“Why do I need to learn this?”
“Math is so boring!”
“I hate math!”
Have you ever heard any of these from your children (or perhaps even thought them yourself)? Why does math
seem to come so easily for some people,
while for others it just doesn’t “take”? Research suggests that the depth at which
concepts are taught has much to do with
a student’s competence, confidence, and
general attitude toward math. What does
it mean to “go deep” in math, and what
does that look like for homeschoolers?
In the late 1990s, Norman Webb, an
educational researcher, proposed the
Depth of Knowledge (DOK) model of
learning. Webb divided learning tasks
into four levels of complexity: recall and
reproduction, skills and concepts, strategic
thinking, and extended thinking. Let’s take
a simple mathematical concept (
multiplying by 9) and see how it might appear
in all four levels of cognitive depth.
Recall and Reproduction
At this level, a student simply retrieves
information from memory. With the
multiplication facts for 9, for example,
your student should be able to recite
them randomly, at any time or place.
The problem with rote learning, however, is that it doesn’t offer any context to
make it meaningful or interesting. This
is why students need to move to deeper
levels in math.
Skills and Concepts
Students now apply the facts they’ve
learned to familiar tasks. Most math
programs offer simple word problems to
challenge students at this level, such as
CDs are on sale for $9 each. How
much will it cost to buy 5 of them?
First, the student needs to analyze the
situation to determine that multiplication
is required; then he must select the cor-
rect fact ( 9 × 5) to find the answer. Now
students see a reason for math, but this
may not be enough to stimulate interest
or develop critical thinking skills.
At this cognitive level, students use reasoning and planning to tackle a learning
task. This sample problem for the 9 facts
shows how a student at this level can apply deeper thinking.
The number 3,806,784 is multiplied
by 9. In the product, what digit is in
the ones place? Explain your answer.
A student at a lower cognitive level
would most likely need to perform the
actual calculation to arrive at the answer.
At this level, however, a student would
draw on his factual knowledge ( 9 × 4 =
36) and deduce that, since 36 has a 6 in
the units place, the product of 3,806,784
and 9 will also have a 6 in the units place.
In his explanation, the student is required
to reason aloud, thus developing important critical thinking skills.
by Jean Soyke