Constructivist
math educators regularly cite Parrot Math by Thomas C. O'Brien. Although this paper promotes
constructivist "activity-based" learning over direct
instruction, it's primary claim to fame is the open hostility to
memorization. Professor O'Brien rejects "memorization
and parrot-like drill " in favor of "children's invented strategies." He
references a paper by Kamii and Dominick as evidence of "considerable
research" showing that mastery of the standard algorithms of
arithmetic is harmful for children. [See The Bogus Research in Kamii and Dominick's Harmful Algorithms Papers]

Understanding and Thinking Depend Fundamentally on Remembered Content Knowledge

Constructivist math educators reject memorization and practice and thereby severely limit the
student's ability to remember specific math facts and skills. They say they want to maximize "understanding" and develop
"powerful thinking skills," but appear blind to the fact that both
understanding and thinking depend fundamentally on remembered content. Knowledge must first be loaded into the brain before
it can connected to other knowledge and "understood." Explicit memorization is
sometimes the most efficient way to get it there. Without specific remembered math facts and skills, students must
regularly revisit previously covered content ("spiral back") and rely on general content-independent
skills, such as the NCTM problem-solving strategies: 1. use manipulative materials, 2. trial and error, 3. make a list, 4. draw a diagram, 5. look for a pattern, 6. act out a problem, and 7. guess and check.

K-12 math is
the first man-made knowledge domain where children build a remembered
knowledge base of domain-specific content, with each child gradually
coming to understand hundreds of specific ideas that have been
developed and organized by countless contributors over thousands of
years. With sound methods of knowledge transmission, the student
is led, step-by-step, to remember more and more specific math facts and
skills, continually moving higher and higher in the vertically
structured knowledge domain that comprises traditional K-12 math. This
first disciplined knowledge-building experience is a key enabler,
developing the memorizing and organizing skills of the mind, and
thereby helping to prepare the individual to eventually build
remembered knowledge bases relative to other knowledge domains in the
professions, business, or personal life. Although the specific
facts, skills, and organizing principles differ from domain to domain,
genuine domain experts must necessarily remember a vast amount of
specific information that is narrowly relevant to their targeted
knowledge domains, frequently without the possibility of transfer to
other domains.

The Power of Automaticity

It is a profoundly erroneous truism ... that we
should cultivate the habit of thinking of what we are doing. The precise
opposite is the case. Civilization advances by
extending the number of important operations which we can perform without
thinking about them. - Alfred North
Whitehead

Constructivists fail to understand that it's desirable to move to automatic use of knowledge. The mind must be free to think
at higher levels of complexity, without consciously revisiting underlying
details. For example, a key idea of the standard algorithms of arithmetic is that
multi-digit calculations are reduced to single-digit calculations. If
children don't have instant recall of the single-digit number facts, they aren't
equipped with the essential pre-knowledge for easily carrying out multi-digit
computations. For more about the power of automaticity, see The Bogus Research in Kamii and Dominick's Harmful Algorithms Papers.

Activity-Based Learning

Thomas C. O'Brien and his
constructivist colleagues are not offering a different way to teach
traditional K-12 math content. They are discarding most of this content, including all of standard arithmetic. In Parrot Math
Professor O'Brien promotes
"activity-based" learning, but doesn't give any
examples. He misses a nice opportunity when he spends
two pages discussing "the handshake problem," but offers nothing
about how to solve this problem. To learn about
the historic importance of this interesting problem, see How Many Handshakes in How the NCEE Limits Elementary School Math, Chapter 2 in How the NCEE Redefines K-12 Math. .

To better understand "activity-based" learning, see the examples in Chapters 2, 3, and 4 of How the NCEE Redefines K-12 Math Also see Chapter 1, A Summary View of NCEE Math and notice that most of traditional K-12 math content is missing.