2. Designing Effective Mathematics Instruction
• various In-Out Machines. The exercise illustrates that under some
circumstances, there may be the possibility of no answer or multiple answers.
Because most of the questions involve multiplication, this particular exercise
would be suitable for third graders. Exercises could be made up to illustrate
the same points for younger children.
3. Exercise A
• An In-Out Machine takes what you put into it (input) and consistently
produces a result (output) through some invisible internal process.
INPUT OUTPUT
4. • Example: When 7 is put in the machine above, 8 comes out the other end. When
4 is put in, 5 comes out. When 2 is put in, 3 comes out. When 13 is put in, 14
comes out. Summarize the information above in the table below. By listing each
output next to its input, it may be easier to figure out what the In-Out Machine is
doingwhat the invisible internal process is.
INPUT OUTPUT
5. Exercise B
• An In-Out Machine takes what you put into it (input) and consistently
produces a result (output) through some invisible Internal process.
• INPUT OUTPUT
6. • Example: When 7 is put in the machine above, 9 comes out the other end. When 4
is put in, 6 comes out. When 0 is put in, 2 comes out. When 13 is put in, 15 comes
out. Summarize the information above in the table below. By listing each output
next to its input, it may be easier to figure out what the In-Out Machine is doing
what the invisible internal process is.
INPUT OUTPUT
8. Exercise D
• In the In-Out Machine below, two inputs are fed into the machine at a time,
Via machine works on the inputs and a single output comes out.
2
7
5
9. Table A below summarizes the input and
output for this machine:
INPUT OUTPUT
(2,5) 7
(6,3) 9
(1,1) 2
(4,4) 8
(5,1) 6
10. Table B below shows the inputs and outputs of
Machine B.
4
1 3
•
12. Exercise E
• Uonsider the inputs and outputs for In-Out Machine A.
INPUT OUTPUT
CAT 3
GARBAGE 7
BY 2
13. Consider the inputs and outputs for In-Out
Machine B.
INPUT OUTPUT
3 0
8 0
5 0
1 0
14. • A. Previous lesson: Counting square units to determine the area of a
rectangle. The method is illustrated with a rectangle 5 units by 4 units:
Exercise F
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
15. • B.Lesson introduction: "Yesterday we learned how to figure out the area of a
rectangle by counting up all the square units that we could fit inside it:' (Review
the procedure units 5 X 4 unit rectangle c -.1 shown in Step A.) "On your
worksheet (the chalkboard), you will find five rectangles (5 X 6, 5 X 9, 2 X 8, 1 0
X 6, and 10 X 9). Is there a way we can make the job of figuring out the area of
all of the rectangles easiei?“
• C. Informal shortcut: Skip counting
• 5
10
15
20
17. • Routinely question children about their work. Regularly ask children how they
arrived at their answers and whether or not their answers are justified. For
example, have them explain their reasoning by using objects to illustrate their
solution method and answer (Peck, Jencks, and Connell, in press). A steady
exchange encourages children to approach mathematics thoughtfully, to check
the reasonableness of their answers, and to depend on themselves to evaluate
their work (Lampert, 1986). It puts the emphasis on thinking and understanding
rather than on producing answers (Peck, Jencks, and Connell, in press). Challenge
children to think by asking what-if questions about their work. For example, to
help children discover an important property about addition, a teacher can ask:
"When you started with five blocks and added three, you found out you had eight
altogether. What if you started with three blocks and added five? Would you have
the same number or a different number in the end?" What-if questions are
especially useful in prompting children to justify their method andanswer and to
gauge whether they really understand what they are doing.
18. • 5. Create an atmosphere where children are interested in learning mathematics
rather than disinterested in or even afraid of it. A teacher sets the tone. If a
teacher exhibits interestin teaching a topic, then there is a better chance that pupils
wall be drawn to learning it. If a teacher approaches mathematics instruction
unenthusiastically,many children will approach learning it mechanically. Several
suggestions for setting a positive tone are delineated below
• Discuss mathematics with children. In addition to making connections explicit and
encouraging thoughtful evaluation concerning the reasonableness of methods and
answers, discussing mathematical problems sends a clear signal that this is a topic
worth talking about, not something that is boring. In encouraging the discussion
of problems, children are also more likely to pose questions and make a real
attempt to understand mathematics.
19. • Discussing mathematics, then, fosters the belief that mathematics involves more
than memorizing facts and procedures; it involves thinking. Any number of
occasions may present an opportunity to discuss mathematics (e.g., a child's
question or error, a conflict of answers or opinion, something puzzling to a
teacher). Indeed, mathematics touches our personal lives in many ways that are
worth discussing (see Example 2-13).
20. SUMMARY
• Mathematics instruction can be interesting, meaningful, and
thoughtprovoking if properly designed to take into account how children
learn and think. Instruction needs to involve children actively through games,
meaningful activities, small -group discussion, and carefully tailored
explanations and demonstrations. Instruction should be introduced
concretely in terms of counting and meaningfully in terms of word
problems. Work involving written symbolism should be introduced after and
explicitly linked to this informal mathematics.
21. • s. Indeed, written notation can often be introduced as a shorthand for the
mathematics already familiar to children. Instrucion can be better tailored to meet
individual needs by grouping according to ability, ensuring that a child has
mastered previous content before continuing on, introducing work with smaller
numbers, and using a variety of teaching methods. One focus of instruction
should be relational learning such as discovering patterns or connection susing
negative instances, and thinking can be fostered by beginning lessons with a
question. Finally, it is essential to create an atmosphere of inquiry and enthusiasm
by discussing mathematics with children and by fostering constructive beliefs
22. • There is no one way to implement these general guide- The chapters that follow
do provide numerous examples of how that.. ,gnitive principles can be applied to
teach specific concepts and skills. Tables in Chapters 3 through 12 and the
Appendices delineate a recommended instructional sequence for these
competencies. The workbook (Baroody and Hank, 1988) that accompanies this
text details a sequence of games, activities, and exercises for each of the content
areas covered. However, even the specific examples and instructional sequences
are intended as a guide rather than a definitive prescription. Most teachers will
find that they will have to adapt the activities and the sequences to meet particular
nteds of their students and situations.
