Mathematical Mindsets - Ch. 5

The Title. Rich Mathematical Tasks.
To be honest, this is what I have been striving to do every single day since the workshop "Anatomy of Instruction" at ESSDACK many moons ago with Adelyn Soelner.  That series of workshops moved MY thinking more than anything in my career and with the curriculum Math In Context, changed the way I taught students from that point forward.

Anyway, back to rich mathematical tasks.
"Teachers are the most important resource for students.  They are the ones who can create exciting mathematics environments, give students the positive messages they need, and take any math task and make it one that piques students' curiosity and interest.  Studies have shown that the teacher has a greater impact on student learning than any other variable."
This is a LOT of pressure but one that I take seriously as I ponder each day's lesson every evening the night before.  What is the best way for students to learn the material? Will the lesson be engaging?  What questions do I need to be sure to ask?  

I love how this chapter describes mathematical excitement.  Not only have I seen it but I have experienced it through my students as we explore learning math together.
Math excitement combines...
...curiosity,
...connection making,
...challenge,
...creativity,
...and collaboration.

Back to number sense...Bill Gates said the reason we have so many students failing math is because of algebra.  Jo Boaler states that students are failing algebra not because algebra is hard but because students don't have number sense.
How do we instill a greater degree of number sense in our students?
Where does it start?

Case 1 - Seeing the Openness of Numbers
"Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, flexibility, and multiplicity of ideas, the mathematics comes alive for people."
Yes. Yes!  Yes!!
In my notes from reading this section, I have an example that happened around a fishing pond with my family.  We had seven poles in the water but we weren't having much luck pulling any fish in.  In fact, only two poles had even had a bite.  I asked my grade schooler, he couldn't have been more than 4th grade, what was the percentage of our poles that had had a bite.  In my own mind, I quickly did the math.  2/7 = 28%.  The only reason I knew this was because I had memorized what 1/7 was for grading quizzes in school.  1/7 = 14% so logically 2/7 would be doubled.  Well, by the time I knew the answer, he had also piped up with the answer of about 28%.  I asked him how he knew that, while my dad stood nearby listening.  His answer, "I knew that 7x7 is 49, so then I had 14/49.  49 is almost 50 which is halfway, so I just doubled it to roughly get to 100%, and my answer is 28%."  Wow.  Talk about number sense.  He had more number sense at 9 then I had!  I could see that my dad was thinking, so I asked him.  He was standing there on the bank of the pond trying to do long division in his head!  This was his only go-to method...the algorithm of division to turn a fraction to a percent.  What a perfect example of how simple number sense can move our math thinking way beyond memorization or algorithms!

Case 2 - The Power of Visualization
This section just reminds of why the math practices are so important.  The task was essentially a function task where students had to extend the growing pattern of cubes.  After observing several struggling, misbehaving boys become totally engaged in the process they broke it down on why the boys became engaged in this problem.
1. The task is challenging but accessible.  Low floor, high ceiling.  One of my goals to find more of these types of problems.  MP 1 - Perseverance.  MP 2 - Reasoning
2. The boys saw the task as a puzzle.
3. The visual thinking about the growth gave the boys understanding. MP 4 - Make a Model
4.  They developed their own way of seeing the growth pattern. MP - 3 Justification
5.  The classroom was set up to encourage proposals of ideas with fear of making mistakes.
6.  Teach students to respect each other's thinking.
7.  Students were using their ideas and not just following a method from a book.
8.  They were working together.
9.  They were working heterogeneously.

Case 3 - A Time to Tell?
There are times when teachers need to introduce students to new methods and ideas.  Three methods to do so.
1.  The teacher shows the methods, students solved using the method.
2.  Students discovered methods through exploration.
3.  Students given applied problems to work on, even before they knew how to solve them, then they were shown the methods.  Students that were taught in this method performed at significantly higher methods.
So when should we tell...after the students have explored the problem!

Designing Tasks
1.  Can you open the task to encourage multiple methods, pathways, and representations?
...adding a visual requirement
...make sense of the solution
2.  Can you make it an inquiry task?
...change it from reproducing a method to coming up with an idea
3.  Can you ask the problem before teaching the method?
4.  Can you add a visual component?
...draw the problem out
5.  Can you make it low floor and high ceiling
6.  Can you add the requirement to convince and reason
...convince yourself, a friend, a skeptic

Pg 91 has a list of resources, some which I have used or refer to often, that provide mathematical tasks that incorporate one or more of the feature highlighted above.