One Question

One of the great things about teaching with Problem Based Math is that students are able to open up their work and extend their thinking, finding challenges to think harder about.  It is all about higher order thinking, which I do believe every child is capable of, within their zone of proximal development.

We had focused on using tables and looking for patterns, as well as following a five-step process for answering problems.  As an assessment piece, I wanted to see:

- can this child extend a pattern?
- can this child add or multiply (bonus if they multiply, because then they've extended their thinking into an algorithm)?
- can this child properly use a table and label each column in a way that makes sense?
- can this child prove their work?
- does this child give a final answer?

As an added bonus, I used measurements, to see a) if anyone converted when the numbers became large, and b) if they remembered to identify the unit of measurement.

The question itself was made up of manageable numbers.  I didn't want the challenge to be in the adding; I wanted it to be in reading the problem, understanding how to solve it, and showing their thinking/organizing their work.  For kids who were able to handle a challenge, or ones who needed a very easy-to-work-with question, I left the number of years wide open:

Draven was 50 cm tall when he was born.  If he grows 40 cm every year, how tall will he be when he is ___ years old?

You could also change the question by:
- changing the numbers for measurement, giving options, or leaving them (both or just one) blank
- asking how old he will be when he is a certain cm. (given, left blank, or options) tall


Have a look at how this student solved the problem.  It's nothing amazing, but rather average, until you get into the "meat and potatoes" - the PROOF.  This explanation is beyond awesome!

You can see that the student:
- solved the problem
- organized the answer into a table with appropriate headings
- chose an appropriate challenge of "14 years" (rather than 4, 8 or 10, for example)
- explained the answer
- explained how tall he was at birth, and how he knows not only based on the question, but also based on his work, by doing a reverse-operation and subtracting
- clearly answered the question

This student answered in a fantastic way.  This child works well above other students, who also completed the question successfully.

This is a video that shows a student working on the same question, but using cm cubes as counters.  On his own, he counted each block as 10 units to save time, and counted up the groups.  It's not ground breaking to watch, but beautiful to see a child finding their own way.  Later, he transferred the work into a table and answered the question.  I didn't tell him to do any of this, and although he learns differently from other students, he was still able to come out on top.  This is the beauty of open questions.

By getting our heads out of text books and into problems with open possibilities, the kids are able to think deeper, look farther and show everything they're capable of doing.