- Is the mathematics of the task within the students’ depth of knowledge?
- Is the mathematics of the task related to current learning targets?
- Does the task require sense-making?
First, in order for the students to be willing to tackle the task, it must be be in the zone of proximal development discussed above. If the task is too hard, students will be easily frustrated and want to shut down. If the task is too easy, students will gain nothing from the experience.
Second, if the task does not involve mathematics with which students are familiar and are able to do, they will not be successful with the task. However, if the mathematics involves something with which the students have been working, then they are much more likely to continue to work at the task until they have completed it successfully.
Third, in order for students to gain a deeper understanding of mathematics, they must be able to make sense of what they are doing. The selected task is central to this idea. If students cannot make sense of the mathematics, no knowledge will be gained. Growth occurs when students begin to make sense of something they did not initially understand.
Let’s return to the example with Mrs. K from last week’s post and examine her instructional decisions. (Read it again here.)
Mrs. K presented her fifth grade class with the following problem:
Farmer Brown’s niece Angie is in charge of her uncle’s farm while he is on vacation. He gave her strict instructions to make sure none of the animals ran away. When Angie counted the cows, chickens, and sheep, she counted 96 animals. There were three times as many chickens as cows and twice as many sheep as cows. How many sheep did she count?
As soon as Mrs. K presents the problem, the students begin to show signs of struggle and she overhears several students say that they do not know what to do. In order to respond to the students’ cry for help, Mrs. K asks for the class’s attention and then encourages the class to analyze the situation. Together, they begin to complete a KWC chart to identify what they know, what they want to know, and the special conditions for the problem. Mrs. K then begins a discussion to probe the students about what should be done next. After several suggestions have been offered, Mrs. K encourages the students to consider the ideas that were shared and choose a path to explore.
If we analyze the scenario, we can see that the problem is within the students’ depth of knowledge since it only requires knowledge of basic operations in order to obtain the solution. Understanding how to compute with basic operations is central to the mathematics of fifth grade. The algebraic foundation of this problem definitely requires students to make sense of it before beginning to tackle it; so, this becomes the place where students struggle productively. They have all of the tools necessary to solve the problem, but must create a solution strategy that will address the specifics of this task.
Mrs. K understood this need for the students to struggle with the task and proceeded to help them analyze the problem and determine a place to begin and direction to follow. Unlike Ms. S, she did not walk the students through each aspect of the solution strategy process. Instead she offered the support students need to stay in their zone of proximal development so that they could gain the most from the experience.
Read Part 3 here.
- National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.
- Peterson, B. E. (2016). How and why to let students struggle: Thoughts from research. Retrieved from: https://mathed.byu.edu/~peterson/NCTM%202016%20-%20Productive%20Struggle%20-%20San%20Fran%20handout.pdf on June 14, 2016.