Some of my favorite math units/topics are those where I feel I have a handle on how to really get my students to construct their own understanding. This is NOT the way most textbooks operate! Most textbooks have you set a clear learning target:
I can find the perimeter of a rectangle by using the perimeter formula.
Then, the teacher models how to do the problems…talks through them…gives some guided practice…some independent practice–and then assesses student understanding. It works–for some.
Instead, I really try to find ways to put students in situations where they are exploring, looking for patterns, and deriving their own rules. I have found that this type of learning is more engaging, more meaningful, and “sticks” with the learner so much better.
Here’s what this looked like with area and perimeter in my room last week. Had I used our math series in sequence, the progression would have looked something like this:
1. Teach the formula for the area of a rectangle.
2. Teach finding the area of irregular shapes by decomposing into smaller rectangles.
3. Teach the formula for the perimeter of a rectangle.
4. Practice problems with area and perimeter.
I’ve been doing this fourth grade thing a long time–and I know my students aren’t ready for that yet. We still have a TON of misconceptions that need to be worked out before we move to formulas! For example, many students still aren’t crystal clear on the difference between perimeter and area–so I certainly don’t want to teach formulas for concepts that they aren’t confident about!
Also, students aren’t ready to flexibly and correctly use labels (a “precision” issue!) related to units of length (ex. cm, in., m) as opposed to SQUARE units to measure area (we like to call it “squarea” to help with that!)
I also know from the past that students really struggle to even COUNT the squares on the side of a shape and often get confused between “inside” and “outside” of shapes.
There are other things that come up–there always are–but to jump right into formulas certainly doesn’t give students time to explore, get these misconceptions corrected, and allow the time for them to build and deepen their own understanding. Here are a few snippets of what I did BEFORE we tackled some of the work in our math book!
Our first investigation simply involved asking the students to build a rectangle using 12 tiles. Students were able to do this with ease–and then I asked them to measure their rectangles, jot the answer on a sticky note, and come up to the group to share.
We had an amazing discussion about how to measure a rectangle! Some measured only one way (“Mine was 6 squares long.”) and others used two dimensions (“Mine was 3 one and 4 the other.”). I asked if anyone measured theirs and got 12. No one had. This led to a great chat about whether or not we should measure the INSIDE of a shape or the OUTSIDE–until we realized that BOTH could be valuable! We came up with all sorts of real world examples when we would need to measure the outside edge or “rim” (peRIMeter) like fences, wallpaper borders, door frames, and so on. We also then talked about times when we might need to measure the entire area (“SQUAREA”) in units that take up space…like for carpet or tile or planting sod and so on. Once students were comfortable that there is more than one way to measure a rectangle, it was time to roll!
It was so cool to listen to their discussions, the hints they gave each other, and the questions they asked. Check out the shape one student came up with–which led to quite a debate within the group. Can you do this? Can there be a “void” (a student’s word, not mine!) in the middle? I listened to the debate for quite a while and then we agreed to solve it by agreeing that we would NOT count “voids”–and that all shapes had to be solid figures. We called the other groups over to chime in–and then we reached consensus that we would proceed with that new rule.
Colored it…cut it…