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.

Vygotsky, Mind in Society

“Any learning a child encounters in school always has a previous history” (84)

Thinking more about constructivism as a learning theory has brought me to explore the ideas of Vygotsky. His thinking about student's development and learning is directly connected to the ideas I have already explored about inquiry based learning. This discovery has helped me to build a greater understanding of constructive learning as an overarching theme in learning.

Vygotsky introduces the idea of prior knowledge and how it impacts learning. I have thought a lot about this idea with my readings of Geneva Gay, specifically how culture specific prior knowledge impacts learning. Students come with knowledge that can be a beneficial tool to their learning in the classroom.

‘For example, children begin to study arithmetic in school, but long beforehand they have had some experience with quantity-they have had to deal with operations of division, addition, subtraction, and determination of size” (84)

Vygotsky also talks about the Zone of Proximal Development- the area where this prior knowledge supports new learning to reach an attainable goal. 

“We propose that an essential feature of learning is that it creates the zone of proximal development; that is, learning awakens a variety of internal developmental processes that are able to operate only when the child is interacting with people in his environment and in cooperation with his peers. Once these processes are internalized, they become part of the child’s independent developmental achievement” (90)




And finally, perhaps the strongest connection to Dewey, is the impact of collaboration with learning in the ZPD. Social learning comes into play, as students work together and communicate to achieve this learning. 

“It is the distance between the actual development level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (86)

"Knowing is a process, not a product" (Bruner, 72)

Why is it so important to integrate theory of instruction into effective pedagogy?

"But a theory of instruction, which must be at the heart of educational psychology, is principally concerned with how to arrange environments to optimize learning according to various criteria" (Bruner, 37)

Bruner supplies an answer to this arrangement, highlighting the 4 features of effective instruction:

When applied, this instruction method provides a way for students to focus on skill development in order to increase their readiness. Bruner also stresses the importance of applicable learning, and that without an authentic application, "the information is simply useless" (53).

This idea of effective and connected learning connects directly to the major themes and ideas which Professor Diana at WSU expressed when discussing her place based honors level college course. Her goals for the course challenged students to think critically, come to creative conclusions, and take away a greater sense of confidence and involvement in their ability to make effective change.

Most interesting, however, was Professor Diana's perspective and role in her course. Rather than see herself as an instructor, Diana explained her position as more of a 'writing coach'. Serving as a guide, Diana helped her students write strong, effective papers, which attacked arguments and problems generated by her students. Rather than provide material, Diana led her students towards and through new ideas and skills.

As Bruner would most likely agree, the results were powerful. Students left Diana's class feeling empowered to make change. After an end of semester reflection, students reported to feel as if now, they had the skills and knowledge to become active members of their current and future communities. 



During our conversation about students learning in this inquiry model, the idea of anxiety and discomfort arose. Diana reported that some of her students struggled with the lack of structure given in the course; the student's freedom in learning was new and overwhelming.

I couldn't help but think about the clear connection to what I read in Denise Pope's "Doing School". These high school students reported that ""Everybody" does the minimum required to get by and everybody focuses on grades instead of learning the material" (Pope, 28). How much of this anxiety in college is created based on student's lack of opportunity in learning during high school? And, how does this lack of opportunity start to grow in the elementary levels? It seems that, as we get to college, where learning is structured to be so authentic and applicable, we lack the skills necessary to succeed. According to Bruner, these skills are vital; "a curriculum should involve the mastery of skills that, in turn, lead to the mastery of still more powerful ones, the establishment of self-reward sequences" (Bruner, 35). The disconnect of skills and experience, ranging from elementary school to college, shows the impact of neglecting Bruner's 4 essential features, and in this situation especially, effective sequence.

• Where else does this disconnect occur?
• Is there disconnect in the way that the Investigations math program is taught throughout the grade levels?
• What about the connection between a school's mission statement and the math program which they use? Does the mission statement and program goals align?

"Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself" (Bruner, 72)

Constructivist Learning Theorists

Where do the theorists fit in with this connection between community and learning?

Dewey

"The relationship between knowledge and reality is a result of individual and social experiences (Ultanir, 199)

"learner centered and less teacher dominated learning environment" (Ultanir, 201)

"the child own experience must be acknowledged" (Ultanir, 206)


The idea of social experience and its impact on learning is closely related to Bruner's ideas of engagement. Social discussion provides students with authentic motivation and a reason to apply and develop their problem solving skills.

Piaget

"continuous and gradual change of schema" (Jones)

"the most frequent cause of accommodation is the interaction, and especially linguistic interaction, with others" (Glaserfeid, 66)

"motivation to master new problems is most likely to spring from having enjoyed the satisfaction of finding solutions to problems in the past" (Glaserfeid, 181)

This gradual change supports the instructional practice of connecting prior knowledge with new learning, using each lesson and activity to build on a big idea rather than focus on specific content points.

Vygotsky

"social interaction leads to gradual, incremental changes in thought and behavior (adapted from the cultural and societal norms)" (Jones)

A direct connection to the importance of a classroom community.This raises the question of what are these cultural and societal norms? Are they different in and out of the classroom? In school, are we teaching students to norms of our society, or teaching them to seek out norms in order to understand social arrangement and communication?

Bruner

"effective instruction is engaging and stimulating to student minds" (Jones)
"set purpose for learning- motivate further exploration" (Jones)


How do the verbs we use in student objectives define their learning experience?

What are our expectations of education and how does these expectations impact the learning models used in classrooms?

What is the difference between memory and understanding? How do we assess this difference?