The Value of Mistakes/DMI CH 3

After reading through this week's casebook studies present in DMI's chapter 3, there are several errors and identifiable misconceptions present within many of the children's thinking. Coinciding with these misconceptions, there is still absolute logic and reasoning that exists. For example, in Case 11 (Dawn), the kindergarten teacher is using a hundred's chart to track the number of days the children have been in school. The chart is completed up to the number 59 and the teacher poses the question to the students, "What number should I write on our chart today?" One student in particular named Andrew replies "fifty-ten". The teacher records this as 510. Though this answer is incorrect, we can identify the logical reasoning that exists within Andrew's mindset. Page 46 states, "Once again, physical involvement seemed a necessity as Andrew moved up to the chart and dragged his finger across the row with the numbers 51, 52, 53, 54, 55, 56, 57, 58, and 59. Andrew seemed to be making use of the number sequence; thus his response of fifty-ten made perfect sense" (46). In Andrew's mind, the sequence of 5's or 50's as the first numeral should remain constant and carry over into the next space, while the second numeral should increase by one interval in order to continue the pattern in place. Thus, fifty-ten, is a perfectly logical, though incorrect, solution to this problem. In addition, in Case 13(Marie), many students make similar errors and struggle when regrouping numbers and/or accurately representing place value amounts. For example, it states, "Lou showed 99 by laying out 9 rods and 9 units. His written number sentence to represent this amount was 99=9+9" (50). In this example, though Lou is able to use the manipulatives in the correct manner, he is failing to understand that 99 is, in actuality, 90+9. While this is a common misconception found in children, it is one that is logical and makes sense in the minds of young children. The number 99 is quite literally comprised of two 9's thus alluding to the fact that 99 would simply be equivalent to one 9 plus another 9. Other errors found within this particular case study are that of ones and tens placement reversal. For example, "Mary showed 127 as one flat, 2 units, and 7 rods, which she arranged in the same order. Her number sentence was 127=100+27" (51). Similarly, "Darryl's number was 237, which he showed by lining up 2 flats, 7 rods, and 3 units in this order" (51). Both of these students are accurately representing each number with the correct amount of manipulative (for example, in 127, representing the 2 as 2 units) however, they are using the incorrect manipulatives and failing to recognize the place value that the number in the ones and tens place holds. All of these mistakes, though erroneous, are actually quite logical and make sense in the mind's of young children.

The article, The Value of Mistakes, states, "When we teach children that mistakes can be the beginning of the learning process, that they are to be used as stepping stones leading to success, we are much more likely to engender children who say 'I don't get it yet'. As one student shared in a journal entry, 'Mistakes are like little lessons all in themselves. If you face them as learning opportunities rather than reminder of your inabilities, then you're bound to be a better mathematician" (47). All of the children in the case studies can thus use their mistakes as segues for success and take the initiative to deconstruct their thought process in order to determine where the error occurred and how they can improve next time. If all of these steps are taken, students can surely attain academic success.