Sometimes, Always, or Never.

 

I have occasionally run across activities involving statements that students classify as sometimes, always, or never true.  In other words, is the mathematical statement always true, only true some of the time, or never true.  An example might be "A cube has the same length for all edges."  This is always true.  

They were always interesting but I wondered why one should use an activity such as this.  It turns out, there are good reasons to use this type of activity, especially for assessing student knowledge, especially  when students are expected to provide justification for their answers.

First of all, this activity is a good way to determine if a student is over generalizing or under generalizing a mathematical concept based on their justifications.  It also provides them the opportunity to think about their own understanding. In addition, this activity can help students improve their understanding of various concepts. 

When students have to come up with examples or counter examples as they try to prove their answer, it encourages mathematical thinking.  If this is done in a small groups, students have to use mathematical conversation when explaining their choices. 

The sometimes, always, or never can be used at the beginning of a concept to establish how much students know about the topic, or it can be used later on after they've had a chance to learn the material to see how much they really understand about the concept.  If the activity is used before, look at statements that focus on the concepts that will be taught.  If used after the lessons, choose statements that focus on what they learned.

One of the best ways to do this is to give students access to the statements so they can answer the questions individually.  Then place them in small groups so students can discuss their answers using the conversation to come to a consensus on the answers of always, sometimes, never.  Finally, go to a whole group so the students can discuss the statements as a group, sharing their examples and counter examples.

Another way to use sometimes, always, or never is to use it as part of journaling in math. Provide the statement for students to determine if it is always true, sometimes true, or never true.  Let them write their answer in their journal but it must include examples and or counter examples to explain their thinking.

Where the sometimes, always, or never activity offers itself as a better choice over true or false activities because there always seems to be the concept or topic that shows up is sometimes true so its hard to determine whether you want to use true or false.  

It is possible to find or develop statements for all levels and types of mathematics.  It might cover properties, the application of definitions, patterns, and so much more.  This activity is one that can be done as needed and is a wonderful way of having students practice mathematical thinking and mathematical conversation.  In addition, it helps students learn to express their thinking in an understandable way.  Let me know what you think, I'd love to hear.  Have a great day.

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.