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How to Improve Math Course Learning Outcomes

Leonard Geddes 11 Comments

Systems are designed to produce predictable results. In fact, expecting different results from the same system is a recipe for insanity. The relationship between professors and students is an academic system in which their interactions produce particular course learning outcomes.

What if the outcomes are consistently bad? Then the system needs a redesign.

To change the outcomes, we must understand how they and the elements of the course systematically interact with each other. Once we appreciate these exchanges, we can create a new system that produces better outcomes.

The Design Flaw in the Current System

During a recent visit to a college campus, I observed math faculty working through problems with their classes. They got through about three problems per class. Professors and a few students did nearly 90 percent of the cognitive lifting for the class. Most of the students “learned” vicariously rather than actively doing their own cognitive work.

The net effect of this system is that students spend most of their time memorizing rules and methods. They hope to match the correct rules and methods to problems on tests. When they are faced with problems that they’ve not previously seen or in unfamiliar contexts, they are unable to solve these problems.

This experience may leave them feeling treated unfairly because the tests required them to solve problems they hadn’t encountered in class. As a result, they pressure professors to devote even more class time to problem-solving. The cycle repeats and produces consistently poor outcomes.

Students do not sustain this system alone. Professors reinforce the flawed system by working through problems in class without explicitly and strategically covering key concepts. With modest changes, however, professors can position students to produce satisfactory course learning outcomes.

A New System for Better Course Learning Outcomes

Here’s how I define the complex problem-solving required at the collegiate level: using knowledge and cognitive skills to solve unpredictable problems that manifest in unknowable contexts. Students must use a network of complex cognitive skills, such as the “analyzing” and “evaluating” modes of thinking described in the ThinkWell-LearnWell Diagram.

For example, before students can begin solving a complex problem, they must first analyze all relevant information, including explicitly expressed information and implied elements. Then they must decide which concept(s), rule(s), or method(s) are appropriate given the specific context of the problem. Levels four and five of the diagram represent this type of cognitive work. Keep in mind that this activity occurs before students can solve the problem, which is level three of the diagram.

Since professors are currently working through approximately three problems in a fifty-minute class, I created a similar structure with some strategic changes. I call it the 3×10 structure. The objective is to cover three problems in class with a different distribution of labor.

The professor divides class into five ten-minute blocks:

  1. The professor covers the “why” of key concepts. This is level two of the ThinkWell-LearnWell™ Diagram.
  2. The professor works through problem 1 while students observe.
  3. Students divide into groups to work through problem 2.
  4. Students work individually on problem 3.
  5. The professor reviews the key concepts and answers questions.

To learn more about the 3×10 structure and the ThinkWell-LearnWell Diagram, download “Missing Math Skills: Students Just Don’t Understand.

Your thoughts matter! The first 20 thoughtful responses will get a digital copy of the 3X10 Worksheet. This resource will help students solve math-based problems better and more independently.

Comments 11

    1. Post

      Great question.

      1) Doing the progressive steps doesn’t pose any additional challenges than any other type of work in large lectures. Since many students have similar questions and uncertainties, and they make similar mistakes, it is likely that widespread concerns will become evident while other groups are discussing. Thus, the time may be managed the same.

      2) Of course, educators should not expect to go over all of the students’ worksheets in a large lecture class. However, if instructors are interested in gathering insights, then some have found taking samples from different sections of the class useful. This method provides them to glean some insights in a relatively short span of time. I hope these responses address your concerns.

  1. I would suggest that the problem sheets be more formally structured rather than just blank space. When an instructor is presenting a concept (and in the 1st problem which the instructor works through) there should be specific areas for noting: the name of the concept illustrated by this example, any limitations or qualifiers related to this concept and a check that this problem meets required criteria, definitions and unit checks, ***related concepts that either have been built upon or which will build from this idea***, what information is provided (is this sufficient or is a sub-problem needed to obtain required information). flowchart of thought process for more advanced multistep problems and at each stage of the actual work through an area to answer WHY this is the next step to perform on the journey to a solution.

    I would especially stress the area of related concepts – I have run into many students who can only look DOWN the structure of an idea.
    For example in statistics the computation of the Margin of Error is presented as a formula which requires knowing many pieces of information. The Margin of Error later plays a key role in computing a Confidence Interval. Students when presented with a problem to find the Margin of Error flounder at not having all of the necessary pieces for the formula (the build up) without even seeing that they were told the Confidence Interval and could very easily break it down from above.

    1. Post


      Good points. So, I have a more structured worksheet that I use during my consultancies. It has generated mixed reviews. Some educators and students like having dedicated workspaces for specific types of work. However, others prefer the freedom that the blank sheet provides. I see pros and cons.

  2. This process definitely works! I teach college math and I use the same principles in my classes. Not only does it your critical thinking skills but the students build up their communications skills.

    1. Post

      Hi Wilson,

      It’s nice to see that we are like-minded. I too have found that providing a structured way to invest students more in doing the labor building critical thinking skills. Perhaps most importantly is that this method builds several metacognitive skills, such as awareness, regulation, and problem-solving skills, which are essential to functioning as independent learners.

  3. Leonard,
    I appreciate that your insights are both learner-centered, and also systemic. I think this is a really interesting strategy that I intend to use in trainings with my study group facilitators and tutors. Thank you!

    1. Post

      Hi Laura,

      Tutors, academic coaches, supplemental instructors, teaching assistants, and others really like this method because it is an effective way to get students doing the work, rather than wanting others to do the work. I hope it works the same for you!

  4. This feels really doable to me. It’s simple and adaptable. In my faculty, many of the classes are math-based and it is challenging to find dynamic and engaging ways to solve math problems in university. I will definitely be sharing your definition of complex problem-solving and the 3×10 structure with the tutors I mentor as well as the faculty I work with. Setting expectations that learning means being ready for unpredictable circumstances and scaling/scaffolding up the problem process is so helpful. Thanks for another great framework!

  5. I like the idea of combining this with the ideas of interleaving ( and spaced retrieval ( We run a summer math camp for college and too many times I have watched students wait (after the introduction of the problem). They wait because they know that the answer is coming and that their efforts are often futile because the answer will be produced, even if they get to the problem right away, faster than they can answer it themselves. They feel defeated, so they wait. Then, the instructor or a model student demonstrates how to solve it and it “makes sense,” so they think that they understand it. But, as we know, there are different levels of learning – recognizing that a problem was likely completed correctly is very different than examining the problem, understanding what is being asked, and devising a path to get to that solution with the information provided… And, then, following that devised path and getting to the solution!

  6. Hello Leonard,

    I am the Tutorial Coordinator at MVC and this strategy is so amazing, I would love to see students bring this practical application forward as they seek additional support in the Learning Center as well as share the strategy with fellow classmates. Your fresh approach is a long overdue tool for students seeking success in Mathematics.
    Thank you for sharing!!!

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