Relational Thinking/Understanding

MMLA Fall 2006 TrainingElementary Breakout Session

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Richard R. Skemp1919 - 1995

British Mathematics Educator “Relational Understanding and Instrumental

Understanding”printed in Mathematics Teaching, journal of the Association of Teachers of Mathematics, Great Britain,

December 1976 reprinted in Arithmetic Teacher, November 1978

reprinted in Mathematics Teaching in the Middle School, NCTM, September, 2006

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Definitions Instrumental Understanding –

Rotely learned knowledge (rules without reasons)

Relational Understanding – Understanding that is associated with many other existing ideas in a meaningful network of concepts and procedures (knowing what to do and why)

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Example Two teams agree to play football –

one from the US and one from Great Britain. They arrive to play, but find that when on the field it is total chaos. Why?

A teacher and a student can have the same difficulty – using two different approaches to teaching and understanding.

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Music Lesson

Instrumental UnderstandingLesson taught by using paper and pencil – writing the notation only

Relational UnderstandingLesson taught with sounds connected to the notes on the page

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2 4

538=

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Instrumental Understanding: “multiply the two numerators together to

make the numerator of the product and multiply the two denominators to make the product’s denominator"

Relational Understanding:45

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What Mathematical concepts are connected in this problem?

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(a + b)2 = a2 +2ab + b2

A student with an instrumental understanding of this identity would justify it by citing the FOIL method.

A student with a relational understanding might justify it using the illustration below.

a

b

a b

a2

b2ab

ab What Mathematical concepts are connected in this problem?

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Teaching with:

Instrumental Understanding methods:Students often have to learn an increasing number of fixed plans, plans tell what to do at each step of the way; students are dependent on guidance for learning “new” ways

Relational Understanding methods:Students must build up a conceptual structure so that they can produce an unlimited number of plans – schema never complete however, because students are always aware that there are other possibilities

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Relational vs. Instrumental Understanding: PROS

Instrumental

1. Within its own context it is easier to understand (i.e. invert and multiply).

2. The rewards are more immediate and more apparent (i.e. a page of correct answers).

3. Often the correct answer comes more quickly and reliably.

Relational

1. More adaptable to new tasks.

2. Easier to remember – less relearning to do – remember as parts of a connected whole.

3. Can be a goal in itself (reduces need to provide external rewards and punishments).

4. Organic in quality (i.e. act as agent of their its growth).

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Relational vs. Instrumental Understanding: CONS

Instrumental

1. Easier to forget or remember incorrectly.

2. Harder to apply a procedure learned in one context to another.

3. Needs to be reviewed frequently.

4. Necessitates memorizing which method works or doesn’t work

5. Leads to a perception of mathematics as a set of meaningless rules for the manipulation of symbols. The rules bear little relationship to each other or to real life.

6. Results in frustration and anxiety for a large proportion of students.

7. Creates innumeracy.

Relational

1. Harder to learn

2. Often takes more time

3. May appear to be too difficult

4. Skill may be needed in another subject before relational understanding can be developed fully

5. Hard to assess mental processes students have used

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Area of a Triangle Instrumental Understanding

learn the formula A = ½bh

Relational Understanding1. establish the formula for the area of a rectangle2. relate the triangle to the rectangle and develop the area formula3. continue and relate the area of the parallelogram and trapezoid

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Thinking Mathematically

Use of T or F and open statements: 12 – 9 = 3 34 – 19 = 15 27 + 48 – 48 = 27 345 + 568 - 568 = 353 48 + 63 – 62 = 49 674 + 56 – 59 = 671 12 + 9 = 10 + 8 + c

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Activity Rank the following problems from easiest to most difficult. Discuss and Report out.

a. 73 + 56 = 71 + db. 92 - 57 = g – 56c. 68 + b = 57 + 69d. 56 – 23 = f – 25e. 96 + 67 = 67 + pf. 87 + 45 = y + 46g. 74 – 37 = 75 - q

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Activity Rank the following problems from easiest to most difficult. Discuss and Report out.

a. 73 + 56 = 71 + 59 – db. 92 – 57 = 94 – 56 + gc. 68 + 58 = 57 + 69 – bd. 56 – 23 = 59 – 25 – se. 96 + 67 = 67 + 93 + pf. 87 + 45 = 86 + 46 + tg. 74 – 37 = 71 – 39 + q

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Problems around Place Value

56 = 50 + 6 246 = 24 x 10 + 6 240 = 24 x c 47 + 38 = 40 + 30 + 7 + 8 63 – 28 = 60 – 20 + 3 – 8 0.78 = .078 1.95 = 1.9500

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Problems in your classroom

Write a series of problems that you might use with your students to encourage them to begin to look for relations.

Discuss at your table. Select one series from your table to

share with the whole group.

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Student Work

Examine each students’ work. Discuss. What does each example suggest

about the student’s understanding of mathematics?

What specific big ideas of mathematics do you think each example illustrates?

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Kevin

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Ahren

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Lesley

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Eve

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GLCE Clarification Document

While looking through the document keep the idea of Equivalency and Relational Understanding in mind. Locate examples of each