Does "=" equal understanding?

Do your students understand what the equal sign means?

Take a moment to reflect on that question.

Nearly every classroom teacher, when asked this question by the Student Work Study Teacher (SWST), answered “Yes!” or, “I sure hope so!” It’s probably reasonable to assume that student understanding of this concept is something that’s taken for granted by many educators. From an early age, students are shown “=” embedded in mathematics questions: from simple addition subtraction in the primary grades to more complex algebra and numeration in the junior and intermediate grades. In many ways, success in mathematics operations hinges on the idea of equality in a sentence -- simply put, knowing that 2 + 2 equals 4. If students are doing well in math, it reasons that they’ve got a firm understanding of this concept.

To check this understanding in junior and intermediate classrooms, the class is always give the same question. Students are asked to write down the problem on their paper or white board and to solve it.

4 + 8 = __ + 5 or 4 + 8 = __ + 7

Below is a representative sampling of some the answers students gave to this question.

In each classroom this task was given, anywhere from 70-100% of the class wrote an incorrect answer in the blank space. This usually comes as a surprise and/or shock to the classroom teachers:

"After some discussion with the SWST about how students would answer this question (4 + 8 = ? + 5), I was convinced that the majority of my students would ‘get’ this without any difficulty.  I certainly did not believe that many of them would write “12”.

“I asked students to think about the problem I wrote on the whiteboard, without talking to a neighbour or working out any math; just to think about it.  After a minute or so, I asked them to write what they thought would go in the box on their own whiteboards.  I was floored as I walked around the room and saw the overwhelming majority of responses being ‘12’!  I couldn’t understand it!”

After some discussion, some students offered that the answer must be 12 because 4 + 8 = 12.  They completely ignored the + 5 part of the equation."

Clearly, there is a trend: this misconception exists in several grades across multiple schools in this system. As a next step, we asked the students what the equal sign means to them:

“The equation before it needs to be equal to the one after it.”

“The first part before = has to equal second part.”

“The sum (or total) of 4 + 8.”

“Just a way of adding them up.”

“Equal to or the same as.”

“Equivalent equations that have to have the same answer.”

“The total of numbers you’re trying to find the answer to.”

“Equal up to ___.”

“The answer.

“The total.”

“The equal sign means they are equal to each other.”

“This sign means the sum of the question/equation.”

In his article A Balancing Act, Henry Borenson identifies two inherent meanings students can acquire for the equal sign - relational and “calculator”:

The use of the equal sign to indicate the unique numerical result of the sequence of computations that precede it (the “calculator use” of the equal sign) is a valid and necessary use of this sign...However, understanding the relational meaning of the equal sign also is essential for success in mathematics, and particularly in algebra. Whereas such equations as 3x + 5 = 26 can be understood knowing only the operational meaning of the equal sign (because 26 can be considered the result of the operations that appear to the left of the equal sign), examples such as 4 + 3 = __ + 2 and 4x + 2 = 2x + 6 cannot (Kieran 1992). In these examples, the equal sign indicates equivalence between two sets of expressions, each one of which includes one or more operations within it.

Thinking back to our initial question -- Do your students understand what the equal sign means? -- there is sufficient evidence to conclude that across the system students have a range of misconceptions about what the equal sign means in the relational context of the question 4 + 8 = __ + 7. The next question then naturally becomes: What can we do as teachers as a next step to immediately respond to this misconception?

How do you think your students will answer if you gave them the question above as a cold task? We would love to know what happens in your class - let us know in the comments below.

Thank you to guest blogger, Christopher St. Amand, SCCDSB Student Work Study Teacher. He can be found on Twitter at @MrStAmand 

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Next Year is Too Late to Start - One Teacher's Reflection

It really is about learning! 

Recently I have moved from experimenting with Docappender and Goobrics to using them.  These are tools that I use in Google Apps for Education to quickly record observations into separate student files and assess and give feedback to my students on their work.

Like everything, there are stages that I have to go through before I can put something into practice.

Stage One-The Commercial

Stage Two-Reading the Instruction Manual

Stage Three-Unpacking the box

Stage Four-Attempting assembly

Stage Five-Playing with the toys while the children are asleep

Stage Six-Telling a Friend

Stage Seven-Helping a Friend

Stage Eight-Finding every use possible for this new toy!

The first four stages are really all one. Not all occurring in the sequence listed all the time. Sometimes I unpack the box of tools long before reading any instructions. This is often a big mistake for me, because the instructions will often help me find uses for the product (stage 8).

Stages five/six are also more one phase – the fun phase where it is not really productive yet, but becoming more familiar. Stage Six is the consolidation phase. Good pedagogy tells us that if someone can explain it, then the likely have a deeper understanding of it. Although a little out of order, if I explain it to someone who I consider “a safe person” then I am more likely to share the excitement.

I am using Docappender to keep anecdotal comments about student inquiries in science and collaboration and use of class time in my math classes. I think this will help give me a much more accurate set of Learning Skills for the final report card.

My next goal is to 

1. Share the file with my students once I have a habit of using it, 

2. Have them respond digitally by making comments to my entries and 

3. Conference with them face to face.

Next year is too late to start, but not too late to hit the ground running.

More to follow …

Thank you to guest blogger Deacon Raymond Girard, teacher at UCC.  He can be found on twitter @MrGatUCC

Engaged and Excited-Geometry Art

In this activity, students used polygons to recreate one of the stations of the cross. The student was responsible for researching all of the stations and then creating an art piece to depict what one of them was about. 

I found that the students were engaged and excited to guess which station other students were doing. 

In the future I would have the students create all of the stations in order as a graphic text. They could also write from different perspectives for the witnesses and for Jesus for a station of the cross. 

Thanks to Ridgetown teacher, Janice Knowler @MrsKnowler for sharing!