We have recently been working on concepts taken from a new website called Open Middle. These problems can be attempted by anyone (closed beginning) as they normally only require values from 1-9 to be placed on a problem to generate an answer. Most problems have multiple answers (closed ending). They are called Open Middle because students are asked to use their incorrect guesses to move closer to a correct answer. Wrong answers are celebrated!
Of course, students can continue to simply guess answers, but this is a poor strategy and unlikely to generate a correct answer quickly. So we are building students depth of knowledge (DOK) and ability to control and manipulate numbers (fluency).
An example, the grade one DOK 1 problem featured might ask the student to Find the Sum of 44 and 27. This problem requires little deep thinking and can be quickly calculated mentally. It requires no more than a formula or mental chunking strategy.
The DOK 2 requires a little more thinking but can be solved with trial and error. No real strategy is required. One answer might be 14 + 53 = 67. This answer meets the requirements of not using the values 1-9 more than one time.
The DOK 3 question asks the student to generate the values and an answer using the values 1-9 not more than once each. Students need to show an understanding of place value in order to generate their best answer. They use previous answers to build on the next in search of better.
Below are some examples of Open Middle questions we have been playing with in class. There is also an answer scaffold that we use to collect answers. More information about the Depth Of Knowledge matrix can be seen here courtesy of Robert Kaplinsky.