Educators Summer SymposiumJune 8, 2011

Sue McAdaragh

"To teach means scarcely anything more than to show how things differ from one another in their different purposes, forms, and origins...therefore, he who differentiates well teaches well."

- John Amos Comenius (17th Century Educator)

A research-based way of teaching mathematics that embraces and/or encompasses the following concepts:◦ Problem solving in meaningful context with

flexible solution strategies◦ Building mathematical understanding through

questioning◦ Integration of mathematical concepts

*CGI is also a Professional Development program.

The development of students’ mathematical thinking;

Instruction that influences that development;

Teachers’ knowledge and beliefs that influence their instructional practices;

The way that teachers’ knowledge, beliefs and practices are influenced by their understanding of students’ mathematical thinking;

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.

A research-based way of teaching mathematics that embraces and/or encompasses the following concepts:◦ Problem solving in

meaningful context with flexible solution strategies

◦ Building mathematical understanding through questioning

◦ Integration of mathematical concepts

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.5. Use appropriate tools

strategically.6. Attend to precision7. Look for and make use of

structure.8. Look for and express

regularity in repeated reasoning

http://commoncoretools.wordpress.com/about/

Bill McCallum, University of Arizona

CGI Problem TypesJoin Results

UnknownChange Unknown Start Unknown

Separate Results Unknown

Change Unknown Start Unknown

Part-Part-Whole

Whole Unknown Part Unknown

Compare Difference Unknown

Compare Quantity Unknown

Referent Set Unknown

Understanding Procedures

"Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to . . . deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices."

…understand relationships, properties, and procedures

…are able to explain and justify one’s actions on numbers

…are able to use strategies appropriately and efficiently

Marilyn Burns and her colleagues have developed a tool to assess students’ numerical understanding and skills.

These problems and possible solution strategies were shared at NCTM in April of this year.

The following examples for designed for fifth graders.

Be thinking about how this tool could be modified for your grade level.

Which is greater-- 3/5 or 1/2?

How did you decide?

Which is greater—3/5 or 1/2? Converted to common denominators Converted to decimals or percents Explained that 3 is more than half of 5 Described a visual or physical model Gave other reasonable explanation [record] Guessed, did not explain, or gave faulty

explanation

Don’t figure out the exact answer. Without paper and pencil, decide which of

these choices is closest to the answer:

1/2, 1, 2, or 8

Why do you think that?

Rounded one or both fractions to 1, then added.

Compared to ½ (e.g. both are greater than ½ so the answer is greater than 1)

Analyzed choices and chose one that seemed most reasonable.

Gave another reasonable explanation (record).

Guessed, did not explain or gave faulty explanation

Without using pencil or paper, decide if the answer to this problem is greater than one or less than one.

Why did you think that?

Converted to common denominators Explained that 2/3 is greater than ½ so the

answer must be greater than 1 Converted to decimals or percents Described a visual or physical model Gave other reasonable explanation Guess, did not explain, or gave faulty

explanation

There are 295 students in the school. School buses hold 25 students. How many school buses are needed to fit all the students?

How did you figure out the answer?

5 10 20 30

Don’t figure out the exact answer to this problem. Without using paper and pencil, decide which of the choices is closest to the answer – 5, 10, 20, or 30.

Used standard algorithm Rounded then multiplied Analyzed choices and chose one that

seemed most reasonable Gave other reasonable explanation Guessed, did not explain, or gave faulty

explanation

Which is greater – 3/5 or 1/2? Is 5/6 + 12/13 closer to 1/2, 1, 2, or 8? Is 1/2 + 2/3 greater than or less than 1? 25 students can fit on one bus; how

many buses needed for 295 students? Is 3.9 x 4.75 closest to 5, 10, 20, or 30?

A research-based way of teaching mathematics that embraces and/or encompasses the following concepts:◦ Problem solving in meaningful context with

flexible solution strategies◦ Building mathematical understanding through

questioning◦ Integration of mathematical concepts

*

Thanks for attending the Symposium!

susan.mcadaragh@k12.sd.us

http://2011ess.sfinstructionalresources.wikispaces.net/Sessions