Being Taught to Teach

A Partial Analysis of a Student’s Mathematical Understanding
March 29, 2012, 4:38 am
Filed under: My Papers, Student Teaching

My students have been working on developing their strength at counting and thinking in terms of 10s.  The students’ ability as a whole on adding one more or one less to another is extremely strong.  They have more recently begun adding 10 more and 10 less to a number.   This is important for my students to know because a strong understanding of our base tens system in necessary to move on in our math system.  After having a strong understanding of adding or subtracting one, adding or subtracting ten is the next logical step for my students to take.

In addition to this task we have also been working on fact families in fact family houses so students can get faster at recognizing things like missing addends and figuring out subtraction sentences that work for the complete addition sentences in front of them and vice versa.  We are hoping that through fact families our students will begin to see how addition can help with subtraction and vice versa to make them faster at figuring their math facts.

Sharron has recently discovered her voice in mathematics and enjoys verbally sharing about the math that she does.  She has become vocal about how she does her math in class and during our reflections for the class she is often eager to share.  I have been observing Sharron during math time to witness this.   A few weeks ago the students had to take a benchmark assessment for the district.  This was still fairly soon after we had started the idea of our fact families.  In reviewing Sharron’s benchmark she has correct almost every question referring to missing addends, however she has erased some of these “correct” answers and bubbled in a different one.  Unfortunately my students are not yet used to showing their work on tests like this.  I cannot tell why Sharron erased these answers and put other ones.  Because of this I decided to interview Sharron and find out more about her thinking.

Sharron and I went into the hall with some of her familiar math materials.  We had her 10s and 1s blocks, 100s chart, 10s and 1s chart and a whiteboard/marker.  We started off with a familiar activity the students have been doing with my CT, I wrote a number on the whiteboard and asked her to show me the number with her blocks.  The first number was 63.  I asked her to show me 10 more and tell me what it is.  She added a 10s block to those already existing, counted and told me there were now 73.  When I asked her how she knew she explained that you have to add one more, I asked for clarification and she said that you have to add one more 10s.  We tried again with another number, 27.  She showed me the correct combination of blocks and when I asked why she pointed to the 2 in 27 and said that she had to have 2 tens blocks.  I asked her to find 10 less.  She took one 10s block away, counted and told me there were 16.  When I asked her to check she counted in her head again and told me there were 18.  I asked her to count out loud for me and she was able to give me the number 17.  For the next number I asked Sharron to take out her 100s chart.  I asked her to find the number 81.  She pointed to it, but when I asked her to find 10 more she immediately when back to her blocks, made 81 and then made 91.  Although she was unable to do this task on the 100s chart, she immediately gave me the number 91 without counting her blocks.

After these first few questions we moved onto fact houses.  The first fact house that I drew included number 5, 3, 2.  I wrote three of the four problems in, omitting an addition problem.  Sharron was able to quickly identify the numbers of the fact family, indicating that 5 was the greatest because it was at the end of the addition problem.  She was then able to create the missing addition fact.  For the second family I omitted both addition facts.  Sharron was able to identify the three numbers as 7, 5, 2, but suggested that 2 was the greatest number because it was at the end of the subtraction problem she saw.  I did one more fact family that was missing all but one addition problem.  Sharron was able to recognize all the numbers in the family (4, 5, 9) and knew 9 was the greatest.  She was able to come up with all three missing facts.  Because of the discrepancy in her finding the greatest number I asked Sharron to take out her 100s chart and put her finger on 14.  Then I asked her to put a different finger on 12.  I asked her which was the greatest number, she said that twelve was greater than 14.  I asked her to tell me why and she said that 12 is littler and 14 is longer so 12 is greater than 14.  We finished up our interview with a few questions about what Sharron likes about math and what she thinks about mistakes.

I believe that for her conceptions of 10s and 1s blocks Sharron is well on her way to a good understanding.  She enjoys using her manipulatives and this is something that she talked about liking a lot in math.  She is not confident in using her 100s chart to find 10 less or 10 more and this would be an important other strategy for her to get a grasp of.  For a next step we could work further with the 100s chart and how the columns can help us with the concepts of 10s.  Another piece that I would like to make sure Sharron starts doing is drawing out her blocks.  I am very happy that she enjoys using the manipulatives for math; however I want her to also be able to see how she can draw these out when they are unavailable.

For Sharron’s ideas of her fact families I am having some fears about how she is thinking about them.  I am afraid that she has found some particular patterns that make this particular task easy to accomplish, but are not helping her overall understanding.  This shone through when I gave Sharron one subtraction problem and what she had seemed to know got thrown askew.   Because of the “mistakes” that were made I am seeing that Sharron does not have a deeper understanding for how knowing one math fact (such as 7-4=3) can help her quickly know additional facts about those numbers.  She is recognizing patterns which is a key aspect of how the fact families were introduced to the class, however now she needs to be moved beyond this to see how knowing just one of these facts can translate to knowing much more math around the three numbers the family is made up of.  I was excited about her reactions to math.  Sharron is very eager to do and share her thoughts in math.  I was also very pleased that as a learner she is understanding that mistakes are ok to make, but I would also like her to understand why mistakes are good and how we can learn from them.