Always, Sometimes, Never?

I've occasionally seen and used always, sometimes, and never statements but I've never made a practice of it because I've not been sure what they accomplish.  As stated earlier, for the first few weeks of school, I have to send work home for part or all of the week and after reading up on these, I think they will be a good addition to the packet. 

If you've never seen or done one of these activities, they are fairly simple.  Students are giving between two and five statements to look at.  They have to decide if the statement is always true, sometimes true or never true.

When teaching mathematics, statements are made which are only true in certain contexts and usually the context is what is being taught.  This activity allows students to think critically about math to determine when things apply or don't apply.  Furthermore, it promotes mathematical reasoning and introduces the idea of counter examples to prove something isn't always true.  

It is important to have students provide counter examples for sometimes statements because it helps students see the context of when it is true or false and requires deeper delving into the statement as they think about it. Determining if the statement is always true, sometimes true, or never true requires higher level critical thinking skills rather.  In addition, this activity helps  identify misconceptions in student understanding.  

The always, sometimes, never activity also promotes dialog and communications because students are required to explain why the statement is always true, sometimes true, or never true.  It is a good activity to help students learn to "Justify their answer" which is often seen on tests.

Overall, always, sometimes and never activities are considered low floor with a high ceiling because students who have low mathematical reasoning skills can still participate by substituting numbers to see if it seems to always be true or find a counterexample.  It is high ceiling because students have to justify their conclusions. Furthermore, always, sometimes, and never is also a way to introduce logic statements to students without using straight mathematical language and theorems.  

Always, sometimes, and never also helps students develop perseverance as they look for examples or counter examples.  They build arguments while evaluating their logic and the logic of their peers arguments. It also has students engaging in authentic mathematical thinking.

When doing this with students, they often will try one number to see if it works but they will need to be guided into trying several possibilities such as zero, fractions, or negative numbers and not to just accept the first possibility.  Take the statement "Any number added to five gives a number larger than five" which on the surface seems to be true because people automatically assume one or above which makes this true but if you add a negative number to it, the answer will be less thus making the statement, sometimes true.

So if you want a way to help student develop mathematical thinking, this is a good way to do it.  Let me know what you think, I'd love to hear.  Have a great day.