Wednesday, November 26, 2008

What's Next?

November 27th, 2008

Well, I have come to the last few days of my official blogging. (My course is over today!) I have enjoyed this experience as I find it very helpful to reflect, in writing, on my practices and the ups and downs I have experienced in this whirlwind journey. I am no where near where I want to be in terms of teaching through problem solving, but I feel I have come so far since I began. I am still working through some things such as how to ask the right questions, especially in class when you are on the spot teaching, and how to plan for this. Marian Small talks a lot about planning for questioning and has done some research on how to plan your math lessons to ask the right questions. This is something I will do more research on. I also feel I do not always know how to bring students where they need to be in terms of attainment and achievement of outcomes. I am resisting the urge of just telling them, but there are times when I do not know where to go from there.

The biggest change I want to see is for me to throw away the actual units of study that is now guiding me and helping me to choose investigations, and go toward a more holistic approach where I am not doing problems involving number patterns for two weeks, and then geometry for five and then on to something else. I want to be able to pose problems that does not necessarily achieve outcome 1, 2 and 3, in unit 4, for example, but pose problems in a way that at the end of the year students will have achieved all outcomes. In thinking about the logistics of this, I wonder then how would my assessment be different. How would I go about communicating student progress to home? Again, something to think about.

As I continue to learn how to teach math through problem solving, I know I will face more challenges, and more hiccups. I am confident I will work my way through these problems and be a better teacher for it. My students are doing very well thus far, and have come a very long way since September. They are becoming more independent and starting to pose their own questions more frequently. I want them to become better questioners and become more confident in themselves and their abilities to do math. I will continue to blog the happenings in my class, and please feel free to comment when you feel necessary. I would love to know what others think about what I am doing and how I am doing it. Please contact me with any questions or comments. I would love to hear from you.

Problem Solving at it's Finest

November 24th

After spending a few days talking about area and perimeter, I decided it was time to move on to getting students to understand the connections between length, width and area and how one influences the other. At the beginning of class I wrote two questions on the board, the first, a review of what was taught in the last few days, and the second, something a little different. Here are the two questions:

Mrs. John said that she had two different sized patios built onto her house, but they both had the same area. Can Mrs. John be correct? Prove it!

Mrs. John’s desk is the same width as Alex’s desk, but Alex’s desk is three times longer than Mrs. John’s. What can you say about the area of both desks? Use pictures, words and numbers to explain your thinking.

Question number one was just meant to be a means of assessment of the previously taught material and to get a sense of student reasoning and communication.

I would like to spend a few minutes talking about question two. Before today’s class, I did not do any work involving the influence of changing the dimensions on the area of the shape. We did, however, look at what happens to one of the dimensions when there is a fixed perimeter and the other dimension changes in the Changing Garden’s problem. I was hoping students would think back to this problem to help them solve the current one.

As usual, I spent a few moments discussing the context of the problem ensuring everyone knew what they had to do. I demonstrated, using my desk, that the widths of the two desks were the same, but the other desk, Alex’s desk was three times longer than mine. I entertained any questions and then let them go.

I was very pleased to see that all students were able to infer that three times the length required them to multiply the length of my desk by three to obtain the length of Alex’s desk. I seen all of the students using diagrams to sort out their thinking, which was a breakthrough for me as this is something, using illustrations to help solve the problem, I have been emphasizing all year. By using these diagrams students then decided to find the area of the two rectangles. Three students automatically seen the connection that the area was going to triple. They were so excited about this discovery that they went ahead and began questioning what would happen if they doubled both the length and the width. Of course, I let them go ahead and investigate that. The other students kept working on this and many, but not all, seen that the area tripled. The class ended with two students not being able to complete the question and it was left for next class.

I feel this illustrates learning math through problem solving at its finest. I did not tell students what was going to happen, nor did I lead them through conversing or probing or questioning. They discovered these connections on their own and when asked what would happen if the width, for example, tripled, they were quick to tell me that the area would also triple. I then knew they got it. But then I wanted to see how well they actually knew it, namely why the area was doubled when the length doubles. This will be the focus of my next lesson.

Discovery

November 21st, 2008

Today, I began a new unit on geometry. To begin, I posed questions to find out what they knew about length, perimeter and area of rectangles by using a problem called Changing Gardens. This problem can be found in Navigating through Measurement in Grades 3-5. (see reference list for details.) This required students to recognize that when given a fixed perimeter, there are different possibilities for the dimensions of the rectangle, leading them to recognize that the areas can change. At the end of the activity, there were several open ended questions that helped to lead students to make the necessary connections about length, width, perimeter and area of rectangles. Student’s responses to these questions enable me to plan the next activity for this unit. From these questions, I saw that there were still a few students whose ideas about area and perimeter were sketchy. Their concept of area and perimeter were not concrete, not where I thought they should be at grade six, and as a result, I knew I could not proceed in the way I had intended. Instead, I would focus my next lesson(s) on doing activities that would reinforce the concept of area and perimeter.

If I to give students a page or two, asking them to look at a figure, maybe a square or rectangle, and tell me what the area and perimeter was, I am confident they would be able to do this. This would lead me to believe, and themselves, that they understood what these concepts were. Hey, there is no real thinking involved, once you know how to do it. It wasn’t until I asked them to describe how they knew the area they had calculated was correct, or why, when the length of the rectangle decrease, the width had to increase with a fixed perimeter, that their ‘real’ understanding of these concepts came to light.

What would have happened if I did not give them these types of questions, or had them complete this activity? I would have continued with the lessons as planned and assumed that because they got the right answer; they knew what was meant by these terms. They would still be playing the game, and I would not have known the difference.

Communicationing and Refining

Nov. 17th, 2008

Student understanding of mathematical concepts is what teachers strive for, or at least, what teachers should strive for. Classrooms that see students interacting, communicating and helping each other solve problems, is what I would argue, a classroom where students actually understand the mathematical concepts being taught. Marian Small also agrees. Small (2008) notes that “it is through interactions with other students as well as with the teacher, and with the opportunity to articulate their own thoughts that students are able to construct new mathematical knowledge” (p.4). Small (2008) goes on to note that it is in these classrooms, one in which she calls constructivists classrooms, “that students are given the opportunity to develop richer and deeper cognitive structures related to mathematical ideas and students’ development of a level of mathematical autonomy”(p.4).

Often times, what you read in a book as theory, may seem really out to lunch. “yeah, it may be alright for them to say that, or do that in their classroom, but it would never work for me.” Well, I would like to demonstrate what this looks like as I take you, once again inside my classroom.

Please keep in mind we have only been engaged in learning mathematics through problem solving for a little less than two and a half months. The particular class I am about to describe was a concluding class after doing investigations with using patterns to explore division of 0.1, 0.01 and 0.001.

The question was posed at the beginning of class that went something like this : Frank divided 42.8 by 0.1 and got an answer between three and found hundred. How do you know his answer is incorrect? Show your thinking using pictures, numbers and words.

I talked a little about the problem, making sure everyone understood what the question was asking. I then let them go and begin solving the problem. What happened next, illustrates what Marian Small has just described.

Students began talking through the problem with their partners, they used things like Base Ten blocks, calculators (yes, I allow them to use calculators), graph paper and pictures to begin making sense of the problem. Most, if not all used Base Ten blocks to work out their thinking. I was hearing students correcting others thinking and explaining how they know in their own ways, talking about their own strategies in solving the problem. Once I seen that everyone had some type of answer written, I brought the class together as a whole and began discussing what they had found out.

I began by asking students to explain the problem, telling me what was required. I asked particular students to tell me how they thought about the problem. One student said she thought of it as money and you wanting to split $42.80 into dimes. Another student said that they thought of it by using Base Ten blocks and said that because they were dividing the number (42.8) into tenths, they would use a rod as a whole one and the unit block would represent a tenth. At this point I seen some students were a little lost, so I stopped. I asked another student, the one with a blank face, to explain what was just said in his own words. He began trying to make sense of what the other student was saying and as he was playing with the base ten blocks in his hands, things were beginning to make sense. Other students were helping this process of understanding evolve, understanding the concept of using a unit to represent one tenth and the rod would then equal one whole. With the support of the class and the use of the base ten blocks and conversations, the reluctant student understood why a unit was used as a tenth to represent this problem. He was then able to recognize that he would need 42 rods and 8 units to represent this number. I got him to come to the overhead (which is a part of every math class) and show the class what he had just discovered. But there was a problem. This student quickly realized there was not 42 overhead Base Ten blocks to put on the overhead. He looked at me so innocently and said, “but Miss, there’s not enough.” I laughed and said, you are right, so what are you going to do about it. He quickly recovered and said, well I know that a there are 10 rods in a flat, so I could use 4 flats. He then places 4 flats on the overhead. I asked him to look at the number he had on the overhead and say what it was. He said 400. Before I got a chance to respond, another student quickly corrected him and explained to him where he made the mistake. She pointed out that because he was using the unit as a tenth, the rod had to be a ‘one’ and the flat was now 10, so four flats would be 40. (Bingo, I now know that she understood the concept!) Then the reluctant student, who is now demonstrating a better understanding of this place value concept, placed two more rods and 8 units on the overhead and confidently said, “there you go miss, 42 and 8 tenths.” I looked at him as seriously as I could and said, prove it to me. After a little look, he went into detail about how he was able to come up with this arrangement. Once his explanation was complete, I looked at the class and asked if anyone agreed with him or if anyone would like to challenge him. I could see that everyone had the same base ten arrangement and they all agreed that this student’s explanation was correct.

So, from here, to ensure they really understood this, I posed the question: Why are we using the unit block as one tenth and why the rod would be equal to one? One of my weaker students, looked at me and simply told me because we are breaking the number into tenths miss, and a unit is one tenth of a rod, so that means all we have to do it count how many units or tenths there are and we have our answer. Well, I was a little blown away, to say the least. If my weakest student could think of it in that way, I knew something right was happening.

To keep the momentum going, I then asked the class, what would Frank’s answer be if he did divide 42.8 by 0.1. At once, they said 428. Eureka! One student stated,’ well miss we knew it was going to be that because dividing by tenths is just like multiplying by ten and 42.8 ‘times’ ten is 428. Horray!!

For the sake of time and space, I did not include all the conversations that was had during this class. More students were asked to come to the overhead in front of the class and explain how they went about solving the problem, or to prove that what was just said was indeed, true. The point I want to make is that through much interactions, conversations and student talk, students were given the opportunity to think about this problem, to talk about the problem and to talk through the problem. We used students’ lack of understanding as a springboard to help them, and the rest of the class, to have a deeper understanding of the problem and the concept being taught. I, as their teacher, did not intervene or disrupt this thinking, even if it was incorrect. They worked together to collaboratively ‘fix’ any misconceptions about this concept and used their own invented strategies and thinking to explain how they seen it to others. This type of classroom can be achieved, and this is proof that teaching through problem solving do work.

Reasoning and Communication

November 13th

Today I posed two different questions, an open and a closed. I wanted to once again see how these two questions compared in terms of what information it gave me about student understanding. I am noticing also, that open questions are giving me a lot of assessment information on student reasoning and communication skills, something that closed questions do not give me. Here are the two questions:

Mrs. J has a cell phone. She has a plan that allows her to talk for 1000 minutes a month. She gets her bill at the end of the month, and as she is scanning through, she spills coffee on it, blurring out some of the numbers. She notices that she talked for 345 minutes the first week, 210 minutes the second week, and 534 minutes the fourth week. The amount of minutes she was on her cell phone for the third week was missing. How many minutes was Mrs. J talking during week three? If she pays 5 cents a minute for her phone, how much was her bill?

#2. Mrs. J’s cell phone plan allows her to talk for 1200 minutes. As she is scanning her bill she notices that she talked the most in week four, the least in week two and about the same in weeks one and three. If Mrs. J has about 100 minutes left at the end of the month, what are some of the possible times Mrs. J could have talked in those four weeks? Mrs. J’s cell phone bill totaled 129.95. Knowing that she only pays 5 cents a minute, how did she know this was not correct? Explain.

I took about ten minutes to explain the two problems and entertained any questions students had. I gave students the entire class to work on the problems, where I collected their work and then compared their answers. As expected, the open question tasks allowed students greater access to the problem as every one of them was able to come up with appropriate answers, some even went as far to play with the numbers to see how many different combinations they could come up with. Students also did well with the closed question, but four out of ten students did not come up with an appropriate answer to the problem. Many, perhaps six of them had difficulty deciding what it was they had to do and they became frustrated, which may have contributed to the incorrect solutions. Using the open questioning technique, it required students to show me how they went about figuring out the problem. As a result of this, I was able to assess their level of thinking, reasoning and communication. I found that only three of my ten students were performing at a level four based on the rubric used in the CRT’s. All other students were scoring a level three with one student scoring a level two. This concerned me, but after doing this activity, I now see where I have to focus my teaching for this time being. I need to work with these 7 students to help them build their communication and reasoning skills. I intend to do this by using more open ended questions having the content scaled down a little so they will only need to focus on communicating their thinking and not on how to actually do the problem. This type of information would not have been gathered, nor would I have been able tole to realize I needed to work more closely with seven of my students to improve their reasoning and communication skills, if I had to only rely on the closed question.






November 7th, 2008

Today, I decided I would revisit multiplication to get a feel on how students’ understanding was with the concept. Throughout the last month, students have been using multiplication strategies to solve and work out various problems. Today, I wanted to use open ended questioning and assess their understanding of this very important concept. I began class by posing a question – I asked students to tell me a time in the last two weeks when they used multiplication outside the classroom. Some of them literally turned white, while others immediately raised their hands waiting to be called on. I called upon those that volunteered an answer and we discussed ways in which they had used multiplication. Some of their responses were good as it truly represented a multiplication situation, while others did not. I was convinced the activity I had planned was worthwhile.

I asked students to get in a group of two and take a piece of chart paper and a marker. They were then asked to think about everything they had learned about multiplication and write a really good multiplication question on their chart paper. I told them it could not be something simple, such as Bill has two dolls, each doll has two feet. How many feet? I encouraged them to think about their lives to come up with a question that they would have to face in the real world that would involve using multiplication to solve the problem. Then, I let them go. I did not intervene or suggest or guide. I wanted to see what they could come up with. I felt this activity was going to give me the necessary information about whether they really understood the concept of multiplication. Giving an example and coming up with their own problem, to me, is a good way to assess their real understanding.

Most groups were successful, with one group posing a problem that primarily involved division. I did not interject here as I wanted to see if other students would pick up on this. I then stapled each piece of chart paper to different places on the wall and had the pairs visit each question and answer it in a different way than was done previously. As students approached the division question, most recognized that it was not multiplication. Some began working it out by multiplying, but when they looked at their answer, they seen it was not reasonable. Some knew how to go about answering the question, and two groups even did it as an open frame by writing the multiplication sentence with the missing data as a part of the factors.

Students completed all questions, then the bell rang.

I found this really distracting. I wanted to engage discussion about the various problems and about the ways in which students solved the problems. Leaving it until tomorrow, in my mind, would be of little benefit as it is now fresh in their minds about how they went about solving the problem. I know how important it is to discuss activities at the end of the problem, but when you only have certain time slots allotted for math, and especially when students have to leave to class to go to another, it is very hard to have the much needed conversations.

My own research

November 5th, 2008

Well, after discovering that students did not perform as well on the procedural question as they did on the conceptual question, I really began thinking. I really started to wonder where these kids were with regard to their performance in math compared to where the class I had last year been at this time. This led me to decide I was to give a test. Yes, a test. At this point, we had covered outcomes from the first two units. I decided to take the two unit tests I gave last year and mesh them together to create one. I looked at the final product and was not happy with what I saw. The two tests that I gave last year were about 85% procedural with 25 questions that were closed, (multiple choice, fill-ins, and short answer questions) with only two conceptual questions. Oh my gosh! What was I thinking? Those poor kids. How was I really able to assess their UNDERSTANDING? So, I took some of the more procedural questions and made them more open, using the different strategies as identified previously. I did this to ensure I had an even number of procedural and conceptual, as I wanted to see how they performed on the procedural compared to conceptual as well as how they compared to last year’s class. (I know this is not a real valid test, but I think the results say something.) I gave the test without students knowing about it before hand, and ironically, and honest to God, the class average of this year’s grade 6’s was the exact same as last year’s class.

Let’s take a closer look into this. Last year, students completed many questions that were very similar to the test questions; they were given an assignment, a study guide and two hours for review. This year’s class was not given any of this and they were still able to perform as well, percentage wise as last year’s group. Actually, I would argue that this years class knew more of the concepts than last year, given they had no review or notice of the test.

Given this information, I then studied the test more closely. I wanted to see how students did procedurally versus conceptually. I looked at the assessment and calculated the percentage correct on procedural tasks and then calculated the percentage correct on conceptual questions. I then broke it down further into boys and girls. Here are the results:

Boy

Girl

Procedural Correct

69%

79%

Conceptual Correct

67%

80%

From this table you can see that girls are outperforming the boys both procedurally and conceptually. This coincides with Boaler (2002) in her study as she suggests that a problem based approach to teaching math may in fact favor girls’ disposition and quest for understanding math, and not just playing the ‘math game’, as boys sometimes are seen to do. Boaler (2002) discusses the dissatisfaction of girls at Amber Hill School, where math is taught in a more traditional approach. Boaler (2002) state that “women tend to value connected knowing, characterized by intuition, creativity, and experience, whereas men tend to value separate knowing, characterized by logic, rigor and rationality” (p.138). Boaler (2002) then goes on to note that as a result of doing many studies and other data sources she is convinced that “it was this desire to understand, rather than any difference in understanding, that differentiated some of the girls from the boys” (p. 140). I see this in my classroom as the girls are always engaged in their tasks and always trying to understand why this is or that. One girl in particular has said to me on numerous occasions that she likes math a lot better this year because she finally understands what is going on. The boys, on the other had, also resemble what Boaler is talking about. I see them wanting to just finish their work and to them, it seems that it doesn’t matter if they understand or not. I asked a student (male) one day if he understood why the pattern worked in the particular way it did, and he replied “Not really miss, but I have it done, and now I don’t have to do it for homework.”

Am I concerned that the boys are not performing as well as the girls? Am I worried that using this approach is favoring one gender over the other? Yes I am, but according to Boaler (2002), she insists that although the boys in her study stated that they did not enjoy the open approach, “there were no significant differences in the achievements of girls and boys” (p.150). I have to keep believing this approach will work for both boys and girls.

Conceptual vs. Procedural

October 30th, 2008

Many studies, such as the Amber Hill and Phoenix Park study conducted by Jo Boaler, examines different teaching approaches and their affect on student learning. Boaler (2002) suggests that students who are taught in a problem based context “do not have a greater knowledge of mathematical facts, rules and procedures than students taught in a more traditional classroom. This concerns me, due to the fact I am teaching Grade Six and as we all know, Grade Six = CRT’s, but I believe done in the right way, students will learn the necessary procedure and rules through their investigations and problems throughout the year. That being said, I would like to share an observation I made while students were working on two different types of pattern problems, one being an open type and the other, a more closed type of problem. It was through these two different questions that I wanted to investigate for myself if my students were lacking in their procedural knowledge.

I gave a closed question that involved them looking at a number pattern; find the rule and extend it. The other question, the open questions, involved them showing me to find as many ways as possible to show how the number 25 could go through an input-output machine and come out as 77. To my surprise, 80% of my students were able to show me at least three different ways to get 77 from 25, whereas only 30% of the class was able to extend the pattern correctly and give the appropriate rule. So, my class was able to answer the conceptual question with a higher accuracy than the procedural problem. I thought this was very interesting. This leaves me with questioning my own beliefs about what I think is most important – kids being able to remember facts, procedure and rules or to be able to understand they why’s of math and be able to apply it to other situations. Ultimately, in the end, if students understand the material, they will be able to use it with or without knowing the exact procedures and rules, won’t they?

October 28, 2008

We have been working on patterns over the last few day, how to extend patterns, finding the pattern rule by investigating patterns in a problem based context. For this particular outcome (solving problems using patterns), I decided I would group my class by ability and get three different problems for them to work on. Once the groups solved their problems, they were to compile their work and present it to the whole class explaining the problem and how they went about solving it.

I put Brad* in a group on his own and got him to work on patterns in the hundreds chart. I thought this would work well for him as he was having a lot of difficulty with multiplication, factors and even odd and even numbers. I introduced the problem to him, one where he had to find patterns in the chart and design a quilt based on the patterns (this lesson can be found in the Navigation Series – Navigating through Algebra in Grade 3-5 by NCTM, see side bar for reference information). For my other group, I gave them a problem involving a growing worm where they had to figure out the pattern of how it was growing. I wanted to challenge the other group. I chose the problem “Tiling a patio” which involved them working with square numbers, area and perimeter. This was the first time that everyone in my class was not working on the same problem. I thought this worked really well as I was able to target their abilities and give them a problem that I knew they would have success with, but also, one that challenged them. They liked the fact that they were working on different problems and I made sure they knew that I chose that particular problem with them in mind.

One particular thing I noticed about this set up was with a student I will call Kerry*. Now,Kerry is what I would consider an average student. It may take her a little while to catch on to the concept, but once she got it, she got it. She was in the middle group, in looking at the growing worm. I chose Mike* and Ben* to be in this group with her, as they were all on the same level of understanding. Kerry, along with her partners worked on this problem and finally, she solved it and seen the pattern. She was able to explain to me in good detail why the problem worked the way it did and how she came about solving it. At this point, the two boys still did not understand. I took this opportunity to have Kerry take the lead and try to help the boys understand the problem. I thought by giving her the leadership role in this situation it would help her confidence and communication skills. Instead, she got royally ticked with the boys and their inability to ‘get’ the problem. She was frustrated with them that they just did not understand. I stood back from the situation to see where it would go. She eventually gave up trying to explain it to the two boys and wanted to get out of the group. In a journal writing activity completed later, I asked students to talk about one thing they really liked about math this year, and one thing they didn’t like, and ironically, Kerry, in talking about what she didn’t like about math, talked about being put into a group that she felt was not ‘as good in math as her.’ She talked about how she felt really frustrated to be in that group and how much it bugged her that she was made to try and explain it to the boys. I thought it was interesting to see how Kerry reacted when she was put into the opposite situation that she normally finds herself into. I think this may have been a good lesson for her, I guess we’ll wait and see.

Patterning

October 28, 2008

We have been working on patterns over the last few day, how to extend patterns, finding the pattern rule by investigating patterns in a problem based context. For this particular outcome (solving problems using patterns), I decided I would group my class by ability and get three different problems for them to work on. Once the groups solved their problems, they were to compile their work and present it to the whole class explaining the problem and how they went about solving it.

I put Brad* in a group on his own and got him to work on patterns in the hundreds chart. I thought this would work well for him as he was having a lot of difficulty with multiplication, factors and even odd and even numbers. I introduced the problem to him, one where he had to find patterns in the chart and design a quilt based on the patterns (this lesson can be found in the Navigation Series – Navigating through Algebra in Grade 3-5 by NCTM, see side bar for product information). For my other group, I gave them a problem involving a growing worm where they had to figure out the pattern of how it was growing. I wanted to challenge the other group. I chose the problem “Tiling a patio” which involved them working with square numbers, area and perimeter. This was the first time that everyone in my class was not working on the same problem. I thought this worked really well as I was able to target their abilities and give them a problem that I knew they would have success with, but also, one that challenged them. They liked the fact that they were working on different problems and I made sure they knew that I chose that particular problem with them in mind.

One particular thing I noticed about this set up was with a student I will call Kerry*. Now,Kerry is what I would consider an average student. It may take her a little while to catch on to the concept, but once she got it, she got it. She was in the middle group, in looking at the growing worm. I chose Mike* and Ben* to be in this group with her, as they were all on the same level of understanding. Kerry, along with her partners worked on this problem and finally, she solved it and seen the pattern. She was able to explain to me in good detail why the problem worked the way it did and how she came about solving it. At this point, the two boys still did not understand. I took this opportunity to have Kerry take the lead and try to help the boys understand the problem. I thought by giving her the leadership role in this situation it would help her confidence and communication skills. Instead, she got royally ticked with the boys and their inability to ‘get’ the problem. She was frustrated with them that they just did not understand. I stood back from the situation to see where it would go. She eventually gave up trying to explain it to the two boys and wanted to get out of the group. In a journal writing activity completed later, I asked students to talk about one thing they really liked about math this year, and one thing they didn’t like, and ironically, Kerry, in talking about what she didn’t like about math, talked about being put into a group that she felt was not ‘as good in math as her.’ She talked about how she felt really frustrated to be in that group and how much it bugged her that she was made to try and explain it to the boys. I thought it was interesting to see how Kerry reacted when she was put into the opposite situation that she normally finds herself into. I think this may have been a good lesson for her, I guess we’ll wait and see.

Multiplication PROBLEMS

October 21th, 2008

Today I wanted to get into multiplication strategies. I read a book entitled “Amanda’s Amazing Bean Dream” which is all about a girl, named Amanda, who loved to count everything and finally realizes that using multiplication can help her to count faster. A lot of my students have trouble with the whole concept of multiplication and I thought this would be a good place to start with getting their conceptual knowledge caught up with their procedures in multiplication. I read the book and we discussed different multiplication scenarios in the book. I used different open questions like asking particular students to look at the pictures and tell me a multiplication story and why it is multiplication. Most were getting good at that, and then I broadened it to telling me a multiplication story involving the classroom. Up until this point, I thought they all, finally, knew what multiplication meant. Until, they could not give me a multiplication scenario that involved them or the classroom. Oh, oh!

So what did I do? My mind went to Boaler’s (2008) book, specifically, to the excerpt by Sarah Flannery, about the importance of conversations in math. I was letting students talk about math, multiplication in this situation, but not in the right way. All they were doing really was giving back to me what I had asked for, a simple multiplication sentence that involved a picture in a book which, on every page were similar. There were rows of books on so many shelves, how many books? There were rows of windows on so many floors, how many windows? There were 4 wheels on a bicycle, so many bicycles, how many wheels? There was no thinking needed. They could simply regurgitate the answer that I wanted.

So, I put the book down and we began talking. We talked about the book, but it wasn’t about the pictures of rows or columns, it was about the math that was going on there. We talked about how Amanda loved to count and how she did not want to use any other method of counting, only the way she already knew. We talked about counting and the different things that we counted each day, money, points in a game, step to the bus, minutes until the bell rang and so on. From there we talked about how as numbers started getting bigger Amanda was faced with the challenge of counting these larger numbers faster. We then talked about their experiences with wanting to count faster or using larger numbers. They came up with things like estimating how much money we spent on school supplies (this one made me happy as they were using one of their problem solving strategies – think of a similar problem), how many meals we have eaten in our life times and how many days we have been alive. (These last two made me happier as it gave me ideas for the next day). Then we talked about how Amanda finally realized that multiplying numbers together did make things easier. This lead into a discussion about how in order to multiply you must have equal groups of something. I then asked the students to think of a multiplication question/situation that could be used in the classroom and asked them to write in on a piece of paper.

* Each and every one of those students was able to come up with an appropriate question that involved multiplication.

This class today opened my eyes to how important conversations and discussions really are. As I am going through my problems and investigations, I sometimes don’t take enough time to talk about it after. I am realizing the power in these discussions and how it is through these discussions that can help students make the necessary connections and understandings to the new material that was learned in the problem. This is something I am going to have to work on.

Talking, is it important?

October 18th, 2008

I am very pleased with how well things are going. I think I have come a long way since September when I first began teaching through problem solving. I am not sure who is learning more, the students or me!

As I experiment with open questioning more and more, I see the benefits of this type of questioning and how it fits into teaching through problem solving. I am also realizing how important conversations and discussions are. It is through conversing with one another and with me, that students are finding the connections they need to make sense of the concepts. I find it really helpful to encourage a student to talk through his/her answer or thinking, especially when they have come to an inappropriate conclusion. Most times, as they begin talking through or explaining their thinking, they realize their mistake and are able to come to correct conclusions. It is really good to see this happening as I, as their teacher, can see how they are thinking about something and how they were able to reach the appropriate solutions. This makes instruction easier as I do not have to guess or assume students know something, but I can plan my next lesson knowing exactly where student understanding on the concept is. This is something, I feel can not be achieved by teaching math in the traditional way, nor by using closed tasks and questions. In traditional classrooms, students are just expected to produce the correct answers. In my classroom, I expect students to go beyond giving me the correct answer and explain how they were able to come to that solution. They need to prove to me what they are saying is true. I require students to show their thinking using numbers, pictures and words, and it is through this explanation that their true understanding of the topic, concept can be seen, and corrected, when needed.

So many ways to plant a garden!




October 16th, 2008.

The planting a garden activity resumed as students picked up where they left off. I gave them about 20 minutes to finish up and then brought the whole class together to discuss their findings. I got each group to come forward and explain their methods and procedures and any conclusions they came to. This was the hard part for me, to give up taking the lead in the discussions and step back to encourage student lead discussion.

As students began explaining their methods and procedures, they began writing the numbers on the board that they investigated. It happened so naturally and so beautifully – without them realizing it, students began isolating the prime numbers from the composite numbers on opposite sides of the board. Once the presentations were over, we were left with a very detailed list of composite numbers and prime numbers. At the very end of this lesson I introduced the words factors, as being the length and the widths and prime as being the numbers or area that there was only one way of ‘doing’ it and composite as being the areas or numbers with multiple ways of getting it. I then used closed questioning to ensure they had understood this by asking them to tell me, if for example, 15 was prime or composite, or to list the factors of 30.

I really enjoyed seeing students work through this activity and for the first time this year, seen how an open ended task can lead to students entering it in different ways. Each of the five groups thought about and attacked this problem in a different way, but all came to the same intended understanding about factors, and prime and composite numbers. Some did have a more deeper understanding, but everyone demonstrated and communicated their understanding of these outcomes. One thing I am learning through this whole process is the importance of making the tasks open and equally accessible to all students. Boaler (2002) discuss Phoenix Park’s belief that in order to achieve equity for all, teachers must work with students to ensure they understand the problem, as they decided how to support student’s needs. Boaler (2002) notes that teachers never left students alone to understand what the text based problem and always made sure everyone in the class knew how to at least start the problem. By explaining the problem, as I did in this case with the potato garden, all students were able to take it and run with it without being caught up in trying to understand what they question was semantically asking.

AHA!

October 14th, 2008

Reviewing some of the articles on using and creating open ended tasks, and thinking about what I have been learning about teaching through problem, I come across the suggestion to make the task in such a way that allows students to solve or explain the problem/solution in more than one way. Hmm, factors involve multiplying two numbers; composite numbers are numbers that have more than two factors, what in math do students already know that involves multiplying two numbers together that can be done in more than one way. Then it hit me, AREA! Create a problem that involves students looking at different areas and the dimensions (length and width) that make up that area. The areas that have only one set of factors would be prime and area that have more than one set of factors would be composite where the actual dimensions, length and width would be the factors.

So, now a problem. Well, it is late fall, almost Thanksgiving and people are digging vegetables, how about looking at possible dimensions and areas for a vegetable garden. So here is the question I posed to my grade six class:

My dad would love to grow vegetables. He really wants to get into gardening but is not sure how to go about planning the garden. He knows he would like to have a rectangular or square shaped garden, but does not know how big to have it. He has a few choices of places where he could plant his garden and would like to know all the possible dimensions his garden could be. He particularly wants to know about the areas that do not give him a lot of choice into how he can create the garden. Can you help him?

I modeled the problem on the board to ensure everyone knew what they had to do. I used the number 64 and questioned if dad wanted a garden with an area of 64 meters squared, would he have different ways of planting the garden. Students recognized that you could get different ways to plant a garden of area 64 meters and we began listing the factors, which at this point I did not identify them as factors, but as lengths and widths.

I then let students go about looking for the different areas of gardens and the possible dimensions. I paired students by ability and let them work anywhere in the classroom. I provided them with different maniplulatives such as multi-link cubes, square tiles, wooden sticks of varying lengths, graph paper, calculators and base ten blocks.

As students began their investigations, I noticed each group entered the problem at different levels and in different ways. One group was haphazardly choosing numbers randomly and finding all the possible dimensions for that area through trial and error of multiplying. Another group was starting at the area of 1 and working through each number finding possible dimensions. Yet another group was using graph paper to draw out the different possibilities of areas and using the calculator to check their answers. Another group was using the square tiles to figure out how to build rectangle and squares and recorded their findings. Soon, I started seeing and hearing more talk about how multiplication and division could be used to help find the factors. Others were finding out that many odd numbers, and no even numbers except for 2, only had one way of getting that area. (BINGO – prime numbers)

My task was to observe mathematics take hold of these students as their curiosity and engagement lead them to understand how to find factors of numbers and what a prime and composite number was without them even realizing it.

Problems, and not the good kind!

October 12th, 2008

I am now faced with perhaps, one of the most challenging tasks thus far this year. How am I going to pose an engaging problem that teaches the concept of factors and prime and composite numbers? I know that I want to have students see how these are connected, but how? Teaching through problem solving has students construct their own knowledge and understandings, having them invent their own strategies. As a teacher I am to pose a problem and in letting them work through this fairly independently, they will come to see these connections and learn the intended concepts. I decide I will sleep on this and see if osmosis allows me to see how in the world I am going to do this.

Warm up Equals Whole Class

October 10th, 2008

Since I have been diving into the research about open ended problems, and have been experimenting with them in my classroom, I decided I would again try to start my math class with a warm up activity that involved an open ended question. Again, I thought I would spend maybe ten minutes on this and then carry on with the intended lesson. Again, I was in for a surprise. I read it somewhere in the mountain of literature I have been swimming in lately that using this type of approach and asking open ended questions may often lead to places you would never dream of, to areas of the curriculum that is at the moment, untouchable. That is exactly what happened today.

I began class with an opening question on multiplication. I posed the question: Tell me what you can about 13 X 52. I wanted to gauge students understanding about number and the concept of multiplication of two, two digit numbers. Asking students to tell me everything they know about this sentence, I am opening it up so that everyone can enter the problem and everyone can offer some input. Compare this with asking students to find the product of 13 x 52. This open-ended problem allows all students to engage, which will give me vital assessment data into what and how much they know about important concepts. This assessment data will then guide my instruction as it allows me to see where exactly students are with their understanding and will guide me to plan my instruction beginning with where they are and leading them to where they need to go. I will use this to ensure the lessons I teach and the problems I pose in the coming days are right for the student, open so everyone can enter the problem and experience success, but also open enough so that I can extend the problem, or change it in some way to challenge the students.

So, once I posed the problem I gave students a cue card to write their responses on. As always, I gave everyone time to at least get something down on paper and then, as a class, we began looking at some student responses. I invited students to come to the board and tell the rest of the class what they knew about this multiplication sentence. I called on the more reluctant students first to ensure they would have something to contribute. They stepped to the plate and offered some very insightful responses. Some said things like, the product is greater than 100, it is an even number, it is a three digit number, and so on. When one of the kids said it was an even number, well, of course, I couldn’t resist. I then asked the question of how they knew it was an even number. The response, and I remind you this came from a low average student, was something along the lines of they seen that the ones digit was a two and knew anything multiplied by two was even. Holy smokes! This led into a discussion about if this was always true, which lead us to writing a conjecture (which I explained to students). They wanted to prove that whenever you multiply an even number with an odd number, it will always be even. I said go for it. I decided to leave that conjecture to the students and gave them the opportunity to take it home and work on it to try and prove or disprove it.

The discussion then turned to the student response of the product being a three digit number. Again, I asked how they knew this and some were not sure. This spawned a great discussion about place value and conditions of what the numbers had to be for the product to be a three digit number. This lead to writing another conjecture. Students wanted to know if when you multiply two- two digit numbers will the product always be a three digit number.

From here, which was now about 35 minutes into the class, we discussed this conjecture and began talking about numbers and how to go about multiplying 13 X 52. We discussed things like breaking the numbers down and multiplying the number parts to get the product. Before long, and in the middle of this discussion, a hand goes up – this caught my attention as this particular student does not usually offer input or comments in the middle of class. When I addressed her, she simply stated that she had disproved the second conjecture. I asked her to come forward and show the class how she had done this. At this point I did not care that she was not engaged in the conversation about multiplying parts of numbers! She came to the board and wrote out a multiplication sentence that read – 99 X 99. She quietly said in a matter of fact way that the conjecture was not correct because if you take the two largest two digit numbers and multiplied them together, the product would be greater than a three digit number. Kind of taken aback, I quickly asked her why she chose the two largest two digit numbers. She, just as quickly said “well, miss, if this conjecture was true for all two digit numbers, the greatest two digit numbers when multiplied together would give you the highest product, and this product is more than three digits because 99 can be rounded to 100 so 100 X 100 is greater than 999, so that’s what I did and I know that this does not work.

So much for a ten minute warm up! The point I am trying to make is this – if I were to simply ask the students to calculate the product of 13 X 52, I would not have been given this wonderful opportunity to engage in this level of discussion with this class, nor would they have wanted to talk about, let alone, work through this level of content. This is just another example of what using open questioning in math can do. These students walked away with a stronger sense of number sense and hopefully in multiplication. I walked away with achieving so many other outcomes than intended, and came away with very strong evidence of where students thinking is and a clear sense of where my teaching needs to go.

Representing 56 in 56 Ways!





October 8th, 2008

Today, I wanted to begin my lesson with a warm up activity to get students thinking about numbers and how to represent numbers. I asked students to think of as many as they could to represent the number 56. I gave them five minutes and then we began talking. I gave students the opportunity to come to the board and write some of their representations. I was expecting that after the first four or five students had written their responses, most of their answers would have been written. Well, was I surprised? An hour later, and a board full of math, and we called it a class.

After about the tenth or eleventh representation was drawn, written, colored, or symbolized, students started coming up with more and more ways to represent this number. The more students wrote and talked about the numbers, the more ways students came up with. We talked about the different ways students represented that number, got students to explain their reasoning behind their representations and once all was said and done, we were able to represent the number 56 in 56 different ways!!

I used one of the suggestions on how to create open ended tasks to come up with this activity. I was not expecting students to be as engaged as they were. They were actually racing to the board to grab a marker so they could write yet another way to show how they thought about the number 56. At one point, I think almost all students were at the board. What was interesting to note about this particular activity, was how little I did to facilitate this conversation. I stood back and let things unfold. I noticed that students were thinking about how others had represented the number and some even questioned these representations. For example, one student drew a digital clock with the time 8:04. One student looked at this and asked how this represented 56. The other student answered by saying that there is 56 minutes left before 9:00. Settled! The questioning student then drew an analog clock using the same reasoning as the time read 8:56, 56 minutes after 8:00.

We all agreed that this was one of most enjoyable math classes yet!

Creating Open Ended Questions

How to do it?

So, the verdict is in. Open questioning is the way to go. So how do I do it? How do I go about creating questions that are open enough to allow multiple entry points so all my students can experience success?

There is no shortage of resources out there discussing how to go about creating open ended tasks and questions. There are free web sites available that offer already made open ended questions, such as www.heinemann.com/math. No matter how many open ended questions sites like this offer, there will come a time when it is necessary to create unique, authentic open ended tasks that suit the students in your class.

Open Assessment in Math (2008) suggest different ways to go about creating open ended questions, which is also consistent with Marian Small’s ideas about creating these types of tasks.

One suggestion, as outlined by Open Assessment in Math (2008) is to get students to come up with or create a situation or an example that meets certain criteria. For example, if you want to assess student’s understanding on multiplication, as them to pose a problem that involves multiplying as a way to solve it.

Open Assessment in Math (2008) continues by stating another type of open ended questions involve asking students to explain why, or how they know something or someone is correct or incorrect. This could easily transform a traditional, procedural based question into an open one. Let look at the question – what is 7 X 6? This could be turned around and look something like this – Mary said that when she multiplied 6 by 7 she got 42. Bill said that when he multiplied 7 by 6, he got 36. Who is correct? How do you know?

Small (2008) suggest to ask students to think about how things are similar and different, or the alike and different strategy, is another way to create open ended questions and tasks. An example of this could be to ask students to tell you how a triangle is alike or different from a square, or how the number 35 is alike or different from 45.

Yet another effective strategy to creating open ended tasks, as Small (2008) suggests is to use the ‘use these digits’ strategy. Here you would ask students to create some type of situation of choice, addition, multiplication, where students would have to use specific numbers somewhere in their sentence to make it correct. For example, create a multiplication sentence where you would use the digits 2, 7, and 15 somewhere in the question.

Asking students to solve the problem in more than one way is another way to create open ended tasks as suggested by Open Assessment in Math (2008). By doing this, you are requiring students to think of other ways to solve the problem. One thing to remember here, though, is to ensure there is more than one way to solve the problem and that students are able to come up with more than one possible way. Representing numbers, and asking student to show situations such a how to get a product and sum could be examples of how to use this particular strategy.

Small (2008) also suggests that by increasing or decreasing the value of the numbers in the various problems that are created can really open the problem up for a broad range of abilities. Making the value of the number greater than the initial problem would challenge students, whereas decreasing the value of a number may be just what someone else would need to access the situation.

October 7th, 2008

October 7th, 2008

As I reflect on what I have been learning about open and closed questioning, I decide that in order for things to work in the manner they are supposed to I am going to have to try to gear my teaching around using open ended questioning. Boaler (2002) states that “Phoenix Park teachers, those that taught math through a problem based approach, believed that the open-ended approach they used was valuable for all students and that it was their job to make the work equitable accessible” (p.60). I want and need all my students to be able to have access to the problem and for them to enjoy success. Boaler (2002) states that students, who were taught in a problem based approach, had very different ideas about math, than those that learned math in the traditional way. Student who were taught through problem solving thought math was active, and “did not regard math to be a rule-bound subject involving set methods and procedures that they needed to learn” (p.77) I need my students to think this way as well, so they can begin to apply what they know, and use this knowledge to construct new mathematical knowledge.

One important point I think I need to make here is that in order for all of my students to learn math through problem solving, they all need to be able to actually ‘enter or access’ the problems. They all need to be able to understand the problem, go about it in some way to offer a solution, and from here, learn what it is they need to learn from the problem. I think this is what Boaler (2002) is trying to get at here, as was seen in the above quotes. As Boaler (2002) talks about the experiences students had in math at Phoneix Park, she emphasizes that the “openness of the activities teachers chose to use at Phoenix Park enabled the provision of differentiated opportunities” (p.57). This is very important because I want all of my students, regardless of their abilities, to reach their highest potential in math, and ultimately in life. By choosing the appropriate tasks, open tasks, I can provide different access points that are geared towards the different students in my class that would enable all of these students to work at their own individual levels.

How they felt about this...



October 6th, 2008

As a follow up activity to asking an open and closed ended tasks, I had my kids to reflect on the two problems I posed regarding comparing fractions and had them to explain which of the two they liked the most and which one they thought showed their thinking more. Most, 8 out of 10, said they preferred the first, open ended question as they could show me their thinking better and they understood it better. I thought this was interesting.


October 5th, 2008.

At the beginning of class, as I had done yesterday, I posed a question. This time it was a closed ended question. I wrote four fraction inequalities on the board where students had to tell me which fraction was greater using the appropriate inequality signs. This, I considered to be a closed ended questions due to the fact that there was only one specific answer. I, once again gave them a cue card where they had to record their answers. After they completed this, I collected the cards. I, again, was surprised. One student out of ten was able to give me the correct answer for all questions. ONE. Ten percent of my class could tell me which fraction was greater and use the appropriate inequality sign. I would be more distraught, but I, instead laughed. I finally seen and understood what the literature was saying. I knew, from the open ended tasks that I posed the day before, that my students had a general understanding of how to compare fractions. If I were to base my judgments (and assessments) on this one closed ended tasks, I would have come to a very different conclusion. I would think that only ten percent of my class could compare fractions. When, in fact, 100% of my class is able to do it. I think, finally, I am getting somewhere.



October 4th, 2008

Today, I decided I would try to pose an open ended question as a warm up activity to start the class. I gave students a cue card and asked them to write one fraction in the top, middle of their card. I then asked them to look at that fraction and write any three other numbers/fractions that they knew was greater than their first fraction and to explain how they knew the other three fractions were greater. I let them work for about ten minutes then collected the cards. I was pleasantly surprised with the results. All ten students were able to write a fraction and then correctly write three others that were greater than the first chosen fraction. What differed was their explanations of how they knew the other three were greater All students were able to tell me, how they knew the other three fractions were greater, some better than others. I was able to see and understand (for the most part) their thinking, but also was able to assess their reasoning. It gave me information about how much they knew about fractions and fractional pieces, which will benefit me as I plan my unit on fractions. I was really pleased with this activity and I can begin to understand the power and benefits of asking open ended questions.