Differentiate in the Math Classroom with Learner Agreements

The readiness levels of our students vary year to year, chapter to chapter, skill to skill. Learner agreements, also known as learning contracts, can help address the various readiness levels throughout a unit of study. Curricular resources and supplemental resources were used to craft this learner agreement. This learner agreement is not the instruction that takes place rather the independent work where students are asked to show proficiency of the standard. Click on the image below and take a look at the learner agreement that was created for fifth grade.

   Graphics by Anchor Me Designs
Learner agreements encourage students to be regulators of their own learning. Students need to be asking themselves...Am I getting this? Do I understand the skill enough to apply it to more complex tasks? Learner agreements provide those students who may be grasping a concept quickly with a more engaging task that they can work through more independently. Rather than wait through repetitious instruction that is at a pace that is not appropriate for their readiness levels, students can be exited out and begin a task that will help address their needs.

So how do learner agreements work? Preassessments can be given prior to a unit of study to get a pulse of where the students are reading to learn. Daily observations and formative assessments also can be used to guide student learning. Learner agreements take some planning and foresight. Standards need to be looked at, student readiness levels need to be considered, materials need to be gathered, and tasks need to be decided upon. It is only then that the learner agreement can be created. However, once created, they are a perfect way to differentiate learning on the spot.

The learner agreement can be viewed as a blueprint for learning. Tiering the tasks allows for multiple entry points for learning. After instruction and depending on the readiness level and the comfort level of each student, students can be assigned a differentiated task to complete. Those students who are more proficient at a particular skill may enter the learner agreement at the progressing level. If they encounter difficulties, they can advocate for themselves and go back to look at/do some of the learning-leveled tasks to ensure foundational understanding is established. Or if a student is demonstrating proficiency at the progressing level, s/he may bump up to the extending-leveled tasks. If students are struggling at the learning-leveled tasks, a small group can be formed with teacher support while other students work at their readiness levels as guided by the learner agreement.

It is important to keep movement through the leveled tasks fluid and based on where students are ready to learn at that moment in time. Keeping in mind the zone of proximal development for students, we do not want tasks to be too easy or too difficult. Some students face tasks that are already in their comfort zone and are not getting the challenge they need. Other students may face tasks where they simply are not developing the understanding necessary.  Tasks need to be "just right." For learning to take place, there needs to be a certain level of challenge. That challenge has to allow for the development of new learnings and skills.

Is this a strategy that might work with your students? What other ways do you differentiate for students in the math classroom?


Concrete, Representational, Abstract (CRA) - Sequence of Instruction

As we work to help students develop deeper conceptual understanding in math, CRA can be helpful. CRA is a system of instruction from the concrete to the abstract where students are first introduced to a concept through concrete experiences. In the area of math that means using concrete objects or manipulatives. Instruction then moves to representational experiences where drawings are used. After the concrete and representational levels are practiced and understood, then students are asked to work with abstract symbols, notation, and numbers in math. The abstract level is what we know as standard algorithms or standard notation. Through the instructional experience of CRA, students can make stronger connections and develop deeper understandings.

When we use concrete experiences with our math students, we help them to develop foundational understandings of a skill or a concept. Think about when students are first introduced to measurement. We do not put a ruler in their hands right away. Students often have experiences measuring with counters, cubes, or links. Using these concrete manipulatives helps to develop an understanding of length. Concrete experiences should be embedded in instruction when new concepts/skills are introduced.

The representational level, sometimes referred to as semi-concrete, is when students draw what would be represented by the concrete objects/manipulatives. For example if students are using counters to make groups when learning about multiplication at the concrete level, students could then draw Xs, dots, or even use mini-stampers (What fun!) to create a visual representation of  counters. The representational level helps to bridge understanding from the concrete to the abstract for students.

Following representational experiences, students can work with a math concept/skill abstractly. Symbols, numbers, and notation are loaded with meaning in math. If students jump to the abstract level before having the concrete and representational experiences there can be a breakdown in understanding. Consider the different standards below and how the activities/strategies focus on the different stages of the CRA framework. Click on the image to grab a copy.

When considering using technology with students, choose programs that are at the appropriate readiness levels for students. View the chart below to see how you can structure technology use at the different stages of understanding for a concept/skill. Click on the image below to download a copy. Visit some of the sites to see the progression of understanding from the concrete to the abstract.

How do you weave these different levels of instruction into your daily routines?


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! 


    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


    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.


    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.


    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.


    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


    Chapter 4: Increasing Comprehnsion by Asking Questions: Book Study (BMC)

    "Questions lead children through the discovery of their world..."  (Keene and Zimmerman, 113)
    Yet...as children progress through elementary school, something squashes this curiosity.
    "Children enter school as question marks and come out as periods." (Postman, 116)

    These opening thoughts in Chapter 4 definitely left me with something to think about. How can we continue to pique the curiosity of our students and foster that endless stream of questioning that was second nature to them in earlier years? Easier said then done, yet questions really are at the heart teaching and learning (117).

    I wonder why... What if... How can... are just a few of the questions we want our mathematicians to ask in order to develop deeper understandings. Another interesting point made was "it may be more important to find the right question than to find the right answer (118)." How can we flip our students' thinking to this mindset? A key take-away was that students NEED to be the ones creating the questions, not the teachers. Students need to be explicitly taught to ask viable questions that will enhance mathematical comprehension.

    Laney Sammons goes onto explain the types of questions and how they are related to math: Right There, Think and Search, On My Own. Sound familiar in the reading world? Teaching students the difference between thin questions and thick questions also can improve the quality of student questioning (128).

    Skill: multiplying a whole number by a fraction
    Thin question: Is the product greater than or less than the original whole number?
    Thick question: Why is the product less than the original number?

    When teaching about thin and thick questions, Sammons recommends using different size sticky notes for students to record their questions. 3" x 3" sticky note for thick questions; 1/2" x 2" sticky page markers for thin questions. Another recommendation was to have students write in a thick marker when recording thick question or a thin marker when recording a thin question. The goal is to help students become more automatic and independent at creating questions that develop deeper thinking and understanding.

    Questions that linger also were mentioned. How often do we ask questions that motivate students to continue to ponder and revisit those questions? It these types of questions that leave a trail of "math residue" that leads to invaluable learning (129).

    Laney talks about strategy sessions and how they differ from a typical mathematics lesson (131). During these strategy sessions the focus needs to be on helping students to develop rich mathematical questions that enhance mathematical comprehension. To use as part of a bulletin board or as a bookmark for students, I created a few visuals called "Get to the point..." to help in the scaffolding and building of students' independence. If you think your students can benefit from these tools, click on the image and grab a copy.

    Check out an earlier post on questioning in the math classroom for these color coded question cards.

    The next chapter is on visualizing mathematical ideas. Come back after July 3 and see what Sammons has to say about visualizing. In the meantime, feel free to comment or link up and share your ideas about student questioning in the math classroom. Don't forget to click on the schedule below and visit some other blogs hosting this chapter.


    Chapter 3: Making Mathematical Connections (BMC Book Study)

    As I was reading this chapter, I thought whoa...it is important not to lose sight that teaching students in the mathematics classroom to make connections is not the learning goal, rather the means to an end. The end being activating schema and opening entry points of learning to solidify mathematical understanding (105).

    I like how Laney Sammons described making mathematical connections as building bridges from the new to the known (100). To many students math is viewed as a textbook with isolated units of study and not necessarily a part of their everyday lives. We need to take where students are at and build from there. Sammons spends some time talking about schema. Schema being the students' prior knowledge. Throughout the chapter, Sammons makes it clear how important it is to link new knowledge to existing schema in order to provide for a richer learning experience. Prior knowledge, or schema, consists of:
    • attitudes: How do students perceive math? 
    • experiences: What role does math play in the daily lives of our students and the world around them?
    • knowledge: What mathematical concepts do students understand and know? (88-89)
    There are three types of connections in math much like reading: math to self, math to math, and math to world (92-94). Math to self connections focus on one's own life experiences involving math (allowance, time, measuring). Math to math connections focus on connecting past mathematical concepts and procedures to current units of study. Students need to see how math builds on prior math. Math to world connections dispels the misconception that math is only taught in school with a textbook. Math to world connections help students to understand the bigger picture of math.

    Once again explicit instruction is needed to help students learn this process of making connections so it becomes automatic for students. As in reading, Sammons goes on to explain how important it is to focus on meaningful connections and not the distracting ones. This, I know from reading, is sometimes hard for students. And when introducing making connections in math, I feel caution needs to be taken so that the making connections does not steer learning away from the math concept at hand.I do feel making math to math connections is a place I want to focus on next year to help students see the bigger picture.

    I took some of Sammons ideas and used them to create these reminders for thinking stems/questions related to making connections. I can make a copy, cut them out, and put them on a ring as a visual reminder to use them during modeling and think alouds. I also can make them for students as bookmarks to help them practice making meaningful connections in the context of what we are currently studying in math. Click on the image above and see if you think this might be useful for your students.

    As students begin to develop an understanding that math impacts and exists in their daily lives, incorporating real world problems create authentic learning experiences. It is through these experiences that math becomes more meaningful and relevant (104). This reminded me of an activity I did during a geometry unit, Architectural I Spy. Once we started talking about architecture and the different shapes we could find in the different architectural structures, I am telling you students no longer looked at buildings in the same way. Talk about connecting to geometric shapes in the real world. Check out the activity below by clicking on the image if you think you could use something like this with your students. Just something to think about...

    What are your thoughts on making connections in mathematics? Leave a comment or link up and share your thoughts. The next chapter focuses on asking questions. Asking questions is indeed an art in itself. Hope you stop by after June 27 to see what new ideas I find. In the meantime, don't forget to visit the other blogs who are hosting this chapter to get some of their ideas too: Teaching with a Touch of Twang and Smiles and Sunshine. You can click on the schedule below to see other blog posts from previous chapters as well!

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    Chapter 2: Recognizing and Understanding Mathematical Vocabulary (BMC Book Study)

    The language of math, you gotta love it! This chapter begins by Laney Sammons emphasizing that explicit instruction is needed to teach the specialized vocabulary of math. Sammons notes a key difference of vocabulary acquisition in reading versus math. For reading, students have incidental exposure to words through daily conversations and reading, whereas in math specific vocabulary is rarely used during everyday conversations (81). If you take a look at the chart below, you can see the different categories of math words and how they can hinder math comprehension if not explicitly taught in the context of math.

    Clipart by Phillip Martin
    Vocabulary instruction DOES NOT mean copying definitions. Sammons refers to Beck and Marzano who explain that the most effective type of instruction that increases vocabulary instruction needs to be robust. It needs to be thought-provoking, playful, interactive, and REVISITED. Who ever knew that learning vocabulary could be so much fun?!

    Sammons goes on to include Marzano's eight research-based characteristics of effective direct vocabulary instruction:
    • Vocabulary instruction does not rely on defintions.
    • Representation of knowledge should be shown through linguistic and nonlinguistic ways
    • Gradual shaping of word meanings occurs through multiple exposures
    • Teach word parts (milli-, centi-)
    • Different instruction for different words - in context and through concrete experiences
    • Verbal practice of word usage in authentic contexts
    • Play with words - KEY!
    • Focus on words that have a high probability of enhancing student success (50-53)
    How do you know which words to teach? This is never an easy question to answer. One starting point Sammons recommends is teaching those words that have a high probability of enhancing student success. Choose words that are mentioned in the standards. These are the must knows that students will need to understand the concepts, skills, and practices for that academic year (57).

    Immersion in math vocabulary is key. Whether that means involving parents to "talk math" at home (60), offering time in math huddles to talk math in authentic contexts, or writing math (63-66), multiple exposures in multiple settings will help help students understand and grasp the language of math.

     Other recommendations by Sammons...

    Math Word Wall (67-68) Make it a "living word wall." Keep it current. Revisit it. Have students use the words to talk math.


    Graphic Organizers (69-76). Graphic organizers help students to organize their thinking and show conceptual understanding of math vocabulary. As was Sammons recommendation in the previous chapter, it is important to model and do teacher think-alouds when introducing a new graphic organizer. Keep in mind the gradual release of responsibility back to the students. You might find reading this previous post interesting if you want to read more about graphic organizers. Do you have a graphic organizer that works well in math?

    Games and Word Play (77-80). Games and word play are motivational ways to help your students become more word conscious. If you click on the image below you can find a few activity cards with word play recommendations from Sammons along with a few others I have used to bring in some kinesthetic practice when reinforcing vocabulary usage. I cut these activity cards out and ring them so if I have a minute, we can look to the math word wall and engage in some vocab play.

    Well, that is vocabulary in a nutshell...sort of! How do you teach vocabulary in the math classroom? What activities have you found to be helpful. Please comment or link up and share your thoughts. Don't forget to visit some of the other blogs to learn more about Building Math Comprehension: Carol from Still Teaching after All These Years (Carol has posted some very helpful downloads.),  Beth from Thinking of Teaching, and other bloggers who have linked up below. The next chapter focuses on mathematical connections. Don't forget to stop back after June 21st to see more ideas.


    Get to Know Your Students as Mathematicians

    Getting to know our students as mathematicians is important. Many of us may give our students reading surveys so we get to know our students as readers. Why not survey our math students to get to know them betters as mathematicians? Here are two different versions of a math survey that can help you to learn more about your students as mathematicians. Have students complete these survey charts by  filling in the height of the bars that best reflect them as mathematicians. My Math-o-Meter can be used by students to rank their comfort level with each of the concepts. To show what I know in math, I do best when I... can be used to tap into your aspiring mathematicians' learning styles. Use these charts in planning different learning opportunities for your students. Have your students tap into areas of strength and venture into areas that may not always be in the comfort zone.

    If you click here you can find a post that includes a multiple intelligence survey similar to these. Start building learning profiles of your students early on and revisit them often to find multiple entry points of learning during different units of study.

    What are some ways you get to know your students as mathematicians?

    Chapter 1: Comprehension Strategies for Mathematics (BMC Book Study)

    With the CCSS standards it has become more apparent that all teachers are READING teachers. Content area teachers are reading teachers, PE teachers are reading teachers, librarians are reading teachers...and yes, math teachers are reading teachers, too.

    In Chapter 1 Laney Sammons discusses the global achievement gap in mathematics - a gap between what our students are taught and what is needed to be successful in our ever changing world. It goes on to define mathematical literacy as "...an individual's capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments, and to use and engage with mathematics in ways that meet the needs of that individual's life as a constructive, concerned, and reflective citizen." (Organisation for Economic Co-operation and Development 2010, 18). Bottom line, our students need to be able to functionally use mathematics.

    Understanding mathematics is much more than just number crunching. Mathematicians have to construct meaning just like readers do. Characteristics of "good" mathematicians are similar to characteristics of "good" readers (22).
    • They use prior knowledge to help them to tackle "new" concepts and problems.
    • They use multiple strategies to tackle a problem.
    • They demonstrate mathematical fluency.
    • They monitor and fix up their understanding of concepts.
    • They reason and defend their thinking to others.
    So how can we help our students construct meaning as they learn math concepts and solve problems? Well, we can borrow from what has worked well during reading instruction. Through explicit instruction (31-34), we teach strategies to our math students...but with a twist on mathematical content and processes. We need to explain "what" the strategy is, "why" we use the strategy, and "when" the strategy should be used. Learning opportunities need to include teacher modeling, guided practice where students work in small groups or with a partner, and then independent practice. As in reading, we want to gradually release the responsibility of learning back to the students to help them develop a deeper conceptual understanding of mathematics. One interesting recommendation made by Sammons (33) is that during the initial modeling of a strategy, it should be soley the teacher talking. Student participation during this time should be avoided so that the focus remains on the teacher's thinking. Something to think about...

    A take-away from something I read from Beth at Thinking of Teaching was her idea to incorporate math text into guided reading. Not necessarily a book, but rather reading a math problem. This would give students the opportunity to peel away the layers of a math problem much like the way they peel away the layers of a more commonly used guided reading text. This reminds me of an activity I did last year with a math task. I didn't do it in a guided reading setting, rather as whole group "close" read. See the post here if you are interested in reading more.
    The instructional strategies and terminology that reading teachers use so successfully in teaching reading comprehension should be utilized in the math classroom as well. The next chapter is Recognizing and Understanding Mathematical Vocabulary. Math is a language all its own. Come back June 15 and see some ideas how to help our students talk the language of math.

    One of my passions is math. I would love to hear thoughts and ideas. Comment or link up below and share your take on math comprehension. Don't forget to visit some of the other blogs hosting the chapters. Click on the schedule below and happy reading.


    Get Your Math On ~ Summer Book Study

    Button by Brenda from Primary Inspired
    Starting June 8 a book study will begin for Building Mathematical Comprehension. The coordinators for this book study, Brenda from Primary Inspired and Beth from Thinking of Teaching, have gathered together a group of bloggers who will be hosting different chapters throughout the summer. Click on the image to get a copy of the upcoming schedule.

    Whether you want to follow along or lurk to get some new ideas, link up and join the blog hop. Check out the different chapters throughout the summer by visiting the different blogs. I will be talking about Chapter 8: Synthesizing Information in July.

    You can take a peek at the book at Amazon by clicking on the image.
    Hope you will join us!


    Just a Quick Pep Talk

    With Teacher Appreciation Week behind us... And the end of the year upon us... 

    A Pep Talk from Kid President


    Measure Up

    We are starting a unit on measurement. To prime our brains and get us thinking for this unit, we are just working on getting our minds wrapped around the two systems of measurement: customary and metric. And...we are looking at the units we use to measure length, capacity, and mass. Students cut apart the word slips and sort. Use this activity as a preassessment to see what students already know. Use it during your unit of study as a formative assessment piece. You can grab this freebie here if you want to try this Measurement Sort with your students.
    Take a look at this YouTube video that helps students to see there is a difference between the meanings of mass and weight. The focus of the video uses metric units. The realization that these two words do not mean the same thing, in reality, sometimes is rather shocking to students.


    Order Up! ~ Fraction Math Task

    To culminate our study of fractions, students created "pizzas." Each student was given an order for a pizza. Based on their understanding of fractions, they "baked" a pizza made to order. Seeing that the goal of this activity was math related, I did have some of the ingredients and the "pizza dough" ready for students to use so they could focus more on the math aspect of the activity. There are different ways to set up this activity. You know your classroom best.

    Now, if you look at the sample orders, you will see some of the orders are more complex. I differentiated the orders based on my students' readiness levels. Students first had to interpret what the order meant before they could actually start making their pizzas. If 2/3 of the pizza has mushrooms and 1/3 has mushrooms and tomatoes, does that mean that only one of the two thirds section also has tomatoes? The questioning was necessary before clarity prevailed for our pizza makers.

    After the pizzas were made, students checked each others orders. We realized some "customers" would not be happy unless we fixed their orders. A real world problem for sure :)! Students then completed the Task Questions. The questions address the following I can... statements.

    • I can partition wholes into various amounts to solve a real world problem.
    • I can explain what the numerator and denominator of a fraction represent.
    • I can use models to show and explain equivalent fractions.
    • I can use models to compare fractions.

    Students can create their own "personal" pizzas rather than follow a given order. They then can use their own "personal" pizzas to complete the Task Questions. Click on the two images above for your free copies of Pizza Orders and Task Questions.  Bon Appetit! 


    Symbaloo Webmix of Math Websites and Resources

    Take a peek at some of the math resources posted on this Symbaloo webmix. You might find something you can use with your math students.

    What other math websites or math web tools do you use?


    Flip for Rational Exponents

    I was doing a lesson on rational exponents with my 8th grade Algebra students. I wanted to do more than an example walk with them to introduce the concept. I created these flippables for students to use. Even 8th graders like to cut and fold :)! Students simply folded along the solid line and then cut slits along the dotted lines.

    The first flippable was the basic understanding of the lesson. What happens when you have a positive unit fraction as an exponent? Each additional flippable was designed to build upon previous understandings. What would happen when a fractional exponent was not a unit fraction? What would happen when the exponent was negative? I copied each in a different color to refer back to in future lessons.

    The goal was for students to look at the problems on each flippable and identify patterns, connect to previous learnings, and ask clarifying questions. Through discussion and exploration, students identified the patterns and recorded the answers under each flip. They were able to check for accuracy through collaboration and/or referring to their textbook. After each flippable, we synthesized our learning. Students would then apply the skill to more complex problems.

    This strategy can be used with any grade level and any math skill. Using different colored flippables while trying to build a skill is one way to differentiate while having students strive for deeper understandings. Students can refer to these flippables as anchors of learning in future lessons. Students also can use these flippables as a review. Click on the image below and find a teacher copy and a student copy of the four flippables we used during this Algebra lesson.