My work with the AIMS team began last month after 20 years in public education, first as an elementary teacher and later as a mathematics coach. I have spent a lot of hours in TK-12 classrooms, walking alongside teachers as they explored ways to make their classrooms places where authentic mathematics learning could happen. In all of this work I have come to understand teaching and learning as innately human activities that connect us to one another through conversation.

Conversations happen in many settings. Think about the last time you gathered around the dinner table with friends. One person starts to tell a story. Another friend chimes in. Soon everyone starts nodding and laughing in agreement. Then one person tells a slightly different version of the story, revealing details from a different perspective. Discussion follows. Perhaps the story is revised further. More laughing follows. Then there’s the quiet lull that indicates collaborative satisfaction. The interpretation of a story is negotiated through contributions around the dinner table.

I’ve been deeply inspired by Paulo Freire, the notable Brazilian educator and philosopher. Freire was committed to dialogue that affirmed the human construction of meaning. “Dialogue is a way of knowing,” Freire wrote. “Dialogue cannot be reduced to the act of one person’s depositing ideas in another, nor can it become a simple exchange of ideas to be consumed… Because dialogue is an encounter among women and men who name the world, it must not be a situation where some name on behalf of others… Dialogue further requires an intense faith in humankind, faith in their power to make and remake, to create and re-create, faith in their vocation to be more fully human.”

At the AIMS center we have faith in children. We have faith in their innate, human ability to make and remake, to create and re-create. And we have faith in teachers and their ability to listen to children. Because of this faith, we’re committed to dialogue about how children come to know. Our Early Mathematics Team is particularly interested in how 3-5 year olds develop early counting and number concepts. Much of our work involves conversations with young children–not conversations in which we “deposit” ideas into children, but rather conversations through which we begin to think like children, to seek to understand their approaches, to explore what their mathematical knowledge might be like.

As we spend time in preschool classrooms and observe children interacting within their physical environments, it’s very clear that early learning involves dialogue. We know from our efforts to understand the ideas of Piaget and other constructivists that children learn from dialogue with others. In fact, it has been suggested in this work that interactions with others are among the most frequent causes of learning.

So, let’s engage in conversation. We’ll learn more about each other and ourselves as a result. It is, after all, a very human thing to do.


Further Reading

Freire, P. (1970). Pedagogy of the Oppressed. Myra Bergman Ramos, trans. New York, NY: Continuum.
Freire, P., & Macedo, D. (1995). A dialogue: Culture, language, and race. Harvard Educational Review, 65(3), 377-403.
Von Glasersfeld, E. (1995). Radical Constructivism: A Way of Knowing and Learning. Studies in Mathematics Education Series: 6. Bristol, PA: Falmer Press, Taylor & Francis Inc.

Less than a month after I read Ilana Horn's thoughtful post on teacher professional development, "Professional Development is Broken, But Be Careful How We Fix It," I came across this article from Education Week:

Here's a brief definition of micro-credentialing from the article:

​The article goes on to say that sometimes these badges are accompanied by salary increases worth several hundred dollars.

While this trend isn't likely to replace traditional professional development, it strikes me as an interesting response to the need for teachers to develop new and improved skills.

Fundamental to the idea of micro-credentialing is the belief that there are specific competencies that are crucial to teaching, and that these skills can be learned and measured. The folks at University of Michigan's TeachingWorks would agree. They've defined what they call high-leverage instructional practices and are working to create a National Observational Teaching Examination which uses on-demand performance assessments to measure teacher readiness. Researchers at Massey University, New Zealand, explain, "Routines capture the certainties within teaching, and as such can be anticipated and can become part of a knowledge base for learning how to teach."

Here's what I like about micro-credentialing for developing these routines:

​1. Jim Knight says, "Goals that others choose for us seldom motivate us to change." Micro-credentialing could allow for self-organized cohorts of teachers to select a teaching practice, connect with an instructional coach, explore the teaching practice together, interact with students, and collect evidence of their implementation. That's sound practice.

2. Micro-credentialing could place the burden of proof in the hands of the teacher: collect student work, create a video of your practice, utilize peer observation. In other words, you choose the evidence. 

3. Teachers who participate in this process are encouraged to implement, submit for approval, and then serve as reviewers for colleagues. This could generate a culture of adult-learning that could be contagious.

Inherent in any system that assigns rewards is the understanding that much of what goes on will be unrewarded. In my work with teachers, a consistent tension is present between developing proficiency in specific competencies and developing a general adaptability to students in the classroom. From Anthony and colleagues at Massey University: "Signifying adaptive expertise, they (teachers) pursue the knowledge of why and under which conditions certain approaches have to be used or new approaches have to be devised."
​

Teacher: I adapted to my students today. Do I get a badge now?
Reviewer: Sorry, there's no micro-credential for that.
​

Now on to some concerns about micro-credentialing:

1. External reward systems obscure and often negate internal rewards. Are external rewards necessary because we've done such a poor job of helping teachers recognize the indirect, sometimes hidden, rewards in the lives of their students? As Parker Palmer writes, "As important as methods may be, the most practical thing we can achieve in any kind of work is insight into what is happening inside us as we do it."

2. Much like online safety-training modules or traffic school courses, computer-based "quests" that issue a micro-credential upon completion can't guarantee that any real learning has taken place.

3. Who determines what competencies are worthy of micro-credentials? How are reviewers selected? How do schools or districts ensure that all teachers have equal access and opportunity to participate in the process?

In the article from Education Week, Brent Maddin, provost of Relay's Graduate School of Education, questions whether micro-credentials atomize teaching to a fault. He asks, "Is there something powerful about how multiple techniques, or moves, or strategies, or competencies move together that are an even better indication of what a teacher can know and do in the classroom?"

The short answer is yes. But there's already a credential for that.

Sources:
Anthony, G., Hunter, J., & Hunter, R. (2015). Prospective teachers development of adaptive expertise. Teaching and Teacher Education, 49, 108–117.
Knight, J., & Learning Forward. (2011). Unmistakable impact: A partnership approach for dramatically improving instruction. Thousand Oaks, Calif: Corwin Press.
Palmer, P. J. (1998). The courage to teach: Exploring the inner landscape of a teacher's life. San Francisco, Calif: Jossey-Bass.

Here's a salute to one of the oldest manipulatives still in use in today's digital classrooms.

Source: Mathematical Thinking: Supports for English Language Learners

Source:  ​Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies​

Tangram Folding (PDF)
Source: AIMS Center for Math and Science Education

Source: ​http://www.relativelyinteresting.com/how-many-times-can-you-really-fold-a-piece-of-paper-in-half/
​And an interesting blog post on exponential growth by Jason Zimba 

The bow hasn't been out since, and I think I know why. "We should make a target," Aaron said. I agreed. This activity would certainly be more fun and more rewarding with a target. And we could instantly tell if we were getting better at shooting.

In his book, Unmistakable Impact, Jim Knight suggests the use of a "Target," a simple, one-page document that clearly states a school's goals for instructional improvement. Most improvement plans fail, Knight suggests, because they are too long, too complex, and too unrelated to instruction. This reminds me of arrows never flying, falling to the ground after a short flight, or soaring over the fence in wild flight.

I have found that at times my own work with teachers has lacked focus. There hasn't always been a clear target in view. So I'd like to offer a few questions that I am pondering related to our work. What would a precise Target look like for mathematics teaching and learning? What elements for teachers and students would it include? How might it be comprehensive yet concise?

There have certainly been efforts to identify best practices in teaching. I've started to pull together a few source lists to start with:

1. Setting up Positive Norms in Math Class. Jo Boaler. (PDF)
2. CCSSM Standards for Mathematical Practice
3. SMP Look-fors (ems&tl Project, 2012) (PDF)
4. NCTM Principles to Actions (PDF)
5. Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for ELs Aligned with the Common Core. Judit Moschkovich, University of California, Santa Cruz. (PDF)

If you're familiar with these resources, I think you'll agree that they are top notch. Each certainly deserves time and attention on its own. However, focusing on a few key elements from these and finding ways to connect them might make for an all-star target list.

Knight's approach to improving instruction is built around four critical instructional areas: 1) community building, 2) planning content, 3) delivering instruction, and 4) developing and using formative assessments. When everyone involved in the educational community can agree on a concise target, efforts to improve teaching and learning take on new focus. Pulling from my above source lists, I'd like to suggest the following target for teachers (T) and students (S).

Community Building
(T) Develops socially, emotionally, and academically safe environments for mathematics teaching and learning
(T) Works collaboratively with colleagues to plan instruction, solve common challenges, and provide mutual support as they take collective responsibility for student learning
(S) Understand that everyone can learn math to the highest levels
(S) Construct viable arguments and critique the reasoning of others

Content Planning
​(T) Establishes clear mathematics goals to focus learning, situates goals within learning progressions, and uses the goals to guide instructional decisions
(T) Recognizes and supports students to engage with the complexity of language in math classrooms
(S) Understand the mathematical purpose of a lesson and how the activities support their learning
(S) Connect their current work with the mathematics studied previously

Instruction
(T) Implements tasks that promote reasoning and problem solving
(T) Allocates substantial instructional time for students to use, discuss, and make connections among representations
(T) Facilitates meaningful mathematical discourse, poses purposeful questions, builds procedural fluency from conceptual understanding, and supports productive struggle in learning mathematics
(S) Make sense of problems and persevere in solving them 
(S) Ask their own math questions
(S) Choose and apply representations, manipulatives, and other models to solve problems

Assessment for Learning
(T) Elicits and uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning
(T) Provides students with descriptive, accurate, and timely feedback on assessments, including strengths, weaknesses, and next steps for progress toward the learning targets
(S) Know how their personal learning is progressing
(S) Reflect on mistakes and misconceptions to improve their mathematical understanding

Building this list was not easy, but I think that's part of the exercise. It takes effort to look discerningly at teaching and learning. I invite any comments or suggestions you may have, perhaps in response to a few questions:
​
1. Are there teacher or student actions that are completely missing that should be added?
2. Is there a way to make individual actions simpler and more precise? There are currently 18 actions on this Target, and I think that's close to the maximum.
3. Is there redundancy in teacher or student actions? If so, let's pare the list down and reduce distraction.