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.

I used this task from NRICH at a teacher workshop several years ago to give teachers an opportunity to practice making conjectures, reasoning, and justifying. It caught my attention again today as I visited the NRICH website in search of rich tasks. I think it's held up quite well over the years and after reading about Alan Parr's implementation with students, I'm impressed with the multiple opportunities for extension this task provides.

Start with a few key questions:

• Which envelope shall we try first? Why?
• What could be in this envelope?
• Are there any numbers which you know definitely aren't in this envelope? Why?
• Are there any other solutions?

Provide students with number cards to try out their ideas. As they work together, ask questions about what they are noticing and wondering. You might find that this recording sheet can assist with keeping track of student observations. These resources from Max Ray of the Math Forum can help guide students toward more productive problem solving.

When students have found and shared some methods and solutions, give them an opportunity to make up some of their own problems. Alan Parr shares some student-generated problems that might be a good starting point.

This post describes a few assessment strategies several colleagues and I piloted with a group of middle school students. One of our main goals was to provide students an opportunity to learn through the assessment process, to revise their initial work, and to make generalizations that reach beyond the boundaries of a given task. The impetus for this work stems from the Mathematics Assessment Project and their emphasis on students' use of mathematical practices throughout task completion.

MAP Task Security Camera found at http://map.mathshell.org/materials/tasks.php?taskid=273&subpage=expert.

When students reached question 3, we turned to ActivePrompt to collect realtime responses from students and share those responses with the class. We found that students were engaged with this portion of the task from the outset primarily because they were given a chance to use their intuition and make a guess.

Give this link to your students: http://activeprompt.herokuapp.com/SPPFK
Go here to see their responses: http://activeprompt.herokuapp.com/TQPJB

Teams explored the task further using the following GeoGebra exploration tool.

Many thanks to the teachers who were willing to pilot some new assessment practices. Our debrief surfaced several key questions that will help to guide our future implementation:

• How well do students adapt based on responses from peers?
  
• How do students learn to become better "guessers" while developing a greater trust in their own intuition?
  
• How can the right tools enhance the assessment process?
  

I recently prepared a series of number explorations for a 2nd grade classroom. The students in this classroom have been working with base ten concepts and place value and are now beginning to compare numbers. In this post I'll share a menu of activities that can be used to support deeper understanding of these concepts. The goal here was to help students reason about the relative size of numbers and quantities without relying on an algorithm to compare numbers. I've also tried to help students build some bridges between various representations of numbers.

1. We started with Andrew Stadel's Estimation 180. While many of his explorations might be targeted at an older audience, I found the following extremely well-received in this 2nd grade classroom.

First huge moment: seeing licorice on the big screen at 12:45 in the afternoon.
Second huge moment: getting to guess and seeing those guesses on the board.
Two equally huge final comments: "I was sooo close!" vs "Wow, I was not even close!" These are both satisfying to me.

Students had opportunities to consider the relative size of numbers as each classmate shared a guess. We discussed our smallest and largest guesses and any that appeared more than once. It was quite evident that students were wondering whether they should guess a really big number, and if so, what that number would be in this scenario.


2. We followed this up with some number line estimation. I used the cards below with the students seated on the carpet for a brief number talk. Along with estimating the location of the red dot, students answered the question, "How did you decide?" This line of questioning encouraged students to use the numbers given as landmarks and do some broad thinking about tens and hundreds.

3. Next we pulled from the deep constructivist work of John Van de Walle. I was first introduced to John Van de Walle's work in my teacher credential program back in 1996. These activities are great work for both teacher and student and make mathematical reasoning about number the focus.

4. I pulled this next task from Illustrative Mathematics as a progressive step toward more challenging estimation with number lines. I drew a long number line across the board at the front of the room and labeled every hundred up to one thousand. Students teamed up and together discussed and decided where to place their number.

5. The final selection in this menu of activities is from the National Library of Virtual Manipulatives. I used the Practice mode with one dot in which you are asked to drag the dot onto the number line to match the value shown. If the location you choose is not close enough to the actual value, you are asked to zoom in and be more precise.

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:

1. Make a list of the first few primes: 2, 3, 5, 7, 11. Give students a composite number, say 6. 
   
2. Now ask, "Can you make the number 6 using multiplication and only numbers in this list? You can use a number more than once."
3. When they come back with a solution, point out the obvious. "You didn't use any other numbers? Only primes?"
4. 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.
5. 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:

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 tree-like 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.