Tuesday, July 30, 2013

Chapter 10: In the Guided Math Classroom (BMC Book Study)

 "You cannot talk a child into learning or tell a child to understand." (Marilyn Burns 2000)
The power of learning comes from within!

Well this is the last chapter of Laney Sammons's book, Building Mathematical Comprehension. A big thank you goes out to Brenda from Primary Inspired and Beth from Thinking of Teaching for organizing this book study. I am hoping you have enjoyed reading the different posts from the different bloggers and have a few takeaways to add to your teacher's toolbox for the coming school year.

The comprehension strategies outlined in previous chapters can be utilized in conjunction with any instructional approach in the math classroom. For those teachers who utilize the Guided Math approach, you can see how these strategies can support the foundational principles of Guided Math (275).
  • All children can learn mathematics. Yes, they can!
  • Learning at its best is a social process. Let the math talk begin.
  • A learning environment that encourages modeling, think-alouds, guided/independent problem solving opportunities, and purposeful conversations supports mathematical growth.
  • Learning math is a constructive process.
  • Ultimately, children are responsible for their learning.
Students need to be immersed in a world of mathematics. They need to be careful observers who view their world through a mathematical lens in order to investigate and recognize relationships and generalize about their mathematical experiences.

Components to consider when implementing Guided Math:
  • A Classroom Environment of Numeracy. Students should use manipulatives, compute, compare, categorize, question, estimate, solve problems, converse, and write about their mathematical thinking. All students should be expected to engage in making meaning of the world mathematically (281). Have you ever read the book Math Curse by Jon Scieszka? This book is a great way to bring math to life and show that math is indeed everywhere!
  •  Math Stretches and Calendar Board Activities. These activities can require students to review concepts already covered and mastered, relate to concepts currently being explored, or preview what mathematical concepts are to come. (282) When starting a unit on measurement, this Measure Up: Measurement Sort could be used as a Math Stretch to preview what is to come. Click here to view a description of the activity and click here for a copy of the Measurement Sort. This activity requires students to think about what they may already know related to measurement units. The activity can promote mathematical thinking where students then can share their ideas in a Math Huddle. Student thinking can evolve during the Math Huddle and while the unit on measurement unfolds. For calendar math, check out this site. If you click on one of the numbers in the left grid, it will give interesting facts about that number. What a cool way to hook learners and add a little something different to calendar math. What are some activities you do for calendar math? Feel free to link up and share your ideas.
  • Whole Class Instruction. This is the time when all students get the same message and engage in the same activity at the same time. Mini-lessons, modeling, think-alouds, and activating strategies can be accomplished during this time. Caution must be taken when using whole class instruction knowing that some students are hesitant to talk in a large group setting, not all students will necessarily have time to participate, and inattentiveness may sneak up on some students (283). One activity I have done during whole class instruction is Number Talks. Click here and check out this previous post to see how it works. This is just one way you can do it. Have you tried this before? Might this work with your students?
  • Guided Math Instruction with Small Groups of Students. It is imperative that small groups are kept fluid and change based on the readiness levels of students. More time is given to each individual student and observations of students can help drive/guide instruction during small group instruction(284). Click here to find a Small Group Instruction ~ Record Keeping sheet. This sheet can help in recording data that can be referred to when making instructional decisions. Click here to read a previous post about small group instruction.
  • Math Workshop. It is here where students take responsibility for their own learning. It is a time for students to show what they know. Monitoring student work and providing feedback is key to ensuring this time is maximizing student learning (284). Learning contracts and menus can be used to design mathematical experiences for students to work on during math workshop time. Click here to see a Fractions: Thinker Keys Menu.
  • Individual Conferences. Conferences can be used to assess student understanding, identify and clarify any misunderstandings, and to extend/refine student understanding. Conferences should be brief with a targeted goal in mind (285). Have you ever visited Dr. Nicki's Guided Math Blog? Over on her site she has some conference templates you might be able to use when you conference with students. Dr. Nicki's post on Individual Math Conferences can be found here.
  • An Ongoing System of Assessment. Effort needs to be made to ensure there is a balanced system of assessment. Observations, discussions, formative assessments, summative assessments, and student reflections are all essential in a balanced system of assessment. I have used a Lesson Recap as a formative assessment tool to help me gauge my students' understanding. Click here to see a copy of the recap. You can easily adapt it to a skill/concept your students are working on. To read a little more about the Lesson Recap click here. You will find the description towards the bottom of the post.
Whether we incorporate all of these components or some of them in our math classrooms, it is with hope that we are teaching our students to become mathematicians!

Hope you found some of the links posted here helpful as we wrap up this book study. I would love to hear your comments and feedback. Link up or your share thoughts. Don't forget to stop by some of the other blogs to learn more! 

    Wednesday, July 24, 2013

    Chapter 9: Monitoring Mathematical Comprehension (BMC Book Study)

    Students need to understand that a breakdown in mathematical comprehension is not a sign of failure, rather it is a part of the learning process. When students monitor their mathematical understanding, they are calling upon the full spectrum of comprehension strategies (273). It is our job as teachers to make sure students realize the importance of being aware when their understanding breaks down. To help students recognize when there is a breakdown in comprehension, Laney Sammons offers this list of signs:
    • Your internal voice is not interacting mathematically with the concept or problem.
    • Your mind wonders away from the mathematical task at hand.
    • You are unable to visualize the mathematical concept or problem.
    • You are unable to recall the details of a math idea or problem.
    • You cannot answer questions asked to clarify meaning. (259)
    How do we help students gain the confidence and skills necessary to rethink and switch strategies for a fresh start when there is a breakdown in mathematical understanding? Students need some fix-up strategies in their mathematical toolbox.
    • ask questions
    • connect to other mathematical concepts
    • draw a picture
    • use manipulatives
    • make inferences
    • pause/refocus
    • reread/rethink
    • collaborate with a peer
    Check out this earlier post to read about this strategy, Need a Hand? Try This! I have used this tool in the classroom.

    Bottom line...students need to monitor and KNOW WHEN THEY KNOW and KNOW WHEN THEY DON'T KNOW. Is it a "HUH?" moment? or a light bulb moment? (263) The Color-Code Metacognition Math Stretch (266) reminds of the red, yellow, green buckets I use in my classroom as a Ticket Out the Door.

    After a lesson or at the end of class, students can put their names on a slip of paper in the colored bucket that best reflects their understanding of the day's lesson. Red: I need help. Yellow: I am getting there. Green: Got it! Another way I have used these buckets is to give students a problem to do as a Ticket Out the Door where they record their responses on an index card. After solving the problem, they place it in the bucket that best reflects how they feel about the problem. This can be used as a quick formative tool to identify which students may need additional help and which students are ready to move on. The buckets allow students to metacognitively reflect on their own personal understanding. The data from the buckets should be used in conjunction with teacher observation.

    Don't forget to check out the other bloggers who are posting about this topic. Link up or comment on ways you help students monitor comprehension in the math classroom. Stop by on July 30 for the last post for this book study: Chapter 10 ~ In the Guided Math Classroom.

    Chapter 8: Synthesizing Information
    Chapter 7: Determining Importance

    Friday, July 19, 2013

    Chapter 8: Synthesizing Information (BMC Book Study)

    The Building Mathematical Comprehension book study is beginning to wind down with only three chapters left. A shout out goes to Brenda from Primary Inspired and Beth from Thinking of Teaching for organizing this book study. I am one of the hosts for this chapter. Happy reading!

    Similar to the ripples caused by throwing a rock into a pool of water, meaning expands and "simple elements of thought  [are] transformed into a complex whole."
    (Miller 2002, pg. 227)

    According to Laney Sammons, synthesizing may be the most complex of the comprehension strategies with having to merge together the other strategies in order to generate an entirely new and original idea, perspective, or line of thinking (227). Synthesizing is the catalyst for the construction of mathematical meaning (229).

    Sammons goes on to explain what students need to know about synthesizing:
    • Mathematicians may change their mathematical thinking with each new mathematical experience.
    • Mathematicians construct new mathematical meaning through the synthesis of new and existing mathematical knowledge.
    • Mathematicians know that mathematical knowledge is constantly evolving.
    • Mathematicians can explain how synthesis helps to create new understandings in math. (230)
    As with the other strategies presented by Sammons, explicit instruction through modeling and think-alouds is necessary in order for students to"see" how to effectively apply this strategy to math. Students need to see how the many facets of mathematics intermingle to form larger mathematical concepts-the big ideas (233).  Not necessarily an easy task...

    How can we make this rather abstract strategy more concrete for students? One recommendation made by Sammons was to use nesting dolls. After lining them up, Sammons recommends asking students how the dolls represent their thinking. Then after stacking them from smallest to largest, students should be asked to reflect how the dolls now represent their thinking. The goal here is for students to recognize that big ideas are made up of smaller ideas that build and change over time. (235-236) After reading this, I thought, this does make sense!

    Conjectures: When students synthesize they take new mathematical ideas along with what they already know to create new understandings. After observing patterns and relationships that appear to be true but have not been tested, students can form conjectures, or informed guesses and predictions (237). Laney Sammons goes on to explain that students rarely make conjectures unless the process has been modeled and encouraged. Sammons goes on to recommend the following categories that can offer opportunities to model and create conjectures.
    • Properties of number operations (identity, zero, inverse, commutative, associative, and distributive)
    • Characteristics of special types of numbers (odd/even, prime, improper fractions)
    • Procedural rules (regrouping, multiplying a decimal) (239)

    Support or Disprove This Conjecture Stretch (240): Sammons recommends this stretch where students are asked to add evidence that would support or disprove a given conjecture. You can begin this stretch by posting a chart and having students share evidence, verbal or numerical. Using sticky notes to post ideas is one recommendation Sammons makes. I thought the next step could be small group/independent work where students can ponder a conjecture, record their thinking, then share out with others using the template below. I chose the puzzle template to help remind students that synthesis requires students to put together different mathematical ideas in order to create new understandings or revise mathematical thinking. Click on the image below to see some examples.

    I am going to leave you with this statement from Sammons: 

    Students may be "taught" the mathematics curriculum, but unless they are able to recognize the big mathematical ideas, see how the details relate to the big ideas, and apply these to real-life situations, they are not mathematically literate (246).

    How do you help your students synthesize the big ideas in math? Please link up or share your comments below. Don't forget to visit some of the other bloggers listed below. Stop back July 24 to see the next chapter: Monitoring Mathematical Comprehension.

    Monday, July 15, 2013

    Chapter 7: Determining Importance (BMC Book Study)

    When trying to determine the importance, it can occur at three levels: word level, sentence level, idea level. The ultimate goal of learning is determining the importance at the idea level (201).

    Mathematicians determine importance based on:
    • mathematical purpose.
    • background knowledge.
    • knowledge of text features and structures.
    • ideas shared during discussions (202).
    Problem solving. Helping students to better understand the typical text structure of word problems may help them to identify the important facts necessary to comprehend and solve the problems (204). Think about it, in most instances it is not until students read the final question at the end do they know what parts of the text are essential to solving the problem. Skimming the text first to get to the question, and then rereading the text and deciding what is important and not important is a process that should be modeled and practiced.

    Color coding with red and green is one way to draw visual attention to importance. Red indicates those pieces of information where one should STOP and take note since it is IMPORTANT information. Green indicates those pieces of information where one can let GO of the information since it is NOT essential to solving the problem. Might this strategy work with your students?

    Laney Sammons gives the recommendation to pose a story problem with plenty of facts, but no questions (213). Ask students what information is important. It should quickly become clear that there is no basis to filter the information since there is no answer to find. Click on the image to grab your copy.

    The next step is to give groups of students different questions to answer based on the story. Have each group identify what information is important in answering their given question on chart paper. Once all groups are done post the different charts. Do all the charts have the same information? Why or why not? Discuss how importance is determined by the purpose of the task. Below is a sampling of questions that could be used with the problem above. What other questions come to mind? Click on the image to grab your copy.

    Helping students become more cognizant of how to filter out important information can help them to build mathematical comprehension and become stronger mathematical thinkers (224). How do you help students with determining importance? Link up or comment below. Don't forget to visit some of the other bloggers by clicking on the schedule. Stop by on July 19 to see what Sammons has to say about Synthesizing Information.

    Tuesday, July 9, 2013

    Chapter 6: Making Inferences and Predictions (BMC Book Study)

    Inference is a mosaic, a dazzling constellation of thinking processes... 
    Inferences result in the creation of personal meaning.

    Even though students often make inferences and prediction regularly, they are often subconsciously made. It is important for students to become aware of this process and know when and how to use it (173).

    When students infer, they can...
    • draw conclusions
    • make reasonable predictions (link to prior schema yet describes something in the future and can checked)
    • make connections 
    • gain insight to what might no be explicitly stated 
    • make critical and analytical judgments. (172)
    One idea presented by Laney Sammons to help students make predictions and inferences is to use a Word Splash. To create a Word Splash when introducing a concept or topic, pick key and important words related to the topic. Then have students make predictions/inferences when discussing how the words are related/connected (187). Record student thinking on an anchor chart so as the unit progresses student can revisit and revise their inferences and prove/disprove their predictions.

    I have used Word Splashes in the past. I like to add images if possible as well. See one here that can be used to introduce 3.MD.2-Measurement to 3rd graders. Click on the image if you think you can use this with your students.
    Another idea Sammons talked about was What's the Question? Stretch (189). Give students a short scenario and have them generate questions that can be answered using the information presented in the scenario. Students will have to infer in order to generate these questions. Be sure to have students share their varied questions. This activity offers students the opportunity to extend their thinking and make connections with other mathematical concepts. The beauty of this task is that it is open-ended and there are multiple answers. Take a peek at the football scenario below. What questions come to mind? What might your students have to infer? Grab a copy by clicking on the image if you think you can use this with your students.


    Building student fluency in predicting and inferring can help students when doing problem solving. This is definitely a strategy that is not as clear cut for me in terms of explicitly teaching it in the math classroom. I am always looking for student friendly ways to bring this strategy to the forefront for my students. I would love to hear any tips and tricks you use in the math classroom to help students in this area.

    The next chapter will be on Determining Importance. Don't forget to stop back July 15 to see what Sammons has to say about this comprehension strategy. In the meantime, do not forget to visit some of the other blogs who are talking about this topic. Comment or link up and share your thoughts on making predictions and inferring in the math classroom.

    Wednesday, July 3, 2013

    Chapter 5: The Importance of Visualizing Mathematical Ideas (BMC Book Study)

    Visualization is essential to understanding, so how can we get our math students to visualize mathematical concepts and problems? How can we get our students to visualize multiple representations and then be able to evaluate which one is the most appropriate for a given purpose?

    We probably have heard of picture walks in reading. Laney Sammons recommends doing "picture walks" to build the capacity to visualize (161). Mathematical picture walks can be used with textbooks, online screen shots, or even children's literature. Asking students why a particular image was chosen to represent a mathematical concept in a text can help them to build their own capacity of options to use when visualizing. Two questions that I feel can really lead to a deeper understanding the purpose behind visualizing are:
    • How effectively does this representation promote greater understanding of the concept?
    • Are there other ways that this concept or idea can be represented? What are they?
    Another strategy recommended by Sammons was "Visualize, Draw, and Share (162)." This activity can help students to create mental images from verbal statements. The teacher gives a statement about a mathematical idea.
    • I'm adding ten plus five.
    • A rectangle with an area of 18 square units.
    • Two thirds
    • What do you visualize when you think about _____? (multiplication, decimals, a foot)
    The students then are asked to create a mental image and then transfer it to a pictorial representation. Students then share their pictorial representations with classmates where they can discuss each representation and its effectiveness at improving mathematical comprehension.

    You also can flip this idea and start with a representation and have students explain what the representation might be trying to explain. The example Sammons uses is an array with X's in a two rows by three columns representation (165). Answers may include:
    • 2 x 3 = 6
    • Two children have three cookies each. How many cookies do they have altogether?
    • Repeated addition
    Sammons recommends the use of children's literature to bring in real-life examples to help students visualize (166). One recommendation was Basketball Angles: Understanding Angles (Wall 2009). Has anyone ever used this book before? It looks like an interesting book that certainly shows the importance of angles in the real world. Using nonfiction literature that explores real world math concepts can help create visual anchors for students. Poetry is another recommendation made by Sammons to help students visualize math concepts. One poem that I have used in the past is Smart by Shel Silverstein. A short little poem that teaches an important lesson about money. Students can be asked to "visualize" the trades to better understand why the title of the poem is quite ironic. Click the image to read the poem.
    One key take-away is that visualization in math does not necessarily need to be a drawing. It is being able to represent a math concept in more than one way: mathematical symbols, real-life examples, model/diagram, and/or explain with words (163). Click on the image to grab the freebie.
    MyCuteGraphics, Creative Clips, Hello Fonts
    What are some activities you use to help your students visualize in math? Comment or link up and share your thoughts...even if you haven't read the book. Don't forget to stop by some of the blogs on the book schedule below to get some more ideas about visualizing in math. The next chapter is on making inferences and predictions. Stop by July 9th to check out inference, a mosaic of thinking, in the math class.


    Thanks for going me on this Throwback Thursday. Don't forget to visit The Teaching Tribune to see some other "throwback" ideas!

    Chapter 2: Math Vocabulary