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Chapter 2Exploring What It Meansto Know and Do Mathematics

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What Does It Meanto Do Mathematics?

Doing mathematics means generatingstrategies for solving problems, applyingthose approaches, seeing if they lead tosolutions, and checking to see if youranswers make sense.

Doing Mathematics

Science of Pattern& Order

 Science is aprocess of figuring out or making sense

 Math should make sense!

ClassroomEnvironment

 Language of Doing

 Setting for Doing

Ideas

Choice of methods

Mistakes

Logic and structure

An Invitation to Do Mathematics

In order to create an appropriateenvironment, you must experience doingmath for yourself!

Verbs of Doing Mathematics

explore explain represent

investigate predict formulate

conjecture develop discover

solve describe construct

justify verify use

Brainstorm some examples of

doing mathematics using the

verbs on the list above.

Constantly ask yourself, “Are the students inyour classroom doing mathematics?”

Four Features of a Productive Classroom Culture

1.Ideas are the currency of the classroom. Ideas, expressed by anyparticipant, have the potential to contribute to everyone’s learning andconsequently warrant respect and response.

2. Students have autonomy with respect to the methods used to solve problems.Students must respect the need for everyone to understand their ownmethods and must recognize that there are often a variety of methodsthat will lead to a solution.

3. The classroom culture exhibits an appreciation for mistakes as opportunities tolearn. Mistakes afford opportunities to examine errors in reasoning, andthereby raise everyone’s level of analysis. Mistakes are not to becovered up; they are to be used constructively.

4. The authority for reasonability and correctness lies in the logic and structure ofthe subject, rather than in the social status of the participants. Thepersuasiveness of an explanation or the correctness of a solutiondepends on the mathematical sense it makes, not on the popularity ofthe presenter.

Let’s Do Some Mathematics!

Start and Jump Numbers: Searching for Patterns

You will need to make a list of numbers that begin with a“start number” and increase by a fixed amount we will callthe “jump number.” First try 3 as the start number and 5 asthe jump number. Write the start number at the top of yourlist, then 8, 13, and so on, “jumping” by 5 each time untilyour list extends to about 130.

Examine this list of numbers and find as many patterns asyou can. Share your ideas with the group, and writedown every pattern you agree really is a pattern.

Two Machines, One Job

Ron’s Recycle Shop started when Ron bought a usedpaper-shredding machine. Business was good, so Ronbought a new shredding machine. The old machinecould shred a truckload of paper in 4 hours.

The new machine could shred the same truckload in

only 2 hours. How long will it take to shred a

truckload of paper if Ron runs both shredders at the

same time?

One Up, One Down

For Grades 1–3.

When you add 7 to itself, you get 14. When you make the first number 1more and the second number 1 less, you get the same answer:

↑ ↓ 7 + 7 = 14 has the same answer as 8 + 6 = 14 It works for 5 + 5 too:

↑ ↓ 5 + 5 = 10 has the same answer as 6 + 4 = 10

What can you find out about this?

For Grades 4–8. What happens when you change addition tomultiplication in this exploration?

↑ ↓ 7 × 7 = 49 has an answer that is one more than 8 × 6 = 48

It works for 5 × 5 too:

↑ ↓ 5 × 5 = 25 has an answer that is one more than 6 × 4 = 24

What can you find out about this situation?

Can this pattern be extended to other situations?

The Best Chance of Purple

Three students are spinning to “get purple”with two spinners, either by spinning first redand then blue or first blue and then red. Theymay choose to spin each spinner once or oneof the spinners twice.

Mary chooses to spin twice on spinner A; Johnchooses to spin twice on spinner B; and Susanchooses to spin first on spinner A and then onspinner B. Who has the best chance of gettinga red and a blue?

Where Are the Answers?

Providing solutions only encouragesstudents to seek the “right” answer, theteacher’s answer

What Does It Meanto Learn Mathematics?

How does doing mathematics relate tostudent learning?

The answer:Look to current theory and research

Constructivist TheoryJean Piaget

Creation of Knowledge

Assimilation and Accommodation

Reflective Thought

Sociocultural TheoryLev Vygotsky

Learning depends on:

Learner (working in his or her ZPD)

Social interactions

Culture

Implications of ConstructivistTheory and Sociocultural Theory

There is no need to choose betweentheories

These theories are not teaching strategiesbut should instead guide teaching

Implications

Build newknowledge fromprior knowledge

Provideopportunities to talkabout mathematics

Build inopportunities forreflective thought

Encourage multipleapproaches

Treat errors asopportunities forlearning

Scaffold newcontent

Honor diversity

What Does It Meanto Understand Mathematics?

Understanding is the measure of quality and quantityof connections between new ideas and existing ideas

Knowing  Understanding (students may knowsomething about fractions, for example, but notunderstand them)

Richard Skemp named the ends of the continuum ofunderstanding

What to do and why

Relationalunderstanding

Just doing it

Instrumentalunderstanding

Mathematics Proficiency

Conceptual and procedural understanding

Five strands of mathematical proficiency

— conceptual understanding

— procedural fluency

— strategic competence

— productive disposition

— adaptive reasoning

Implications for Teaching

The need to replace the question “Doesthe student know it?” with the question“How does the student understand it?”

Early number concepts

Computation

Benefits of a RelationalUnderstanding

Effective new concepts and procedures

Less to remember

Increased retention and recall

Enhanced problem-solving abilities

Improved attitudes and beliefs

Multiple Representations toSupport Relational Understanding

Examples ofmodels

Technology-based models

Models andmanipulatives

Ineffective use ofmodels andmanipulatives

Connecting the Dots

This is the foundation for the content ofmathematics

Construction of individual understandings

Connections between theory and practice