Building Mathematical Comprehension Chapter 10

Guided Math Classroom

Last summer I read Laney Sammons's Guided Math book. I was so eager to get started and hopeful to implement in my classroom. Unfortunately, I was not quite prepared and did not fully see the whole picture. Now that I see the light, and am more fully prepared with effective tools I am so excited to get started.

I so appreciate how flexible the instructional framework for Guided Math is. There are specific components but within each component there is lots of flexibility. The components include:

1. A Classroom Environment of Numeracy- (daily) a classroom filled with numeracy provides students with lots of opportunities to engage with math. Students are encouraged to use manipulative, talk, problem solve, question, generate ideas, etc.
2. Math Stretches and Calendar- (daily) "warm-up" math activities provide students with a preview of future math concepts or a review of what has been learned. Math Stretches allow students the opportunity to use the 7 strategies they have been taught (the ones in this book ;). The Math Huddle is a great idea. Students can share their thoughts and engage in a mathematical discussion. How awesome is that?! The Calendar provides so many opportunities to preview and practice skills that have been learned. 
3. Whole -Class Instruction- (your choice) My inner voice keeps saying Modeling and Think Alouds. Good, it has been engrained in my brain.
4. Small-Group Instruction- (your choice) make groups homogeneous yet flexible to change. Here is where strategies are reinforced and individual student's needs are met. 
5. Math Workshop- (your choice) students can work alone, in pairs, or in groups. Teachers provide anchor charts, organizers, and feedback. Students responsibility is to complete tasks and demonstrate their mathematical understanding.
6. Conferencing-(daily) one-on-one conferences are brief conversations that assess students understanding and guide their thinking. 
7. Assessment- (daily) use formative and summative assessments. This can be done by observation, students can share orally or in writing.

Along side this framework a teacher must motivate and encourage students to "become mathematicians." Sammons shares what "becoming a mathematician" looks like and how teachers can create an environment where this is possible.

Within each component of the framework there is flexibility and structure. I think that allows for teachers to be creative and resourceful in the use of materials they have. For instance, Wendie has been working diligently to create our "Critical Crumbs." -->Critical Crumbs are individual math tasks that focus on the four critical areas of third grade math in the common core standards, namely:

• multiplication/division relationships
• unit fractions
• area of two-dimensional regions
• properties of two-dimensional shapes 

Critical Crumbs provide a daily practice and warm-up activity that focuses on these four critical areas. I can see how they can fit in during Math Stretches or Math Workshop. Here is an example of our Critical Crumbs.

 Also, our Math Anchors and Math Jumbles could work nicely as Math Workshop activities. Check them out and let us know what you think. 

 

 

Building Mathematical Comprehension Chapter 9

Monitoring Mathematical Comprehension

How will our students monitor their comprehension?  They need to have a wide variety of comprehension strategies in their tool belt in order to do this. All the strategies we have been reading about will come into play when students are monitoring their comprehension.

First step: Conceptual Understanding

Students need to become active, engaged participants in their thinking. Yes- they need to "think about their thinking." Sammons provides for us great "Fix-Up" strategies that become the students responsibility to use when they do not understand. It is important for students to be responsible for this process. Only they know when their understanding breaks down. Explicit questions to pose: 

1. How do you know whether or not you understand?
2. What do you do if you are confused?

Second Step: Problem Solving
Have you ever had a student who would start a problem and attempt just about everything, but never really stop to understand what the problem was about? I have. This is often true of our struggling readers who focus on a word or two and start trying to problem solve. Sammons emphasized the importance of spending time helping students to look at how to find the "overall meaning" to problems.

Then, students need to develop a plan of attack. They need to be aware of connections between things and how to link it to the unknown data. So, they need to understand the problem and how the mathematical concepts are related.

Sammons shared a "Comprehension Checklist" that will help students monitor their comprehension.
I took it and made it my own, but it is all Sammons. Click pic to download.

Here are some signs that students should be aware of to help them know that their comprehension has broken down.

• Your internal voice interacting with math concept or problem
• You are unable to visualize math concept or problem
• Your mind wanders away from work at hand
• You are unable to recall details of math idea or problem
• You cannot find answers to questions asked to clarify meaning

I had to create something handy dandy to keep these close and readily accessible

Sammons suggests that students brainstorm "Fix-Up" strategies and create a chart they can refer to. She also provides lots of suggestions. Click pic to download.

I am so thankful that now I have additional resources to help my students who struggle. The result of modeling this strategy and doing think alouds consistently will be that they now have resources that will allow them to be successful.
Other suggestions to continually have students monitor their comprehension are:

• Ticket out the Door- students can explain their understanding and also rate their understanding
• Comprehension Constructor- provides scaffolding for students fix-up strategy and overall plan
• Color-Coded Metacognition Math Stretches- students reflect on a word or concept and use a color to represent their understanding

Amazing strategies for our students to surpass those "comprehension roadblocks."

Building Mathematical Comprehension Chapter 8

Holy Moly! Synthesizing.

This chapter on synthesizing was a toughie. Mainly, in the sense that it is difficult to come up with a way for my students to think about their process of putting together new mathematical ideas, previous learning, and reformulations.

Synthesizing is definitely one of the more challenging strategies to teach. It requires lots of think alouds and teacher modeling. The ability to synthesize first requires teachers to begin with the concrete models (nesting dolls and baking a cake) and then to move to the process of synthesizing.

The process is where it gets tricky. One way to do this is by making conjectures. Conjectures are described as informed guesses and predictions. Students can make conjectures by being given open number sentences where they have to decide if they are true or false and explain why.

In order for our students to practice synthesizing, the problems we choose need to allow students to find patterns and relationships, have more than one solution, and allow for questions (the what ifs).
I honestly had such a difficult time with, how do I help my students "see" their thinking process? I put together an organizer I am going to try out this year. I want to share it with you, but please know it is a work in progress. Click pic to download.

Building Mathematical Comprehension Chapter 7

Determining Importance

I found this chapter to be dense with Amazing information. The focus of this chapter is helping students identify relevant information. This will help them not get sidetracked with all the facts and details of a math problem.

Sammons refers to levels of determining importance:

1. Word Level- math vocabulary, bold print, italics, highlighted words, etc.
2. Sentence Level- typically most important information is found in the middle of a math problem. Knowing structure of mathematical word problems is helpful.
3. Idea Level- the Big idea. Students will benefit by practicing how to identify the overall meaning of word problems. Word level and sentence level help get to idea level.

Sammons share several strategies to teach determining importance. Here are three:

• Overviewing- students skim over text and try to find important words, sentences, and ideas
• Highlighting- students decide what is worthy of highlighting
• Read a Little, Think a Little- students read one sentence at a time to find important information

Like I said, it was a dense chapter and I want to focus on a bit at a time. Take highlighting for instance. Put a highlighter in the hands of my students and they will highlight just about everything on the page. I need to explicitly teach them how to distinguish between interesting information and what is important information. After reading this chapter I know have better tools on how to do this. I created a chart that will help me focus on what is interesting information and what is important information and one that will limit the amount of irrelevant highlighting my students are prone to.
Let me know what you think and just click on the pic to download.

It just hit me that I should be compiling a list of children's books that supports each strategy. For determining importance these books will help.

Caps for Sale
The Grapes of Math
Mind-Stretching Math Riddles

Teaching my students to recognize information that is important and useful is essential for them to understand mathematical concepts and problems in deeper way. This is the road to becoming a critical thinker.

Building Mathematical Comprehension Chapter 6

Making Inferences and Predictions

Yet another learning filled chapter. With all this knowledge I am absolutely eager to start teaching math.

I am so thankful that I have been able to see inferring and predicting in a new way. I can see the depth of knowledge I have gained and how my students will benefit.  My hope is that gone are the days that we felt the need to rush through standards in order to expose our students to everything we were suppose to cover. As Sammons mentions these strategies require time for students to understand and begin using on their own. If I expect my students to give me reasonable and justifiable inferences, I need to be ready for it to take awhile. 

Sammons mentions one-to one conferenences often and I so guilty of not doing this during math. I went into last school year hoping to make this happen and I failed. But I have to say after reading this book I have a clearer picture of what I need to do to reinforce learning one to one or in a small group. Thank you Laney Sammons!

This chapter was filled with such valuable ways to practice inferring and predicting, I had to put it into a tangible way for me to use. I have seen some great freebies created for this chapter that I know I am totally going to use. Just click on the pic to download and please let me know what you think.

Building Mathematical Comprehension Chapter 5

The Importance of Visualizing Mathematical Ideas

I have used the visualization strategy for so long and not considered it while teaching math? In such simple, but profound ways Sammons's book is helping me rethink the way I teach math. Sammons shares that cognitively, if deeper learning and understanding is to be made students, they  must be able to process information in two ways. One being linguistically and the other nonlingustically. Combining the two when I teach will increase my student's comprehension. I have to say, when I think about my lessons that went very well I was naturally (intuitively) doing both.

Now, in order to help my students use visualization as a true strategy I need to be systematic and explicit in my teaching they use of the strategy. Sammons suggestions of what mathematicians do with this strategy is beneficial. Mathematicians...

• use their schema to create a mental image- remind students that we all have different schemas and that it is okay if they are unsure. That is common in learning something new.
• are motivated and engaged as they visualize- tell students that combining prior knowledge and new learning is essential to create images.
• revise their mental images as they discover new information- as students receive new information students need to learn how to incorporate that information into their mental images. I love this one! 
• understand the value of sharing their mental images- this not only helps build verbal skills, but it helps students process their own thinking. It will also help them with the revision of their mental images.

Pretty powerful stuff! Another amazing section was how to concretely help students who struggle with visualization. I can definitely visualize myself using this 7-step sequence in a small group. Is everyone as excited as I am?!

 This chapter also reiterates the power of modeling and Think- Alouds. I know that I cannot assume my students will naturally understand visualization on one try. My think-alouds need to go beyond the introduction to my lesson. I need to embed them throughout the lesson; in large group, small groups, with students help, in writing, one-to-one conferences etc.

I cannot do this chapter justice. I created an organizer that I hope will help me be more direct and explicit in teaching students to draw upon what they know and create a mental image. Just click on the pic to download. Let me know what you think.

Building Mathematical Questioning Chapter 4

Asking Questions

This chapter was so dense with information, I am not sure where to begin. Let me start by saying that I am so glad to see a renewed interest and emphasis on student generated questions. I know that this is an area that I can continue to grow in.  Having taught grades 1st though 6th, I agree with Sammons that it is concerning to see how students begin to ask fewer questions as they get older. So, what can I (we) do to improve this?

• Teach students to ask question. 
• Asking questions is a skill that can be developed. No one is hopeless :)
• We don't always have to have the answers. Good, because sometimes I really don't have an answer.
• Ask questions that have depth and purpose. This is on the part of the teacher- it requires content knowledge and planning.
• Spend more time looking for the right question than the right answer.

I was reminded, that I must be intentional about the questions I ask and more explicit in explaining to my students the importance of asking questions. I love that more and more of my students see themselves as mathematicians. Because they are! But I want them to believe it to their core. Sammons shared specific ways how mathematicians use questioning. I already envision an amazing anchor chart :) Mathematicians ...

• are purposeful and spontaneous in their questions. They ask questions 24/7 before, during, and after math
• ask questions for many reasons
• keep thinking about a question even after they have one right answer
• understand further exploration of a question may be needed 
• understand that collaboration inspires new thinking and learning

Wow! Isn't math awesome!! Sammons offers such relevant strategies in this chapter- I have to encourage teachers to read it and find out how fantastic this book is.

As I was reading this chapter, I though of how I want to use thinking stems more consistently this coming school year. I work with a large English Language Learner (ELL) population and having been and ELL myself I know that I often struggled with the words to get my point across or generate a question. So, I created a "Fan Tags Book" with Thinking Stems that each of my students can use as a resource for asking questions. I hope they will become part of their math journals. This past year, I made a concerted effort to use math journals in a way that it would truly serve as a resource to  to my students. I have a manila folder stapled to the front where students store their math manipulative for easy and quick access. Just click on the pic to download.

Tanny McGregor, author of Comprehension Connections has great quotes in her book that encourage the art of asking questions. Here's one:

The important thing is not to stop questioning. 

-Albert Einstein, physicist

Building Mathematical Comprehension Chapter 3

Last summer I participated in a Guided Math Book Study by the same author. I have had a tough time keeping up, but I am caught up now and I am so thankful I didn't quit.

This chapters focuses on the importance of making connections in order to build understanding. The more students see connections the better their understanding. That makes sense. Sammons mentions the importance of having students see connections within the discipline of math, but also in other content areas and within their daily life.

I have my master of science in School Psychology and I remember the analogy that some of our students brains are making connections like traveling down the freeway with no traffic and other students connections are like traveling down a road with lots of turns. They will both arrive to their destination - one just makes it their faster. Sammons states,

...teachers must also explicitly teach learners how to recognize connections between their new learning and their existing background knowledge (p. 86).

One of the things that I am lovin' about this book is that it gives us some real tangible ideas and examples of what we as teachers can do.

Teachers can build schema by facilitating mathematical connections using:

• Math-to-Self (M-S) connections between own life experiences and math
• Math-to-Math (M-M) links between past and present learning
• Math-to-World (M-W) relationship to current events

Modeling Think -Alouds are essential:

• They should be genuine and conversational in manner, so proper planning on the part of the teacher is important
• Be authentic- use life stories that can be related to math. For example, "On Saturday, I baked chocolate chip cookies. The recipe was for one dozen cookies but I only wanted to make six cookies. So, if the recipe called for two cups of flour ... Is the amount of flour I use important?" (You get the idea)
• Be precise and concise- Use a set of sentence stems

I have not used Math Stretches in my class and am eager to try them out this coming school year. I created a few organizers that you can use too. Just click on pic to download!

I think the organizers could also be modified and made into anchor charts. On to Chapter 4.

An Amazing Math Book Study

Like I said, we love book studies. We have been teaching for quite some time and we haven't come across a book that just seems to change the way we completely look at math. We look forward to reading, Building Mathematical Comprehension and sharing our insights and hearing from others as well.