Mathematical Mindsets - Ch. 4

Creating Mathematical Mindsets.
Yes.
This is what I want...what I need...to help my struggling students.

To encourage our students, we need to allow them to play with numbers and shapes, thinking about what patterns and ideas they can see.  Successful math users search for patterns and relationships and think about connections.
I can see this happening some in my classroom, but not to the extent that it probably needs to because of the lack of time and the amount of curriculum we need to cover.  
My other concern or question is...am I too late?  Have the misbeliefs of math been so engrained that the joy of playing with numbers is lost?  Will they instead be thinking what is the ultimate goal and have I reached it?  

Maybe I need to start out the year with a 3 act math problem from Dan Meyer to get the students to start asking questions as outlined in chapter 3 and to get them playing with numbers to see relationships and patterns.  Hmmm???  What would that problem look like?

Number sense means working with numbers flexibly and conceptually.
I do see in my students that if they don't know how to do the "rule" that they learned previously, they don't have the number sense skills to continue to work with the problem.  Fractions is a huge part in their frustration as they struggle with the conceptual understanding of a fraction.  

Mathematics is a conceptual domain.  It is not, as many people think, a list of facts and methods to be remembered.
This is my basis to my "Move Your Thinking" philosophy.  We are not just in class to learn what to do and how to do it, but more importantly why does it work.  
"Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches.  But once you really understand it and have the mental perspective to see it as a whole, there is a often a tremendous mental compression.  You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process." (Thurston, 1990)
"The brain can only compress concepts; it cannot compress rules and methods."
Yes and yes!  This totally confirms my philosophy and what I continue to strive to reach in my classroom!

"The most powerful learning occurs when we use different pathways in the brain."
Students love to give their different strategies and are usually completely engaged and fascinated by the different methods that emerge.
And this is why I put so much emphasis on multiple methods to finding solutions.  My students struggle when new methods are introduced.  Is this because they are not used to playing around with numbers?  Their number sense is not fully developed?  

Practicing Math
It is important to revisit mathematical ideas, but the "practice" of methods over and over again is unhelpful.  It is helpful to reinforce new ideas and hte best way to do this is by using it in different ways.

When learning, sometimes it is best to use non-examples along with examples that are not necessarily the simplest version of the concept.

There is a lot of evidence that homework of any form, is unnecessary or damaging.
   ~ Spend more time reflecting about what we learned (written response), and less time doing more math.
   This statement stands in support of our interactive notebooks that we are going to incorporate this next year.
   ~ Questions that ask students to think about errors or confusions are particularly helpful in encouraging students' self reflection.

The lowest scoring students in the world are those you use a memorization strategy.
The highest scoring students are those who approach mathematics looking at and thinking about the big ideas and the connections between them.
Does this need to be a goal for our district???