Journal Summaries #1

Going with the Flow: Challenging Students to Make Assumptions 
Felton, M., Anhalt, C., Cortez, R. (2015). Going with the flow: Challenging students to make assumptions. Mathematics teaching in the middle school, 20 (6).

        This article discusses a unit to introduce modeling to prospective teachers. The unit is focused on the Water Conservation task, a task that is suited for middle school students. The goal of this lesson was to advance prospective teachers understanding in the modeling process. The prospective teachers thought that models started out with mathematical concepts and then representing that in multiple ways. Actually models are the opposite. Models start with a real-world phenomenon and then you determine what mathematical problems could you use to understand the phenomenon, and then coming back to the original phenomenon. Modeling can be an application problem or be a way of teaching a new mathematical concept. The preservice teachers were given a problem to figure out if people use more water bathing or showering. When they worked through the problem they figured out that modeling involves making assumptions.
       I thought that the article was informative. Like the prospective teachers I thought modeling was using a mathematical concept and then representing it. I didn't know that modeling was starting off with a real-world problem and then deciding what mathematical problem to use to figure out the problem. I also realized that modeling involves making assumptions. Reading this article I learned what modeling really is. Modeling is something I can use in the class to have students apply what they know about mathematics or to even teach a new mathematical concept. Modeling will allow for students to stretch their minds and use critical thinking skills to solve the problem. If there is real world problems happening in the community then we can do a model problem in the class.

The Story of Kyle 
Dyson, N., Jordan, N., Hassinger-Das, B. (2015). The story of kyle. Teaching children mathematics, 21 (6).

        Kyle is a kindergartner from a low-income family. Kyle can complete a "nonverbal" calculation activity. When Kyle is read a story problem aloud and or a number sentence he couldn't get the answer correct. He couldn't perceive the relationship between the nonverbal problem and the conventional story problem and number combinations. Children in low-income families typically show that they have a hard time making these connections. In this article they developed a program called number sense intervention program (NSI) for kindergartners like Kyle, who are at risk of failing math. The NSI program is based on numbers,  number relations, and number operations. NSI has 24 lessons that last for 30 minutes. An important aspect of NSI is part-part whole understandings to story problems and number combinations. The lessons are fast-paced and often in a game format. The lessons build on each other. This program put Kyle on the right path to be successful in first-grade mathematics. Catching students weakness in numbers early is important so that you prevent more serious difficulties down the road.
      I thought this article was a good article. I thought the NSI program was a great program. It seemed like a good intervention program for students who are at risk for failing math. I liked that the program was at the kindergarten level so that you can intervene at an early age. The program can also be adapted for older students. I think that I could use this program if I had a kindergarten class. It seemed to be an effective way to help improve mathematical skills in children. It helped the students connect symbolic representations to their developing knowledge of quantities.  This is important for students to be able to connect these. If students don't connect these then in the future students will struggle with mathematics.

Video Analysis - Word Problem Cues

Planning - During the planning section of this video Tracy reflects on how her student solve word problems. She notes that many students see two numbers in the word problem and then add them together without knowing why they did that. She wants to teach her students that they need to read the problem and then explain why and how they got their answer using pictures and words. She is going to do this by having student do two problems and then reengage the students and have the students work on the problems again. She then is going to have to students work on the last two problems on their own to see if they use their critical thinking to solve the last ones. I think that this is a good strategy to do to teach these students.

Lesson - During the lesson Tracy demonstrated a lot of good teaching skills that I noticed. At the beginning of the lesson Tracy had the student sit on the carpet and review word problems from the other day. She had student's answers to the word problems. Some were correct and some were incorrect. She had a discussion with the students about what they saw and how they think they got the answer. I thought it was a good strategy to have to students talk in partners and then talk as a whole class. I also liked that Tracy never gave the answer to the question, she always had the students figure it out for themselves. She prompted them to get the right answer. Then Tracy had the students go back to their desks and get their word problems back. She first had the students talk in partners about what they did and how they got their answers. This was good because this forced students to describe why they did what they did. If they didn't know why they did it then this allowed them to try to think why they might have done that and if that was right. Tracy then gave students a pen to allow them to make corrections on their papers. She also wanted them to make corrections on questions 3 and 4, which they had not talked about in the whole class. This would show if the students are understanding how to do the word problems and can use their critical thinking skills. I liked how she walked around the room and helped students, but still never giving them a straight answer. I liked how she used demonstrations to help her teach. I thought it was a good idea to have the worksheet where they couldn't erase. 

Debrief - During the debrief the Tracy discussed what she thought went well and what she thought didn't go well. After hearing what she wished would have happened it makes sense. She wishes she could have shown more of the posters so that students could see different ways of answering things. She would have also liked more time because she ran out. The things that went well is that the students were willing to participate. This is very important in a class so that you can understand as a teacher what the students needs help with and what the student understands. The students participating made the class go very well. Tracy also explained her rationale for picking the answers for the posters. As a teacher you need to explain why you do what you do. 

Reflection - I thought this video was good to watch. It showed an example of a good teacher teaching students on how to go through the process of word problems. I liked the introduction and the debrief because you got to see what the teacher was thinking before and afterwards. I was able to see how she planned the lesson and if it ended up the way she wanted it to. This shows a good example of what I will have to do as a teacher. In the debrief she reflected upon her lesson and reflecting on lessons is very important so you know what to do in the future and where to take the lessons next.

Enhancing Students’ Written Mathematical Arguments Journal

Lepak, J. (2014). Enhancing students' written mathematical arguments. Mathematics teaching in the middle school, 20 (4)

This article gave good information on how to enhance students' mathematical arguments through peer-review activities. Mathematical arguments take an important practice of justification and reasoning. Students can justify their arguments through writing or speaking. The reformed curricula, common core, reflect this by having students 'explain why', 'convince', and 'justify'. Students need to understand which mathematical resources to draw from such as a graph, symbols, tables, and pictures to justify a conjecture.
One teacher, Ms. Hill, used peer-review activities invaliding rubrics to communicate mathematical resources to draw from when justifying a claim. The teacher found that on-going feedback and practice was essential for students to understand what type of statements could be used in justification. Ms. Hill introduced students with a mathematical problem. As she introduced the problem she drew a triangle on the board a labeled each part with words, pictures, and symbols. The students arguments must consist of all three parts of the triangle. Many students didn't provide a complete mathematical, justification to the claim. So she decided to use peer-review activities. She provided the students with rubrics to how the arguments were to be graded. Students gathered in groups and looked at each others arguments. Students had trouble understanding what other peers were trying to explain. The teacher explained that if you don't include everything and you don't think of the audience then the reader will not understand the claim. Ms. Hill found peer-review activities an effective way to teach how to communicate mathematical resources to justify a claim. Students' arguments became more coherent and strong.

Important Points of CCSSM 3 and 5

Standard 3: Construct viable arguments and critique the reasoning of others.
This standard explains that students who are mathematically proficient construct an argument by understanding and using stated assumptions, definitions, and previously established results. These students can also justify their conclusions and communicate them to others, and also be able to respond to others arguments. Mathematically proficient students are also able to compare two plausible arguments and be able to identify what is logic information and what information is flawed.


Standard 5 Use appropriate tools strategically. 
This strategy explains that mathematical proficient students can identify the available tools the student has and considers what to use to solve a problem. Students can identify what tools are appropriate to their grade level and how to use them and when they are useful. Mathematically proficient students are also able to identify when there are possible errors by using estimation and other mathematical knowledge.