Since I teach High School, I decided to look for additional ways to help high school students increase their number sense. Students who are low performing or perhaps English Language Learners need scaffolding so they develop a better number sense.
While looking for additional information, I discovered this 17 page pdf with some really good suggestions including the idea of what most teachers mean when they say "show me your work."
In a sense, I am guilty of what the author charges but at the same time, its hard to break with traditional ways of teaching the way I learned things. I do like some of the activities such as the one on representation which shows four different ways of approaching the same problem. This appears to be a sample of a much longer book but its enough to help the teacher approach the topic in a different way.
Here is another article from Yale on ways to improve number sense in the high school. The author discusses in detail various topics related to number sense and includes a wide set of practice problems. Each set of practice problems focuses on a specific skill. I have high school students who are unable to subtract if they have to borrow. I am going to teach students to count back rather than try to borrow because it is actually finding the difference.
I looked at a presentation on improving number sense in the middle school and one line stood out above all the rest. "Always put number in context" Instead of saying "What is 20 \ 1.79?" Ask it as a word problem such as "How many composition books priced at $1.79 can you buy with $20?". This puts the problem in a more realistic setting while working on helping students learn to work with word problems. Furthermore, many of the new tests are more likely to use the problem as it appears in the second example rather than the first.
The same site discusses ways to talk about numbers. Start with smaller problems to get students thinking from a variety of perspectives. Discuss a strategy offered by another student. Limit talks to no more than two minutes and remember it is fine to put a strategy on the back burner till another time.
According to a different website, you can tell how well as freshman student has done previously in math by asking him or her if they can estimate the number of objects in a group. Such as simple test, yet I didn't know about it. I often have no idea what my incoming students know and this will help.
So enjoy and let me know what you think?