This post originally appeared 11/14/16 at http://www.aimsedu.org/2016/11/14/conversation. 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 TK12 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 recreate, 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 recreate. 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 35 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), 377403. 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.
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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 microcredentialing 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 microcredentialing 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 highleverage instructional practices and are working to create a National Observational Teaching Examination which uses ondemand 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 microcredentialing for developing these routines:
1. Jim Knight says, "Goals that others choose for us seldom motivate us to change." Microcredentialing could allow for selforganized 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. Microcredentialing 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 adultlearning 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 microcredential for that.
Now on to some concerns about microcredentialing:
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 safetytraining modules or traffic school courses, computerbased "quests" that issue a microcredential upon completion can't guarantee that any real learning has taken place. 3. Who determines what competencies are worthy of microcredentials? 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 microcredentials 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: JosseyBass.
Here's a salute to one of the oldest manipulatives still in use in today's digital classrooms.
1. Investigating Area by Folding Paper2. Algebraic Models3. Tangrams4. How many times can you fold a piece of paper?
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, onepage 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. In a typical week, a school leader will receive dozens of emails and several catalogues of teaching practices, and over a year will be presented with literally thousands of options for instructional improvement. 
The eight Common Core Standards for Mathematical Practice help define what it means to do mathematics. Worthwhile math tasks can help shape classrooms into practicecentered environments. "Rich" tasks are those that invite students to

3 units PD credit Start at any time Selfpaced No materials needed 
Students love these problems. They're neat, there's one possible solution, and Hey!  they look like trees.
No need to discuss prime factorization, because a) students don't understand those terms, b) what students really want to know is how to solve the problem, and c) it's much easier (and cuter) to say, "Just make a factor tree.”
What we have effectively done here is reduced one of the most fundamental theorems in all of mathematics to a Christmas tree ornament. This isn't the first time we've done this. We define prime numbers for students, we tell them the value of pi, we calculate the sum of the triangle's angles for them. We sell all the secrets, reveal all the magic, and spoil all the endings.
Granted, we've all tried to play it up big in the classroom in an attempt to generate some enthusiasm. We soon discover that not all students are as interested in significant mathematics as we are. I'm not saying we should attempt to surprise students for the sake of enthusiasm levels. Besides, students who are struggling to understand something may not feel any sense of surprise. Significant mathematical surprise requires solid intuition  enough to appreciate the differences between intuition, expectation, and fact. But if I can get a "Wow, that's kinda cool!" or a "I didn't expect that!" from a student, I'll take it.
So here's my attempt to return to students some opportunity for discovery:
What are students wondering now? Are they thinking about which numbers are possible? At this point do they expect that every composite number can be written as a product of primes in a unique way? Do they understand that they have just stumbled upon the Fundamental Theorem of Arithmetic that was first proven by mathematician Carl Gauss in 1801? I can't expect surprise, or even appreciation of this discovery. But students are making conjectures and testing them as thoroughly as they can, and that's worthwhile.
George Pólya gave ten commandments to teachers. This seems fitting:
Granted, we've all tried to play it up big in the classroom in an attempt to generate some enthusiasm. We soon discover that not all students are as interested in significant mathematics as we are. I'm not saying we should attempt to surprise students for the sake of enthusiasm levels. Besides, students who are struggling to understand something may not feel any sense of surprise. Significant mathematical surprise requires solid intuition  enough to appreciate the differences between intuition, expectation, and fact. But if I can get a "Wow, that's kinda cool!" or a "I didn't expect that!" from a student, I'll take it.
So here's my attempt to return to students some opportunity for discovery:
 Make a list of the first few primes: 2, 3, 5, 7, 11. Give students a composite number, say 6.
 Now ask, "Can you make the number 6 using multiplication and only numbers in this list? You can use a number more than once."
 When they come back with a solution, point out the obvious. "You didn't use any other numbers? Only primes?"
 Give a few more to try: 12, 14, 20, 45, 28. "See if these numbers are possible." Or ask students to volunteer composite numbers between 1 and 50.
 Ask, "What other composite numbers do you think you can make?"
What are students wondering now? Are they thinking about which numbers are possible? At this point do they expect that every composite number can be written as a product of primes in a unique way? Do they understand that they have just stumbled upon the Fundamental Theorem of Arithmetic that was first proven by mathematician Carl Gauss in 1801? I can't expect surprise, or even appreciation of this discovery. But students are making conjectures and testing them as thoroughly as they can, and that's worthwhile.
George Pólya gave ten commandments to teachers. This seems fitting:
9. Do not give away your whole secret at once – let the students guess before you tell it – let them find out by themselves as much as is feasible.
I'll agree that a factor tree is an efficient tool for grinding out prime factorization. However, it is unfortunate to see absolutely no connection between the tool and the discovery. Why not wait until students see the need for such a tool? When discussing ways to reach the number 48, I have seen students start with factors 6 and 8, then represent these using primes. These students needed a way to organize and keep track of this process and were able to invent their own treelike structure. Save the trees for art class and let your students do some real thinking. You'll be helping to spread the joy of discovery.
Here's a short video introducing a new course I'm offering with the Center for Professional Development.