2. Conceptual vs. Procedural
⢠Conceptual ⢠Procedural
understanding means: understanding means:
Knowing Knowing
â What to do â What to do
and
â Why
3. Procedural understanding âŚ
⢠âRules without ⢠Examples:
reasonsâ â âborrowingâ
⢠Multiplicity of rules â âcarryâ the 1
⢠Usually easier to â âturn upside down and
understand (to follow) multiplyâ
⢠Rewards are more â âtake it over to the
immediate other side and change
the signâ
⢠Less knowledge
involved
⢠Get answer quickly
4. Conceptual understanding âŚ
⢠Fewer principles
⢠More general
applications
⢠Adaptable to new tasks
⢠Easier to remember
⢠Enjoyable = goal in
itself
⢠Natural growth = active
seeking of new areas
(like tree extending its
roots)
5. ⢠Procedural = learning of an increasing
number of fixed plans⌠to go from
starting points to finishing points
⢠Conceptual= building up a conceptual
structure⌠go from starting to finishing
points in unlimited number of ways
6. Different kinds of math?
⢠âWhat constitutes Mathematics is not
the subject matter, but a particular
kind of knowledge about it.â
⢠Conceptual math = Mathematics
⢠Procedural math â Mathematics
7. What Does It Mean
to Understand Mathematics?
⢠Knowing â Understanding
⢠Understanding is the measure of
quality and quantity of connections
between new ideas and existing ideas
8. âUnderstanding is the key to
remembering what is learned
and being able to use it
flexibly.â
Hiebert, in Lester & Charles,
Teaching Mathematics through
Problem Solving, 2004.
9.
10. Computational Fluency
I th o u
0 gh
i s 14 6 25âs - t seven
x 20 8 is 5 Then thatâs
7 I need 175.
x or 21. se
and 7 0 is 196 So th ven 3âs
e an
+ 14
is 175
+ 21 = swer
56 196
7 x 28
I did 7
Thatâs x 30 fir
210. T st.
off sev hen tak
en 2âs e
So itâs or 14.
196.
12. What is Computational Fluency?
Computational fluency is having and
using
efficient and accurate methods for
computing with understanding.
13. What is Computational Fluency?
Fluency demands more of students than
does memorization of a single procedure.
⢠An understanding of the meaning of the
operations and their relationships to each
other
⢠Number relationships, including
facts
⢠Understanding of the base ten number
system
14. Procedural vs. Conceptual
Knowledge
Objects and names of objects
are not the same as
relationships between objects.
15. Implications for Teaching at
ISM
The need to replace the question
âDoes the student know it?â
with the question
âHow does the student
understand it?â
16. Concrete Place Value
T U
2 6
XXXXXX
26
Partitioned
Words / Numbers
20 + 6 = 26
2(10) + 6 = 26 Twenty six
26
18. The value of a tool is in its
usefulness
⢠Being able to do pencil-paper
computation will not serve students
without the ability to interpret a
problem, analyze what needs to be
done, and evaluate the solution.
19. Models
⢠Concrete -- Pictorial -- Abstract
⢠Multiple models help solidify thinking
and deepen understanding
20. Why draw?
⢠Computational practice, but much more
⢠Notation helps them understand the
question.
⢠Notation helps them invent new
solutions.
⢠Notation helps them undo the solution.
⢠But most important, the idea that
notation/representation is powerful!
21. Manipulatives
⢠any concrete object which can be moved
and used in a way to represent abstract
concepts in a physical fashion.
⢠commonly implies a touchable and
movable object which can provide
children something real to reflect on.
⢠their importance lies in being able to
represent mathematical situations which
are generally abstract.
PURPOSE: An example of studentsâ strategies developing different models for multiplication that support the development of computational fluency SPEAKING POINTS Students exhibit computational fluency when they have flexibility in the computational methods they choose, understand and can explain the methods, and efficiently produce accurate answers. These methods for students in grades 3-5 should be based on the structure of the base-ten number system, properties of multiplication, and division and number relationships. Fluency with whole-number computation depends on fluency with basic number combinations single-digit addition and multiplication pairs and their counterparts for subtraction and division. Fluency develops from understanding the meaning of the four operations and focusing on the development of strategies based on understanding. REFERENCES Principles and Standards: pp. 148-156
PURPOSE: An example of studentsâ strategies developing different models for multiplication that support the development of computational fluency SPEAKING POINTS Students exhibit computational fluency when they have flexibility in the computational methods they choose, understand and can explain the methods, and efficiently produce accurate answers. These methods for students in grades 3-5 should be based on the structure of the base-ten number system, properties of multiplication, and division and number relationships. Fluency with whole-number computation depends on fluency with basic number combinations single-digit addition and multiplication pairs and their counterparts for subtraction and division. Fluency develops from understanding the meaning of the four operations and focusing on the development of strategies based on understanding. REFERENCES Principles and Standards: pp. 148-156
Fluency rests on a well-built mathematical foundations with three parts Mathematical memory vs. memorization
The most important message is that there *can* be a more powerful notation than words, and that some mathematical acts become easier if we find a suitable notation. Too often, algebra is just âanother thing to learn,â not at all a favor to kids. At this early age, algebra should be a convenient way to record what they already know, and to help see the results of processes. Syntactic manipulations of algebra, at this age, are rarely appropriate -- kids canât use algebra to derive or prove what they donât already know -- but they *can* use the symbols of algebra to record what they *do* already know. It is a language, and kids are great language learners, when the language is used sensibly in context!
