Direct Instruction v. Direct Construction

Direct Instruction is contrasted with self discovery.  But if self discovery is guided enough it can become direct construction?  A sort of happy middle ground that may be difficult to find for each actual specific person?

The following paper was linked to previously, but is worth relinking for this post.

Direct Instruction vs. Discovery:
The Long View

Search on

Dean, D., & Kuhn, D. (2006). Direct instruction vs. discovery: the long view

The top search results on this search are rather interesting looking.

Is Direct Instruction an Answer
to the Right Question?
Deanna Kuhn
Teachers College
Columbia University


Get student buy in to learning math.  Applications are done as a way for that currently.  Math foundations is not used.

One approach to math foundations is to tell students they can learn the Peano Axioms of natural numbers and then learn the definitions of addition and multiplication of natural numbers.  The successor function is the foundational function in the Peano Axioms and in arithmetic.  In effect, the foundation of arithmetic is the successor function.  By learning this function and to use it, the short road to understanding the structure of natural numbers is exposed. This means they can learn the logic of natural numbers in a succinct and compact way.

This gives them the structure to learn fractions as ordered pairs of natural numbers.  The successor function and the structuring of natural numbers through the function concept prepares the way to see fractions as functions on natural numbers and then a new type of number in their own right because their properties as functions on natural numbers lets us define addition, subtraction, multiplication and division for them using their internal data.

When we do math this way, we have a short logical structure from counting by one to fractions as numbers.  This then lets us apply fractions to geometry to define length and area of point sets of fractions.

The current way is a seemingly random set of Khan Academy videos with one procedural lesson after another that seem to stretch on to infinity.  Khan is adding more videos than you have time to watch them, so that you can never get to the end.

Pure proceduralism presents the never ending array of procedures. Since you can never learn all the procedures, you never feel you get to the end.  So you don’t feel you understand math.

Math foundations is a form of chunking.  It is logical chunking.  There are important parts that have ends.  You get to a meaningful understanding of natural numbers when you prove the commutative and associative laws of addition and multiplication and the distributive law.  This also leads to a meaningful use of letters to stand for numbers and thus of algebra.

Doing this requires a textbook that does it.  You can’t just remember it.  There is too much memory load not to have a textbook.

Constructive methods that are not guided by math foundations will impose loads and go in directions that don’t make structural sense even after the lesson is completed successfully.  Structural understanding of math gives you the understanding points or conceptual nodes to link knowledge to.

You know where you are in learning elementary math by having a structural guide from the axioms and basic definitions of natural numbers through fractions.

The Double-edged Sword of Pedagogy: Modeling the Effect of Pedagogical Contexts
on Preschoolers’ Exploratory Play
Elizabeth Bonawitz1*, Patrick Shafto2*, Hyowon Gweon1, Isabel Chang1, Sydney Katz1, & Laura Schulz1

Although the Peano Axioms may not seem relevant to pre-school, they still give guidance to the teacher and parent.   The Peano Axioms are the logical meaning of counting by one.  They also give the meaning of keeping your place in the count.  Not getting lost in the count is part of what is being learned in pre-school.  This is a major part of what the Peano Axioms encapsulate, is not getting lost in the count.

Two other searches I want to explore later are

“radical constructivism” “foundations of mathematics”

“reading ahead” “flipped classroom”




About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s