Tuesday, July 9, 2013

Minds on Math Chapter 3 - Tasks

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Finding worthy tasks...

In this chapter of Minds on Math we focus on finding tasks requiring a high cognitive demand.  Ward Hoffer offers suggestions on where to find high-cognitive demand tasks and suggestions for modifying existing tasks to increase the level of cognitive demand.

Ward Hoffer opens the chapter with a vignette about two seventh grade girls each rolling a dice, and determining whether the sum of the numbers was odd or even.  At the outset I wondered why a seventh grader would be concerned about odds and evens.  Even given the fact that they were looking for the odd/even pattern of sums, I thought that was an elementary task.  At one point during the class period, a student asked the teacher what even meant.  The teacher responded to check the list on the board (which said ends in zero, two, four...)  Ward Hoffer's point, I think, is not whether or not this task was appropriate for a seventh grade classroom, rather, she was pointing out that the task did not have a high level of cognitive demand.  The task did not require students to understand what odd and even meant, they just had to memorize or remember that even numbers ended in zero, two, four, six, or eight.

As I read this chapter, I thought about some of the times I've been guilty of this.  Namely, working with formulas.  Our state (Texas) gives students a mathematics chart to use on the STAAR test.  All formulas are shown there.  Wanting the students to be familiar with the tools available to them, too often I tell the students to refer to their mathematics chart and follow the formula.  While I think there are valid reasons for students to be able to use formulas, I think there's also work that needs to be done to help student understand the formulas.

One of the areas that is very procedural (and is a good argument for students learning to follow rules) is order of operations.  We can work with students to see that performing the operations in different orders results in different answers, but at the end of the day, students need to know that there is a specific order we need to follow so that we all arrive at the same conclusion.

So, those high cognitive demand tasks...what are they and how do we get them?  Ward Hoffer says that tasks should have multiple entry points, various approaches, higher-order thinking required, opportunities to synthesize, and justification and explanation (p. 40).  These are certainly important to keep in mind because our current textbook offers very few of these types of problems.

One of Ward Hoffer's suggestions for finding tasks was to modify existing tasks.  She offered an example problem and then modified it six different ways!

  • increase complexity
  • introduce ambiguity
  • synthesize strands of mathematics
  • invite conceptual connections
  • require explanation and justification
  • propose solutions

I loved that she was able to take one problem and change it in so many ways.  My favorite was introducing ambiguity.  Students hate ambiguity and I think this is the one that could be the most fun to work with because it really gets students thinking about all of the possibilities.

I'm off to find some high cognitive demand tasks for my first unit!

Until next time...


  1. Melanie, I agree with your argument about order of operations. Sometimes we need to follow rules. I've done that activity where I've given a multi step problem and the students all get different answers. We then discuss the need for Order of Operations.

  2. We have Carnegie Learning curriculum and I have to say, that I really think they give us great problems to work on with students. They seem to scaffold nicely for group work. What I need to focus on is picking the best of the best. We have more problems than we could every possibly do, so it's picking and choosing what will provide the best conceptual understanding with productive struggle. Thanks for linking up!


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