Closing Thoughts with Piaget...

Circling back to the basics, I investigated some of what Piaget has to say about prior knowledge and its impact on learning. Just as I read through his work, I read along with some of Ball's research on the impact of teacher's expectations of mathematics on the effectiveness of their instruction. This connection helped me to make one of the most important discoveries in my research- how both student and teacher are learners in the classroom, and the prior knowledge that both parties bring to the lesson impact their expectations and learning at all moments.

First, Piaget, who starts with young students and their first learning experiences with means-ends actions. As children grow, they learn that specific actions can lead to a more desirable second action. They create schemas and expectations. As a result, their learning from this desired action is shaped by the previous action- prior knowledge and experience.

“perform one action in order to do a second action…the second action is usually the more desirable action” (Piaget, 27)

“Means-ends sequencing is essential to adult intellectual functioning because it’s what allows us to make plans about the future” (Piaget, 27)

“Old schemas aren’t activated for their own sake, they are activated for the sake of making other schemas possible” (Piaget, 32)

And then, with Ball, and the expectations of teachers in mathematical education. Teachers approach math with ideas that impact their teaching- the concept is difficult, students will struggle, etc. Just as with students prior knowledge, the teachers enter their classroom with experiences and opinions that impact their teaching.

“Mathematics does not have much relationship to the real world and most mathematical ideas cannot be represented any way other than abstractly, with symbols” (Ball, 4)

“prospective teachers also bring ideas of what mathematics is—what it is about, what it is good for, where it comes from, and how right answers are established—that shape their understanding of and approach to the subject” (Ball, 10)

Growth Mindset in Mathematics

Encouragement through Community Learning

“the key growth mindset message was that effort changes the brain by forming new connections, and that students control this process” (Boaler, 144)

This effort is present when students are learning in a responsive environment, where the focus is on the problem solving skill development, not on finding the correct answer or memorizing the method.

“The awareness that ability is malleable and that students need to develop productive growth mindsets ha profound implications for teaching. Teachers and schools constantly communicate messages to students about their ability and learning through their practices in which they engage and the conversations they have with students” (Boaler, 145)

“When tasks are more open, offering opportunities for learning, students can see the possibility of higher achievement and respond to these opportunities to improve” (Boaler, 146)

Rather than restrict students in ability grouped learning models, allowing for growth gives students the chance to develop their thinking. When students have the chance to constantly learn and improve their understanding, they connect and build upon new and old skills, allowing for more naturally fluid instruction.

Culturally Responsive Classroom

-respectful of students learning supports and strengths

-mobility achievable

-supported risk taking

-focus on thinking

-respect and emphasis on making mistakes

VS

Ability Grouping

-unchallenged

-competitive atmosphere

-students division is known

-rigid levels

-supported accurate problem solving

-emphasis on the answer

Conceptual vs Procedural Understanding

Preparing our students for authentic problem solving

How do we make mathematic education accessible for students?

How are students encourage and able to apply mathematical processes in authentic, problem solving situations?

Quick recap- Boaler observed and researched behavior of teacher and student in two different classroom settings. The first, where student's math education was based on procedural understanding, and the second, built on conceptual understanding. I tracked the differences between the two settings below:

Findings suggest that authentic, self-directed, project-based learning teaches the skills that allow students to think critically and apply mathematical knowledge to new contexts

SKILLS

1. Willingness and ability to perceive and interpret different situations and develop meaning from them and relation to them

2. A sufficient understanding of the procedures to allow appropriate procedures to be selected

3. A mathematical confidence that enabled students to adapt and change procedures to fit new situations

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.