Merylhawley's Weblog

Products and Sums Video

merylhawley — Tue, 18 Nov 2008 18:22:19 +0000

The video “Products and Sums” showed a 4th grade classroom working with patterns and relationships of the products and sums. The teacher used an indirect method of teaching the students about these relationships and had them discover the patterns on their own with her guidance. The students clearly knew their basic facts of addition and multiplication and showed their knowledge of this while creating their model. The students were then asked to come up with patterns that they saw regarding all of the numbers. The teacher was very patient and accepted all of the answers that the students came up with. Several of the students did not seem to be on the path that the teacher was steering towards, but she embraced each pattern that was brought forth. This activity encouraged the students to really be creative and work together to come up with the different relationships. Some of the patterns, I had never even thought of.

I found it a great idea that the teacher was going to have the students also write about what patterns they had discovered. Writing about math will help with the students’ communication skills and allow them to sort their thoughts and truly display what they know or learned. I did think the assignment was a bit vague though and many of the students clearly were not coming up with the patterns she had hoped. So as a teacher, I woud try to explain what I am looking for to assist in the students’ thought process. Perhaps her point was not to arrive at a common pattern but to just get the students thinking and discovering ideas on their own. If that is the case then I wouldn’t change much about the lesson. I think the teacher did a great job. My only negative thought was that at the end of the lesson, I wasn’t really sure what she was trying to teach specifically about patterns.

Regrouping/Renaming in Math

merylhawley — Mon, 17 Nov 2008 20:16:12 +0000

After reading the articles, “Children Inventing Their Own Addition Procedures”, “Do My Students Think Flexibility? Do I?”, and “Creating Thinking in Subtraction”, I see how many different ways there is to solve addition and subtraction problems. I had no idea that students would come up with all of the ways presented in the articles. Many times students just go through the motions but do not really comprehend what they are doing. By having them come up with their own way and then explain it, truly heps ensure that they understand. Until coming into this program, I never really heard the terms “regrouping” or “renaming”. I had always learned to just do pencil and paper addition and subtraction without truly thinking of easier ways to come to the answer in my mind. That is probably why I am still pretty bad at mental math.

I decided to try out the activity performed in the articles. I gave 5 of the 3rd grade students I work with some multi-digit addition and subtraction problems to work out in their head. They then had a few minutes to figure it out and they had to describe to me how they reached their answer. Every single one of them used regrouping and renaming to answer it. None of them became frusterated or questioned how to do it. A few of the students came up with different ways of regrouping, but they all had the same ideas. It was neat to see them work these problems out and they seemed very excited to explain how they did so to me. We had worked on this first thing in the morning when they are unpacking and getting ready for their first class. When the bell rang for announcements we had to stop, but they begged to do more tomorrow in the morning. Seriously??? I was shocked that they wanted to do extra math problems. This is when I am very thankful for these new ways of learning and teaching math. I am seeing students repeatedly get enthusiastic about math and I never thought that was possible. I think I’ve commented on this a few times in my blogs this semester, but everytime I go through one of these moments, it truly amazes me.

5 Frame and 10 Frame

merylhawley — Tue, 04 Nov 2008 15:59:49 +0000

After playing around on the 5 Frame and 10 Frame activity, I can see where these would be very helpful. They do not force the child to go right into addition, but they provide the students with activities for recognizing “How Many” and “Building”. Once they understand those concepts then they can move on to the addition activity. By having these different levels of learning within one activity, the student is able to master one skill and then building on that one, they move to the next. These activities assist the students in truly understanding the values of the numbers and also provides a visual tool so that they understand what each number represents.

On the down side, I know that they are called “5 Frame” and “10 Frame” because they stay within those numbers, but the activity became very repetitive and could wind up boring a child because they are limited by those numbers. It would be nice if, within the same activity, they could click a buttom and advance to other number sets (i.e. counting by 2′s, 5′s, or 10′s or adding larger numbers). By not having advanced activities the program is limited to a portion of students and could be easily outgrown. I think that it would help students when they are first learning or those that are struggling learning, but does not accomodate those students that need something more challenging.

On another note, I finished implementing my 5 day lesson plan last week! I think that it went really well and turned out to be a lot of fun. In addition, the students truly grasped the concepts I presented in graphing. It was a fun lesson to teach because there are so many activities involved in that. With it being Halloween week also we definitely incorportated candy in the mix. We were able to get through Circle Graphs, Pictographs, Bar Graphs, and Sorting. Since Circle Graphs is what they are tested on at the end of the year, we spent 2 days on that. I think they enjoyed and benefited from having the individualized time with a teacher and were able to have all of their questions answered without fighting for the instructor’s attention. It was great to see the concepts click for them and to see them enjoying math.

Implementing My Lesson Plans

merylhawley — Mon, 27 Oct 2008 18:20:18 +0000

I started implementing my lesson plans this week; I am doing 3rd grade Graphing. The 4 students seemed very excited to be chosen but 2 of them definitely had some trouble with raising their hand and continually speaking out of turn or goofing off. It started off a bit rough because they were all at different levels of understanding so I had to go a bit slower for 1 of the students and really explain the concepts a bit more then the others needed. I will have to make sure that I have someting to keep the others on task if this happens during the upcoming lessons.

Today’s lesson was introducing circle graphs and primarily going over what 1/2, 1/4, and less than a quarter meant. We started off with tile cubes before even looking at a circle graph. This really helped the students understand those ideas. They aced the concept of one half right away but struggled a bit with one quarter. We stuck with it though and between the tile cubes and the sample graphs, they really grasped it by the end. I felt so proud! They were enjoying the lesson and I could already tell a growth in their understanding from the beginning of the class to the end. One girl even said that this was her favorite day in math so far! It felt great to hear that because, to be honest, I was so nervous for today. It’s one thing to be in a classroom observing or assisting, but to be the instructor, even to just 4 students, was intimidating (but exciting at the same time). I am just happy that the lesson went smoothly and the time frame of everything went as planned. We had a few kinks that we worked out, but I will be more prepared for managing the class the rest of the week. I am looking forward to tomorrow’s lesson because now that they understand the lesson from today, tomorrow’s will be a bit more fun for them.

I will let you know how those go!

Figuring out Shapes

merylhawley — Tue, 21 Oct 2008 16:17:03 +0000

The first article I read was “Too Skinny, Too Pointy, Going the Wrong Way”, and I found this very typical of a classroom learning about the different types of triangles. It is so true that, as early as babies, children are introduced to the shape, but very rarely does that triangle shape vary from the equilateral triangle. This weekend I tested this and searched through my 17 month old niece’s toys. She had at least 5 different toys that were dedicated to learning shapes. All of them had a triangle shape, and as anticipated, all were equilateral triangles. The article involved 3rd graders, which means that, these students went about 7-8 years of learning that there is 1 type of triangle, and now, they are finally being introduced to the many sizes and directions a triangle can sit. I would imagine it would be difficult for these students to learn these new concepts. This article teaches me that, not only do students need to learn their shapes prior to 3rd grade, but there needs to be more toys that represent shapes in the diverse way that they are. If triangles can be represented more acurately when a student is a baby, then perhaps these issues will not arise when the children are being introduced to shapes again in elementary school.

The second article I read was “Parallelograms or Not Parallelogram?”, and I have to admit that, after reading the article, I had to look up online what the acurate definition of a parallelogram actually is. The children seemed so confused coming up with their own definition and the teacher showed such patience allowing them to continue to figure it out. Although it was a struggle for them, I bet in the end, they will understand what a parallelogram is more because they learned by creating their own observations, rather than the teacher just telling them. I feel I may struggle with this as a teacher. I have witnessed the teacher I am working with show this kind of patience and in the end the students found the desired result. It almost hits an uncomfortable level though before that happens and that is usually the point where I break and give the answer. I now try to bite my tongue and give them extra time to truly make their own results. I find that ends up being more beneficial than me giving the answer. It just takes some restraint to allow them to keep with it.

Measuring Area

merylhawley — Tue, 07 Oct 2008 20:30:06 +0000

It was intereseting to read that article, Fostering Communication About Measuring Area in a Transitional Language Class and see the steps taken to ensure the students not only understood how to measure the area, but how to communicate their knowledge through discussion and the written word. When the insructor presented the material and worked together with the class initially, that helped to eliminate or prevent common mistakes that they might make when they advance to working on the geoboard on their own. That also helped the students become more familiar with the objects being created and measured. I thought it was smart having the teacher acting more as a facilitator, guiding the students to their own conclusions and justifications, rather than telling them all the answers and how to get to them. The students were able to share and compare their solutions with their peers. The teacher even had the students evaluating the other students using mathematical explanations and arguements. Many times as humans, we learn more when we teach something to someone else. So by having the students assessing one another, they might have learned something more themselves.

I liked in lesson 2 the idea of having the students choose their favorite polygon. This helps them relate to the assignment and forces them to take ownership for the lesson. When going through the different students’ polygons, the teacher posed questions for the students to think about (i.e. “Do I agree/disagree?” and “Do I have any questions?”). By having them think about these will help the students engage in a discussion and be able to formulate their thoughts and their knowledge on the subject.

One of the other aspects of this assignment that I thought was effective, was the diversity of learning methods that were presented. The lesson was discussed out loud, the students participated in hands-on learning, and the ideas and conclusions were even written down. By covering all of these grounds, it made it easier for the students to comprehend the lesson in whichever way they found more comfortable to learn.

Comparing 2 Types of Math Classrooms

merylhawley — Tue, 30 Sep 2008 18:34:46 +0000

Since I have been working in the classroom this semester I have noticed a couple things that are consistent in every lesson in their math class. One, that neither the students nor the teacher ever open a math book and two, that manipulatives are involved in every lesson. When I asked the teacher I work with about the textbooks, she responded that the students have never even seen the book. She comes up with her own lessons while working with the other 3rd grade teachers and the Pacing Guide to come up with a curriculum. She denounced the textbook and feels that the students understand the lessons better when they can learn it through the use of concrete manipulatives, games, or other hands-on materials. Since starting this program at UNCC, this concept has been stressed to us and always makes me a little nervous that without that textbook I will have diffculty coming up with appropriate and effective ways to teach the lessons required. I worked in a 4th grade classroom a year ago in California and the textbooks were a staple in their lessons. The teacher literally followed word for word, example for example, the whole day’s lesson out of the teacher’s manual. I had never seen the students in that classroom get excited about math or cheer for more problems to do. In the classroom I am working in now, I see the excitement on a regular basis. They seem more engaged and actually look forward to the class. They ask to be challanged and look forward to explaining their answers. In the “textbook” classroom that I worked in, manipulatives were non-existant and worksheets were a daily assignment. The boredom on these children’s faces was what I had always associated with the look of anyone in a math class. Now seeing this Investigations method being used is so refreshing. Although I am nervous that I will struggle to teach this way, especially at the beginning of my career, I also look forward to using it because of the differences I have seen it make in the classroom. My hope is that through this year I will become more comfortable with it.

Open-Ended Questions

merylhawley — Wed, 24 Sep 2008 02:55:04 +0000

Open-ended questions seem like a great way to assess a student’s comprehension of a particular topic. It forces the students to explain and put in their own words the answer, how they came up with it, or why it is so. Many students can memorize and regurgitate what they have placed in memory, but that doesn’t truly capture what they learned. I was a student who memorized everything when I was in elementary school. I dreaded anything in math where I had to explain my answers because I wasn’t really sure how to. Immediately that was an indicator to my teachers that I needed some extra help in those areas.

Not only are open-ended questions a good indicator on a child’s understanding of a lesson, but it also helps highlight where a student may be struggling if they are producing the wrong answer. If you simply were to get an answer but no reason as to how they came up with that answer, a teacher would not know which step they struggled on or where they miscalculated. But if given an open-ended question where they had to figure out how to come up with the answer, the teacher can pinpoit which step they were incorrect on or where their thought processes were headed. If there are common errors in multiple students’ work, the teacher can then use these open-ended questions to help focus on which areas need to be re-addressed or re-emphasized.

Mathematical Tasks

merylhawley — Thu, 18 Sep 2008 18:40:00 +0000

After reading both articles, “Teacher As Architect of Mathematical Tasks”, and “Selecting and Creating Mathematical Tasks: From Research to Practice”, I have a better understanding of the importance of “good mathematical tasks”. It is the teacher’s job to create the good task, ensuring it is grade level appropriate and of a high level. I was surprised of the stressor that was put on using a high level task when introducing a new lesson and the mentioning that low level tasks “almost never result in high-level engagment.” (Smith) One of my worries of when I become a teacher or even when I am doing my student teaching is that I present material that is too tough and I later have to back track with a classroom full of confused students, ultimately destroying a whole lesson. I thought that I was best off using simple thinking in presenting new material and gradually working to higher level thinking, but these articles seem to take the opposite view.

I also found it interesting and completely agree on how much the spontaneous lessons can have an impact on the students. I find that people enjoy finding out fun facts about themselves and when relating a math lesson to the child rather than just making up story problems of random unknown people, the student is more likely to remember and comprehend what the lesson is about. I loved the example the author used about the children finding out how old they were in months. The students learned a new math lesson while still engaging in something fun and meaningful.

Another common theme in the articles was encouraging the students to explain their answers. This ensures that they understand the lesson and forces them to think and reason. It helps bring together mathematical thinking with mathematical concepts and skills. By students adopting the habit of explaining their answers and how they reached then, teachers can make a better assessment of the student’s progress and students can use that skill in future high level tasks.

Both articles stressed the importance of a “good task” and that the teacher is responsible for creating and presenting them. While the lessons that need to be taught to the students is determined by the state, the teacher is accountable for how they are presented. Coming up with “good tasks”, while sparking the students curiousity and challenging them to think and reason, with aid in a child’s academic success and help a teacher become a great teacher.

Learning Theories in Mathematics

merylhawley — Tue, 09 Sep 2008 00:53:19 +0000

After reading the articles and the chart on the learning theories, I definitely see how important it is to improve our math curriculum and have our nation reach the math standards that other power countries have. In the article on Computational Fluency by Susan Jo Russell, she points out different scenarios of students figuring out problems and how they came to their answer. There was always one student that figured out the correct answer but in a more time consuming or inefficient manner. She even suggests that if a teacher comes across a student answering like this, then there is a problem that must be addressed. I completely related to that student and felt that I was taught that those methods were fine. I didn’t know any other method. Until taking the courses here at UNC Charlotte, I would never have imagined a different way of learning or teaching mathematics. I would categorize myself as a product of the Information Processing perspective. I memorized theories and methods and learned by changing the knowledge in my memory. In the end, I do not think this proved to be effective. I never became strong at math and, throughout elementary, high school, and college, I struggled in math courses. I never had a teacher to show me a different method that might have been conducive to my way of thinking or learning. I was never shown a way to visualize or comprehend the actual math function. I merely memorized the symbols and steps that were told to me.

In the past few months I have been exposed to the Constructivist perspective and, although it’s been a struggle re-learning everything with the new wording, concrete manipulatives, and different approaches, I am finding that this method will prove to be more beneficial for students today and in the future. It truly helps visualize what the function means or is asking, and allows the student to build on what they already know. When observing the math camp at Providence Day school this past summer, I was blown away by the students’ ability to breeze through math problems that I originally believed to be multiple grade levels above them. To see kindergarteners easily completing 3 digit multiplication tables was proof that the constructivists method works.

One of the questions that Susan poses in her article to help assess computational fluency is, “Can the student explain why the steps he or she uses works?” I find this to be a very important question because many times we see students go through the motions but are not really able to explain how they got there or what they are trying to do. I believe if we can teach students in a way that will help them understand this, then we will be a step closer to improving our mathematics program in the United States.