problem solving Archives - Page 2 of 7 - Mathematics for Teaching

Regular Polygons Problems

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One of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.

Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.

Problem 1

The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.

Problem 2

What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?

Problem 3

What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.

Problem 4

What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.

Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.

You may want to view another similar lesson on quadrilaterals.

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

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

Reasoning and Proof

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

Communication:

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

Connections

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

Representation

Emphasize a variety of mathematical representations including written descriptions, diagrams, equations, graphs, pictures, and tables;
Select, apply, and translate among mathematical representations to solve problems;
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.

Image from 123RF

Mathematics education

Bloom’s Taxonomy and iPad Apps

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The original Bloom’s taxonomy include Knowledge, Comprehension, Analysis, Synthesis, and Evaluation. I was introduced to this when I was in college and I must admit it was not of much help to me in planning my math lessons. I just couldn’t fit it. The pyramid image was not of help at all and I think even created the now much ingrained deductive method of teaching. I think teachers must have unconsciously looked at it as a food pyramid so they give a dose of those of knowledge-acquisition activities first before providing activities that will engage students in higher-level processes

Lorin Anderson, a former student of Bloom, revisited the cognitive domain in the learning taxonomy in the mid-nineties and made some important changes: changing the names in the six categories from noun to verb forms and slightly rearranging them. The new taxonomy reflects a more active form of thinking of Creating, Evaluating, Analyzing, Understanding, and Remembering. I also like the inverted pyramid as long as it is not viewed like there is a strict hierarchy of the categories. In fact in my own experience I just make sure that all these are covered in a lesson as much as possible. The way to do this is to teach mathematics through problem solving or engage students in mathematical investigations. Still, the best framework will still be one tailored to mathematics. For me its my list of Mathematical Habits of Mind.

In searching for Bloom’s taxonomy I came across the image below – Bloom’s taxonomy for iPad. It’s a collection of iPad apps classified according to Bloom’s taxonomy. I found it cute so I’m including it here. This will come in handy once I have my own iPad and start creating math lesson for this device.