Representations of Ten

Last week, while I was walking around helping the first grade students in my main placement with the math problems they had been assigned, I noticed that many of them were struggling with the “draw a picture” portion of their worksheets.

The lesson had been on determining the number of “tens” in a two digit number and the students seemed to be able to choose the correct answer from the listed choices, but then had difficulty representing it with a picture.  I puzzled over this as I heard the teacher explain to more than one student that they hadn’t finished the problem until they had drawn a picture to represent their answer.  Many of them had correctly identified that 7 “tens” were in the number 70, but they were having difficulty drawing seven towers of ten.  One student had small squares scattered across the allotted space.  Another had seven leaning towers of various heights.  Still another had drawn a tower of stacked squares, but none of them were the same size.  This puzzled me because I wasn’t sure if this was indicative of them not understanding the concept of the number of tens in a number or just of them “not being able to physically draw a picture to represent it”.    

 It seemed to me that that if the students were able to draw these groups of ten “correctly” it would help them determine their answers.  They had studied skip counting with tens and if they were able to draw distinct groups of ten they would be able to “count them” to arrive at their answer.  In my prior experience, drawing a picture had brought clarity.

Later, my master teacher and I discussed this and wondered if we should consider doing a “warm up” exercise where she’d model drawing towers of ten, giving the students step by step instructions.  Then we would give them an opportunity to practice this skill.  We had possibly moved too fast over this portion of the lesson and needed to slow down a bit, given the outcome of the assessment. 

Another idea we discussed, to see if they understood the concept of “tens,” was to have the students build the towers with manipulatives instead, rather than draw them.  Their ability (or inability) to do this would certainly shed light on their understanding of this concept.   This exercise of “troubleshooting” with my master teacher reinforced the idea that teaching math is definitely a dynamic process, an art rather than a science.  It reinforced the idea that there isn’t a “one size fits all solution” when teaching.    

Studio Day

A few weeks ago, during our Studio day in my mathematics class, I had the pleasure of meeting with two students from Maywood Hills Elementary, a second grader and a fifth grader, to talk about their mathematical thinking.   This was a very eye opening experience for me because it proved to be a lot more challenging than I originally thought it would be.  I seemed to understand the fifth grader’s thinking pretty well because he was in the Recall stage on the CGI trajectory.  He used algorithms to solve the problems from the interview task card and I was familiar with this method so we seemed to speak “the same language”.  Occasionally, I’d have to ask for clarification, but for the most part, I understood the method he was using to solve the problems.

Working, with the second grader, however, proved to be much more challenging.  I really had to listen to his strategy because I wasn’t familiar with the way in which he was solving the problems I gave him.  He, for example, when adding two, 2-digit numbers, started from the left side of the problem, with the tens, rather than the right side, with the ones.   He then represented the tens by writing lines (10-sticks) and circles to represent the ones.   After this was done, he’d count the 10-sticks and/or circles (ones) to see if he could form groups of ten.  If he did have any groups of ten, for example, ten 10-sticks, he’d draw a circle around them and write a 1-stick to the left of the tens column to represent 100. 

I later found out, toward the end of my session, that my second grade student had used base 10 blocks to learn addition.  Once I found this out many of the things he had told me started to make a lot more sense.  He, for example, had drawn a square under his circled grouping of ten 10-sticks.  I now realize this represented a 100-square from the base ten blocks.  

 I so enjoyed working with this student and once  I understood his method I was actually able to help him find a mistake he had made.  This exercise truly showed me the challenges I’ll be facing in my future classroom, to figure out the thinking of my students.

Moving From Assessment to Instruction - Reading Lesson Ideas

T enjoys reading.  He usually reads every night, at least 20 minutes.  He likes adventure/fantasy books.  Currently he’s reading a book called, Sasquatch by Roland Smith.  He told me that this book is easy for him to read but he’s enjoying it.  He wasn’t sure when he’s going to be finished reading it.  He estimated a couple of weeks (less than a month for sure).  T has good fluency when reading and uses some inflection but he doesn’t always pay close attention to the punctuation.  He often reads from one line to the next without pausing at the end of sentences and/or at commas.  He also reads very fast, but this doesn’t seem to affect his comprehension.  He a great speller, only missing one word from the spelling test I gave him during our first meeting.

T tested at the independent level on the Examiner Word Lists I gave him up through the seventh grade.  The only word he missed was desert, pronouncing it dessert.

The first reading passage from the Qualitative Reading Inventory book that I gave T was Pele.  He had never heard of him before but he knew the game of soccer very well.  He also knew what a professional athlete was.  He scored only a 67% on the concept questions for this narrative but he answered 7 of the 8 questions correctly at the end of the passage, scoring at the Independent level.  He had no miss-cues when he read this passage.   I later gave Thien two more passages to read from the Qualitative Reading Inventory book.  Overall, T seems to have pretty good comprehension when reading but I believe he would benefit from a lesson that would require him to clarify his thinking.  At times, he has difficulty articulating his ideas.  I also believe he would benefit from a lesson that would provide practice determining the most important idea an author is trying to make.          

I also believe T would benefit from a lesson on homonyms.  (He had previously confused desert with dessert).  The teacher may want to try dividing his students into two teams where a representative from each one take his or her place at the board in the front of the classroom as the teacher challenges them to write sentences using words that have more than one spelling and/or meaning (For example; Our principal was really testing the teacher’s principles.)  The two teams would compete with one another.  Having the students play a game would make this activity both fun and educational. 

Reflection on peer blog responses to my Writing Analysis

The feedback I received from my writing group was constructive, specific and timely.  Brian mentioned that I might want to consider incorporating technology into my lesson.  In response to this suggestion I found a great online crossword puzzle activity dealing with synonyms (for the words good, nice and bad).  I could possibly use this as an extension activity or possibly assign it as homework.  This site would provide a fun and interactive way for students to get additional practice finding synonyms. 

Brian also mentioned that I might want to have my students draw a picture, based on the descriptions written by others in the class.   This would show the students if their writing was providing enough detail by the drawings produced.   This would also work well as an add-on activity. 

Thirdly, Brian mentioned that I might want to provide some instruction on how to use a thesaurus before expecting my students to use one.  This suggestion reminded me of the importance of not just assuming my students know how to do something. 

Jessica also provided some important suggestions.  She indicated the importance of ensuring that my lesson follows the “gradual release of responsibility” model to optimize student learning. Explicitly thinking and writing about these stages helped clarify my lesson.   (I had followed this model in my write- up but hadn’t explicitly included this type of language.)

Lastly, Jody commented on the importance of this lesson (of finding synonyms) and suggested that this type of lesson be done frequently.  We both agree that teaching students to add meaningful descriptions and details in their writing is an important skill to learn.