Error Blog

This was somewhat of a difficult activity as we were to assess student errors with working with mathematical problems, but were not able to actually have access to students that we could question for their reasonings that were not particularly displayed in the work. This activity did make me realize once again, however, how difficult it is to understand students' work and errors. We have practiced with work like this in the past while working with the NAEP problems and rubrics, but it is still noted that this is something to be practiced a great deal before going into the classroom. This activity was also a reminder that a teacher must understand the students' thought processes to give quality feedback that can be followed up with all of the student in the classroom to deepen understanding.

Manipulative Reflection

How do you know if students deepen their understanding while using manipulatives?
Students deepen their understanding while using manipulatives because they have something physical and real to relate their ideas to while working with a mathematical concept. If you were to simply give students an example on paper and ask them to apply the concept they have just been taught moments ago without any real-life relation or physical manipulative work, it would be extremely hard for the student to understand what they were writing and working with. Using manipulatives helps give students real and tangible proof that the concept they're working with has reason and meaning.

How do you know if the students can transfer their understanding from manipulatives to other situations?
Teachers can begin to check on their students' abilities to transfer their understanding from manipulatives to other situations by slowly working away from the manipulatives towards real-life situations. For example, a teacher could start with 3D shape manipulatives while working with area, then move to having the students draw the shapes themselves or use the Smart Board, and then finally having students find objects of these 3D shape in or outside of the classroom to work with for a real-life connection.

How can you assess that understanding or growth?
Teachers can first assess the understanding and growth by asking students to give reasoning and proof of why and how they are using their manipulatives. We can also assess growth beyond this by having discussions about what their manipulatives can relate to in the real-world, and the importance of understanding that connection and the connection to the mathematical concept they are working with. If students are able to relate to these ideas, they will have shown their understanding and growth beyond working with simple manipulatives. 

When students work in groups, how do you hold each youngster accountable for learning?
An activity that I like to do with students working in groups is having students assign themselves a "job" for that group discussion/activity. Whether it be simply through discussion or writing down what is happening in their group, each student will have the opportunity and responsibility to participate. For example, if I had students in a group of 3 working on problems with manipulatives, one student may have the job of discussion leader, another might be a connector, and the other student may be asked to represent the work, etc.
When students work in groups, how do you assess each youngster's depth of understanding?
As I had stated before, if students were to have specified jobs in their groups, it would be beneficial for the students to come up with some written work of what they accomplished during the group work as somewhat of a reflection and checklist for understanding. Once the students had done their own write up or reflection, there could be a discussion or group write up of how their parts all came together and what they understand of the overall concept that would be beneficial.

How are you improving students' problem solving skills with the manipulatives?
Manipulatives greatly help students with their problem solving skills as it gives them physical objects to work with while learning and practicing various mathematical concepts. It also helps students begin to make connections with the manipulatives to real-world situations, and thus will lead them to work with this practice throughout higher levels of mathematics.

Curriculum Plan Reflection

Creating this curriculum plan was by far the most challenging project of the semester. However, it was also the most beneficial to my learning as a future teacher and brought to light a lot of personal strengths and weaknesses I had not noticed before. When our group started the curriculum plan we were extremely overwhelmed. Once we really began looking through different activities and projects that we wanted to include for each grade level, however, things started to progress a lot more. We were really able to get into the standards for our 3 grade levels and look at what things were repeating and what things were building off of each other from the previous years, and this led our group to some wonderful ideas about review activities for the start of each school year. Overall, working with this project was a good experience for our group and was especially helpful in getting us fully involved with the standards for grades 3-5.

Watching the other groups' curriculum plans was really great. It was tremendously obvious that our group was a little technologically challenged after watching how great everyone's videos were. One thing I did notice, though, was how it was somewhat difficult to really understand everyone's curriculum plans form just watching what they did in the video. We had to go back to Sakai and really look through the written plans to get the whole picture and see how it was connecting to the other grade levels. Once we did this, we could really connect a lot of the ideas from grades prior and after ours together and see how a lot of standards we building off of each other and how some were very redundant. Once again, this was a great experience to see how the standards were fully worked in to the other grade levels that we did not focus on.

Standards and Classroom Changes to Deepen Math Learning Reflection

Throughout this course, we have been consistently working with the Standards of Mathematical Practice and the NCTM Standards. The utilization of these standards is in regards to changes made in the K-8 curriculum by introducing the Common Core Standards for mathematics and language arts into the schools. This change in the curriculum is extremely beneficial for both students and teachers as it allows both parties to utilize many different ways of learning, as well as tie in the process standards into our everyday learning.

NCTM Process Standards:

• Problem Solving
• Reasoning & Proof
• Communication
• Connections
• Representation

The new Common Core Standards easily connect to these 5 process standards and have the instructor focusing on fulfilling each of these to deepen mathematical understanding in the classroom. For example, a 4th grade teacher may be working with her students on problem solving with fractions. Instead of incorporating just one way of looking at this concept with just one process standard like we might have seen in the past, the Common Core Standards ask the teacher and the students to show reasoning and proof behind their thought process while problem solving, communicate with peers about ideas, connect these ideas to real-life situations and past mathematical concepts, and finally represent their ideas and mathematical skills in many different ways besides simply filling out a worksheet.

Assessment Reflection

There are many different types of assessments that teachers can use in the classroom. What is most striking to me is how much the idea of what kinds of assessments to use has changed over the years. When I was in elementary school, and even somewhat into high school, a lot of my teachers were solely focused on formal assessment. However, as I am now going through classes, I am getting the opportunity to research, read about, and practice the many different types of assessments that can be much more beneficial than old-school formal assessments. For example, there are many different authentic assessments that can be utilized in the classroom. Authentic assessments "ask students to read real texts, to write for authentic purposes about meaningful topics, and to participate in authentic literacy tasks such as discussing books, keeping journals, writing letters, and revising a piece of writing until it works for the reader." These types of assessment can help teachers better realize how much their students are actually understanding on a real-world level and work to grasp their students' though process behind the work they have completed, just as much as the finished product. Authentic assessments such as these also open up the floor to much more of an opportunity for discussion amongst peers and teacher-student discussion. As we have seen throughout this semester, whether it be through videos, articles, or simply through discussing our own experiences, children tend understand and grasp a concept more when there is communication happening in the classroom. Likewise, this communication can be a form of assessment on its own. 

Additionally, I still believe that creating a form of structure to assess the students it necessary, as it helps not only the teacher stay aware of what is to be looked for, but it guides the students in the right direction as well. Even with authentic assessments, some form of rubric should be provided for the students, especially in mathematics. Using rubrics in assessments will also help us as teachers to be able to go back and reflect on what the students have done, give feedback, and follow through with that feedback to create a richer learning experience and more beneficial assessments that both measures and strengthens understanding in the classroom.

Technology Reflection

Throughout this course, we have been shown and had to research our own ways to use many different forms of technology in the classroom. Using technology in today's classrooms is extremely important and it is important to stay up to date with how to use the different technologies. One thing that I really loved about this course is that we worked with the Smart Board a lot and in many different ways. This was especially important for me because I was not used to the Smart Board and had the opportunity to learn a lot of useful techniques that I can take to my future classroom. Having us as the students research Smart Board activities was also very helpful because we could learn from our colleagues as well as teach ourselves. Another activity that involved technology that I know was extremely beneficial was allowing us to research math apps and applets. So many of us brought wonderful ideas to the table and we were all able to explain how to use these in the classroom. I know that I would be able to use every one of these technologies in my future classroom. Overall, this class was extremely helpful in teaching many different technologies to take to the classroom and I believe this is so important for us as future teachers to continue working with.

Moving Beyond Brownies and Pizza

"A lack of fractional understanding is a well-documented obstacle to student achievement in upper elementary and middle school math" says the National Center for Educational Statistics. This article begins by pointing out their main focus as being about helping 4th graders to use number lines to develop a rich understanding of comparing fractions. The article then goes on to discuss that there are 3 main goals they are focusing on:

1. Build students' understanding of fractions as numbers with a definite magnitude
2. Increase students' understanding of measuring with fractions
3. Develop fraction number sense by avoiding early introduction of traditional fraction algorithms

Then, as they move onto trying to reach these 3 goals, they tell us that they will be focusing on 6 different types of lessons that identify ways that 2 fractions might be related to one another:

1. Both fractions are one unit fraction more than a half
2. Both fractions are one unit fraction less than a half
3. Both fractions are one unit fraction less than one
4. Both fractions can be used in the context of money
5. More of a bigger unit fraction or less of a smaller unit fraction
6. One fraction can be expressed in terms of another

There was then 2 different examples from students and their thought process about different perspectives of the same number line. Even though both students thought differently about the problem, it was clear that the number line was central to their thinking. The same process was shown for different students a few more times, as well. After the lessons, the teacher was able to see that all of the students in the class had made progress in relation to the original 3 goals. 

I had some extremely good thoughts about this article. I really liked the overall idea of getting the students to build an understanding of fractions as numbers with a definitive magnitude. Also, I think this teacher's example of her plan for the class was very well thought out and put into motion the best way it could have been. It also really got me thinking about the troubles students commonly have when being introduced to fractions. It's hard to realize that once some misconceptions or misunderstandings are made right off the bat, it is difficult to help the students get back on track and fully understand the concept they're working with. And this is common with any mathematical concept. This is the main reason I liked this article and it really brought that idea to life.