One of the benefits of being a substitute teacher is that you get to see a lot of classrooms. some of them are beautiful and inspiring, but other are counter-examples of what you want to achieve. Today I have been in a classroom that could be the epitome of the latter category. I am quite sure I could not teach in a classroom like that. And I actually had problems getting into theaching mood this morning.
Besides the fact that the room had only three walls, the fourth one being replaced by a courtain that separated it from another classroom, it was utterly dirty, disorganized and quite depressing. The walls had a few posters and some art displays with no rhyme or reason, the teacher's desks were covered in papers and other things, under the teacher's desk there was a dirty food container, there was a pile of paper scraps and tissues on the floor next to the sink and the room was always noisy because of the students on the other side of the curtain.
What kind of teaching and learning takes place in a classroom like that? To me it looked like a place in which only soulless and dull activities could be taking place. The walls and the charged tables were shouting "I don't care!".
Friday, November 30, 2012
November 29th, 12
Yesterday I had two jobs, at two different elementary school. In the morning I was in a half day kindergarten class and in the afternoon in a 5th grade classroom. Huge difference of age and abilities, as you can imagine!
Although I have been subbing for two months already it was my first time in a kindergarten class. I was nervous before I got there, not knowing how it would turn out to be. I've always been intimidated by kindergarten because of the children's age and inexperience. I truly admire kindergarten heroes for their ability to work with and organize children so young and that are (some of them) for the first time in an organized institution, with a complex set of rules, procedures and a fixed (and long) schedule. I don't know if I could do it. I probably could, but I find it very difficult and slow. I am more of a fast paced person and I enjoy working with older kids, that are a little more independent and can think and act on their own.
It turned out that I was being nervous without reason, my kindergarten group of 20 was made up only of sweet kids, very quiet and obedient. Not only did I not have any problems with them, but I have actually enjoyed my time with them a lot.
The story was different in the afternoon, with the 5th graders. This time I was not nervous and I was expecting quite an easy afternoon (especially after I saw the lesson plans and the teacher assured me - several times - that she has amazing kids and that I will have no problems with them). They were very talkative and at times silly, especially when they had to work with partners. They were pretty good during individual work and read aloud.
I want to talk about this discrepancy that I do not understand between what the teacher told me to expect and what I actually got. Was it me? Did I do something wrong? If YES, what? Was the teachers expectation unrealistic? How come?
First of all I have to say that I had had this class before, once, when I substituted for the PE teacher in the same school. That particular day this group was my first and by far the most unruly that I had that day. I blamed it then on them being the first group of the day (I wasn't familiarized well enough with the lesson plan) and on my not giving them the instructions that they needed before we started.
Yesterday, when I was in front of the same classroom the students saw me together with their teacher, discussing the plan. They heard her telling them to support me and to help me. She even picked two helpers among the students and introduced them to me. Has this "introduction" to the class had anything to do with the way the afternoon went? I think it is possible. The teacher might have given them the impression that I need help, that I am not good enough to handle things by myself . Or it might be that I did not introduce myself properly and I did not talk at all about my expectations. But again, this situation was brought about by the weird situation in which I found myself, being in the room with the teacher while they were getting ready for lunch, being introduced by her in a hurry etc. How do you act in situations like this? How do you make things better for yourself? How do you approach a class when you substitute for half a day, in the afternoon? And finally: how do you handle misbehavior issues and frustrating interuptions when they do appear?
Let me try a few answers:
To begin with, I think the introductions is really really important. It is that first impression that counts a lot for the rest of the time you are going to spend with that group of students. So as a substitute I need to make sure that I get that right every time. I usually introduce myself by saying a few things about me, my strange name, its origin, my accent and its origin. Then I tell them what my expectations are and it is usually something very general, like respecting the same rules that they usually respect in their classroom, being respectful, helpful, participative etc. I talk about the plan of the day, always mentioning that it has been prepared by their teacher and I mke it clear that the day will not look exactly like it does when their teacher is there because I am a different person and I might do certain things differently.
I try to use the same classroom management techniques that the teacher uses in that classroom (if the teacher has left notes about it). Otherwise I use a lot of positive reinforcement. I always have some type of rewards on me but I rarely use it. I do a lot of praising, instead. Also, I make sure that I teach the students the signal that I use when I want their attention. It usually is CLASS, CLASS, CLASS (answer YES, YES, YES and silence). Students of all ages like it and it works very well. These are the corner stones of what I do every day in another classroom. I wonder what I might be missing and what I could do better.
The question that I need to answer in a more satisfying way than I do now is the last one: how do I handle misbehavior/interruptions when they appear? Yesterday, because the disruptions and the silliness took place during partner work I broke out partnerships and/or moved people around.
What else can I do?
One of the things I can always do and should have done better yesterday was positive reinforcement. Other options:
-Set clear goals when you talk (individually) to the students that are being disruptive. Explain in simple words and short sentences what they have to do, in how much time and what the consequence is going to be if they don't.
-Give choices: it is either this or that.
-Count backwards to the moment when the behavior is going to switch.
One idea that I encountered in my reading and that I liked a lot was that the consequences we should use with the students should always be related to what the student did wrong. For example if the student had the priviledge to work with a partner, outside the classroom and he/she misbehaved the consequence will be that the student is no longer allowed to work with a partner outside the classroom. He will be working by himself, in the classroom, under the close supervision of the teacher. So I have to keep my "eyes" open for consequences of this type, that are not meant to punish, but to fix a problem, to "reward" the student according to his behavior.
Another thing that might have had good results yesterday is having a short and private discussion with each student that was acting out. I could even use a system of conference tickets with the substitute teacher. If I start handing them out other students might feel compelled to behave.
I could have also used some type of table group points system. At the end I would have rewarded the table group with the most points - some type of material reward (although I did not have anything worthy of a fifth grader) or something like 10 minutes free choice time, 5 minutes early release, mention in note to the teacher etc.
In conclusion there are many things that I could have done and that I can do in similar situations. This free writing/reflecting experience helped me a lot and gave me hope that I can find answers to all my questions if only I dedicate them a little time.
Although I have been subbing for two months already it was my first time in a kindergarten class. I was nervous before I got there, not knowing how it would turn out to be. I've always been intimidated by kindergarten because of the children's age and inexperience. I truly admire kindergarten heroes for their ability to work with and organize children so young and that are (some of them) for the first time in an organized institution, with a complex set of rules, procedures and a fixed (and long) schedule. I don't know if I could do it. I probably could, but I find it very difficult and slow. I am more of a fast paced person and I enjoy working with older kids, that are a little more independent and can think and act on their own.
It turned out that I was being nervous without reason, my kindergarten group of 20 was made up only of sweet kids, very quiet and obedient. Not only did I not have any problems with them, but I have actually enjoyed my time with them a lot.
The story was different in the afternoon, with the 5th graders. This time I was not nervous and I was expecting quite an easy afternoon (especially after I saw the lesson plans and the teacher assured me - several times - that she has amazing kids and that I will have no problems with them). They were very talkative and at times silly, especially when they had to work with partners. They were pretty good during individual work and read aloud.
I want to talk about this discrepancy that I do not understand between what the teacher told me to expect and what I actually got. Was it me? Did I do something wrong? If YES, what? Was the teachers expectation unrealistic? How come?
First of all I have to say that I had had this class before, once, when I substituted for the PE teacher in the same school. That particular day this group was my first and by far the most unruly that I had that day. I blamed it then on them being the first group of the day (I wasn't familiarized well enough with the lesson plan) and on my not giving them the instructions that they needed before we started.
Yesterday, when I was in front of the same classroom the students saw me together with their teacher, discussing the plan. They heard her telling them to support me and to help me. She even picked two helpers among the students and introduced them to me. Has this "introduction" to the class had anything to do with the way the afternoon went? I think it is possible. The teacher might have given them the impression that I need help, that I am not good enough to handle things by myself . Or it might be that I did not introduce myself properly and I did not talk at all about my expectations. But again, this situation was brought about by the weird situation in which I found myself, being in the room with the teacher while they were getting ready for lunch, being introduced by her in a hurry etc. How do you act in situations like this? How do you make things better for yourself? How do you approach a class when you substitute for half a day, in the afternoon? And finally: how do you handle misbehavior issues and frustrating interuptions when they do appear?
Let me try a few answers:
To begin with, I think the introductions is really really important. It is that first impression that counts a lot for the rest of the time you are going to spend with that group of students. So as a substitute I need to make sure that I get that right every time. I usually introduce myself by saying a few things about me, my strange name, its origin, my accent and its origin. Then I tell them what my expectations are and it is usually something very general, like respecting the same rules that they usually respect in their classroom, being respectful, helpful, participative etc. I talk about the plan of the day, always mentioning that it has been prepared by their teacher and I mke it clear that the day will not look exactly like it does when their teacher is there because I am a different person and I might do certain things differently.
I try to use the same classroom management techniques that the teacher uses in that classroom (if the teacher has left notes about it). Otherwise I use a lot of positive reinforcement. I always have some type of rewards on me but I rarely use it. I do a lot of praising, instead. Also, I make sure that I teach the students the signal that I use when I want their attention. It usually is CLASS, CLASS, CLASS (answer YES, YES, YES and silence). Students of all ages like it and it works very well. These are the corner stones of what I do every day in another classroom. I wonder what I might be missing and what I could do better.
The question that I need to answer in a more satisfying way than I do now is the last one: how do I handle misbehavior/interruptions when they appear? Yesterday, because the disruptions and the silliness took place during partner work I broke out partnerships and/or moved people around.
What else can I do?
One of the things I can always do and should have done better yesterday was positive reinforcement. Other options:
-Set clear goals when you talk (individually) to the students that are being disruptive. Explain in simple words and short sentences what they have to do, in how much time and what the consequence is going to be if they don't.
-Give choices: it is either this or that.
-Count backwards to the moment when the behavior is going to switch.
One idea that I encountered in my reading and that I liked a lot was that the consequences we should use with the students should always be related to what the student did wrong. For example if the student had the priviledge to work with a partner, outside the classroom and he/she misbehaved the consequence will be that the student is no longer allowed to work with a partner outside the classroom. He will be working by himself, in the classroom, under the close supervision of the teacher. So I have to keep my "eyes" open for consequences of this type, that are not meant to punish, but to fix a problem, to "reward" the student according to his behavior.
Another thing that might have had good results yesterday is having a short and private discussion with each student that was acting out. I could even use a system of conference tickets with the substitute teacher. If I start handing them out other students might feel compelled to behave.
I could have also used some type of table group points system. At the end I would have rewarded the table group with the most points - some type of material reward (although I did not have anything worthy of a fifth grader) or something like 10 minutes free choice time, 5 minutes early release, mention in note to the teacher etc.
In conclusion there are many things that I could have done and that I can do in similar situations. This free writing/reflecting experience helped me a lot and gave me hope that I can find answers to all my questions if only I dedicate them a little time.
A new type of blog
Up to here I had posts from the period in which I have been a student and an intern. I've been substituting for two months and a half now, almost every day, and I have been starting to feel the need for a journal, a place where I can record my experience in the classroom. I think it will be very useful to be able to write a short blog at the end of every day of subbing. There are so many things that I can learn and improve if only I would take the tyme to reflect for at least half an hour. So here it goes, a new type og blog, the one about my substitute experience!
Saturday, March 10, 2012
Things that are Different in my Classroom
During the last quarter I had the chance to observe several other teachers besides my CT. This helped me see how my CT's practice differs from that of other teachers.
One of the most noticeable things about my CT's classroom is the insistence on whispered voice. My CT demands of her students to keep their voices down most of the day and she reinforces this rule all the time. As a consequence, her classroom is one of the quietest in which I have ever been. In other classrooms the teacher is more flexible with the level of noise that she/he can tolerate. In this classroom the tolerated level of noise is very low. It is impressive to see 1st grade students working as quietly as this class does and for many students it is a very beneficial aspect. It makes the students feel safe, it calms them down and it helps them stay focused.
Another thing that is different in my main placement classroom is the way in which the teacher announces the learning target. At the beginning of most lessons she tells the students what the learning target is and she might show it written on a white board and then she asks the children to whisper the learning target to at least two neighbors. I have seen classrooms in which the teacher has the learning target written on the board or announced out loud to the whole group, but this is the only classroom in which the students get to repeat the learning target (at least twice). I think it is a good technique, as it insures that students are paying attention and will be able to remember during the lesson what they are learning and why.
A third aspect that is quite original in my CT's classroom is the way she reinforces the classroom rules by using a system of cards: green, yellow and red (green being for good, appropriate behavior; yellow for less so and red for repeated misbehavior). The cards by themselves would not be very useful, but they are made strong by the star chart that the students have to color in at the end of the day. The color of the card will be represented on the star chart and the chart has to be seen and initialed by a parent every day. The teacher writes a comment next to each star that is a different color than green, explaining to the parents what happened that day in class. I think it is a great way to keep students on task and at the same time to keep the parents informed about the children's behavior.
One of the most noticeable things about my CT's classroom is the insistence on whispered voice. My CT demands of her students to keep their voices down most of the day and she reinforces this rule all the time. As a consequence, her classroom is one of the quietest in which I have ever been. In other classrooms the teacher is more flexible with the level of noise that she/he can tolerate. In this classroom the tolerated level of noise is very low. It is impressive to see 1st grade students working as quietly as this class does and for many students it is a very beneficial aspect. It makes the students feel safe, it calms them down and it helps them stay focused.
Another thing that is different in my main placement classroom is the way in which the teacher announces the learning target. At the beginning of most lessons she tells the students what the learning target is and she might show it written on a white board and then she asks the children to whisper the learning target to at least two neighbors. I have seen classrooms in which the teacher has the learning target written on the board or announced out loud to the whole group, but this is the only classroom in which the students get to repeat the learning target (at least twice). I think it is a good technique, as it insures that students are paying attention and will be able to remember during the lesson what they are learning and why.
A third aspect that is quite original in my CT's classroom is the way she reinforces the classroom rules by using a system of cards: green, yellow and red (green being for good, appropriate behavior; yellow for less so and red for repeated misbehavior). The cards by themselves would not be very useful, but they are made strong by the star chart that the students have to color in at the end of the day. The color of the card will be represented on the star chart and the chart has to be seen and initialed by a parent every day. The teacher writes a comment next to each star that is a different color than green, explaining to the parents what happened that day in class. I think it is a great way to keep students on task and at the same time to keep the parents informed about the children's behavior.
Friday, February 17, 2012
Word Problems, Modelling and Inquiry
Another thing we tried with our third grade buddies was to introduce word problems and to help them think about its solving strategies before actually starting the solving process. It was a completely new approach for me. What surprised me most was the idea of covering the numbers in the problem and asking the students to think about the problem without numbers. I can understand this approach, as many times all that students do is look at the numbers and decide what operation to do. Sometimes they just pick an operation that they are familiar with or that they have been practicing lately in class, without thinking about the meaning of the problem.
It was a difficult task for students, they did not know where to start and how to approach the problem without numbers. The things got much more easier when we showed them the numbers, but instead of telling us what they would do first to solve the problem (as we would have liked), many of them started solving the problem: drawing groups of five objects, counting by 5s etc. How can you stop them and engage them in deep mathematical thinking about the meaning of the problem and the logical steps to be taken if they have never worked this way before? These techniques probably require some modeling and some practice. We need to show the students what we mean by "thinking about the problem without solving it".
The discussion that we had afterwards: modelling vs inquiry, touched upon very important points that we should take time to consider. For me, direct modeling doesn't come naturally, it is not something that I am used to do. I am much more inclined to introduce and conduct a lesson using the inquiry model. I find it more engaging for students and I think that it gives them the chance to learn more, by exploring and discovering by themselves. However, I am becoming more aware of the fact that there needs to be a well thought balance between inquiry and modelling. Not everything can be taught using the inquiry model and not all students benefit from it at all times. Direct instruction is important if we want to level the playing field. There will always be students that understand the new concepts fast and find it easy to work/discover/explore by themselves. But then there will also be students that need time, explicit teaching and a lot of practice in order to get it right. It is about them that I need to think every time I consider whether I should be using direct instruction or inquiry. The question should always be: what works best in this case, for these students?
It was a difficult task for students, they did not know where to start and how to approach the problem without numbers. The things got much more easier when we showed them the numbers, but instead of telling us what they would do first to solve the problem (as we would have liked), many of them started solving the problem: drawing groups of five objects, counting by 5s etc. How can you stop them and engage them in deep mathematical thinking about the meaning of the problem and the logical steps to be taken if they have never worked this way before? These techniques probably require some modeling and some practice. We need to show the students what we mean by "thinking about the problem without solving it".
The discussion that we had afterwards: modelling vs inquiry, touched upon very important points that we should take time to consider. For me, direct modeling doesn't come naturally, it is not something that I am used to do. I am much more inclined to introduce and conduct a lesson using the inquiry model. I find it more engaging for students and I think that it gives them the chance to learn more, by exploring and discovering by themselves. However, I am becoming more aware of the fact that there needs to be a well thought balance between inquiry and modelling. Not everything can be taught using the inquiry model and not all students benefit from it at all times. Direct instruction is important if we want to level the playing field. There will always be students that understand the new concepts fast and find it easy to work/discover/explore by themselves. But then there will also be students that need time, explicit teaching and a lot of practice in order to get it right. It is about them that I need to think every time I consider whether I should be using direct instruction or inquiry. The question should always be: what works best in this case, for these students?
Math with Choral Counting
This week I got the chance to rehearse choral counting again, in a third grade classroom. I led the choral counting with another colleague. It went better than the first time, as the children we worked with were more engaged and task oriented.
This time we chose to count by 6s. I expected it to be a little difficult for the students and I was right, the counting part was more difficult than counting by 3s or 4s. However, when it came to exploring patterns, students were as eager and as surprising as ever in discovering a multitude of patterns. It doesn't cease to amaze me how many things they see in a list of numbers.
Our first stop during the choral count was at number 24. At this point, with only 4 numbers written on the whiteboard, the students were able to make generalizations and predictions. They noticed that all the numbers in the ones place are even and that they follow a pattern involving subtracting 4 on one row and adding 6 on the next row. A surprising aspect was that while I had difficulties seeing this pattern and understanding what the student was talking about, the other students in the group had no problem understanding his explanation. This shows the value of having the students explain their thinking in their own words and letting them rephrase other people's answers. Children have a common language and a way of thinking that sometimes eludes us, the adults. It is important to give them the chance to discuss and to work together to explain new concepts.
By the time we had finished the second row of numbers the students had noticed that the fifth number in each row is a multiple of ten and it is the result of adding 30 to the last number in the previous row. They had also noticed a pattern of even, odd, odd, even, odd for the tens numbers that is reversed in the following row and repeated in the next one. The students were also able to predict if a certain number will be on the list and to name the number on a predetermined position in our list.
The choral counting gave us the chance to rehearse questioning and re-voicing. It was also very useful to be observed and recorded by a group of peers that were able to give feedback at the end.
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This time we chose to count by 6s. I expected it to be a little difficult for the students and I was right, the counting part was more difficult than counting by 3s or 4s. However, when it came to exploring patterns, students were as eager and as surprising as ever in discovering a multitude of patterns. It doesn't cease to amaze me how many things they see in a list of numbers.
Our first stop during the choral count was at number 24. At this point, with only 4 numbers written on the whiteboard, the students were able to make generalizations and predictions. They noticed that all the numbers in the ones place are even and that they follow a pattern involving subtracting 4 on one row and adding 6 on the next row. A surprising aspect was that while I had difficulties seeing this pattern and understanding what the student was talking about, the other students in the group had no problem understanding his explanation. This shows the value of having the students explain their thinking in their own words and letting them rephrase other people's answers. Children have a common language and a way of thinking that sometimes eludes us, the adults. It is important to give them the chance to discuss and to work together to explain new concepts.
By the time we had finished the second row of numbers the students had noticed that the fifth number in each row is a multiple of ten and it is the result of adding 30 to the last number in the previous row. They had also noticed a pattern of even, odd, odd, even, odd for the tens numbers that is reversed in the following row and repeated in the next one. The students were also able to predict if a certain number will be on the list and to name the number on a predetermined position in our list.
The choral counting gave us the chance to rehearse questioning and re-voicing. It was also very useful to be observed and recorded by a group of peers that were able to give feedback at the end.
Sunday, February 5, 2012
Valuable Feedback
For us, as student teachers, all the feedback that we get is very valuable. However, I often find myself forgetting what that feedback was if I do not write it down quickly or if I realize that it is something that I already knew about me and that I am already working on. I think the feedback that works best for me, that I internalize best, is the one that I get after I had some time to reflect on what I did.
One such example that I remember was the feedback that I received from my cooperative teacher after one of the first lessons that I taught by myself. It did not take place right after the lesson, but the next day, after I had plenty of time to go over what I did, to analyze my actions, my words and to criticize myself. By the time my CT asked me what I thought about how my lesson went I was ready with my answers. To my surprise, her interpretation of the lesson was quite different from mine.
While I had been very focused on content and I had tortured myself thinking that I hadn't managed to explain the new concepts as well as I had wished, she was much more interested in the classroom management aspect of the lesson. She told me that the lesson went pretty well, but she told me that I should intervene more quickly when students are not paying attention or when they are doing something else. Her observation opened a new window in my mind. I confessed to her that during the lesson I had been so focused on what I was teaching, and I was so preoccupied with explaining the concepts well and making sure that everybody will get them that I did not notice the students that were talking at the back of the class or the ones that were playing with their whiteboards. She understood what the situation was and she told me that it was quite normal for beginner teachers. However, she advised me to keep the classroom management aspect in mind too while teaching, as it is very important.
I remember her feedback very well because it stems from a completely different interpretation of the lesson. Now, every time I teach I am consciously trying to stay aware of what every student is doing. I am focused not only on what I am teaching and on students' understanding, but also on what each one of them is doing. In short, I am trying to see more than I used to see, to be more perceptive. It is starting to work, but it takes practice.
One such example that I remember was the feedback that I received from my cooperative teacher after one of the first lessons that I taught by myself. It did not take place right after the lesson, but the next day, after I had plenty of time to go over what I did, to analyze my actions, my words and to criticize myself. By the time my CT asked me what I thought about how my lesson went I was ready with my answers. To my surprise, her interpretation of the lesson was quite different from mine.
While I had been very focused on content and I had tortured myself thinking that I hadn't managed to explain the new concepts as well as I had wished, she was much more interested in the classroom management aspect of the lesson. She told me that the lesson went pretty well, but she told me that I should intervene more quickly when students are not paying attention or when they are doing something else. Her observation opened a new window in my mind. I confessed to her that during the lesson I had been so focused on what I was teaching, and I was so preoccupied with explaining the concepts well and making sure that everybody will get them that I did not notice the students that were talking at the back of the class or the ones that were playing with their whiteboards. She understood what the situation was and she told me that it was quite normal for beginner teachers. However, she advised me to keep the classroom management aspect in mind too while teaching, as it is very important.
I remember her feedback very well because it stems from a completely different interpretation of the lesson. Now, every time I teach I am consciously trying to stay aware of what every student is doing. I am focused not only on what I am teaching and on students' understanding, but also on what each one of them is doing. In short, I am trying to see more than I used to see, to be more perceptive. It is starting to work, but it takes practice.
Monday, January 16, 2012
Teaching Math with Quick Images
In my work with first graders and other young students I've had the chance to observe the strategies that students in this age group use to make sense of their math problems. It is sometimes funny and sometimes frustrating to see how much they rely on counting. Some of my students still have to count their fingers on each hand, although presumably they know by now that they have five fingers to each hand. The hard part is that many times counting lets them down because it is easy to make mistakes, especially with big numbers. This is why I was very excited to learn about the Quick Image method. I had not thought about this before, but helping students see groups, and not individual objects, is a very useful tool indeed. I enjoy the opportunity to hear and see different students' thinking when using quick images. The moment when you realize that there are so many ways of seeing and thinking about things in a classroom is very revealing. It is helpful for me, as a teacher, but also for the students, because they get to experience and understand things from a different perspective and they might find new and easier ways to solve problems. I wish I had this opportunity when I was a student.
When we tried Quick Images with our third grade buddies I was surprised to see the variety of answers that we received. A group of four offered an average of three ways of finding the right solution. I was also surprised to see that they were using multiplication quite proficiently, although they are just starting to study it in class.
There was one solution that took me completely by surprise. One of the students used a previous problem and his knowledge of multiplication to solve a problem. Let me illustrate.
First we showed the quick image for groups of three and he saw it this way:
Of course, there were other answers too, like these two:
Then, the next Quick Image that we used was the one for groups of four. The answer that surprised me was this one:
The student used his solution for the previous problem: he noticed that each group has 3+1 dots. He also noticed that there are 6 groups of dots, exactly like in the previous problem. So 3 x 6=18, the previous result, plus the extra one dot for each of the six groups: 18 + 6 = 24.
There it was: a student teaching me to see things in a different way. I loved it!
When we tried Quick Images with our third grade buddies I was surprised to see the variety of answers that we received. A group of four offered an average of three ways of finding the right solution. I was also surprised to see that they were using multiplication quite proficiently, although they are just starting to study it in class.
There was one solution that took me completely by surprise. One of the students used a previous problem and his knowledge of multiplication to solve a problem. Let me illustrate.
First we showed the quick image for groups of three and he saw it this way:
Then, the next Quick Image that we used was the one for groups of four. The answer that surprised me was this one:
There it was: a student teaching me to see things in a different way. I loved it!
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