Structure 1: Open-Ended Tasks

My first “ah-hah!” regarding differentiating instruction came through using some of the open-ended tasks in the Investigations in Number, Data, and Space (Dale Seymour Publications, 1995) curriculum which includes work with “How many of each?” problems. Children are posed questions like “There are eight vegetables on my plate. Some are peas and some are carrots. How many of each could I have?” They are asked to show their work in pictures, numbers, or words. Because this sort of question prompted a wide variety of responses, I realized that an open-ended problem-solving situation allowed both older and younger students to find an appropriate level of challenge. Here are some of my favorite work samples from children that demonstrate the different ways children can access the same type of problem.



In response to the peas and carrots question, Travis, one of my youngest first graders, took crayons and drew plates of green peas and orange carrots, then counted them up to determine that there could be five peas and three carrots, two peas and six carrots, and so on.

This piece of work is in response to a question about 10 pets, some of which are dogs and some are cats. Maggie began like Travis did, by drawing pictures and counting. As she worked though, she began to grow tired of drawing so many legs and whiskers and tails. Before long, she realized that she could use her fingers to find her answers, and she stopped drawing each cat and dog and simply wrote down number combinations with one little cat or dog illustrating each answer.

Sarah had more experiences than Maggie or Travis with formal addition so her answers “5 peas and 3 carrots = 8,” “4 carrots and 4 peas = 8,” “7 peas and 1 carrot = 8,” took on a more traditional feel.

Megan’s work looked a lot like algebra as she created abbreviations in her number sentences. In recording combinations of dogs and cats she wrote equations like “4d + 4c=8” and “5d+3c=8.” She could also be pushed to find more and more combinations.

When encouraged to find all the possible combinations for “How many of each?” problems, Sam developed a chart with numbers counting up in one column and numbers counting down in a second column. When other students saw this strategy, it quickly caught on amongst a small group and they wound up working together to develop charts for questions with three or more variables.

I love that the same question can be posed here to the whole group, but that students at different developmental levels can each access it in an appropriate way. As a teacher, I can challenge each child individually as well-- pushing them to try a new strategy or to solve the problem more completely.