## Meaningful Mathematics … Learning Mathematics

Among the pushes from policy makers is ‘meaningful math’ … make it applicable to student interests. Of course, this is not a new idea for us in mathematics; we’ve been using ‘real world applications’ to guide some reform efforts for decades. #realmath #collegemath #learningmath

Some current work in reforming mathematics education in college is based on a heavy use of context, where every new idea is introduced with a situation that students can understand. We know that appropriate contexts with meaning to students helps their motivation; does it help their learning?

Before I share what I know of the theory and research on these approaches, I would like you to envision two types of textbooks or instructional materials (print, online, or whatever).

- Opening up the start of a lesson or section, repeatedly and randomly, generates a short verbal introduction followed by formal mathematical symbolism related to the new idea(s) in almost all cases.
- Opening up the start of a lesson or section, repeatedly and randomly, generates a variety of contextual situations and mention of a mathematical idea or tool that might be used.

Many traditional textbooks are type (1), while a lot of current reform textbooks are type (2). Much of the change to (2) is based on instructor preferences, and I am guessing that much of the resistance to change is based on instructor preferences for (1).

Let’s take as a given that contexts that are accessible to students improve their motivation, and that we have a goal to improve student motivation; further, let us assume that we share a goal of having students learn mathematics (though that phrase means different things to different people). There seem to be a number of questions to answer dealing with how a context-intensive course impacts student learning of mathematics.

- How uniform is the impact of context on different learners? (Is there an “ADA-type” issue?)
- Do students learn both the context and the mathematics?
- Is learning from a context more or less likely to be used and transferred to new situations?

I know the answer to the first question, based on research and experience: The impact is not uniform. You probably understand that there are language issues for quite a few students, perhaps based on a class taught in English when the primary language is something else. However, most of the contexts have a strong cultural factor. For example, a common context for mathematics work is “the car”; there are local cultures where cars are not a personal possession, as well as cultures outside the USA where cars are either generally absent or relatively new (and, therefore, people know little). The cultural problems can be overcome with sufficient scaffolding; is that how we want to spend time … does it limit the mathematical learning? There are also ADA concerns with context: a sizable group of students have difficulties processing elements of ‘a story’ … leading to problems unpacking the context into the quantitative components we think are ‘obvious’.

The second question deals with how the human brain processes different types of information. A context is a type of narrative, a story; stories activate isolated memories and create isolated memories. To understand that, think about this context:

You are standing on a corner, and notice a car approaching the intersection. When the light turns red, the car applies the brakes so that it stops in about 2 seconds. You estimate that the distance during the stopping process is about 100 feet, and the speed limit is 30 miles per hour. Let’s look at the rate of change in speed, assuming that this rate is constant through the 2 second interval.

The technical name for a story in memory is “episodic memory”. This particular story might not activate any episodic memories for a given student; that depends on the episodes they have stored and the sensory activators that trigger recall. Some students will respond strongly and negatively to a particular story, and this does not have to depend upon a prior trauma. More of a concern are students who have some level of survival struggle (food, shelter, etc); many contexts will activate a survival mode, thereby severely limiting the learning. Take a look at a report I wrote on ‘stories’ http://jackrotman.devmathrevival.net/sabbatical2006/2%20Here%27s%20a%20story,%20Ignore%20the%20Story.pdf

What happens to the mathematics accompanying the story? If we never go past the episodic memory stage, the mathematics learned is not connected to other mathematics; it’s still a story. In the ‘car stopping at an intersection’ story, the human brain might store the rate of change concepts with the rest of this specific story, instead of disassociating the knowledge so it can be used either in general or in new ‘stories’ (context). Disassociating knowledge is another learning step; many context-based materials ignore this process, and that results in my biggest concern about ‘problem based learning’.

Using knowledge (Transfer … question 3) depends upon the brain receiving sensory input that activates the knowledge. This is the fundamental problem with learning mathematics … our students do not see the same signals we see, ones that activate the appropriate information. For this purpose, traditional symbolic forms and contextual forms have the same magnitude of difficulty: the building up of appropriate triggers to use information, as well as creating chunks of information that work together. We need to be willing to “teach less mathematics” so that we can focus more on “becoming more like an expert with what we know”. For more information on learning mathematics based on theoretical (and research-based) points of view, see http://jackrotman.devmathrevival.net/sabbatical2006/9%20Situated%20Learning.pdf and http://jackrotman.devmathrevival.net/sabbatical2006/6%20Learning%20Theories%20Overview.pdf

I can’t leave this post without mentioning a companion issue: Contextual learning is often done by ‘discovery’. Some reform materials have an extreme aversion to ‘telling’, while traditional materials have an extreme aversion to ‘playing around’. From what I know of learning (theory and research), I think it is safer to take the traditional approach … telling does not provide the best learning, but relying on discovery often results in even more incomplete and/or erroneous learning. Just for fun, take a look at http://jackrotman.devmathrevival.net/sabbatical2006/8%20Telling,%20Explaining,%20and%20Learning.pdf

Looking for a brief summary of all of this stuff?

Contextualized learning comes with significant risks; use it with caution and a plan to overcome those risks.

Mathematics is, by its nature, a practical field; all math courses need to have significant context used in the process of learning mathematics.

These ideas about context are related to the efforts of foundations and policy influencing agents (like CCA). We need to keep the responsibility for appropriate instruction … including context.

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