Puzzles - Try These

Relevance to teaching and learning mathematics?

Task: There is always a letter on one side and a number on the other. You must verify whether or not the following rule is true:If there is a vowel on one side, then there must be an even number on the other side.

What is the minimum number of cards you must turn over to test the truth of this rule?

A 3X2

The answer is that you must turn over the A card

and the 3 card. Most people choose the A card, but most fail to select the 3. You must choose the 3 because if there is a vowel on the other side, then the rule has been violated.

Something like 20 percent of college undergraduates get this problem right. The percentage goes up very little even when subjects have just completed a one-semester course in logic that includes learning the logical form Modus Tollens, on which this problem is based (Cheng et al., 1986).

You are a border official. Each card represents a traveler. One side lists whether the traveler is

entering the country or just in transit. The other side shows exactly which vaccinations the person has received.

4 cards: Entering, Transit, Cholera-Mumps-Typhoid, Flu-Mumps

Task: You must make sure any person who is entering is vaccinated against cholera.

About which travelers do you need more info? That is, which travelers must you check?

Many more get this correct.(Even if given separately and independently

of the first puzzle.)

Why?

You need to check the "entering" person (to ensure that he has the cholera vaccination), and you need to check the "flu mumps" person (to ensure that he is in transit).

Compare the two puzzles: How are they different? How are they the same?

Same underlying mathematical structure:A --> BCard 1: ACard 4: not B

Vowel --> Even NumberCard 1: vowel; Card 4: not even

Entering --> Cholera VaccinationCard 1: entering; Card 4: not cholera

Application to Math Teaching“Findings from the field [Cognitive Science]

that are strong and clear enough to merit classroom application.”

Willingham, Daniel T. How We Learn: Ask the Cognitive Scientist. American Educator, Winter, 2002.

Finding 1“The mind much prefers that new ideas be

framed in concrete rather than abstract terms.”

READ pg. 21 in PSSMStudents can learn mathematics with

understanding.What is the teacher’s role in understanding

mathematics?

Finding 2

Rote Knowledge, Inflexible knowledge, and Deep Structure

Which is bigger: 100 or .001 ?

A real answer from a college student:.001

What knowledge does the student have?

Rote:An informal definition of rote knowledge might be

"memorizing form in the absence of meaning."

Example:

Q: What is the equator?

A: A managerie lion running around the Earth through Africa.

Anguished English

Inflexible Knowledge:• Deeper than rote knowledge, but at the

same time, clearly the student has not completely mastered the concept.

• Understanding is somehow tied to the surface features.

• Meaningful, yet narrow.• The student does not yet have flexibility.

(Knowledge is flexible when it can be accessed out of the context in which it was learned and applied in new contexts.)

Deep Structure Knowledge:

• Deeper than inflexible knowledge• Transcends specific examples• Knowledge is flexible -- it can be

accessed out of the context in which it was learned and applied in new contexts

• Knowledge is no longer organized around surface forms, but rather is organized around deep structure

43

-1726

3

What is 43 – 17?

What knowledge does this student have?

Finding 3 How To Develop Deep Structure Knowledge

• Direct instruction of deep structure doesn’t work.

• Use many examples, from many contexts.• Work with the knowledge, to increase the

store of related knowledge.• Practice• Don’t despair of inflexible knowledge, and

don’t confuse it with rote knowledge

How Does the Use of Curriculum Materials That Embody Teaching Mathematics through Problem Solving Affect Students’ Learning of Mathematics?

Students exhibit greater conceptual understanding.

Students perform at higher levels with respect to problem solving.

(as compared to students in more traditional curricula)

Effects of Teaching Mathematics through Problem Solving on Students’ Learning of Mathematics:

Gains in conceptual understanding and problem solving did not come at the expense of those aspects of mathematics measured on standardized tests.

Students do better on assessment tasks that cover topics and approaches about which they have an opportunity to learn.

Effects of Teaching Mathematics through Problem Solving on Students’ Learning of Mathematics: Students are more likely to have positive

and broad attitudes about math. Traditional: Students come to view math as

a meaningless collection of rules to be memorized.

PS approach: Students regard math more as a “doing discipline” and “sense-making,” a subject in which they use their own thoughts.

Effects of Teaching Mathematics through Problem Solving on Students’ Learning of Mathematics:

Students do not just gain more or less knowledge in these two approaches (PS or traditional), but they acquire knowledge, beliefs, and understandings that differ in important ways.

Teacher implementation decisions are essential to the success of any curriculum.

Problem-Based Instructional Tasks

Help students develop a deep understanding of important mathematics

Accessible yet challenging to all students Encourage student engagement and

communication Can be solved in several ways Encourage the use of connected multiple

representations Encourage appropriate use of intellectual,

physical, and technological tools