tag:blogger.com,1999:blog-20497741339905636232024-02-18T17:56:41.469-08:00Number Talks in High School: Dilemmas and SuccessesMelissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-2049774133990563623.post-78686421365322477862012-11-28T10:36:00.002-08:002012-11-28T10:38:23.677-08:00Tara: Weeks 10-1210th grade<br />
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During these next few weeks, I continued with multiplication for my 10th grade students, but crossed into double-digit numbers and even some triple digit multiplication. </div>
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It was interesting to see how the students reacted the first time I put up a double-digit multiplication problem. You could literally hear sighs in the classroom! However, as I continued to do them, I got less and less push back on the difficulty. What I found is that the students who offered their answers were able to transfer strategies from the earlier number talks to these more difficult multiplication problems. </div>
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One struggle that I still have is that I believe a few students are really struggling with their basic math facts and this is seriously inhibiting their ability to participate in the number talks. I can think of at least 5 students who do not have their basic multiplication facts memorized (like 6x8). I'm not quite sure how to have them still participate fully during this time and at the same time continue to push the other students in the class to think more deeply about multiplication. </div>
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Another thing that I ran into during these number talks was that students were not distributing when they were breaking up numbers. For example, the number talk 15 x 13. There were some students who wanted to break up the 15 into (10 +5) and the 13 into (10+3) and then they wanted to multiply. However, they were not distributing when they multiplied so they were only performing these operations: 10 x 10 + 5 x 3. </div>
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This brought up a good discussion in each class and I was able to introduce the area model for them to see why we had to distribute. </div>
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Below are some of the number talks with two digit multiplication. </div>
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1st period:</div>
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2nd period:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLz4KqvWM1IVOBy1zE1Z-73hbktFT7vkO35UDaWxrSF_D2QODzVPwB10SQxDJthEk3Bj84l07sNX8Hwy9Pj5AnVoPyxQp7LicZ5NvNHoFd7LP63UMqAFJGgNbbugNKkbz4ofSc8I09jcCw/s1600/mult2dig2nd.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLz4KqvWM1IVOBy1zE1Z-73hbktFT7vkO35UDaWxrSF_D2QODzVPwB10SQxDJthEk3Bj84l07sNX8Hwy9Pj5AnVoPyxQp7LicZ5NvNHoFd7LP63UMqAFJGgNbbugNKkbz4ofSc8I09jcCw/s320/mult2dig2nd.jpg" width="320" /></a></div>
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4th period:</div>
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1st period:</div>
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4th period:</div>
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I am planning to stay on multiplication throughout the rest of the semester, however am transitioning to using money to have my students think about bigger numbers. For example, I will do problems like $1.99 x 6.<br />
I am hoping that they will use strategies they have already seen to think about these types of multiplication problems and I am hoping they will also come up with new strategies because of the different context of money.<br />
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Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com0tag:blogger.com,1999:blog-2049774133990563623.post-2113351945104960772012-11-25T13:30:00.001-08:002012-11-25T13:30:20.193-08:00Melissa: Weeks 11 and 12I started to see a little more success during these weeks. There was more participation from students that do not typically share. The students seem to be gaining confidence and there were even positive remarks from the students. <br />
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First Period: All Freshman Geometry. This class was given to me several weeks into the school year so we are still working on providing other methods besides the traditional algorithm. <br />
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Second Period: 10th-12th grade Geometry. This class is one of my more challenging classes. The class is predominately male and it is not "cool" to have high academic status. So, assigning competence has proved itself to be challenging. Specifically, David F. has very interesting and valuable ways of thinking, but he tends to act out. So, I was pleasantly surprised to see David volunteer his method. Luis and Guadalupe rarely volunteer their ideas during whole class discussions, as well. And, Garrette regularly shares and he has high academic status, but he was not listening during the number talk and he shared Guadalupe's strategy. Hence the "Not paying attention" note.<br />
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5th Period: 9th Grade Geometry.</div>
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6th Period: 10th-12th Grade Geometry. This class' culture is very similar to my second period's. Again, I was pleasantly surprised by the students that shared their thinking. Valentin is an EL senior student that seriously struggles in math. While he did show the traditional algorithm, I was happy that he was the first one to share his ideas. Amanda is also a senior and she has an IEP. And, Denzell is a junior that has high social status. I haven't done number talks with this class because they are my most challenging class. I have six students with IEPs in this class and the classroom management has been a constant struggle. After this number talk, several students mentioned that they had never thought about "borrowing" like Denzell and Amanda did. Most of the students did the traditional algorithm when I polled them, but they were very enthusiastic about seeing the other strategies. After this, I decided that this class needs to do number talks just as much as the other classes. And, when I did a number talk the following week, more students raised their hands for methods besides the traditional algorithm.<br />
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Looking forward, I am going to start percents. I am anticipating more challenges, but I am going to remain positive. Because the high school I work at serves a very rural and low income demographic, I am still thinking about how I am create a number talk that will build on their experiences (like we did last year with the milkshakes - my students do not go off campus for lunch and the town they live in is in the middle of the desert so there are not really any common restaurants that I can use). I am going to talk to another teacher to see if there is something he thinks I can use. I will update on how it goes!Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com0tag:blogger.com,1999:blog-2049774133990563623.post-77530342117326261202012-10-22T14:30:00.001-07:002012-10-22T14:30:42.899-07:00Melissa: Week 10Before I begin my reflection on Week 10 I want to talk about what happened during weeks 5-9. I inconsistently did number talks during those couple of weeks. There were a few weeks where I did not do a number talk at all or other weeks where I did a number talk in one class and not in others. I wish there was a good reason for why I did not do them, but there isn't. I think a lot of the reason is because of time. During those weeks, I had the number talks on my lesson plan but ended up not doing them. I guess I did not make it a priority. Also, I felt like my students did not see the benefit and had negative attitudes about number talks. So, I saw it as another challenge I would have to deal with in the period. Rather than facing the challenge and trying to overcome it, I guess I ran away. However, I still see great benefits in number talks and I want my students to enjoy them so I did one number talk last week (109 + 26). I wanted to give the students a problem that all students would feel successful on, but that was enough of a challenge for the students that have been successful earlier.<br />
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<b>Reflection</b><b> </b><br />
In all of the classes, the traditional algorithm was the most used strategy. However, it is not a strategy the students choose to share in front of the class. When polling the students on which strategies they used, the traditional algorithm gets the most hands. However, there are many different methods students use that break numbers down and are flexible ways of thinking. Because I had a long lapse between doing number talks, I am not surprised at the amount of students that still do the traditional algorithm. In the same vain, I noticed that the same students are the ones to volunteer. However, more students raised their hands to share out the answer and I did get a couple new kids to share their strategies. The main issue I faced was that students did not seem to see the value of hearing other people's strategies. The students I work with are mainly motivated by getting a "grade" for work they have done. In general in my class they will ask, "Are you collecting this? Are we going to get a grade?" They only want to do work if they will get a grade for it. So, it is hard to motivate them to participate and be involved in things that are not graded. Going into this next week, I am going to continue on with addition because the students are still not confident sharing and I want more students to try other methods besides the traditional algorithm. I am going to consistently do number talks, regardless of challenges that may arise.<br />
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<b></b><br />Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com3tag:blogger.com,1999:blog-2049774133990563623.post-44307930451024881342012-10-14T21:15:00.002-07:002012-11-28T10:16:40.269-08:00Tara: Weeks 4-9Wow! As the first quarter of teaching wraps up for me at school, I have finally had some time to sit and reflect further on number talks.<br />
I have been consistently doing them anywhere from one time to three times a week depending on our schedule. I have had several challenges and several successes during the past 5 weeks and I want to share some of these now.<br />
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Number Talks with Addition</h3>
I have taken my number talks in two different directions depending on my classes. As I have discussed before, I began with addition problems, mainly focused on problems that could employ strategies such as rounding, moving numbers flexibly (for example 19+ 34 is the same as 20 + 33), and breaking down numbers into tens and ones.<br />
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Here are some highlights of the boards from some of the number talks with addition. I spent 6 weeks doing addition with my students. By the end, students were getting much more comfortable using strategies besides the traditional algorithm and were bringing up the same methods during each number talk. On the other hand, I would say only about 60% of my students have shared their strategies. I have been pushing new students to try and explain their methods, but it seems like the same students are constantly wanting to share. This is still a challenge that I am facing and I hope that I can work towards pushing these quieter students to eventually share their ideas.<br />
9th grade class:<br />
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10th grade, Period 2<br />
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10th grade class, Period 1<br />
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10th grade class, Period 4<br />
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10th grade classes--Multiplication Number Talks</h3>
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In my 10th grade classes, I decided to take my number talks in the direction of multiplication. I decided to do this for a few reasons. First, in these classes we are doing a unit on area and volume and I thought that doing problems with multiplication could fit nicely into making number talks fit a little more closely with the curriculum. This coming week I plan on introducing the area model for multiplication as I think that it can connect some of the things we have learned about area with the number talks. </div>
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Secondly, I decided that multiplication would be a nice transition from addition--it can be a little more challenging, but I was hoping that students would be able to use some of the strategies that they had for addition with multiplication--primarily seeing numbers as tens and ones and being able to break up numbers to make the problems more simple. </div>
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So far, I have seen students using strategies such as those mentioned above and have been impressed with students' thinking about the multiplication problems. In my first set of number talks on multiplication, not one student mentioned the traditional algorithm, although I'm sure some students still used this method and it has come up several times since then. I presented the problems such as </div>
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Below are some of the boards with the number talks with multiplication. </div>
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10th grade class, period 1</div>
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10th grade class, period 2</div>
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So far I have not used multiplication problems with two double digits and that is my plan for this coming week. I think that this will lend itself to introducing the area model for multiplication as well. </div>
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<h3>
9th grade class: Percentages</h3>
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For my freshman class, I decided to do percentage problems with number talks. The reason I chose this is because they are currently in a unit on probability. I thought this would be a good way to strengthen their idea of percentages and flexibility in working with percentages. Additionally, when we did percentages last year at Fremont, my students seemed to be scared of percentages initially, but I think that my students definitely had a breakthrough in thinking about percentages during these number talks. I am excited to see how my current students are thinking about percentages and if they have a similar experience. </div>
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My freshman were pretty nervous when they first saw me put up a percentage problem and there were many groans around the room. However, I think that some students have really started to catch on to some of the strategies of their classmates. I had one number talk where 5 different students shared their ideas (see below). Again, a challenge is still getting all students to participate in sharing their thinking. I have not yet done a pair share with my students and I think this is something I want to try this week. Or, another idea would be to have them record their own strategies and collect them so that I can see how individual students are thinking about these problems as well. </div>
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Below is one of the number talks done with percentages:</div>
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Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com1tag:blogger.com,1999:blog-2049774133990563623.post-73848819752905521562012-09-17T10:24:00.001-07:002012-09-17T10:24:25.969-07:00Melissa: Weeks 4 and 5These past two weeks have been hectic. But, I managed to do 4 number talks. <br />
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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.<br />
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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. <br />
<br />
I do not have questions, at this point, but I do have a couple goals for myself:<br />
- 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."<br />
- 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.<br />
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<br />Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com2tag:blogger.com,1999:blog-2049774133990563623.post-20533739466965054872012-09-04T20:15:00.004-07:002012-09-04T20:15:49.191-07:00Tara: Week 3<br />
<h2>
Reflection on the Week:</h2>
<div class="MsoNormal">
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. </div>
<div class="MsoNormal">
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.</div>
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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. </div>
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<h2>
Questions that I still have:</h2>
<div class="MsoListParagraph" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]-->1)<span style="font-size: 7pt;">
</span><!--[endif]-->How am I going to proceed with my first two
classes?</div>
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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. </div>
<div class="MsoListParagraph" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]-->2)<span style="font-size: 7pt;">
</span><!--[endif]-->For my other classes, should I keep going with a
string of similar problems to the ones that I have already done? </div>
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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. </div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]-->3)<span style="font-size: 7pt;">
</span><!--[endif]-->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. </div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]-->4)<span style="font-size: 7pt;">
</span><!--[endif]-->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. </div>
<div class="MsoListParagraphCxSpLast" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]-->5)<span style="font-size: 7pt;">
</span><!--[endif]-->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. </div>
Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com0tag:blogger.com,1999:blog-2049774133990563623.post-44126048741732642802012-09-03T11:47:00.001-07:002012-09-03T11:47:46.499-07:00Melissa: Week 3To 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.<br />
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<span style="font-size: large;">Reflection</span><br />
<span style="font-size: small;">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.</span><br />
<br />
<span style="font-size: small;">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 <i>correct</i> 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. </span><br />
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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.<br />
<br />
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.<br />
<br />
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.Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com2tag:blogger.com,1999:blog-2049774133990563623.post-11543750872468645362012-08-28T18:31:00.000-07:002012-08-28T18:31:47.512-07:00Tara: Week 2<h2>
Reflection on the week:</h2>
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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. </div>
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For my first number talk of the week, I did problems of the family: 19 + 31. Below are some screen shots of
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<div>
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! </div>
<div>
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.</div>
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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). </div>
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<div>
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. </div>
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Period 4:</div>
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Period 2:</div>
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Period 1:<br />
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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.<br />
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Period 3:<br />
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Period 1: </div>
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Period 4:<br />
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<h2>
Questions as I go into next week:</h2>
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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. </div>
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Questions I am left with are: How can I keep up their enthusiasm and positive attitudes about number talks?</div>
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How can I help students who have not yet participated feel comfortable sharing?</div>
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How can I begin to have students build upon other methods they are seeing?</div>
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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. </div>
Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com0tag:blogger.com,1999:blog-2049774133990563623.post-16989581021659251522012-08-22T19:19:00.001-07:002012-08-22T19:19:54.857-07:00Melissa: Week 1I 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.<br />
<span style="font-size: large;"><b>Introducing Number Talks</b></span><br />
<span style="font-size: large;"><span style="font-size: small;">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. </span></span><br />
<span style="font-size: large;"><span style="font-size: small;">I also gave my students two goals.</span></span><br />
<span style="font-size: large;"><span style="font-size: small;">1. Actively listen.</span></span><br />
<span style="font-size: large;"><span style="font-size: small;">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.</span></span><br />
<span style="font-size: large;"><span style="font-size: small;">Basically, I am trying to get the students to see the importance in having a variety of strategies and approaches.</span></span><br />
<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><b>Reflection</b></span></span></span><br />
<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><span style="font-size: small;">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 :)</span></span></span></span><br />
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<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><span style="font-size: small;">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.</span></span></span></span><br />
<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><span style="font-size: small;"><b><span style="font-size: large;">Questions</span></b> </span><b> </b></span></span></span><br />
<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><span style="font-size: small;">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.</span></span></span></span><br />
<span style="font-size: large;"><span style="font-size: small;"><span style="font-size: large;"><span style="font-size: small;"> </span><b> </b></span> </span><b> </b></span>Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com1tag:blogger.com,1999:blog-2049774133990563623.post-61361579421329997132012-08-16T21:21:00.002-07:002012-08-16T21:21:41.864-07:00Tara: Week 1This 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.<br />
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Introducing "Number Talks"</h3>
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:<br />
1) Push themselves to offer their strategies and methods out loud in class<br />
2) Work on actively listening to their peers' methods as a way to help build their flexibility with numbers<br />
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.<br />
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Reflection on the week</h3>
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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. </div>
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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. </div>
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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. </div>
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Below you will see pictures of the blackboards for the first dot talk.</div>
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Period 1 (10th grade):</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtBs_OJ83ajl2PLv-tgKKLFaDl66pyC-Tj_9JxYmelSOJ7qIqlEyq1_401IaBmM7Y-iCquPKlr4ZE9Ms5lkGVyCLKUw_Zk_tf2nonKEQkVux97VRBcB4Mj4FM8a4-3zxGGMBXuB1KdHScy/s1600/week1_per1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtBs_OJ83ajl2PLv-tgKKLFaDl66pyC-Tj_9JxYmelSOJ7qIqlEyq1_401IaBmM7Y-iCquPKlr4ZE9Ms5lkGVyCLKUw_Zk_tf2nonKEQkVux97VRBcB4Mj4FM8a4-3zxGGMBXuB1KdHScy/s320/week1_per1.jpg" width="320" /></a></div>
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Period 2 (10th grade):</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKc1uK42TtuHnbrX8Z84TgAA_Elx09v5dpZplUGOukVO8rmKkZLIrd75TqBEvlXMJhsN3f7hDLtqnzsVST1aYbR8BogYlEPmSdfC9t_UlLWnSg6lTI5Ull-Q_8JPWAkfhWOAxC1qkqBX9f/s1600/week1_per2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKc1uK42TtuHnbrX8Z84TgAA_Elx09v5dpZplUGOukVO8rmKkZLIrd75TqBEvlXMJhsN3f7hDLtqnzsVST1aYbR8BogYlEPmSdfC9t_UlLWnSg6lTI5Ull-Q_8JPWAkfhWOAxC1qkqBX9f/s320/week1_per2.jpg" width="320" /></a></div>
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Period 3 (9th grade):</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNNLVWPq6WTE7Zg7tjEqBQubWFiIvyjpP_kO1jVn8BhHKkbOhzlvKEfOyxyhceKfq0OM6EjQ-ORyVvtmFwzX_qIMBrUppdRWRl-qE704xSDkUSmhZHMU_6JCBTsWF7J_v135qb2Pvu2uuo/s1600/week1_per3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="238" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNNLVWPq6WTE7Zg7tjEqBQubWFiIvyjpP_kO1jVn8BhHKkbOhzlvKEfOyxyhceKfq0OM6EjQ-ORyVvtmFwzX_qIMBrUppdRWRl-qE704xSDkUSmhZHMU_6JCBTsWF7J_v135qb2Pvu2uuo/s320/week1_per3.jpg" width="320" /></a></div>
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Period 4 (10th grade):</div>
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Questions as I go into next week</h3>
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How can I maintain the enthusiasm of my 3rd and 4th period classes as we go into doing number talks with arithmetic?</div>
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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? </div>
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Additionally, the native language of over 90% of my student population is Spanish. Could this be affecting some students' willingness to share their ideas?</div>
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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. </div>
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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. </div>
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Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com3tag:blogger.com,1999:blog-2049774133990563623.post-72840733679728903962012-07-27T20:13:00.000-07:002012-07-27T20:16:07.201-07:00Knowles Conference Poster<b>Tara</b><br />
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.<br />
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Information presented on my poster: </div>
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<a href="http://www.scribd.com/doc/101264458" target="_blank">http://www.scribd.com/doc/101264458</a></div>
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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.<br />
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Study by Eddie Gray and David Tall and about the importance of flexible thinking skills in student achievement in mathematics: <a href="http://www.scribd.com/doc/101263006" target="_blank">http://www.scribd.com/doc/101263006</a></div>
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Article in NCTM Mathematics Teacher by James Shultz and Michael Waters (2000): <a href="http://www.scribd.com/doc/101265088" target="_blank">http://www.scribd.com/doc/101265088</a><br />
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A very useful and interesting book Cathy suggested that Melissa and I read, is <i>Building Powerful Numeracy for Middle and High School Students</i> 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!</div>
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Questions and thoughts that I have following my conversations:</div>
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<li><span style="background-color: white;">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?</span></li>
<li>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!)</li>
<li>How can I structure conversations or use sentence starters for ELL students? (In my classroom I will have a large percentage of ELL students)</li>
<li>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</li>
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<span style="background-color: white;"> 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.</span></div>
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<li>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? </li>
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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?)</div>
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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!! </div>
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<br /></div>Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com0tag:blogger.com,1999:blog-2049774133990563623.post-41803188365556444672012-06-22T12:09:00.000-07:002012-06-22T12:31:52.089-07:00<span style="font-size: large;">When 5 x 12 is More Than 60: Exploring Number Talks in High School, </span>Presentation at Stanford Teacher Education Program Conference<br />
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These are the slides from a presentation we gave as part of out teacher education program. The slides have note several studies that show the benefits of number talks for high school students. Additionally, the slides lists some of the dilemmas that Melissa and I faced when implementing 10 number talks in our student-teacher placements this past year.<br />
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<a href="http://www.scribd.com/doc/97954719/Number-Talks-in-High-School-STEP-Conference" target="_blank">STEP Conference Presentation</a>Melissa + Tarahttp://www.blogger.com/profile/06784380737392701963noreply@blogger.com1