Ball and Boaler- How to Teach Mathematics

In keeping a culturally responsive classroom, what is the role of the teacher?

What actions are made by the teacher in order to provide for an environment for students to take supported risks in learning?

How does capacity of mathematical knowledge impact the effectiveness of math instruction?

Thoughts...

“Develop an inquiry relationship with mathematics, approaching math with curiosity, courage, confidence, and intuition” (Boaler)

This approach encompasses both the student and teacher perspective necessary for inquiry based learning to be effective in mathematics. It is interesting how the two positions in the classroom become a mirror for the other- this equality in approach and goal maintain both the teacher and student as learners, opening the discussion for divergent thinking processes.

“Are assessed formatively- to inform learning- not summatively to give rank with their peers” (Boaler)

Here, Boaler speaks specifically to a teacher's responsibility in the classroom- assessment. Formative assessment holds students accountable for their learning in an authentic way, as the teacher takes note simultaneous to student learning and development. This type of assessment allows the teacher to jump right in and address misconceptions and guide building ideas. In addition, the teacher learns, constantly adapting instruction with questions to guide deeper thinking.

"Mistakes are valuable, they encourage brain growth and learning” (Boaler)

“A mathematics algorithm invented by a student, for example, must not be diagnosed as a mistake simply because it deviates from the teacher’s way of arriving at a solution” (Ball, 8)

Duckworth, Boaler, and Ball enter the same playing field here- when students and teachers are both learners in the classroom, mistakes are encouraged and supported. With a classroom culture that appreciates mistakes, students and teachers take opportunities of missed problems as a chance for positive thinking development.

“My purpose is to encourage them to talk about what they think is going on to get them to figure out the idea of permutations for themselves” (Ball, 6)

Talking- what I've learned to be vital to an inquiry community, and the backbone of effective mathematics instruction. When students get talking about their thinking, they are able to make connections and fill in the gaps in what they know. The ability to explain and teach is ranked high in Bloom's taxonomy, and is threaded within all aspects of the Mathematics Common Core standards for mathematical practice. Students and teachers get talking, question and learn from each other, and develop their thinking.

“By the end of our time together, I had learned not only how valuable questions were to teaching, but I realized that how I asked them and when I asked them made a big difference…I could steer her with the word “why?” and although it was very subtle, it made her look deeper” (Ball, 13)

And here, how teachers make that talking possible- asking effective questions as a coach, helping students to use language while explaining their understanding of new mathematical concepts.

“if you know your subject matter well, (inside and out), it is easier to find different explanations and examples. You can’t be tied down to just one way of doing a problem” (Ball, 11)

Finally, some accountability for math teachers understanding content. So many times I've heard teacher candidates say they chose early elementary to avoid math, or they don't have to worry because they are 'just teaching 3rd grade math'- WRONG! These math lessons are the ground work for math problem solving for the rest of our students lives. When teachers have an incredible hold on this knowledge, they are able to fluidly address students questions, understanding student-invented algorithms, and uncover misconceptions. A high level of mathematical understanding is vital in effective math education.