Last December I tweeted this from the MAV Annual Conference (the Mathematical Association of Victoria):
4+7=__+3=__+10=?. Fill in the gaps. What did you get for ? #MAVCON
— Amie Albrecht (@nomad_penguin) December 3, 2015
What did you get for the gaps and for the final result?
If you said ‘4+7 = 8+3 = 1+10 = 11’, you are, of course, right. But plenty of people don’t get 11. Stop for a moment and wonder what they do instead. (I’ll tell you soon.) Unfortunately I didn’t record the source of the problem, or the data supporting the claim that plenty of teachers get it wrong, so I committed on Twitter to try it with my primary and middle pre-service teachers this year.
Four weeks ago I got a chance. I used the question as an opener for a lecture on equations. From 20 pre-service teachers, I got only two responses:
- Two-thirds got the right answer of 11.
- One-third got an incorrect answer of 24.
(Actually, the proportion of students who got it right was higher than I expected!)
So what goes wrong to get an incorrect answer of 24? It stems from the interpretation of the equals sign. Many students — at all levels — interpret ‘=’ as ‘find answer to previous expression and keep working’. This gives: 4+7 = 11+3 = 14+10 = 24.
But this post is not about why this misconception exists. Rather it is about how we might help students make sense of this misunderstanding for themselves.
Recently, at NCTM 2016 in San Francisco I recounted this story separately to Andrew Stadel, Jana Sanchez and Tracy Zager (clang!). I idly wondered what the response might have been if I had instead asked: ‘4+7 = 3+_ = 10 + _ = ?’. My conjecture is that many more students would get this revised problem right. I suspect that finding a number already in the spot where you plan to write ‘the result so far’ is just enough to cause you to pause and reflect. In that moment of reflection, the ‘=’ sign becomes more meaningful.
Andrew asked what happened when I asked the students to try this revised problem. Good question! I hadn’t tried it because I only thought of it while reflecting on my experience out loud (that’s another blog post right there).
We quickly sketched out a plan:
- Ask students the original problem: ‘4+7=__+3=__+10=?’.
- Without any discussion, ask students the revised problem: ‘4+7=3+_=10+_ =?’.
- Ask if anyone wants to revise their answer to the original problem.
- Share experiences and discuss what happened with their thinking.
This simple revision to the lesson plan, from ‘try, correct, move on’ to ‘try, revise, reflect’, means students are more reflective about their own learning. I’m excited to try it out next time I introduce equations, which will probably not be for at least another year.
This misconception is well-documented. But, in wondering, discussing and reflecting on it with others, it became a whole lot more meaningful to me.
So, have you had experience with this misconception? Where do you think this misconception comes from? Do you have ideas how to help students correct this misunderstanding? I’d love to hear if you try the sequence outlined above. Please share in the comments.