Doing Mathematics (Kuliah)

7/28/2019 Doing Mathematics (Kuliah)

1/27

MTE 3107


2/27

"Doing math" involves all kinds of mental capacities:numerical reasoning, quantitative reasoning, linguisticreasoning, symbolic reasoning, spatial reasoning, logicalreasoning, diagrammatic reasoning, reasoning about

causality, the ability to handle abstractions, and maybesome others.

Definition: Reasoning present participle ofreason(Verb)

1. Think, understand, and form judgments by a process oflogic: "humans do not reason entirely from facts"; "thereasoning behind the review".

2. Find an answer to a problem by considering variouspossible solutions.


3/27

1. Number sense. This includes, for instance, the abilityto recognize the difference between one object, acollection of two objects, and a collection of threeobjects, and to recognize that a collection of threeobjects has more members than a collection of two.Number sense is not something we learn. Childpsychologists have demonstrated conclusively thatwe are all born with number sense.

2. Numerical ability. This involves counting andunderstanding numbers as abstract entities.3. Spatial-reasoning ability. This includes the ability to

recognize shapes and to judge distances, both ofwhich have obvious survival value for many animals.


4/27

4. A sense of cause and effect. Much of mathematicsdepends on "if this, then that" reasoning, an abstractform of thinking about causes and their effects.

5. The ability to construct and follow a causal chain offacts or events. A mathematical proof is a highlyabstract version of a causal chain of facts.

6. Algorithmic ability. This is an abstract version of the

fifth ability on this list.7. The ability to handle abstraction. Humansdeveloped the capacity to think about abstractentities along with our acquisition of language.


5/27

8. Logical-reasoning ability. The ability toconstruct and follow a step-by-step logicalargument is another abstract version of item 5.

9. Relational-reasoning ability. This involvesrecognizing how things (and people) are relatedto each other, and being able to reason aboutthose relationships. Much of mathematics dealswith relationships among abstract objects.

All nine capacities are basic mental attributesimportant to our daily lives.


6/27

Materials/ manipulative Discourse

Making conjecture


7/27

children should be actively involved in doingmathematics.

Educational research indicates that the mostvaluable learning occurs when studentsactively construct their own mathematicalunderstanding.


8/27

One way to facilitate this is to provideopportunities for children to explore,

develop, test, discuss, and apply ideas.Extensive and thoughtful use of physical

materials, particularly in the primary grades,is conducive to the concrete kinds of learningthat lay a satisfactory foundation for thedevelopment of this mathematicalunderstanding.


9/27

Although the value of manipulatives has beenrecognized for many years, some teachers

and parents have been reluctant to includethem in their lessons.They may lack confidence in their ability to use

them successfully or underestimate theirvalue to real mathematical understanding.


10/27

Manipulatives such as buttons, dried beans,cubes, and animal counters can help youngpeople develop a rich understanding of

number which forms the basis for counting,arithmetic, and real-world applications. They can be used for comparing as well as

sorting and classifying. For example, childrencan use any of these objects to make, count, andorder collections of objects. (E.g. Give me fourbuttons. How many beans are there? Put thesegroups in order from smallest to largest.)


11/27

Sets of attribute logic blocks with theirdifferent shapes and colors are particularly

effective for classifying, sorting, andordering. (E.g. Find all the pieces that belongin the set labeled small and square. Look atthe three shapes in the loop. How are theyalike? How are they different?) Attributeblocks can also be used to explore size,fractional relationships, and area.


12/27

Base ten blocks, Cuisenaire rods, Unifix cubes, and currency canbe used to help children learn a wide range of number concepts,including place value, addition, subtraction, and multiplication.With the base ten block manipulatives, cubes represent ones, rodsrepresent tens, flats represent hundreds, and blocks representthousands; so a combination of manipulatives can be used torepresent twenty-five or two groups of ten and five ones, etc.(E.g. Can you trade ten cubes for a rod? Two rods and five cubes isthe same as one rod and how many cubes?)

By building number combinations with these materials, childrenare helped to understand the logic of the concept of carrying orborrowing or regrouping as used in the paper-and-pencilcomputations for addition and subtraction.


13/27

The term classroom discourse refers to thelanguage that teachers and students use to

communicate with each other in theclassroom. Talking, or conversation, is the medium

through which most teaching takes place, sothe study of classroom discourse is the studyof the process of face-to-face classroomteaching.


14/27

Students write in their journals about theirmathematical reasoning or processes.

A student states, I see a pattern that I think will always

work, because each number is 3 more than the onebefore it.

A group of students discuss the mathematicalconditions in which an idea will or wont always work.


15/27

The teacher provides instructions to the class about anactivity they are about to engage in.

The teacher provides a counter example to a method

posed by a student. A student asks a question about nonmathematical

procedures related to an assignment, such as when theassignment is due, whether students need to show their

work, etc.


16/27

Mathematical classroom discourse is aboutwhole-class discussions in which students talkabout mathematics in such a way that they

reveal their understanding of concepts. Studentsalso learn to engage in mathematical reasoningand debate.

Discourse involves asking strategic questionsthat elicit from students both how a problemwas solved and why a particular method waschosen. Students learn to critique their own andothers' ideas and seek out efficientmathematical solutions.


17/27

Discourse can be used at any time during aunit of work.

It can: be used to determine what students are thinking

and understanding in order to build bridgesbetween what they already know and what thereis to learn;

offer opportunities to develop agreed-uponmathematical meanings or definitions andexplore conjectures.


18/27

Limitations Students may not arrive at an agreed-upon answer during their

discussion. The teacher has to decide when to step in and providean explanation, when to model, and when to ask pointedquestions that can shape the direction of the discourse. One wayto overcome this is to ask "If someone from the classroom nextdoor said '..' what would you say?"

The teacher needs to be able to anticipate responses and respondspontaneously to students.

The teacher needs to develop a deep knowledge of mathematicsconcepts and principles in order to understand the reasons behindstudents' errors. A teacher needs to have one eye on theunderlying mathematical concepts while the other eye is focusedon the current understandings of the students.

Some students may have difficulty explaining their reasoning.


19/27

The teacher of mathematics should promote classroom discoursein which students-

listen to, respond to, and question the teacher and oneanother;

use a variety of tools to reason, make connections, solveproblems, and communicate;

initiate problems and questions; make conjectures and present solutions; explore examples and counterexamples to investigate a

conjecture; try to convince themselves and one another of the validity of

particular representations, solutions, conjectures, andanswers;

rely on mathematical evidence and argument to determinevalidity.


20/27

Mathematics PhilosophyThe primary goal of the mathematics curriculum is togive students experiences that promote their ability tosolve problems. Students are expected to makeconjectures and conclusions. Students willcommunicate their reasoning in words, both writtenand spoken; with pictures, charts and graphs; andwith manipulatives.

Students will be engaged in discovering mathematics,not just doing problems in a book. They will haveopportunities to explore, investigate, estimate,question, predict, and test their ideas aboutmathematics.


21/27

Students will explore and develop anunderstanding for mathematical concepts.

The students will have opportunities to lookat mathematics in terms of daily life and tosee the connections among the differentmath standards. Students will use availabletechnologies to solve mathematicalproblems.


22/27

MATHEMATICS MISSIONMathematics Program ensures that all students arecompetent mathematicians who solve real worldproblems.

We Believe... 1. Students should have numerous and varied experiences

to help them appreciate the value of mathematics.2. Students need to view themselves as capable of usingtheir growing mathematical power to make sense of new

problem situations in the world around them.3. The development of each students ability to solveproblem is essential for him or her to be a productivecitizen.


23/27

4. Students must have an opportunity to read, writeand discuss ideas in which the use of the language ofmathematics becomes natural.5. Making conjectures, gathering evidence, andbuilding an argument to support mathematicalreasoning is fundamental to doing mathematics.6. A certified teacher must engage all students in dailymathematical instruction for a minimum of 45minutes a day.7. Ongoing staff development in the area ofmathematics is essential.8. The principal will ensure the implementation ofmathematics instruction.


24/27

Definition of ConjectureConjecture is a statement that is believed to be true

but not yet proved.

Examples of Conjecture

The statement "Sum of the measures of the interior

angles in any triangle is 180" is a conjecture.

The statement "If two parallel lines are cut by atransversal, the corresponding angles are

congruent."


25/27

Solved Example on ConjectureSam proposed that '3n + 1 yields a prime number

for any even number, n.' and called it a

conjecture.He explained his conjecture using the numbers 2,4, and 6. He got 7, 13, and 19.

Is his proposition a conjecture?Choices:A. YesB. NoCorrect Answer: B


26/27

Solution:Step 1: A conjecture is a statement that is believed

to be true but not yet proved or disproved.

Step 2: Sam's proposition is not a conjecture,because for n = 8, 3n + 1 gives 25, which is not aprime number.

Related Terms for Conjecture

StatementInductive ReasoningProof


27/27

In mathematics, a conjecture is amathematical statement which has beenproposed as a true statement, but which noone has yet been able to prove or disprove.

Once a conjecture has been proven, itbecomes known as a theorem, and it joins the

realm of known mathematical facts. Untilthat point in time, mathematicians must beextremely careful about their use of aconjecture within logical structures.
http://www.mathdaily.com/lessons/Mathematicshttp://www.mathdaily.com/lessons/Mathematical_proofhttp://www.mathdaily.com/lessons/Theoremhttp://www.mathdaily.com/lessons/Theoremhttp://www.mathdaily.com/lessons/Mathematical_proofhttp://www.mathdaily.com/lessons/Mathematics

Documents

Doing Mathematics (Kuliah)