Understanding Our Students’ Thinking

Case Study Reflection

Case Study #11

In the first case study that we read about this week Andrew’s knowledge of rote counting reflects the understanding of many young elementary students. He has mastered the skill of understanding and counting to 10 in order. He sees that all the numbers in the row of 50 begin with a 5. He observes that the second number is in order from one to nine and therefore makes the connection that 10 comes after nine and this is the reason he thinks that “fifty ten” comes after fifty-nine. He understands that the ones are going in a consecutive order; however, he does not understand that when you have more than nine the 10 becomes a single unit of ten. He has not grasp the concept of making a group of 10 and where it belongs in the place value system.

Case Study #15

In Danielle’s case study she is asking her students if “anyone knows how to write the number 195 using numerals?” She records these answers that the students have come up with.

1095

10095

195

1395

1295

In the first response I would say that the students has some knowledge about what they number 100 looks like. However, they do not demonstrate a full understanding because they only give one 0 in between the 1 and the 95. The second student has a full understanding of how to write the number 100, but doesn’t understand how to record numbers bigger than 100. They are demonstrating literally putting together 100 and 95.  They have put these numbers together to math what they think forms 195. The last two response I am not completely sure about. I would say that these students were just guessing. I am not sure where the 2 or the 3 came from.

I enjoyed Nathan’s (which is the student that provided the correct answer) explanation when asked “how he figured out how to write 195 correctly, using numerals?” He said that he knew when you write 101 you replace the last 0 with a 1 and as you keep going up you replace the 0’s so that is how he came up with 195. I think that this is a very clever observation and demonstrates that he has some understanding of bigger numbers and their different values. This explanation could help other students see similar patterns and relationships between numbers.

Case Study #14

I enjoyed the discussion that this 2^nd grade class took part in about the number zero. They are genuinely  thinking and trying to make sense of this number for themselves and for their classmates. They refer to zero as being different kinds of zeros.  Lamont says “There are two ways to make zero. This is the 7 for the tens and it (the zero) makes 70.” He understands that depending on where the zero is placed in the number will determine what the number is. If the 0 is before the 7 then the number is just 7, but if the 0 is after the 7 then that make the 7 seven groups of 10 making the number 70. If this concept was just explained by the teacher without allowing students to discover and come up with understandings of what zero is, then I can see where many students would easily become confused. By allowing them to think through the problem and listen to other students ideas they engaged in a deeper understanding of the number zero.

Case Study #12

The the activity presented in case study 12 is a great way to get students thinking about the patterns that can be observed when grouping sets into 10. This is a pattern that I think we take for granted as adults. I loved this activity and am eager to try it in my classroom. I think that it is perfect for introducing first grade students to place value.

The math skills that students are practicing through this activity are rote counting, skip counting, grouping, problem solving, representation, and reasoning. As they participate in this pre-place value lesson they are thinking about connections and patterns that I believe will lead them to a more comprehensive understanding of place value. This frim understanding will lead to a firm foundation that will carry them through their mathematical career.

The activities presented in these case studies were very meaningful to students and allow students to being thinking critically about math. I enjoyed thinking about the ideas presented and loved how students were engaged in math conversation. I am eager to try some of these activities in my own classroom.