Encouraging our children to “go deep” in math can increase motivation, help them learn how to think and reason, and yes, even make math fun!
“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?
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 that they’ve learned to familiar tasks. Most math programs offer simple word problems to challenge students at this level, such as the following:
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 correct 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.
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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.
This is the deepest cognitive level and can also be the most fun! Student-designed projects generally fall into this category because they require pulling together multiple ideas and skills into an experiment, survey, model, or investigation. To utilize the multiplication facts for 9, for example, a student might be asked to plan a party for 9 people and present a detailed list of the costs, using newspaper ads or websites to find prices. (For an extra challenge, the student could be given a spending limit.) This task obviously draws on the 9 facts but also incorporates many other skills, both mathematical (adding) and practical (finding the best price).
Looking at these examples, you may be able to see how rich mathematical learning can be. You are probably wondering, though, “What does this mean for me as a parent?” I’d like to suggest three possible applications:
1. Determine the depth of learning that is best for a particular task. Not all mathematical learning needs to involve complicated problems or projects. In fact, it’s generally best to keep the activities varied to maintain interest.
2. Determine how well you are able to coach your child in “going deep.” You may have never been given the opportunity to think past the second level yourself, and, even then, you may have struggled. Being aware of your own abilities will help you with the next item.
3. Choose the math program that offers the best support. If you function well at the deeper levels in math, you can supplement any curriculum by planning instruction and learning activities similar to the ones presented as examples. If you are not comfortable thinking this way, you may want to seek out programs that include the types of problems and activities that encourage deeper thinking.
As parents, we want to ensure that our children acquire basic math skills and are able to apply them in advanced study and everyday life.