One of the many electives offered here at Westwood Middle school is an elective called Success in Mathematics. The idea of it is to challenge our mathematicians in ways they don’t often see in our general math classes. Use of team work, problem solving, and communicating about mathematical processes is the core of this class. Students quickly find that this isn’t just another math class but something to look forward to, as the challenges and questions asked are possible to do by all, but the journey to get to the answer the exciting part to share out to the classroom.
A simple example could be a “number talk” we have done:
40-19
Students would solve this mentally and need to come up with 2 strategies to solve and share to the classroom their thinking. At surface value this seems easy, and it is; the learning happens when we as a class find there are many ways to break apart numbers and solve in ways other than the standard stack and subtract method. If you’re reading this, try to solve mentally and if you solved is there a second method to solve?
6th graders over the year, have gotten answers like 39, 29, 31, 21. Why? Take the answer 39. How can 40 take away 19 be 1 less??? When you analyze the thinking, you see mistake! They stacked and subtracted in their head, 0 minus 9 is 9 in their head, and 4 minus 1 is 3, so it must be 39. The teacher accepts the answer, keeps it on the board and would then have students discuss if the answer would be reasonable, or where in the description that the student gave a mistake was made. Others share and give different ideas to solve.
In the 4 examples from the picture, you can see all sorts of thinking, some more efficient, some raising more questions. In particular the purple example where you add 1, to get an easy 40 – 20, but then as a class discussing if we didn’t know the answer was 21, why if we add 1 to 19 do we again add 1 to 20? Why not subtract the 1 we added at the beginning??
Finally, I will leave anyone reading this a wonderful question we wrestled with this week. I gave this question, and all students proceeded to struggle through it for the entirety of the 45 minute class. Watching and working alongside 6th grade students as they formulated their own strategies, found new information that we’d share to the class then adjust our previous methods to try and solve just this 1 question for the entire period, was truly an enjoyable moment as a math teacher. Here’s the question!
“A book has 600 pages which are numbered 1,2,3 and so on. How many times does the digit 5 appear in the page numbers?”
As you can see some students took an extreme method in solving! Many started by dividing 600 by 5, but then we dove into how many 5’s are between 1-100? Well 50-59 has 11 5’s, counting 55 twice for each individual 5. Students heads started to burst after that, and then the problem of the 500’s realizing that each number in the 500’s has a 5, on top of 555, which has 3 5’s. Trying to figure out a clean equation was too difficult and many succumbed to writing every 5 digit on white boards!