Building Mathematical Comprehension, Chapter 5

   
     Chapter 5 is about visualizing mathematical ideas. An advantage Sammons points out is students today are surrounded by visual images through technology and media. The disadvantage is these images are given to them-they do not have to create them on their own. Students have to create their own when reading or problem solving. Comprehension is increased when students are able to create these mental pictures automatically. The opposite is also true, students who struggle with creating mental images on their own struggle with problem solving.

     Skilled mathematicians use visualization to help them during the problem solving process. Students need explicit instruction to understand the importance of visualization in math. They need this strategy modeled through think alouds and then the opportunity to implement this strategy with teacher support before they are held responsible for using this strategy on their own.

     Students need to be able to visualize multiple representations during problem solving. Students use their background knowledge and new information when creating mental images. When their new information disagrees with their background knowledge and they are able to work through that, their understanding of math is increased. The more representations they are able to generate means the greater their ability to decide which representation best represents the problem solving situation.

     Sammons gives 7 steps you can use to teach students to create mental images from words.

• Create mental images of observed concrete objects. 
• Create elaborate mental images of imagined concrete objects.
• Envision familiar objects and settings from their own experiences.
• Add familiar acts and events, then relationships and settings. 
• Picture characters, settings, details, and events while listening to a story read or told aloud. 
• Study text illustrations. and use them to create internal images. 
• Create mental pictures independently.

     Sammons also suggests taking a picture walk through your math textbook or other math related book so students can add those mental images to their bank to recall at a later time. I think this would be great to do when introducing a new concept so as you begin to get into the specifics of of that concept students have an image to build on. She also discusses using the visualize, draw, share strategy. The teacher reads over a list of statements about a concepts, students visualize, draw their images, and then share with each other. This gives students another opportunity to stretch their understanding and add images to their mental banks by seeing and discussing each other's images. She also suggests having students use a multiple representations graphic organizer. You can grab the one I created by clicking on the image.

   
     The two suggested math stretches for visualization are:

1. What do you visualize when you think about _____________? You fill in the blank with your concept, for example: multiplication. Then students write on a sticky their image of multiplication. They could draw an array or area model. They could write that multiplication is repeated addition or a real life example: My mom bought a box of pop tarts last night. They are 4 pouches in the box and 2 pop tarts in each pouch. There are 8 pop tarts in the box. They could also write an equation 3 X 3 = 9. Emphasize that their contributions should be unique. They add their sticky to the chart, and then as a class discuss the different images for the concept. 
2. What does this representation mean to you? This is opposite of the first stretch. You draw an image on a chart and they record what the image represents. You could draw two groups of apples with 3 apples in each group. They could write 3 X 2 = 6 or 6 / 3 =2. They could write 2 girls each have 3 apples. They have six apples together, etc. 

     Sammons ends the chapter by suggesting you use children's literature and poetry to help students create mental images. She outlines a daily plan for implementing poetry in the classroom which is very similar to what I did during the poetry section of shared reading when I taught kindergarten.

• Day 1: Read, modeling fluency. Then read together for enjoyment. 
• Day 2: Focus on vocabulary, especially math vocabulary and take time to correct misconceptions.
• Day 3: Highlight a mathematics skill related to the poem.
• Day 4: Come up with and perform actions for the poem.
• Day 5: Visualization practice occurs. Students create mental images and share out. Then they illustrate their poem and add it to their poetry notebook or journal.

Building Mathematical Comprehension, Chapter 4

     Chapter 4 is about using the strategy of asking questions before, during, and after problem solving tasks. I thought an interesting point Sammons brings forth is encouraging questioning even when the questions do not have answers. It is more important to find the right question than the right answer.

     She applied the Question Answer Relationships to mathematics. 

• Right there questions can be answered using the textbook glossary, math word wall, or the word problem. Examples: What does sum mean? How many cookies were eaten? 
• Think and Search questions can be answered using schema or prior knowledge and the problem solving task. Examples: What situations do you use divisions in? What information do I need to solve the problem?
• On my own questions are answered using the student's knowledge. Examples: When do I use division in my life? What strategies can I use to solve this problem? 

     Sammons suggests introducing strategies in small group strategy sessions where teachers can explicitly teach and model by thinking aloud while using the questioning strategy. Then students are given the opportunity to apply and discuss the strategy while the teacher provides immediate feedback. It ends with the charge to apply the strategy to their mathematics work. She also suggests a mini lesson where students brainstorm generic questions stems to use when applying the questioning strategy before, during, and after problem solving. Then you can revisit the chart throughout the year and add to the questioning stems as students become proficient in the ones already posted. 

     Some tools that were suggested for use with the questioning strategy are as follows.

• Wonder Wall-Students write their questions before, during, and after problem solving on sticky notes and sign. Then they add it to the wonder wall. The teacher takes time to review and discuss the questions throughout the problem solving process. It helps to students to work through the problems and may inspire questions of their own. I am thinking about designating a space on one of my walls for this activity and incorporating it as part of our daily routine at the beginning of the school year at least until students become proficient. I use Brenda DeBorde's TEKSing Towards STAAR spiraled practice at the beginning of class. It consists of three different questions about three different concepts. I'll designate one of the problems for students to share their questions about. 
• Question Journal-Students are given a sheet with four columns-Question, Before, During, or After?, Predicted Answer, and Final Answer. It is a tool that gives teachers insight about the student's thinking throughout the problems solving process. I would introduce and practice this in class and then make this apart of their homework assignment as a way to monitor their thinking when they are working on their own. 
• Question Web-Students can use the question web two ways. The first way the a specific question is put in the middle of the web and students/class record possible answers around the web. Then once the final conclusion is reached, it is recorded at the bottom of the paper. They can also be used to generate questions about a mathematical concept such as fractions or multiplication. Students record the concept in the middle of the web and then record questions about the concept around the web. This can be done whole group with students recording their questions on stickies, signing their name, and place it around the web. 
• Math Stretches-Two math stretches are suggested. The first one is questions for understanding. Students  are given a concept and must brainstorm a question that will help other students understand the concept better. This one can be done as a whole class chart like question web, written on a sticky, signed, and then discussed as a class during the math huddle conversation. The second stretch is the what's the question stretch. Students are given a generic problem solving situation that does not have a question. Then they brainstorm a question, record it on the sticky, sign their name, and add it to the chart. Then the questions are discussed. Students should write a question that is not already on the chart. 

     Students should be taught to think about the questions that are helpful during the problem solving process. They gave an example of a graphic organizer that students could use to think through the questions and record their answers. The chapter ends with suggestions for using children's literature. On book that was suggested was Counting on Frank by Rod Clement. Frank is obsessed with measurement in his life. It was suggested that you could have the students brainstorm the questions that lead him to the measurement tasks he takes on in the story.

Building Mathematical Comprehension, Chapter 3

     I'm a couple of days late posting. I spent last week at Student Life church camp in San Angelo, Tx with our high school students. I had so much fun and enjoyed being in God's Word with our students. I meant to take the book with me so I could read chapter 3 on the bus, but I forgot to pack it.

     I loved this chapter on making connections. Sammons emphaisized the importance of activating prior knowledge and accessing schema in math. I'm planning on incorporating the the schema roller activity to teach what schema is at the beginning of the year (page 97). I also liked the idea of incorporating math stretches as part of activating their schema and/or background knowledge. When introducing a new concept, I like to use schema charts. I give each student a sticky to record their schema about the concept. We discuss that schema is what we think we know about a topic. While we are studying the concept, students are given the chance to add new learning to the chart as well as to move their schema sticky to the misconceptions section if they discover their schema was incorrect. Here's an example of a schema chart we made when studying volume. I try to use different colors for schema stickies and new learning stickies. You print off schema chart labels for free from my TpT store here.  I like to laminate my labels and use them again and again. The pictures shows the chart made on chart paper; I also make these charts in a pocket chart as well.

     Making connections is one of my favorite things to teach when I teach reading. I'd never considered teaching math connections the same way I teach reading connections. I also liked the emphasis Sammons put on modeling making meaningful connections. The point of making connections is making connections that aid in their understanding of the concept or their ability to solve the problem, not just making as many connections as possible. I made a sheet for students to record their connections; you can download it for free from my TpT store here.

     I liked the idea of having a current events board, where articles are displayed and mathematical connections are discussed. This could also be done at home with students writing a brief summary of how math was used in the article after they discuss it with their parents. One easy way to get students making connections is to use fiction or nonfiction literature. I like how Sammons emphasized that it doesn't have to be a book written specifically to teach a math concept for students to be able to connect to math. When I taught fractions in kindergarten, I liked to use the book, The Little Mouse, the Red, Ripe Strawberry, and the Big Hungry Bear, when teaching about halves. I would then pass out a strawberry and a plastic knife to each student. Then they would practice cutting their strawberry in half and sharing with their partner just like little mouse did in the story.

     I'm excited to start the next chapter about questions. Don't forget to link up and share your thoughts!

Building Mathematical Comprehension Chapter 2

    This chapter about vocabulary was another "Why didn't I realize that?!" for me. I liked how Sammons begins by discussing how language and learning are intricately linked. Then moves into the importance of explicitly teaching  content vocabulary. She talks about making sure you're teaching the key vocabulary from your grade level math standards and finding ways to involve parents with math vocabulary practice at home. Sammons also gave examples of graphic organizers and games that help students learn and practice key math vocabulary.

     I liked how Sammons said that math vocabulary is not an add on to the math lesson, but should be a way to teach the math content. I'm also going to go back to my TEKS to create a list of the key vocabulary I should make sure my student's understand. I plan on adding vocabulary assessment to my weekly quizzes and unit exams as a means of accountability for myself and my students. Students need to be able to understand and use correctly math vocabulary to fully comprehend and master math content. 

     I did use thinking maps and graphic organizers some in math last year, but not near as much as I did when I taught kindergarten. This is a tool we will definitely use more of this year. A big thing for me will be modeling the graphic organizers for my students, so they can begin to utilize them on their own. I also want to add in some reflective writing time in their journals (with the emphasis that use the content vocabulary in their writing) at the end of class. I will start the year modeling this for the students in a class journal. Then we'll move into shared/interactive writing before this becomes an independent activity. Then I will use that time to have mini conferences or reteach a small group. Here's an example of a chart I made when teaching length in kindergarten. Texas standards have students making direct comparisons using the words longer/shorter/same as.

     I want to utilize a math word wall as well next year. I will not have the ability to keep all the words up all year. I think we'll keep the current concept words up and then move them to a ring that they can pull and look at when they need to. I will also use the be a mind reader and other word wall games to review these math vocabulary terms. I also will add the math vocabulary words to my assignment sheets with parent suggestions for discussion or activities to use to practice these words at home. 

     Make sure to link up and share your thoughts as well!

Building Mathematical Comprehension-Chapter 1

     I am so excited about this book study this summer! I taught 4th grade math this year after teaching kindergarten for 7 years. One thing my students really struggled with was problem solving tasks that involved reading comprehension. I am very familiar with diversity of levels in my class, however, this year I had the most diverse levels of students I have ever had before because I had had a mix of students coming public, private, and homeschool backgrounds with a slightly larger percentage from homeschool and private school.

    The majority of students had little or no word problem experience. As the math and english departments were talking at the end of the year we had come to the conclusion that reading comprehension was playing a hugh role in their lack of success in math. On page 22, Laney has a chart that shows the similarities of characteristics that good readers and good mathematicians possess. I saw this clearly demonstrated first hand this year in my two math classes. As I spoke with the 4th grade English teacher, most students who struggled with reading comprehension also struggled with mathematical comprehension. 

     Laney goes through and explains which comprehension strategies come into play before, during, and after problem solving tasks. As I began reflecting on my own problem solving procedures, I realized I was applying the same strategies in problem solving tasks that I do when I read. It had never occurred to me that I was using the same strategies for both tasks. 

     Another point Laney made in this chapter is that sometimes when we introduce strategies, we assume students will pick up the why and when the strategy should be used. Before modeling of the strategy begins, students need to be taught the what, why, and when of the strategy. This is why explicit instruction in strategies is so important. On page 31, she reviews the six steps of explicit instruction which includes scaffolding and the gradual release of responsibility to the students. This builds their confidence, so they are able to take ownership of these strategies.

    The chapter ends with a framework for planning and teaching strategies which includes points or questions to consider as you move through the explicit teaching process. I like how she broke each phase down into instructional focus and student focus. I definitely plan on referring back to this section next year before and after teaching the strategies for planning and reflection purposes. 

     I strongly encourage you to pick up a copy of this book and join the blog book study fun this summer! I can't wait to read the rest of this book! Make sure to link up with Thinking of Teaching or Primary Inspired to share your thoughts on Chapter 1!