Teacher Forum

We are Community!

Why Should We Work So Hard to Build Community?

Class Math Agreements and Community and Mathodology

Building community beginning on the first days helps to create new relationships and strong bonds that will last throughout the year. Creating a shared vision of the expectations, developing a common understanding of classroom limits, and fostering a love of learning are only a few of the characteristics you might have in mind as desired outcomes. Ultimately, achieving mutual respect and a spirit of collaboration creates an ideal working environment for the classroom.

When community exists, each child feels valued. A sense of shared purpose unites the group and working together to accomplish goals becomes a priority. Our goals are BIG and require the effort of all of our members. The uniqueness that each student provides as a member of the community must be valued and each individual strength will make the community stronger and better. As children develop a sense of duty to the community, self-discipline is likely to emerge more naturally and from the child’s (intrinsic) motivation rather than from external or reward-based methods (extrinsic).

Early in the year, creating purpose in the child’s movement and activity is desired and we balance the freedoms offered within the environment, the needs of the young child to move, and the constraints of the environment. Providing structures and routines will help to create order as well as ensure a safe environment for your children. A strong sense of community is one of the most effective ways to teach how to use individual freedoms.

How do we build community?

We play games and have fun together. We share lunch and work with each other, mixing-up our groups with an emphasis on getting to know new friends. We interview and find out more about each other by sharing experiences, stories, traditions, and the accomplishments we are proud to have achieved. We make time to appreciate each other and learn how to recognize others, as well as ourselves. 

In our community, we learn to problem solve, developing the skills necessary to take care of ourselves and others. When solutions are found and conflicts resolved with little or no direction or intervention by an adult, students feel great pride! Creating a class agreed-upon list of rights and responsibilities with the students allows them to partner in holding others accountable and enforcing your shared vision of community.

Grace and courtesy work also play a role in learning how to act in a community. A firm handshake and smile in the morning set a respectful tone for the day. Allowing students to have the role of a “class greeter” is a great way to have students serve in a leadership role as they create personal and inviting welcomes to the community. Practicing how to greet visitors with a cup of tea and a special chair or preparing a class snack are other ways students can assume responsibility. Modeling ways to ask for help, challenge other student’s ideas and even how to say “no thank you” respectfully are tools your students will need to have in order to work effectively in their community.

A natural extension of building community within our classrooms is to reach outward. The work that starts within our classroom might find opportunities in other areas within the school. Participating in the work of the larger community helps the students feel proud and invested.  Students experience, on a small scale but in a real way, that they can create change. We can act individually or as a group – and we DO make a difference! 

Specific Ideas to try at the beginning of the year might include:

  • Toss a ball in a group to help learn names
  • Learn a favorite food of a new friend
  • Create a scavenger hunt in the room to learn a new environment
  • Share with a friend something you like about yourself
  • Work together to line up without talking
  • Offer lessons on classroom jobs
  • Provide lessons and model grace and courtesy
  • Make a list of “Classroom Rights and Responsibilities” WITH your students and have them initial or sign
  • Have a procedure or place in the classroom for resolving conflicts –create a “Peace Table” or “Peace Corner”

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Lesson Study & Visualization

Lesson Study on Visualization with Mathodology

Follow our Journey. Lesson Study with a Focus on Visualization.

Jugyou kenkyuu, a Japanese phrase gives us the term “Lesson Study”. Introduced in the U. S. in the late 1990s, interest in Japanese lesson study remains strong in the education world throughout the United States. Our Lesson Study this year will focus on visualization and metacognition.

Lesson Study & Mathematics

Lesson study works well across education and in particular, in improving mathematics education. We will wrap up professional summer reading on visualization in September with a look into the routines we create in classrooms that promote visualization. During “Introduction to Lesson Study” in October, we will explore what lesson study is, how it works, how to use it, and best practices with a focus on creating metacognition in students.

Pre-Lesson Study Questions

We engaged our focus group from St. Edward School in Vero Beach by asking the following questions:

What attracted you to this Lesson Study?

Overall the participants felt this lesson study would improve their ability to use visualization strategies in their own classrooms. They felt the experience would allow them to “dig deeper” into learning the best way to improve their teaching skills to build visualization.

What do you hope to learn from this Lesson Study

Participants generally responded similarly, wanting a deeper understanding of the science behind visualization, learning how to integrate visualization into their daily teaching, and using visualization to help students see concepts in a different way.

What is visualization to you?

It is creating a picture in your mind, being able to ‘see’ what you are hearing or reading to help you better understand the lesson, and it brings life to situations, assisting a student in understanding the concepts being taught.

What do you feel you already know about visualization? (before reading)

The response to this question was consistent with all participants. All felt that visualization was a way of seeing something in your mind to better or fully understand it and using it in math as well would bring life to situations and assist students in better understanding the concepts being taught.

Ideas on how to get kids to visualize math?

  • Using various concrete and pictorial models
  • Incorporating color in our board witing to connect ideas
  • Relating ideas especially in the operations
  • Have children create a short movie in their minds with each math concept so they can ‘see’ the process and verbalize it before computing

What questions do you have before we start the lesson study?

  • Can all students visualize?
  • How are other teachers using visualization?
  • Does the brain have any physical limitations with visualizing?
  • How do we teach visualization to students so they use it seamlessly when seeing a math problem?
  • What forces the brain to want/have to visualize?

We will be holding a private Lesson Study at St. Edward’s School, Vero Beach, FL in September.

Follow us through this Lesson Study.

We’ll be at Oak Hill High School in Nashville, TN, October 2, 2019 – October 4, 2019. Seats still open!

Click here to register for this event and for details on this Lesson Study.

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Challenging Word Problems

Advanced Model Drawing Virtual Class Homework #6

Try a bar model and post in the comment section below.

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Challenging Word Problem

Advanced Model Drawing Virtual Class Homework #5

Try a bar model and post in the comment section below.

 

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Challenging Word Problem

Advanced Model Drawing Virtual Class Homework #4

 

Try a model and share in the comment section below.

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Frustrasted with Fractions

Dear Sarah,

I am dying with my 5th grade…they are struggling with fractions (the chapter is hard) and with not a strong Singapore foundation from last year…well…we are on our 4th week! Should I do the 4th grade fractions chapter?? I honestly am at a loss.

Thanks

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Do students need to draw the bars when solving word problems?

Dear Sarah,

I am currently in the unit of Parts to Whole, teaching bar modeling.  Looking ahead, it seems to me that bar modeling is a big part of not only this unit but upcoming units as well.

My wondering and question: I have students who are looking at the problems and can figure them out by stacking numbers then using the renaming strategy which was what they learned in lessons before bar modeling.  Should I be encouraging all of my students to draw the bar model before stacking the numbers to solve the problem? How critical is it for them to bar model? Please know that I am not saying that I don’t want to teach it at all, but I don’t know how much I should be “pushing” those that either don’t understand it or those that feel like they can work the problems using other strategies.

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Why do Students Struggle When Learning Math Facts?

Learning Math Facts

Math facts are simply the basics: addition, subtraction; multiplication; division. They are basic number combinations and calculations we do every day.  So why is it that learning math facts  creates such huge problems for teachers and students in our classrooms?

Subtraction mistake when learning math facts

Common subtraction mistake when children are learning math facts.

Is it that some of us are just naturally better at math than others?  Perhaps.  The good news is you as a teacher can help anyone improve their math skills.  You might be wondering how.  Well, rote memorization alone won’t get us there.

Continue reading

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Meaning of Equality

Meaning of Equality

Often we teach the meaning of equality as the equal sign indicating “put the answer.”  To prepare students for future mathematics, we need to rethink the way we communicate this representation to students.  The equal sign is too important to attach such a limited meaning, especially when students are moving to abstraction.  When moving to abstraction, it is important to use this symbol as a “reader” versus a “speller.”  What do I mean by that? Let me explain.  First, a speller sounds foreign because the student is trying to make sense of abstract symbols that they don’t fully understand. A reader connects meaning that they can comprehend while connecting the story to the symbol.

For example, when teachers record the following equation,

Meaning of Equality

Using a simple addition problem as a reader versus a speller.

 

A speller would say, “Four plus five equals nine”.

A reader would say, “Four apples and five apples is the same as saying ‘I have nine apples’” .

In the journal prompt below, we want students to see the statement as 5 and 4 is the same as 2 less than 11. If we focus too much as a speller, students see symbols as performing actions rather than relating ideas, which leads to misconceptions of the equal sign.

Less is more. Hesitate in thinking math becomes harder by creating larger numbers, rather provide depth early on using children’s intuitive understanding of basic number operations.   A simple journal entry can help assess what students understand about operations and the meaning of equality.  Once you find that a student understands the meaning of addition and subtraction, it is time to create tasks that dive deeper in understanding equality.  These task can be as simple as asking questions such as, 4 + 5 = ____ + 6.  Additional equality tasks can be found in the Algebraic Thinking and Reasoning with Numbers books (Groundwork Series) by Greenes, Carole, et al., (https://www.mheducation.com/prek-12/program/MKTSP-O8302M06.html).

In the book, “Thinking Mathematically,” Carpenter, Thomas P., et al. suggest some benchmarks to keep in mind while moving towards the conceptual understanding of equality.

  1. Establish a starting point.  What are students’ initial ideas regarding the equal sign?
  2. Mix it up.  Don’t always write equations in the form a + b = c, rather c = b + a.
  3. See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
  4. Compare sides without calculations. Encourage students to look for relationships without performing calculations.

 

Listen to students and be aware of where they are in the process. Encourage them to question if it is true.   If so, why is it true?  By listening and looking we may be pleasantly surprised at what we find out.

Below is student work demonstrating various levels of understanding within the concept of equality.

Spiky says that 5 + 4 = 11 – 2 is false.

Curly says that it is true.

Who do you agree with and why?

 

Meaning of Equality

This student understands how to simplify expressions but lacks applying the meaning of equality.

 

Meaning of Equality

This student is confused in understanding what it means to have expressions on each side of the equal sign.

 

Meaning of Equality

This student understands the meaning of equality. What can we ask next?

 

Meaning of Equality

This student demonstrates a shallow understanding of equality. What can we do next?

 

Meaning of Equality

This student can solve for an answer but needs practicing in mixing up the form of an equation.

 

 

 

 

 

 

 

 

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Communicating Home

Often we share our students progress by sending home “Monday” folders with worksheets that were completed, test or quizzes taken the previous week, or maybe we have a site where parents can go and see the current grade in your class.  Are we informing parents on their child’s progress?  Does the grade or worksheet give an accurate picture of student growth?

I often challenge teachers to give feedback on an ongoing basis.  We should not only assess from the papers turned in but from what we are seeing, hearing and collecting on a daily basis.  Use journal entries or a table with your objectives (I can statements) to allow for a more productive discussion at a parent conference.  The following guidelines can help your conversation whether it be in the form of report card comments, parent conferences or the chat in the pick-up line.

-What area is the child is doing well? (Have pieces of work that demonstrate growth.)
-What area does the child struggle? (Show specific concepts not just a broad topic.)
-What are you doing in the classroom to help?(The learning does not stop after a chapter test. It is your responsibility to help the child learn it!)
-What can they do at home to support you and their child? (Parents want guidance on how to help.)

Attached is an article from the Wall Street Journal – 10.18.17 addressing the idea of a student-run conference.  This conference helps children communicate their progress while building the metacognitive process.  Would love to see this happening more in our schools.

Communicating Home

 

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Renaming in the addition algorithm

Many teachers ask how to help students record the renaming process in the addition algorithm.  What language should I use?

I learned to “make trades” or “borrow” when adding and subtracting that require renaming.  Is this still correct?

Renaming in the addition algorithm

How can we help students understand conventional methods?

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Differentiated Instruction without much work

How can we differentiate instruction without putting more work on our plate?

Teachers do not have time to prepare multiple tasks and lessons to meet the needs of each student in the classroom.  The key is giving an anchor task that all students can enter and knowing how to adjust the task based on our observations.

In this class, students were guided to use “Katell’s” method (aka-left to right strategy) to solve the problem 65 – 12.  Notice how the student recorded their thinking process.

How can we differentiate with one task?

During the guided structure component of the lesson, teachers are observing and questioning students.  This student demonstrates and can explain the left to right strategy.  What comes next for this student? During our planning phase, we need to ask ourselves, “What can I do for the student that already knows the answer”?  Looking at this student’s work, how can we deepen the mathematical understanding?

After listening to the student explain his thinking, the teacher commented, I can understand your verbal explanation, but your written work is confusing to me.  I wonder why I am confused? The teacher then walked away, and the student grappled with the idea of equality.  Differentiation is meeting the student where they are at and extending the learning.

Do not try to create more problems for students to practice.  Plan your task thinking of what mistakes students typically make and how you can help extend the thinking. Use your observations to go deeper and differentiate!

What are other ways you could take the task 65 – 12 and extending the task?  Share your thought below.

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Conceptual understanding comes first!

How can we help student refocus on understanding versus just an answer?

Too often when we pose a problem and students shout out an answer.  We need to ask ourselves do students understand the concept or are they obtaining the correct answer by fitting the symbols and numbers into a structure they know?

In this example, the anchor task was to subtract 34 from 87.  The teacher wanted to screen the children first wondering, Do students know what it means to subtract one number from another?  To find out more the teacher posted the following problem, removing the numbers.

Do students have the conceptual understanding or just fit the numbers into a given structure?

Students were asked to set up the expression that could represent the situation.  The student A on the far right was the only student in the class that seemed to understand. When asked to explain his thinking, many observing teachers felt he understood the concept and that he gave us a platform to generate a discussion.

Do students understand the part-whole relationship here?

Following this analysis, the numbers were inserted into the problem.  Subtract 34 from 87.  It was interesting to see student A’s work.  Much to our surprise, Student A who seemed to understand the concept had a hard time transferring that knowledge to another situation.  Notice his equation is not correct but he gets the final answer.

Can student A  transfer knowledge from one setting to the next? Look and listen to help guide students to conceptual understanding.  Looking at answers does not tell us the whole story.

Less is more.  Spend more time on conceptual understanding and listening and watching students versus solving more problems.  The answer does not help assess student reasoning or how we can extend or guide the learning process.

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Report Cards

In emphasizing the core competencies in mathematics, schools have added indicators in giving feedback to their parents.  Here are questions I typically receive.

We are trying to determine how to modify the indicators on our report card.  Have you seen any schools use the indicators/categories from the Singapore Core competencies?  Any insight?

Below is a sample of what one school developed.

 

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Is it possible?

Is it possible?

Participants were asked at the 2017 Fall Mathodology Institute if it was possible to use the digits 0-6 only once to make a true statement?

Is it possible?  Participants worked and worked and did not find a way to solve the problem with the given conditions.  The teacher allowed the task to move on without closure for a purpose.  When students work to become confident problem solvers we want them stating, “this is not possible, I have exhausted every situation.” Listen to their reasoning, and look for a systematic way to prove it is not possible. All too often we focus on finding the solution, and we can deepen the process and allow for practice when we alter the task.

Try giving a task where it is not possible and see how your students handle it.  Follow-up by asking, “if it is not possible, can we change one of the symbols to make it a true statement?”

When we do not provide immediate closure, we allow students to continue to explore.  Many participants emailed after the institute because they would not quit until they figured it out.  Below are a few of the pictures they sent.

Great perseverance by participants who refused to give up?

This is what we want from students…next question for Shelly…can I change any symbol to make it a true statement?  If so is there only one way?

 

Ok…..I’ve worked literally for hours and I’ve come to the conclusion that this problem is NOT possible. If the lesson was to teach us to let kids struggle so they work harder like a ???? I get it. I’ve done at least 100 fraction problems over the weekend. If you would’ve given us the answer I would’ve never thought about this problem again……but you’ve let us struggle and now it’s just painful. Please tell me it’s not possible ???? my sanity is at stake.
-Shelley

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Grade 1 Task

Grade 1 Task- Not enough information

At 2017 Mathodology Fall Institute participants worked on a task from think!Mathematics which required them to use the given shapes to make a composite figure. Figure C posed a problem for many.  Is it possible?  Participants worked and worked and did not find a way to solve the problem with the given shapes.  The teacher allowed the task to move on without closure for a purpose.  When students work to become confident problem solvers we want them stating, “Figure C is not possible with the given shapes.  I would need an additional… to complete this shape.”  We have built perseverance and another way to assess them.  Do they know what to ask?  “I need two more pieces of one shape.”

When we do not provide immediate closure, we allow students to continue to explore.  Many participants emailed after the institute because they would not quit until they figured it out.  Below are a few of the pictures they sent.

Try giving a task where there is not enough information and see how your students handle it.  Will they be confident to state this is not possible, and know what to ask?

Great perseverance by participants who refused to give up?

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Counting Activities

counting activities for grades K and 1 from Mathodology fall institute 2017

 

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Add by Counting On

Students using the think!Mathematics program reinforce the count on strategy.  Games/activities can be used during the guided practice part of a lesson.  Students are encouraged to count on from the larger number.

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Rounding Using the Number Line

Use of number line to show rounding.  In each case what place is the student rounding to?  How do you know?

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Powers of 10 (part2)

Powers of 10 (part 2)

My students are really struggling with multiplying and dividing with powers of 10 (especially division).
Do you guys have any suggestions for reinforcement activities/anchor tasks?

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