Collaborative Math Inquiry: Algebra & Geometry Task

During our last Collaborative Math Inquiry workshop, we began to discuss patterns to algebraic reasoning and uncover some struggles and misconceptions.

Why do students struggle with patterning/algebraic reasoning?

• Junior students struggle with the concept of the equals sign and see it as one-dimensional versus seeing the whole picture.
• Kindergarten students struggle with repeating a pattern because they are only focused on some parts of it but again, don’t see it as a whole set.
• Primary students had trouble with growing and shrinking patterns.
• Difficulty with the language/vocabulary to describe the pattern.
• Intermediate students are struggling at solving word problems involving algebra. They had trouble getting started. They were able to see patterns and solve algebraic equations.

Developing Possible Rich Tasks to assess students’ algebraic abilities

Our focus for these tasks are to uncover where the students' misconceptions are and how we will respond to them. We were given a list suggested tasks from Kindergarten to grade 8 from the resource Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small. With our team, we chose the following Algebra and Geometry tasks:

Kindergarten Task

• Provide a pan balance and linking (snap cubes.Ask students to use the balance to model the equation 10 = 6 + 4. Then ask them to move cubes to demonstrate a different way to show 10.

Grade 1/2 Task

• Provide a selection of equations with missing values, such as the ones below. Ask students to explain how they know the missing value is less than 5 each time.

7 + ___ = 10

8 = 4 + ___

9 - 6 = ___

10 - __ = 8

Today we decided we use our PLC time to pull our target students to do our task in the hallway. One educator will guide the students through the task and the others will use video and notes to document.

Questions we asked while planning for the tasks:

• What are the best materials to provide to do this task?
• What questions we will use to guide their learning?  And in what order?

Kindergarten Task: 

The educator will provide a balance scale and three containers with 10 loose snap cubes in each (red, green, and orange) and ask: Show me how you can make 10 = 6 + 4? Can you show another way of making 10?  If necessary, educator will prompt by providing the written equation a worksheet with the visual  ◽+◽=◽.

Our Findings:

• Kindergarten students needed to have a visual of the equation in order to get started.
• They eagerly used the manipulatives provided to work on the problem.
• They both demonstrated persistence in the given task and were engaged likely due to the manipulatives.
• Both students needed lots of direction to understand the task

What helped the students move forward in their thinking?

• The educator wrote out the equation or encouraged the student to write out the equation to help remember the numbers.
• When the kindergarten student A only put in numbers 10 and 4, the two end numbers of the equation, the educator pointed to the missing number.
• Stacking the cubes into towers for each side of the equation helped them compare the amounts.
• Educators rephrased the questions and broke them into smaller steps.
• For example: Can you may 10 using a different number of the orange cubes?

What do we need to rethink?

• Balance scale was distracting and not perfectly accurate. Perhaps, it should be introduced later on in a different way.
• Students need more exposure to the vocabulary of the equation and the name and function of the symbols : add/plus, equals.
• Educators must be more conscientious in using math terminology consistently when working with students.

Grade 2 Task: 

Students were given the equations 4 + ___ = 9 and 9= __ + 4 and asked to find the missing numbers.

Our Findings:

• Student read the question backwards when it was displayed as follows: 9=5+4
• The student said it was wrong.
• In the end the student made up a new way to show 9 as: 7+2=9 and then wrote 9=7+2 but physically wrote it from right to left.
• The students used their fingers to help count versus the cubes.

What helped the students move forward in their thinking?

• Rephrasing the questions: Can you write the equation in a different way?

What do we need to rethink?

• Students need more exposure to the equations shown starting with the sum.