Number and Operation

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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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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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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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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Sarah Schaefer

 Sarah Schaefer - Student Work

Counting Activities should include matching objects to pictures and different representation of a number.
Games like “Go Fish”, “Concentration”, and “War”, where student have to make a match are great ways to practice the early stages or recognizing numbers and counting.

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Sarah Schaefer Copy

 Sarah Schaefer - Student Work

Can you assess this student from looking at the different ways of thinking? Is there anything as teachers, we need to address?

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What strategy is Victoria using?

Can you follow Victoria’s thinking?  Help me understand.

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Branching with decimals

Do the parts equal the whole?  Explain why you think the student recorded 18.40 in this way.

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Do you agree with my thinking?

When asked to solve a fraction divided by a whole number, the student gave context

(a story), a picture and the solution.  Do you agree with the answer?

If not what does the student need to work on?

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Is this correct?

Take a look at this problem. What was she thinking?  Do you agree or disagree?  Explain.

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Avery

Take a look at how Avery broke the problem about. What was she thinking?

Is it ok to solve the problem this way?

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