The NCTM process standards, Adding it Up mathematical proficiency strands, and Common Core State Standards for mathematical practices are all saying the same thing but why do I get the feeling that the Mathematical Practices Standards is out to get the math teachers.

The NCTM’s process standards of problem solving, reasoning and proof, communication, representation, and connections describe for me the nature of mathematics. They are not easy to understand especially when you think that school mathematics is about stuffing students with knowledge of content of mathematics. But, over time you find yourselves slowly shifting towards structuring your teaching in a way that students will understand and appreciate the nature of mathematics.

The five strands of proficiency were also a great help to me as a teacher/ teacher-trainer because it gave me the vocabulary to describe what is important to focus on in teaching mathematics.

With the Mathematical Practices Standards I had this picture of myself in the classroom with a checklist of the standards in one hand and a lens on the other looking for evidence of proficiency. The NCTM and Adding it Up standards actually said more about math. The ones in Common Core are saying more about what students should attain. I wonder which will encourage ‘teaching to the test’. The day teachers start to ‘teach to the test’ is the beginning of the end of any education reform.

NCTM Process Standards

Five Strands of Mathematical Proficiency

CCSS Mathematical Practices

Problem Solving

  1. Build new mathematical knowledge through open-ended questions and more-extended exploration;
  2. Allow students to recognize and choose a variety of appropriate strategies to solve problems;
  3. Allow students to reflect on their own and other strategies for solving problems.

Reasoning and Proof

  1. Recognize and create conjectures based on patterns they observe;
  2. Investigate math conjectures and prove that in all cases they are true or that one counterexample shows that it is not true;
  3. Explain and justify their solutions.

Communication:

  1. Organize and consolidate their mathematical thinking in written and verbal communication;
  2. Communicate their mathematical thinking clearly to peers, teachers, and others;
  3. Use mathematical vocabulary to express mathematical ideas precisely.

Connections

  1. Understand that mathematical ideas are interconnected and that they build and support each other;
  2. Recognize and apply connections to other contents;
  3. Solve real world problems with mathematical connections.

Representation

  1. Emphasize a variety of mathematical representations including written descriptions, diagrams, equations, graphs, pictures, and tables;
  2. Select, apply, and translate among mathematical representations to solve problems;
  3. Use mathematics to model real-life problem situations.

Conceptual Understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students]
to learn new ideas by connecting those ideas to what they already know.”

Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately.

Strategic competence is the ability to formulate, represent, and solve mathematical problems.

Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

Productive disposition is the
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Mathematically proficient students …

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

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