Category Archives: Task Design

Realistic Math Education: Defining the “real” in real world

Dan Meyer recently posted about how students aren’t easily fooled by attempts to make make tasks “real world” by placing a photo next to them.

I found myself nodding along at what he described.  I once naively asked a fourth grade class to write real world problems about fractions and received the following response:

The Knicks scored 12 ½ points and the Nets scored 13 ¾ points.  How many points did they score all together? 

When I asked the student how it was possible to score ¾ of a point or why you would want to know the combined score of two opposing teams, he looked at me like I was crazy and said it didn’t matter–“you told me to make a problem with fractions so I put fractions in it.”

Even at young ages, students learn there is a game teachers play where they make “real world” problems.  Kids pick up pretty quickly that these aren’t about the world they live in–they are a special type of problem in math.  So they skim the word problems.  They pull out the numbers and assume they should try some mathematical operations they recently did in class.

So I agree with Dan that spreading the “real world” over a task doesn’t fool students–even our younger ones.

I do believe in building off students’ experiences . Although math tasks don’t have to be real world, they need to build on what students know.  An activity they can do.  An experience they had.  An operation they already understand mathematically.

As I was reading the post, I thought about Realistic Mathematics Education (RME), developed in the Netherlands over thirty years ago.  RME is strongly based on Hans Freudenthal’s philosophy that students should be guided in a process that allows them to “mathematize” the world around them.   They should be provided with tasks that allow them to use mathematics to organize and solve a problem.

What’s interesting is that many people incorrectly assumed that this meant real world problems of the kind Dan showed.  However, realistic in RME referred to situations students could imagine, not necessarily something that would happen in real life.  As a result, “the fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student’s mind.”

Making it clear that real is about what is real in the student’s mind can be helpful when thinking about how to select appropriate tasks for students.  Unfortunately, it’s much harder than simply throwing a real world picture next to the problem.

Want to know more?

Check out the Freudenthal Institute’s page:

Read about Freudenthal and RME: Gravemeijer, K. & Terwel, J. (2000): Hans Freudenthal: A mathematician on didactics and curriculum theory, Journal of Curriculum Studies, 32(6), 777-796.

Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational studies in mathematics, 39(1-3), 111-129.


Dan Meyer’s Makeover Mondays: Modifying tasks

As I mentioned in a previous post, not all tasks are created equal.    Different tasks provide different opportunities for students to learn.

Deciding what task to use in a lesson or how to modify an existing task is a decision we make every day as teachers.

Dan Meyer recently highlighted how we can begin to think about modifying tasks with his Makeover Mondays.  Every Thursday this summer, he will post a task on twitter.  The following Monday, he will post his suggested modifications as well as the changes others have suggested.

What I love about this is that we know from the research that teachers are constantly changing tasks.  Tasks are not always implemented exactly as they appear in textbooks.    Sometimes the task doesn’t fit the needs of our students or it doesn’t build on their prior knowledge or sometimes it is just boring.

However, we also know that sometimes when teachers change the task, it can lower the cognitive demand of it.   Researchers have found that when high cognitive demand tasks are implemented in classrooms, the demands were often lowered because of things like time constraints or management issues in the class.    A mathematically complex task became easier as the teacher gave hints or leading questions to help the students proceed.

But what is obvious from the makeovers Dan posted is that the changes that teachers make are far more complicated than simply lowering or increasing the demands of the tasks.

The made-over task Dan posted this week was more cognitively challenging than the original task, but also built on students’ intuition.  The language was simplified but the mathematics was increased so that students who completed the task easily were challenged.

I’d also like to point out another thing I noticed about modifying tasks–the goal matters.  If the goal is application of a concept, the task needs to be modified differently than if the goal is to introduce students to a concept.  If the goal is to have students model different strategies, the task needs to be modified so that it lends itself to multiple ways to be solved.  If the goal is to work on problem solving and not computation, perhaps a calculator needs to be provided.

Tasks can be modified to serve different purposes.  But it’s important to be clear on what your goal for your students is before you think about changing it.

So this summer, take a look at the tasks.  Share a makeover.  I know I’m looking forward to trying my hand at it.

What matters when choosing tasks for students?

“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”  -Stein, Smith, Henningsen, & Silver, 2000

What are we doing today?

One of the biggest decisions we make as teachers is choosing the tasks our students will work on during class.  We use tasks from textbooks and the Internet.  We borrow tasks from colleagues or design our own.   No matter where the tasks originate from, the nature of tasks we select affects our students’ ideas about what it means to do mathematics.  Different tasks provide different opportunities for students to learn mathematics.

One way to think about how to select a task is by looking at the cognitive demand required by it.   Cognitive demand refers to the type of student thinking required to solve the task.   A group of researchers developed different categories of cognitive demands found in mathematical tasks.

Low-level cognitive demand tasks can be solved by memorization or by using procedures that don’t have meaning for students.  High-level tasks engage students in using procedures with attention to the reasoning behind them or in what they label “doing mathematics”–making conjectures, justifying, and interpreting.

After studying over 500 tasks in middle schools, the researchers found that the greatest gains in student achievement occurred in classrooms where teachers used high-level tasks and the cognitive demand of the task was maintained as students worked on the tasks.   It was also noted that although many tasks started out requiring high cognitive demands, the demands of the tasks often decreased as they were implemented in classrooms.

Of course, there is a lot more to think about when selecting tasks.  How do the tasks build on what your students already know?  How do the tasks engage students?   What particular mathematical concepts are fostered by the task?  But considering the level of thinking you want to foster in your students is a good place to start.

Want to know more?   Check out the research below.

Stein, M. K., & Lane, S.  (1996).  Instructional tasks and the development of student capacity to think and reason:  An analysis of the relationship between teaching and learning in a reform mathematics project.  Educational Research and Evaluation, 2(1), 50 – 80.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027.