For those of you who know and believe in the constructivist approach to teaching...the idea that learning is more effective when children are actively engaged in the learning process rather than receiving knowledge passively...you already know the answer to the question. Go ahead and use that US standard algorithm after your students have a variety of strategies to use and have a strong knowledge of the base 10 number system.
What do I mean by the US standard algorithm? It's the one we all did when we were kids...carry the one or borrow from the next door neighbor. Problems were always given vertically requiring little understanding of the place value of numbers. I'm not saying never use this approach. It definitely has its place once an understanding of place value and how numbers work is established. Give students a few years to gain that understanding. Introduce the US standard algorithm in 4th grade, when the CCSS states it should be introduced.
So what are students doing?
The following is a sample of addition and subtraction strategies students discover and use to gain a strong understanding of the base 10 number system and place value:
100 x 42= 4,200
48 x 40= 1, 920
48 x 2= 96
4,200 + 1, 920 + 96= 6,216
Seem confusing? Don't worry. Kids get it. This is how their minds work, and if you take the time to learn these strategies yourself, things start to click. Suddenly math makes more sense and mental tasks become easier.
More Proof
There are two levels of demand in mathematics: lower-level and higher-level. Of course, our goal is to promote higher-level demands. The following information (from Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development by Mary Kay Stein, et al. Copyright 2000 by Teachers College, Columbia University) is a great tool to use when creating lesson plans for mathematical instruction:
Demands on today's students are much higher than in the past. Students now need to pass Algebra II to meet high school graduation requirements. It is going to take a lot more work than memorizing facts and formulas to achieve this, and our students are up to the task. You just have to give them the chance.
What do I mean by the US standard algorithm? It's the one we all did when we were kids...carry the one or borrow from the next door neighbor. Problems were always given vertically requiring little understanding of the place value of numbers. I'm not saying never use this approach. It definitely has its place once an understanding of place value and how numbers work is established. Give students a few years to gain that understanding. Introduce the US standard algorithm in 4th grade, when the CCSS states it should be introduced.
So what are students doing?
The following is a sample of addition and subtraction strategies students discover and use to gain a strong understanding of the base 10 number system and place value:
Adding by Place Subtracting in Parts
349 + 175 = 451-187=
300 +
100 = 400 451-100= 351
40 +
70 = 110 351-80= 271
9 + 5
= 14 271-7= 264
400 +
110 + 14 = 524
An example of one type of multiplication strategy is as follows:
Breaking Numbers Apart by Addition
148 x 42=
100 x 42= 4,200
48 x 40= 1, 920
48 x 2= 96
4,200 + 1, 920 + 96= 6,216
Seem confusing? Don't worry. Kids get it. This is how their minds work, and if you take the time to learn these strategies yourself, things start to click. Suddenly math makes more sense and mental tasks become easier.
More Proof
There are two levels of demand in mathematics: lower-level and higher-level. Of course, our goal is to promote higher-level demands. The following information (from Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development by Mary Kay Stein, et al. Copyright 2000 by Teachers College, Columbia University) is a great tool to use when creating lesson plans for mathematical instruction:
Lower-Level
Demands
Memorization
Tasks
~involve either
reproducing previously learned facts, rules, formulae, or definitions OR
committing facts, rules, formulae or definitions to memory.
~cannot be solved
using procedures because a procedure does not exist or because the time frame
in which the task is being completed is too short to use a procedure.
~are not
ambiguous- such tasks involve exact reproduction of previously seen material,
and what is to be reproduced is clearly and directly stated.
~have no connection
to the concepts or meaning that underlie the facts, rules, formulae, or
definitions being learned or reproduced.
Procedures
Without Connections
~are
algorithmic. Use of the procedure is
either specifically called for its use is evident based on prior instruction,
experience, or placement of the task.
~require limited
cognitive demand for successful completion.
~have no
connection to the concepts or meaning that underlie the procedure being used
are focused on
producing correct answers rather than developing mathematical understanding
~require no
explanations, or explanations that focus solely on describing the procedure
that was used.
|
Higher-Level
Demands
Procedures
With Connections Tasks
~focus students’
attention on the use of procedures for the purpose of developing deeper
levels of understanding of mathematical concepts and ideas.
~suggest pathways
to follow (explicitly or implicitly) that are broad general procedures that
have close connections to underlying conceptual ideas as opposed to narrow
algorithms that are opaque with respect to underlying concepts.
~usually are
represented in multiple ways. Making
connections among multiple representations helps develop meaning.
~require some
degree of cognitive effort. Although
general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual
ideas that underlie the procedures in order to successfully complete the
tasks and develop understanding
Doing
Mathematical Tasks
~require complex and nonalgorithmic thinking (i.e.,
there is not a predictable, well-rehearsed approach or pathway explicitly
suggested by the task, task instructions, or a worked-out example).
~require students to explore and understand the
nature of mathematical concepts, processes, or relationships
~demand self-monitoring or self-regulation of one’s
own cognitive processes
~require students to access relevant knowledge and
experiences and make appropriate use of them in working through the tasks
~require students to analyze the task and actively
examine task constraints that may limit possible solution strategies and
solutions.
~require considerable cognitive effort and may
involve some level of anxiety for the student because of the unpredictable
nature of the solution process required.
|
Demands on today's students are much higher than in the past. Students now need to pass Algebra II to meet high school graduation requirements. It is going to take a lot more work than memorizing facts and formulas to achieve this, and our students are up to the task. You just have to give them the chance.
Marybeth
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