Melissa: Weeks 4 and 5

These past two weeks have been hectic.  But, I managed to do 4 number talks. 

Week 4:  I decided to switch to addition problems part way through the 4th week because the students were not seeing the value of nontraditional algorithm methods.  I figured this may be because they have not thought of multiplication flexibly.  So, I wanted them to have some success with number talks and I wanted to hear from several different students (as opposed to having two strategies for multiplication).  So, I began with 19 + 26.  All of the classes had several different strategies and the attitude towards number talks seemed positive overall.  New students began sharing and students even asked each other questions.

Week 5: I wanted to build off of the momentum from the 4th week, so I had the students try 28 + 17.  This problem gives the same answer as 19 + 26 but it pushes the students a little farther by using 8 and 7.  In fact, in only one class a student shared the traditional algorithm, but they did it a little different.  So, the students seem to be understanding that the the traditional algorithm is not always the best strategy and they are even seeing similarities in each other methods.  Some students have said things like, "His method is almost the same as mine, but I did ______ instead of _______."  I am going to continue with addition and focus the problems on a certain strategy.  Then, I plan to get back into multiplication.  In one class where 2 people tend to be the main "speakers", seven different students shared their methods and I had to have other students hold theirs because of time.  So, I am beginning to see the students' confidence with number talks increasing.

I do not have questions, at this point, but I do have a couple goals for myself:
- I am working on getting different voices, so I have been saying, "I want to hear from someone whose method has not been on the board, yet."
- Quieting the side conversations while a student is sharing their method.  I think the students think my voice is the only one that really matters, so I am working on having the students see value in each other's ideas.

Tara: Week 3

Reflection on the Week:

In my last reflection I talked about how I wanted to get more information about how my students were thinking about many different kinds of arithmetic problems, so I decided to do subtraction problems this week. Due to scheduling at my school this week, I only had time to do one number talk. In my first two classes I decided to give subtraction problems.

The minute I put the problem on the board, I could immediately read my students’ faces—they were not confident and some even chuckled or mumbled curse words under their breath. When the number talk began, I got four or five different answers in each class. In reflecting on this, I am glad that my students feel comfortable providing multiple responses to a problem that they know there is only one answer to. However, I could tell that students were nervous and not confident to share their strategies. When strategies were shared, students had a really difficult time explaining their thinking. Although several students did use the traditional algorithm, many others did not. However, many students somehow got confused in their thinking about subtraction—whether it was taking away too many and not adding some back in at the end, or trying to use strategies that they used for addition, which did not transfer over perfectly for them in this case. I really pushed students to try and make sense of what other students had done in their heads and where their thinking may have gone astray. However, I could tell that my students were not as interested in this number talk as they had been previously.

This happened in both of the first classes that I did this number talk in. However, for the second two classes I decided to go back to addition. I did not want to lose the steam that I had from the previous number talks and I still wanted students to be excited by the math and the number talks themselves. Hence, I went back to addition. The problems that I decided to use were a variation on the previous week’s number talks, except this time using 100s. The problem I used was 109 + 26. This number talk went much better than the subtraction ones. I had more student participation, more student-to-student interaction, and more enthusiasm from my class. I think that while this problem was more difficult for them, and in fact I did get multiple answers for the solution in both classes, it was something that they had been thinking about previously. Because they had previous experience with this type of problem, they were able to follow along and easily figure out where their classmates may have made an error. They did not have to spend time and brain energy trying to understand the problem itself or trying to understand new strategies presented, as in the subtraction number talks , instead they could focus on methods that they had previously seen and why/how someone may have made an error. I am very glad I decided to switch up this number talk because I think that my students were able to get more out of it, and I think I was able to maintain buy in with these two classes.

Questions that I still have:

1)      How am I going to proceed with my first two classes?

 I understand that while my decision to use subtraction may have caused some students to disengage from the problem, I am hoping to bring back addition this week and see how it goes. I am hoping that brining back an addition problem that can employ similar methods as before will allow students to feel confident again with number talks and will help them gain more buy in with them as well.

2)      For my other classes, should I keep going with a string of similar problems to the ones that I have already done?

I think that I do want to continue using similar addition strings, but add in more complexities such as two triple digit number addition instead of a triple digit and a double digit. I think that this adds a layer of complexity in that students now have to keep track of more numbers and decide how they can move numbers more flexibly. I may continue with one more number talk involving a triple-digit number and a double digit number.

3)      Something I am still thinking about and wondering about is how to get new students to participate. I have had many of the same students participate in the number talks in the past few weeks and I want to be sure that other students are able to have their voices heard. I am wondering if I should do a written explanation at some point soon, just so that all students have an opportunity to share their idea and then I can consolidate results and show them how many different methods were used and how many people chose each method.

4)      Another thing that I am still thinking about is how to get students to try methods other than their own. I think that some students have begun to do this but I am not sure about students who do not feel comfortable sharing their ideas with the class and are a little more quiet. I have tried to ask students to raise their hands as a way to show if they used particular methods once they are up on the board, but this is sometimes difficult to tell if that is really the method that they used or if they are just raising their hand to raise it. Also, I have noticed that some students never vote.

5)      I am also thinking about how I can get some feedback from students about his process so far. I think it could be beneficial to see how they are feeling about number talks. In the next few weeks I might provide them with some kind of reflection piece about number talks just to get a feel for how and what they are thinking about the experience so far. 

Melissa: Week 3

To update, I did not do a number talk during week 2.  I was going to wait until the 4th week to start back up, but I decided to start during week 3, instead.  It was brought to my attention that it would be better to build the norm and routine of Number Talks at the beginning of the year (even if new students are added to my classes, etc).  After much thought, I decided to begin with a multiplication problem.  I chose to begin with multiplication because I felt like the dot talks were thought of as childish by my students.  So, I wanted to give them a more challenging, yet accessible problem.

Reflection
The first number talk of the week was 18 x 3.  Periods 1-3 had three strategies shared, period 5 had 4 strategies shared, and period 6 had 2 strategies shared.  In every class, one student shared the traditional algorithm.  I did not push back on that strategy by requiring the students to make sense of it.  Rather, I introduced the strategy as the "traditional algorithm" and told the students that this is how we have learned how to do arithmetic in America.  I mentioned that this strategy will get more difficult as we increase the difficulty of the problem.  I also encouraged the classes to try a new strategy next time we do a number talk.  When polling the classes on how many people did each method, the traditional algorithm was by-far the most popular method.

It is important to mention that the traditional algorithm was not the first method shared.  Instead, students with other methods shared their methods first, then the traditional algorithm was shared.  It seemed like the students felt as though the traditional algorithm was the correct method.  Comments were made like, "That is too complicated.  Why would they do that?" in regards to non-traditional algorithm strategies.  I eagerly welcomed comments regarding wondering why a method worked and there was some enthusiastic discussion in a few classes.

The second number talk of the week was only done in first period.  I had them do 18 x 6.  I was curious whether the students would try new methods or if they would see a connection between 18 x 3 and 18 x 6.  I was disappointed when only 2 strategies were shared (1) traditional algorithm and (2) distributive property.  Over 70% of that class used the traditional algorithm and no one shared a strategy linking the previous number talk's solution.  I decided to not have the other classes do the number talk because first period did not get through the rest of the lesson.  I need to make more time for the number talks so that way I can get through the entire lesson.  So, I am going to do 18 x 6 in week 4 and then I might give the class a simpler two 2-digit multiplication problem.

When introducing the number talks this week, I did not use the phrase "quiet thumbs" and I seemed to get more students to participate.  However, there were a few students that put down their fist as soon as I wrote the problem on the board.  I cannot say why they did this, but it is my goal to get all of the students to do the problem and use the thumb indicator.

Also, I want to get different students to share their ideas.  There are already students who regularly share and I want to help all students to feel comfortable sharing their ideas.  I think this will improve as the classroom culture develops and the students become more comfortable with one another.  However, I am wondering if I could use another strategy to encourage those that are nervous to share while still maintaining the structure of the number talk.

Tara: Week 2

Reflection on the week:

This week I decided to continue number talks with addition problems. I chose addition because after experiencing number talks a little bit last year, I realized how difficult multiplication was for them and thought that starting with addition would be a good way to help them build their confidence with number talks and sharing their ideas. Additionally, because I have so many ELL students, it was also a way to continue to have them build their communication strategies as well as work really hard to listen to each other. I thought that addition might be a better choice for this because there was a higher likelihood that they would understand what their classmates did and how it was being described. 

For my first number talk of the week, I did problems of the family: 19 + 31. Below are some screen shots of 

the different classes' strategies. What I found was very interesting! In two of the four classes, the traditional algorithm did not come up by students who were describing their methods. However, in these two classes, I decided to introduce it because I wanted to see if anyone used it. As I introduced it, I began by saying that this is often the way that we learn how to do arithmetic. However, algorithms are meant to be steps that we follow, but are often difficult for us to understand how and why they work. Also, when get into bigger problems, algorithms can confuse us because it is difficult to keep track of all the different numbers if we don't understand where they are coming from. I also said that it is a strategy, and if they used it, that is okay! 

But, I also said that I am going to push them to understand and try other methods as well. When I went through each of the strategies to ask students to raise their hands if they had used that method, there were quite a few people to raise their hand for the traditional algorithm.

In the other two classes, the traditional algorithm did come up and in both cases I did not put emphasis on the method, but did describe the same thing about algorithms and I did in the other classes. Additionally, I pushed the students who were describing how they used the traditional algorithm to use words like "I carried the 10" not "I carried a 1" and "I then added 10 + 20 + 10" and not "I then added 1 + 2 + 1." (As seen in period 1). 

In addition to the traditional algorithm however, I saw someone in every class move around the 1 from the second number to the first number, making the problem a little more easy. Another strategy I saw often was adding the ones and the tens separately. In one of the classes I saw someone round 11 to 10 to add it more easily as well. 

Period 4:

Period 2:

 Period 1:

 The second number talk I did problems in the family of 19 +36. Below are some screen shots of the classes' strategies. I saw similar strategies in this problem as well. One thing that I do want to mention however is that  many of the students have begun to share their ideas, even if they are extremely similar methods as those already mentioned. I am SOOO excited by this, and I have made it a point that I want to hear similar strategies as well. For example, even if a method seems the same but students added in a different order, this is different and I let them know that I want to honor that.

Period 3:

Period 1: 

 Period 4:

Questions as I go into next week:

Overall, I was extremely pleased with how the number talks went this week. I have not had any push back from students and have not heard any comments like: this is too easy or this is for little kids. I'm not sure if it's the way that I framed number talks in the beginning of the year or if these students are being nice and playing along, but the students do not seem like it is a hassle for them or like it is a waste of their time. 

Questions I am left with are: How can I keep up their enthusiasm and positive attitudes about number talks?

How can I help students who have not yet participated feel comfortable sharing?

How can I begin to have students build upon other methods they are seeing?

With all this in mind, this coming week I am taking some advice from some wonderful resources, I have decided that maybe I need to get to know a little more about how students are thinking about numbers using other operations as well. So, I'm thinking that I will do a subtraction problem to see how students are thinking about subtraction as well. 

Melissa: Week 1

I am posting late.  My first week teaching and doing number talks in my classroom was last week.  I faced many challenges and a few successes.  To begin, I am teaching 3 "regular" Geometry classes (composed of 10th-12th graders) and 2 "Freshman" Geometry classes.  Basically, the freshman classes are supposed to lead to the advanced math track with the students taking Calculus their senior year.
Introducing Number Talks
I decided to introduce number talks, to my students, as a way to develop their ability to think flexibly and efficiently about numbers.  I told the students that number talks are a way for them to perform mental math in a way that is more efficient (we talked about what efficient means).  I decided to tell the students this because the school I work at does not have a culture of accepting multiple methods in math classrooms.  Math is taught VERY traditionally.  Therefore, I wanted the students to buy into doing number talks, and I believed the introduction I gave was a way of doing so.  
I also gave my students two goals.
1. Actively listen.
2. Really explain what you were thinking.  I told them that it may seem obvious, but we need everyone in the class to understand and be convinced of our methods.
Basically, I am trying to get the students to see the importance in having a variety of strategies and approaches.
Reflection
My intention was to do two number talks in the first week of school.  I was going to start with two dot talks for the same reasons that Tara mentioned.  I wanted the students to understand the structure and to build their confidence to share their thoughts.  I did the first dot talk on Tuesday in every class.  The number of people whose ideas where shared ranged from 2-5 students.  Also, I felt like the students were super hesitant to do "quiet thumbs." A good number of students in every class half-heartedly put their fists to their chests and some refused to.  My guess is that quiet thumbs are not "cool" so the students are hesitant to get involved in something that seems elementary.  Also, the students seemed content to hear a small number of strategies.  I think this may be related to the math culture at the high school.  It has not been emphasized that math consists of different ways of thinking.  This is going to be something that I need to create in my classes.  So, for the second number talk, I only did it in one class.  The results will similar to those I saw on the first number talk. Finally, I have not had a consistent roster in any of my classes.  I receive new students and lose current students every day.  I was told that this will continue to happen until the end of the third week of school. Tuesday of my second week, I completely lost my first period "regular" geometry class and was given another teacher's freshman geometry class.  This has made it difficult for me to establish norms because I have new students every day.  Therefore, I have decided to wait to do number talks until the 4th week of school.  That is when the schedules will be finalized and I will have consistent rosters.  When I start back up during the 4th week I am going to start with a two digit addition problem.  My students number sense seems to be lower than the students at Fremont HS.  They really struggle with negative numbers, basic algebra, etc, so I think number talks will REALLY help my students.  I am excited to fully integrate number talks into my class!!  Oh, and, another teacher observed one of my number talks and she said "It was really neat."  She asked if she could see a pile pattern task I was doing, but she came too early and she got to see the number talk as well.  Hopefully other teachers get curious and want to do them :)

Wow!  That all sounds really negative.  I do think that the number talks went well in a few ways.  First, the students did have different ways of seeing the dots and all of the classes there was more than one student that shared.  Also, the day after the number talk, a student in one of the freshman geometry classes wanted to share his way of thinking about a problem (which was different than another student's way), so I felt like he valued multiple methods.
Questions   
I am wondering how I get the students to buy into using the quiet thumbs.  I explained the reasoning behind doing it, but it did not seem to be enough.  Is it normal for students to resist it right at the beginning?  I want to stick with using the quiet thumbs because I believe it is the best way, but I also want the students to fully participate.
   

Tara: Week 1

This week marked the beginning of my first year of teaching, and additionally the first week that I used number talks in my classroom. I will begin by describing what I did in my classroom, followed by a reflection, and questions that I have going into the upcoming week.

Introducing "Number Talks"

I decided to sell the idea of number talks to my students as a way to build their flexible thinking by seeing and trying to understand multiple ways of thinking and doing basic arithmetic. I gave them two goals over the course of the year:
1) Push themselves to offer their strategies and methods out loud in class
2) Work on actively listening to their peers' methods as a way to help build their flexibility with numbers
I began this week with two dot talks. My reasoning for doing this was twofold. First, I wanted to build their understanding of the structure of number talks, without the added pressure that arithmetic often brings. Secondly, I wanted them to get into the habit of sharing, listening, and trying to understand other students' ideas and thoughts.

Reflection on the week

I have four classes that I am doing number talks in. Three of which are sophomore classes; these students are taking IMP2, Interactive Mathematics Program. One of which is a freshman class, taking IMP1. 

I was surprised at how mixed my classes were in their reaction to this activity. In the first two classes, both my sophomore classes, they participated, but less willingly. I had to wait to have students raise their hands, but I did eventually have students share their ideas. In the next two classes, students were very eager to share their ideas and to help me understand their classmates ideas. There were several times where I was unsure about how a student counted the dots and needed some help from other students. They jumped at the chance to explain to me what they thought their peer was thinking. I was so impressed by their ability to listen and focus during the dot talks. 

I do feel lucky that my students are all exposed to used to listening to different methods and I think they have already begun building the idea that understanding and seeing multiple methods can help you understand topics even better. So, I am hoping that number talks can reinforce that idea and help them gain more practice understanding, communicating, and thinking in new ways. 

Below you will see pictures of the blackboards for the first dot talk.

Period 1 (10th grade):

Period 2 (10th grade):

Period 3 (9th grade):

Period 4 (10th grade):

Questions as I go into next week

How can I maintain the enthusiasm of my 3rd and 4th period classes as we go into doing number talks with arithmetic?

How can I try and further the sharing and enthusiasm of my first two classes? Also, are they too nervous to share? Or, have they not bought into the idea of number talks? 

Additionally, the native language of over 90% of my student population is Spanish. Could this be affecting some students' willingness to share their ideas?

I think I want to start with addition next week, but am still having trouble deciding on which problem to begin with. I don't want to give them a problem that is too hard, because then it may further shut students down or curb the enthusiasm of others. On the other hand, I am afraid if I make it too easy, they will think it is too childish. 

Right now I am definitely leaning towards giving them an easier problem because I want everyone to be able to access it, and I don't want them to feel bad if they struggle with it. I'd rather it be too easy and then move to a more difficult problem if they are comfortable with it. 

Knowles Conference Poster

Tara
I was able to present a poster at a Knowles Science Teaching Foundation Summer Meeting about the work Melissa and I had done with Cathy on number talks during the past year. This is some information that I collected in preparation for this presentation and also some thoughts, questions, ideas that I had after talking with other fellows, teachers, and guests at the conference.

Information presented on my poster: 

http://www.scribd.com/doc/101264458

Below are two articles mentioned in the information from my poster on benefits and reasons for doing number talks in high school classrooms as well as one book that was really helpful in our journey this year.

Study by Eddie Gray and David Tall and about the importance of flexible thinking skills in student achievement in mathematics: http://www.scribd.com/doc/101263006

Article in NCTM Mathematics Teacher by James Shultz and Michael Waters (2000): http://www.scribd.com/doc/101265088

A very useful and interesting book Cathy suggested that Melissa and I read, is Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris. Not only does this have a very clear rationale for the importance of number sense, but it also provides strings of numbers for the different operations (+, -, x, /) that might be useful if you decide to try number talks!

Questions and thoughts that I have following my conversations:

• How can I meaningfully integrate number talks into my classroom without having it be separate from the content of the day? Is this necessary? Or is the process itself worth the time?
• What operation should I start with this year? Addition or multiplication? I think subtraction and division are definitely something that I will want to work towards with students, because they are more difficult to explain and record as well as being more cognitively difficult (or at least I think so!)
• How can I structure conversations or use sentence starters for ELL students? (In my classroom I will have a large percentage of ELL students)
• How are number talks beneficial for the teachers who do them?? This is a question that I heard during the poster session and am thinking seriously about! I have, so far, been critical of how number talks can specifically benefit high school students themselves; however, I truly believe that number talks changed the way that I question in the classroom--a skill that I think is essential to creating a classroom of true inquiry and depth. I think that before I started number talks, it was sometimes very difficult for me to think of meaningful questions to ask my students in class and ways to push their mathematical thinking. However, as I progressed in the series of number talks that I did, I realized that I was growing in the types of questions that I was asking and in how I was posing questions. For example, to a student who solved 8 x 15 by splitting the 8 into 2x4 and multiplying 4x15 = 60, then 60 x 2 = 120, I would ask, "Where did that four come from? In the original problem, I see an 8 and a 15, but where does the four come from?" Questions that may have been obvious to the student explaining, but may not have been obvious to me or their peers. Additionally, it helped me build a repertoire of more open-ended questions that I found were great especially if I was confused about a student response. For example, the phrase, "Can you tell me a little more?" I found to be really useful in situations where I was confused and wanted the student to elaborate on their thinking

          In addition to growing in how I asked student questions, I also grew in other aspects as well. For example, I think I was able to really improve in facilitating large group conversations that everyone was supposed to be involved in (a really difficult task). For example, one thing that I learned was to begin getting students to speak directly to one another. In certain number talks there were times when I asked someone else to rephrase or repeat the strategy of another student. During this time, often my students would start with, "Is she/he saying .....?" Instead of turning to that original student to respond to that question, something both Melissa and I started doing was having the student who was repeating a strategy, restate their question, but directed toward the original student. So, instead of "Is she/he saying...." they would now turn to their classmate and ask, "are you saying....?" For me, I think this was a way to begin having students talk directly to each other, reliving some of the belief that the teacher knows all and would be able to answer this question. Instead, they were now relying on their peers to make sense of a strategy instead of on the teacher--for me, an essential idea I want to build into my classroom.

• Are number talks most beneficial to students who are earlier on in their high school careers or are they just as beneficial to older students in  more advanced courses? 

This is a question that I think often about. And I don't know if I have yet formed an opinion about this. However, I think that number talks can offer so many different benefits and can be used to promote specific goals as well. For example, I might see using number talks in a higher level math course to have students see the value in approaching even basic problems from different levels and also to have discussions about efficiency in choosing a representation or method in mathematics more generally. Also, I think they can be beneficial to any class in building discussion norms and giving students practice and the ability to speak in front of their peers as well as practice listening to the methods of others. (This was also a question I got: Can you ensure that all students are following along? What strategies are there for this?)

As I continue thinking about number talks and how I am going to implement them in my classroom this coming year, these are all questions and thoughts that I will revisit and reevaluate as the year progresses. Any comments, ideas, and/or thoughts about any of these questions and ideas would be greatly encouraged and appreciated!!