Well, the answer to that question is taking longer to get down on paper than we thought it would. So next week we will get into *why* we should take the time and effort to dedicate a few grey cells to this pedagogical conundrum.

For now, I have the answers to last week’s 18 x 25 sorting quiz. I also have a new sorting activity that focuses just on addition strategies and models.

So first, the answers to the multiplication strategy vs. model activity:

I’m not going to go over the chart, except to say that seeing multiple examples of a particular model (row) helps us identify the common characteristics of that model. And likewise, studying the same strategy (column) presented on different models can help us solidify just what strategy is, and even what it is not.

Now for the new stuff. Pam has created a similar sorting activity using only addition strategies. I think we can be more specific in that these are advanced addition strategies, useful for multi-digit addition, as opposed to early addition strategies like counting on and counting all.

Study the examples below and determine where they belong on the chart. There will be some blank spots on the chart as not all strategies lend themselves well to particular models. In a similar vein, some cells in the chart may get more than one example.

This might be easier to do on paper, so you can download these sheets to work on by clicking either graphic. Once you have sorted them out, you can download the complete file with answers: Strategies versus Models: 48 + 29 Sorting Chart with Answers. But now, really, spend some time thinking about it before just clicking and getting the answers. We are all on the honor system here, but don’t deny yourself some “productive struggle” first.

Okay, that is all for this week. Stay tuned for next week – the “why this is so important” bit.

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First, let’s take a little quiz. The images that follow are examples of strategies represented on models for the problem 18 x 25. If you were to group them together, which would be examples of the same strategy but represented on different models? How could you group them to show the same model representing different strategies?

One way to do this is to sort the items into the grid below. Put each item into the grid appropriately (click the graphics for a .pdf version to print). Then look at the columns—each column will show the same strategy on different models. Look at the rows—each row will show examples of the same model representing different strategies. Next week, we’ll post the answers and write about why this is an important distinction, one we should pay attention to in our teaching and our professional development.

What do you notice? What do you wonder? Leave your noticings and wonders below!

]]>Pam says a lot of great things, but I have seen many people react to one phrase in particular – “never stop starting”. Whether it is January 1st or January 10th, or even June 10th, we can always start and try again.

So whether you made new year’s resolutions or not, consider trying something new or starting over with something you want to grow into. May we suggest:

- using problem strings to introduce new concepts or models
- using problem strings to cinch learning
- trying a rich problem (investigation) in class
- purposely sequencing your problem strings and rich investigations to build on each other
- using the mini-lesson of Count Arounds
- playing “I Have You Need” with your class
- developing your “non-response” face
- implementing a “thumbs up” policy instead of raising hands

These are just a few of the teacher moves that will support your students in their mathematical development.

We at Pam Harris Consulting wish you a happy new year and hope you never stop starting!

]]>A Count Around is a powerful instructional routine (a mini-lesson) that can be used in many different grades to promote numeracy, especially place-value relationships and connections.

Count Arounds are a great part of a well rounded mathematics curriculum. They are also a great beginning routine for teachers to implement as teachers get their feet underneath them in their journey to teach real mathematizing. In other words, it’s a good place to start. The routine is fairly contained and easy to facilitate and gives teachers a chance to hear student ideas and thinking. It is also an opportunity for teachers to practice responding to students.

A Count Around begins with the teacher selecting a starting number for the class. A student begins at that starting number and adds a specified amount. The class continues to count around, one student at a time, by that same specified amount. As each student says the next number in the count, the teacher records the count. At certain, pre-planned junctures the teacher stops the count and asks noticing, probing questions. Often these junctures are at important place values, where patterns shift or after enough numbers are listed so that students can notice important patterns shifting somewhere in the list.

- The teacher selects starting number. This can be by telling students the starting number. Alternatively, the teacher can ask students to choose a number with restrictions, for example: a number between 1 and 9, any two digit number, or any fraction between 0 and 1.
- If desired, the students could sit in a circle or semi circle. This is especially helpful for younger students because it removes the ambiguity of who is next in the counting.
- The teacher announces the amount to count by.
- A student begins by adding the amount to the starting number, the next student adds the amount to the previous number, and so on.
- The teacher stops the counting periodically to ask important questions.
*What are you noticing? What patterns are emerging?* - The teacher helps wrap up the count around by helping students summarize some of the major patterns that emerged.

For example, the teacher might choose the number 8 and then tell the class to count around, adding 10 each time. The first student starts with 8 and adds 10 to get 18, the next student adds 10 to get 28, the next student says 38, and so forth. As students count, the teacher writes the numbers on the board. Somewhere around 138, the teacher would stop and ask what students are noticing and what patterns are emerging. Then students start counting again until they reach 248. The teacher would again press for description and discussion.

For the kinds of questioning and discussion, watch master teacher and workshop presenter Kim Montague facilitate a Count Around with a group of elementary teachers.

When you watch this video, what do you notice? What do you wonder? We’d love to hear your comments!

]]>Need to see it in action? Play along as you watch expert teacher Kim Montague explain the routine and why it is so important on the Figuring Math channel on YouTube.

- Establish a target number.
- Say, “Total 100. I have 92. You need …”.
- Give brief think time.
- Then cue students to respond “8”.
- Play several rounds at a time, gradually choosing more challenging numbers.
- Alternate between choral response, popcorn response, and down the line response. Keep it moving and keep students anticipating with positive energy. After playing, talk strategy.

When choosing numbers for the “I have” part of the game, consider these three principles:

- choose accessible increments before choosing more challenging increments – for example, work with multiples of 10 before moving to multiples of 5, use multiples of 10 and 5 before using whole numbers, use whole numbers before using rational numbers (fractions and decimals)
- choose numbers in the top half of your range before using numbers in the bottom half of the range
- choose numbers closer to an anchor or benchmark number before using numbers that are more adrift from anchors

Diligently work to adjust to your students. If the problems seem easy, gradually give harder problems. If the problems seem difficult, back up to easier numbers.

Ask students:

- How do you know?
- What do you think about as you figure it out?

Being close to the target number or an anchor number like 50 might influence students’ strategy choices. See what strategies students develop as they play.

When using the target 100, do students deal with the ones first? Do students deal with the tens first? Over time, gradually help students develop the 90-10 strategy, where they fill in the total to 90 and then the ones to 10.

Occasionally choose a number like 73, then immediately choose it’s additive partner 27 for the next round. Ask students if they notice a relationship between the two numbers. Help students realize that they could find the difference up to 100, but also subtract back from 100.

K & 1st: Play with fingers (one hand or two), five-frame cards, ten-frame cards, double ten-frame cards, or number racks, or verbal numbers. As students get used to the game, as you show the card, say the number shown and record the partner as students say it.

1st & 2nd: Play with double ten-frame cards, number racks. As students get used to the game, as you show the card, say the number shown and record the partner as students say it. Alternate between physical representations and verbally saying the numbers.

3rd & up: Play with dollar amounts, coin combinations, fractions and decimals for a total of 1, times for a total of 60 minutes.

You should choose the target number and delivery method to just reach the zone of proximal development of each audience. The general progression of development is to play to a total of 5, then 10, then 20, then 100. By the end of 2nd grade, play for a total of 100. By the end of 3rd grade, play to 1,000. If your students are older, back up to what they need and build them up.

Older students can construct relationships of minutes in an hour and play to 60. As they study geometry and angles, students can play to other benchmark numbers such as 90, 180, and 360. Fourth and fifth graders can play to 1, given fraction or decimal values. As a general rule, students should own the easier target numbers before you move to more difficult numbers. The most important are 10, 100, and 1,000.

Try progressions like these:

- K-2nd grade – target 10 – 9, 7, 8, 10, 5, 1, 0, 2 or 9, 1, 7, 3, 8, 2, 6, 4
- 3rd grade – target 100 – 60, 90, 80, 20, 10
- 3rd grade – target 100 – 65, 95, 55, 25, 15
- 3rd grade – target 100 – 92, 87, 66, 23, 77
- 3rd grade and up – target 1,000- 700, 200, 650, 950, 150, 850, 980, 40, 890, 499
- 4th grade – target 1 – 5/6, 3/4, 11/12, 1/2, 1/4, 1/6, 1/3
- 5th grade – target 1.00 – 0.95, 0.82, 0.54, 0.23

Copyright Pam Harris Consulting, 2017

In Dan Meyer’s blog about Mathematical Surprise, Vince Hoover posted the following:

Dan replied:

To which I replied:

Dan kindly tweeted about it:

Deal. And so here I go, expanding.

The following short bit will not do justice to everything that I believe about teaching and learning, but I will try to state the big ideas.

My work is based on the premise that learning occurs when disequilibrium is settled, when relationships and connections are constructed in the learner’s mind, when the learner has to make sense of what is going on. To maximize the possibilities of learning, we seek to put students in situations when learning can best occur—perplexing, intriguing, interesting situations.

Dan Meyer wrote about such a moment in his blog, where he describes a teacher reacting in his workshop. This “surprise” moment describes the intrigue that piques a learner’s curiosity, that propels one forward toward productive struggle. Dan defined perplexity for the nation a few years ago in his conference talk, where the goal is to put students in a place such that they do not understand something, they want to understand that thing, and they feel like they have the wherewithal to work toward solving that thing. He described how this is different from confusion where students also realize they don’t understand something but they do not feel equipped to figure it out.

Psychology professor, Brene Brown, well known for her TED talk on “The Power of Vulnerability,” said in her book *Daring Greatly* that she tells students every semester:

“If you’re comfortable, I’m not teaching and you’re not learning. It’s going to get uncomfortable in here and that’s okay. It’s normal and it’s part of the process.”

“We believe growth and learning are uncomfortable so it’s going to happen here—you’re going to feel that way. We want you to know that it’s normal and it’s an expectation here. You’re not alone and we ask that you stay open and lean into it.”

Cathy Fosnot, noted elementary math expert, says that we want to “intentionally invite disequilibrium.”

So, when teaching students, we strive to put students in positions of intrigue, where they want to resolve the perplexity and/or the disequilibrium. As these resolve, learning occurs.

What have we done historically to build teachers? We have ** told** them to teach constructively, experientially. How has the coaching/professional development community helped teachers become better teachers? We have

So, how can we ** teach** teaching? How can we find ways to “intentionally invite disequilibrium” (Fosnot) about teaching? How can we put teacher’s in the learner’s seat so that they have an opportunity to

Cathy Fosnot, et al, talk about the need to be able to push up against participants’ notions by rewinding and pushing for justification in the introduction to Learning to Support Young Mathematicians at Work: Teacher Development:

“The decisions and actions teachers make in the heart of teaching are often split- second reactions made in the moment of interacting with their students, and thus inherently are a result of their subconscious (Dolk 1997). They are a product of beliefs about teaching and learning, a result of perceptions, affected by emotions and the climate and culture in schools, and constrained by the particular tasks teachers are called on to do and the assessment policies that are mandated in our institutions. They are even somewhat determined by memories of teachers’ own past schooling. To the point, cognition as it relates to teaching is “embodied” (Varela, Thompson, and Rosch 1991; Thelen and Smith 1994). This fact makes our work as teacher educators and math coaches difficult. “What kind of effect can we have with just a work-shop, or a course or two of study?” we ask.

If cognition is embodied, the context of the elementary classroom is critical for teacher education. So, how do we effectively educate teachers in a workshop setting, or from a university classroom? The traditional approach has been to teach methods courses and send teacher candidates to schools for fieldwork assignments and student teaching. Another approach, more popular in the professional development circle, has been to utilize a “lesson study” model. Here a group of teachers plan a lesson, one carries it out with the group observing, and then a “debrief” ensues.

Both of these approaches, however, leave much to be desired. Perception itself is embodied—we see initially what we expect to see (Thelen and Smith 1994). To effect change, teacher educators need to: (1) bring up participants’ initial observations; (2) analyze and discuss discrepant, even contradictory perceptions; and (3) encourage relooking again, and again, to seek evidence for interpretations. To do these things, we need ways to rewind time, analyze moments in depth, examine the work of a class and individual work of children in that class over time, view instruction and learning over a sequence of activities (not just one lesson), and examine and discuss teachers’ conferrals and their questioning in relation to later work of the children.”

One professional development sequence is outlined below. I have based this on the work of Cathy Fosnot in her *Young Mathematicians at Work* professional development series (out of print). This is my take on her work. The mistakes or immaturities are mine.

I look at developing teachers through two lenses. First is the mathematics and student thinking. Once we have developed a deep understanding of the math and what students are thinking about, we can begin to think about the teaching – the teacher moves.

It is very important that the teachers have the opportunity to analyze the tasks and video through **both** lenses. First, study from the perspective of the mathematics and student thinking about the mathematics. Next, study through the lens of teacher moves, both the moves that were planned ahead of time and the in-the-moment decisions the teacher made. In order for teachers to make sense of the important teacher moves, teachers need to first understand the math and the student thinking. This provided purchase for teachers as they study the sophisticated, and often subtle, teacher decisions.

In my professional development series Focus On Algebra, the above 12 phases comprise the first 1.5 days of a 4-day workshop. During the rest of the workshop we continue to do and then study the next 5-8 days of a sequence of algebra tasks, using video of the same classroom to support certain tasks. These sample sequences of tasks provide teachers the opportunity to study task sequencing—how students can learn the math ** through the sequence of tasks**. In this practice, telling is limited to social knowledge (that which has been deemed so by society, by convention). The rest, the logical-mathematical knowledge, is constructed as the trainer helps teachers make sense of the mathematics through the purposeful use of context, mini-lessons such as problem strings, and purposeful questioning. Likewise, Pam helps teachers make sense of the teacher moves through use of context, mini-lessons such as studying a teacher at work, and purposeful questions.

Please make comments below about this idea of teaching teachers in the same way we believe in teaching students. Let’s have a conversation.

For an example of this process of inviting disequilibrium to promote learner in teachers, check out the free online course: Focus on Algebra: Introducing Linear Functions. This course gives you the opportunity to experience the teaching and learning we are talking about for students. The course is designed to align with these principles.

Come to a training in Summer 2018 that follows these principles of teaching and learning. You will experience mathematical tasks and focus on student thinking before focusing on good teacher moves. We are offering all three Focus on Algebra workshops (Linear, Exponential, and Quadratic functions). Each is a four day “training of trainers” workshop, designed to focus on training experiences for trainers, as well as teaching experiences for teachers.

]]>Now open! A short online course for secondary math teachers written by Pam Harris. The course is FREE*.

Ever heard Pam talk about teaching and learning? Want to see that kind of teaching and learning happening live? Want to experience that kind of teaching and learning yourself? In this course you will:

- watch real students really learning algebra
- see students learn math through sequenced rich tasks
- earn CPE credit (or take the free version)
- do it on your own time, at your own pace.

Sign up with your PLC, department, chair, or administrator.

The course is a study of two tasks that begin a sequence of tasks to teach linear functions. These tasks are appropriate for 8th grade and algebra 1 classes.

In the first phase of the course, you will:

- watch video of an expert teacher launch a rich task in an algebra 1 class
- complete the task yourself
- study video of students working on the task
- then you will study video of the next task—students and teacher working together.

The next phase of the course is to look back at the design of the tasks—what was purposeful and why? Then you study short video segments focusing on the high leverage teacher moves.

The most impactful professional developments in my career have been the T3: Teachers Teaching with Technology workshops and the elementary Young Mathematicians at Work PD series by Catherine Twomey Fosnot.

In this course, I bring the best of teaching appropriately with technology together with the task design and high leverage teacher moves I learned from Fosnot. Of note to leaders is that this course is based on the structure of professional development that got over 100 retweets in a conversation with Dan Meyer. I will be blogging about this PD structure in the coming weeks. Stay tuned!

*The course is free, or you can enroll for $99 for 12 hours of CPE.

]]>Pam and I had this conversation last week and she asked that I share it here. I have a unique teaching situation in that I only teach math, but I teach multiple grade levels in elementary. I have used Bridges for the past two years and LOVE it!

Pam suggested I teach from it for two years before I try to switch it up and start doing my own thing. So I shamefully admit to a community of math education professionals that I am a page-turning teacher.

This fall is the first time I will have a middle school class. I asked Pam for guidance with choosing a 7th grade curriculum. She recommended I check out Connected Math (CMP) and Mathematics in Context (MIC). I bought copies of both and debated probably more than I should have. They both use investigations to develop the mathematics, but when Pam observed that MIC sequences tasks really well, that pushed me over the edge. I decided to go with MIC.

As so often is the case in life, we stress over a decision, but once it is made there is no looking back. One week into a new school year and I can say that I am very pleased with MIC and how my students responded to the text.

But then doubt starts to creep in. Not in my curriculum choice, but in this internal discomfort over how to use the textbook I decided on. I know from professional and parental experience that many math teachers no longer teach from the textbook. Districts pay big bucks for books that sit on the shelf as an occasional place to get some practice questions. With efforts to improve their teaching, math teachers have moved to a norm where they construct their own curricula of bits and bobs that they find on the internet or create themselves.

Teaching from a textbook intimates that a teacher is not really a professional. After all, anyone can read from a book to the kids. If we want to be respected as professional educators and be perceived as worth our salary, then we must do more than follow a script and use pre-made materials.

My oldest child started high school this year. The day before school started we got her schedule and toured the building, meeting each of her teachers. We met her geometry teacher – nice young guy in his first year of teaching. I asked him what textbook they would use, hoping he would whip out Discovering Geometry. No, he dug out some textbook written for Texas, but quickly assured me that they weren’t going to use it very much. Even before teaching his first day in his career, he already knew that the textbook isn’t good enough to use as a basis for the year.

And so here I am, wanting very much to go page by page through both Bridges and MIC. I think they are great curricula based on great pedagogy and with awesome support materials. I know exactly what order to teach concepts in, what investigations to do and what the math associated with them are. Spiraling review is built in, and homework (at least in Bridges) is designed to be of near-mastered concepts instead of new ideas that kids (and parents) will struggle with. Why would I want to re-create a curriculum when I have a great one right here?

So I talked to Pam about it. She helped me see that the reason we have moved to a culture of teacher-as-curriculum-designer is because teachers are given poor curricula and they know they can do better. So YAY! if you are a hunter/gatherer math teacher.

But it’s not right that professional educators have to be professional curriculum designers as well. Those two roles require different areas of expertise. Just like we want to defend the professionalism of teachers by saying that Joe Shmoe can’t just walk in and turn the page to the next lesson to teach it, I recognize that not just anyone can build a great curriculum. I mean there are trained professionals creating the stuff that teachers won’t look twice at. It’s not an easy thing to do well.

Not only are there activities to be figured out, but sequencing! Sequencing the topics of study, but even more important is sequencing tasks to build in multiple exposures. Tasks need to be designed so students can construct concepts when they are ready for them instead of a once-and-done situation. It’s a given that not all students pick up a new concept on the first try. We have to build tasks that are multiple entry/multiple exit so all students can enter and grow. We need tasks that continually push students a little further without completely overwhelming them. This is a delicate balance to strike.

Wouldn’t it be nice if we had good curricula that reflect how students learn so that teachers don’t have to take on that extra role. To quote Hermoine Granger, “Holy cricket!”, wouldn’t it be great if teachers were freed up to focus on good teacher moves and support students as they construct new ideas?

I digress, I’m a page-turning teacher and that is wonderful because I have great curricula. I no longer feel shame that I teach with pre-made materials, that I didn’t reinvent the wheel.

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