Translating among representations

From Thinkmath

The principle of translation within and among modes is used throughout Think Math! Here are just two examples.

Headline Stories

In one of the many varieties of the daily Headline Stories (open-ended problem-solving), students will see a picture of dots or objects, perhaps like this

and are asked to tell a verbal story or make a symbolic summary that relates to the picture. First-graders might respond with any of several symbolic summaries, like 5 + 3 = 8 (noticing the orientation of the stems or how they are boxed), or 3 + 4 + 1 = 8 (reflecting the left-to-right grouping of flowers by color) or 4 + 4 = 8 (reflecting color only). Extracting such variety of translation is important. Students’ verbal statements tend initially to be quite straightforward and simple observations, but teachers are encouraged to help students broaden their statements to include story elements and questions as well. As with symbolic summaries, the nature of the response will vary with the student and also over time, as the teacher encourages the class to find more ways of looking at things.

• Observations come at a variety of levels from such simple ones as “eight flowers” to more detailed ones like “one box has two more than the other box” or “there are just as many pink flowers as purple flowers” or “there are two more pink flowers in that box than in the other box.”

• Questions also come at a variety of levels, allowing all students to participate. Here is a straightforward one, embellished in a story: “Nicquela brought three pink flowers and two purple flowers to class. Curtis brought two purple flowers and one pink flower. How many flowers does that make?” More subtle ones include: “How many more flowers did Nicquela bring?” or “Can this number of flowers be shared evenly among two people?”

From concrete objects to mental representations

(See grade xx lesson xx in Think Math!. See also Developing attention, focus, and working memory.)

Another example of translation is the careful move from base-ten blocks (translation among three representations, written and spoken numbers and manipulative) to pictures of base-ten blocks (with the deliberate intent of getting students to translate from physical objects to paper pictures in preparation for mental pictures) to sounds and hand gestures representing the pictures. Why the latter? Because they are ephemeral: one has to see/hear and store the thunks (the 100 flat), swishes (the 10 rod), and pops (the little units), incrementing each separately, so that “thunk swish pop, thunk pop pop” is recognized as 213 (or 111 + 102, which is the way it was “spoken”). This focus on building mental “buffers” for information, and having children increase their capacity to focus and keep track of information as new information comes in is a feature of the program in many places, and usually involves translating information from one form to another.