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.

Math Talks
February 17, 2012, 5:04 am
Filed under: Student Teaching

In our math course at Mills, Edcuation 303, we are learning all about Math Talks and how to make math exciting and accessible to our students.  Math Talks are to help students realize what they already know about numbers, make connections and hear what others have to say about numbers.

Richard Skemp wrote an article that we read addressing the important distinctions between instrumental and relational math knowledge.  After reading and reflecting on our own educations many of us realized that we had learned math instrumentally, meaning that we had memorized formulas or steps to plug in when we saw the right wording for it, but did not know how it really tied together or why we did it.

Have you ever asked a child to tell you (or write down) everything they know about a certain number?  You should, it is fascinating.  It gives you so much insight into what they are thinking.  The 100th day of school is a big deal to kids and schools nowadays, I don’t remember celebrating it as a child, but to prepare for that I introduced my students to Math Talks on Monday.  We started as a group on the rug and talked about the number 10.  I gave them some types of examples like sentences and equations, but did not want to influence their answers too much.  The students were asked to think in quietly and after a few moments to share with a partner in a whisper voice.  This gave students who did not have an idea a chance to hear others and be able to raise a hand.  After it seemed like everyone had whispered to another person I asked the students to come back together and for a quiet hand with some ideas about the number10.  We created a poster together and then the students were shown a graphic organizer to put together their thoughts on the number 10.  The following day their morning job was to do this same graphic organizer on the number 100.  We did not have a chance to debrief the morning job because it also happened to be Valentine’s Day, but what these students had to say about this big number was truly intruding.  Here are some examples of the student work:

Back at It
January 18, 2012, 3:06 am
Filed under: Reflecting | Tags: ,

Today was the Winter Retreat for the Mills 2012 credentialing group.  It was quite a long day that left us with many things to think about.  We began our day with a Health/Mainstreaming fair that  was made up of twenty different topics that could fall into this sort of fair, we had everything from child abuse to tobacco use, ADD/ADHD to LGDBQ, vision impairments to behavioral/emotional disturbances.  I must say I am glad we dedicated a chunk of time to these topics, how to address them in our school and classrooms and what resources there might be available.  Although they are not all topics I might be encountering my first year or two of teaching, these are great jumping off points for me to begin looking should I need to research them later.

We hit some major topics throughout the rest of the day too, covering an experience panel on learning disabilities which made me question some of my own young learning and where my anxieties actually stem from and then the sombering discussion of what is required of us as (future) credentialed teachers in the state of California on the topic of child abuse.  A credentialed teacher is “mandated reporter” and must report child abuse if they find there to be a reasonable suspicion.  While, I knew somewhere in the back of my mind that this was something that I would eventually be dealing with, I have been so caught up in learning and teaching, writing papers and reading that this subject sort of fell off my radar.  I’m very glad to have finally touched on it and that we had such a good speaker who was able to answer most of our questions.  Although I am sure it will always be difficult to tackle issues like these when they arise, I am thankful that our program allows us a space to get familiar with the process, ask questions and express our worry over these situations.

And so on these notes I begin the second semester of my credentialing year at Mills.  I am excited to be in  my new placement, nervous about PACT and keeping my eyes and mind open for the…well for all of it.



Lesson Writing, Rubrics and Reflection
December 13, 2011, 4:45 am
Filed under: My Papers, Reflecting, Student Teaching

I have been working on creating my own lessons, rubrics and reflecting on them.  Here is one of my first attempts European Diseases Lesson, Rubric and Reflection.  I created a social studies lesson for my 4th and 5th graders on European Diseases that were brought to the Americas during exploration.

Teaching ELLs
December 12, 2011, 5:33 am
Filed under: My Papers, Reflecting

One of my courses asked us to write a keynote speech for our final project.  The speech needed to discuss the ideas we had begun to develop about teaching, and more specifically teaching English Language Learners.  Here is what I had to say:


All of our students are English language learners whether they come to us having spoken English their whole lives or not at all.  Despite this fact how often do you think about language in the lessons you teach in math, science, or social studies?  In reality you need to think about language in every subject, not just during language arts.

That being said, you cannot treat all students the same when it comes to learning English.  It is important to take into account their background and how they have access to English.  As teachers we should be prepared with a toolbox of methods for language learners, as each tool will not work for every child.  We want all of our students to succeed, and in a society built upon inequalities of power and status, giving our students entry to learning about the language of power and how to use it will help them succeed.

The first of these important tools in understanding the language of power that I will talk about is the idea of academic language.  Just as I am speaking to you now in a way that I would not use when talking to a friend or writing a college paper, school has its own set of language and rules.  When you are working with students who are not coming from the dominant culture of power they might be unaware of this.  A key in helping them discover academic language and how to access it is teaching them about language registers.  Even if your students do not use English at home, they will be used to register switching, but not by those terms.  Everyone switches registers, for example my mother can always tell when I answer the phone and it is my grandmother on the other line, my tone, the words I use and my general way of speaking change.  Just as I have learned to change registers, students should learn early on that when they come into school there is a particular way to talk and act.  In order to have access to the culture of power students must learn about it and how to access it.  Teachers must walk a fine line when talking about these issues with their students.  While it is important to ensure that students know the language of power, it is also important to make sure their home languages are given a place of importance and value.  Teachers should encourage students to expand their language and literacy practices to meet the language demands of school, yet retain their existing language abilities.

Another piece of the toolbox that teachers can use to ensure that students’ home culture and language is incorporated into the classroom is the idea of bridging and contextualizing.  Bridging is a key at the heart of student centered teaching and because of this it is often referred to as constructivist by nature.  Bridging is a type of scaffolding for students in which the teacher tries to connect classroom lessons to their everyday lives, experiences and interests.  By providing these connections for students they will gain a richer understanding of the material presented to them.  An example of this can be teaching students to brainstorm before starting a creative writing piece.  By giving them that time to tap into the knowledge they might know about a writing prompt before having them start writing, you are allowing them to connect to the prompt in a more personal way which will result in a deeper tie to the project.

Somewhat linked to this is the idea of contextualization.  Contextualization is another type of scaffolding that tries to link classroom work and learning with each student’s ability and way of learning.  Rather than presenting academic language as a mass of decontextualized jargon that our students cannot access, teachers need to use the vibrant details that students are used to reading in everyday language interactions.  Gestures, images, body language, tone of voice these are contexts that students have grown use to interpreting to interact with people in their world.  Academically they need to be given the tools to interpret the language as well, genre, tone, registers of academic voice, these are all contexts that can be read, written or spoken in the academic setting.  These do not need to be foreign concepts to introduce to children.  In the example I talked about earlier of being on the phone with my grandma, that can be broken down into context and many students will be able to understand and connect to why I speak differently with my grandmother.  The trick is then to get them to see how that applies to academia.  If they know their audience, like my grandmother on the phone, then the papers that they write can also be written as though they have a particular audience.

The final tool I will touch on today is modeling for your students.  Modeling can be as simple as teaching students what a topic sentence is by creating one of your own or showing them how to do a task.  Especially if there are English language learners in the classroom, your students may devote a lot of time to trying to understand directions.  However this can be combated by a teacher’s use of modeling.  If a teacher takes the time to be explicit in modeling the directions for routine, English language learners will be able to redirect the time they would normally devote to understanding directions to actually participating in this familiar routine.  An example of this would be modeling a three way interview.  Three way interviews have been brought up in several group discussions as a good way to have students work with text.  If a teacher takes the time to explicitly explain a three way interview on day one, it can easily be implemented throughout the school year with less stress to the children.

As teachers we want all of our students to succeed.  In order to succeed in our society they need access to the language of power.  Although every student has an end of success to get to, each student is going to take a different route on the road leading to it.  In order for teachers to be successful in helping our students, we must carry a toolbox that allows us to mix and match our classroom techniques to fit our learners.  Knowing which tools fit what situations will come as you get to know your students and make their learning about them.  Not every tool will work for every child, but by keeping things like register switching, bridging, contextualization and modeling in mind while we teach all subjects, the chances of reaching our English language learners will grow.

Sneaky Finals
December 6, 2011, 4:52 am
Filed under: My Papers

Finals have snuck up on me!  I can’t believe that the first semester of my credentialing year is coming to an end, although I am looking forward to having some time to write all the blogs I have been thinking about writing, in addition to taking some very much needed me time.  Finals shouldn’t be that bad because they are mostly papers, however this is the way my schedule is going: last class tomorrow, last takeover morning/last observation/evaluation meeting with my supervisor and CT on Wednesday, 2 finals Thursday including a website based off of a teaching inquiry and a keynote speech (in addition to the other final), lesson plan paper and final on Friday, task paper and final on Saturday, in class discussion question final on Monday, and another lesson plan paper final on Tuesday.  Thus I am taking this moment to procrastinate.

I have just finished my paper for Saturday and let me just say I was really excited to do these tasks and clinical interviews with two of my students.  I will probably be sharing my paper late, but here is a little taste.  Our tasks had to be based off of a game the children play and how they think about it.  I chose chess because two of my students have been regularly playing this at recess.  For the task they had to do some scenarios with me individually so I could ask them about their thinking on the game.

Here is an awesome screenshot from the video of Richard (pseudonym).  He is waiting for me to set up the scenario, so he decided to make the chess pieces fight for the camera…of course…

The Bishop won

Why Can’t I do the Work for My Level?
November 23, 2011, 5:29 am
Filed under: Dilemmas


I student teach in a 4th and 5th grade split.  There are eight 4th graders and seven 5th graders.  The class is extremely racially diverse, however does not represent many second languages.  I have a student in my class who is a 5th grader with abilities above and beyond others in his class.  Most of the students are somewhere on the grade appropriate reading spectrum, for 4th and 5th grade this is between 0-W.  Many of the students are reading at grade level, but in the earlier letters for that grade.  Shawn reads at level X.  In math the 4th graders are learning angles and the 5th graders are reviewing them because of some holes in their learning from last year.  Shawn’s appropriate math level is algebra.  Often Shawn appears quite bored with regular material and as a result of this as well as ADHD (he has recently started on medication for this and it was a factor for the following incident) he tends to act out during lessons.  During math he typically works from his own math work book that is appropriate for his level.


During one particular day the students all learned a game, Angle Rangle.  While the teacher had made the pairings for this math game, Shawn had just come back from being with the resource person at the school.  He had been looking forward to math all morning.  My CT had pulled names of 4th graders and they were asked to pair up with a 5th grader to do their games with.  Robert picked Shawn as his partner.  Because he had been out I explained what the game was that Robert wanted to play with him.  Immediately Shawn was upset.  “Why do I have to do that? I never have time to work on my math.  Why can’t I do the work for my level?”  Shawn tends to be heavily oppositional. In this moment he was not only defying what his teacher was asking of, but making another student, Robert (who could easily be set off in his own emotional disorders about something like this), feel unwanted as a partner.  I tried to get Shawn interested in the activity by re-explaining it and hoping that he would want to play with his friend.  After a moment of trying this, Robert was called to be pulled out by the resource person. My CT made sure that no feelings were hurt and Shawn was set up with his Algebra, but what do you do in those other moments?


Shawn is a case where yes, he is far ahead of some of the other children in the class and is very proud of that, however he does need a lot of work in some subjects as well as social interactions.  How do you balance when a child needs to be working at a higher level with his need to learn to work with others?  Having Shawn work with other children in the subjects he is less proficient in might help, but would that lead him to be poor at working together in a subject that he is ahead and confident in?  How do I make sure a student like this is getting everything he needs?