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What’s Involved in this Math Problem?

k problemA blog this week asked us to guess the grade level for which this math problem was written:

Kristen has four flowers. She gives some to a friend. Now Kristen has two flowers. How many did Kristen give her friend? Draw pictures to help you solve the problem.

It’s listed as a kindergarten homework problem.

If you teach math, you know this problem includes some of the biggest arithmetic concepts there are and you’re not deceived by the use of small numbers.

  • Students need to understand hierarchical inclusion–that 4 includes 2
  • They need to understand conservation–that the number of objects remains the same, no matter how they are arranged
  • And, they need to understand cardinality, that the name of a number relates to a specific quantity–including the huge idea that “two” isn’t the second object, but a set of two objects. This is a major leap in knowledge, often hindered by memorizing names of numbers. Too often, students learn to count to 30 or 100 but don’t understand the concepts involved.
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Aren’t Math Mistakes Beautiful?

It’s official. Here’s the research: having students analyze how math mistakes were made is more powerful than having them solve equation after equation.

I often show math professional learning communities a great film that Lucy West provided to me. Picture an 8th grade math class in an urban school. One girl explains the equation she developed to describe the number of tiles that surround the edge of a swimming pool. Her equation is correct, but instead of saying, “Right!” the teacher says, “Thank you for sharing. Who has a different answer?”

Another boy comes to the front, places his equation and diagram on the overhead projector and says, “After hearing her answer, I think I’m wrong, but I can’t figure out my mistake. Who can help me?” And the other students, not the teacher, analyze the two threads of thinking and determine not just which is correct, but how the second student got off track. The teacher intervenes only to help the students make their diagrams clearer and ask for reasoning.

Powerful, isn’t it.

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Instructional Coaching as Ritualizing

A few weeks ago I wrote about similarities in coaching myself as a cyclist and the strategies of differentiated coaching. A key component is knowing when to help teachers develop a routine or ritual to overcome a persistent struggle.

My problem with biking was remembering to unclip from pedals soon enough to avoid tipping over at stop signs. I needed a routine. To develop it, I identified when to clip in and out, practiced to determine how far in advance I needed to start the process (way sooner than other bikers), and also experimented with whether twisting my feet together or separately was the better way (separately so that I could then flip each pedal and not accidentally re-clip). These routines have so far kept me fall-free.

Teacher Routines

Here are a few examples of routines that helped teachers find flow in their classrooms:

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Coaching to Leverage Teacher Strengths

My blog on bicycling and instructional coaching lays out some key ideas for helping new teachers grow. I’d like to expend on the idea of identifying and using a new teacher’s strengths.

Whether I’m on my bike or in a classroom, one of my strengths is planning ahead. To avoid problems with forgetting to unclip my bike shoes from the pedals before stopping (see the first blog…), I planned out my rides—clip for the uninterrupted one-way paths and unclip where the streets, walking paths and biking trails intersect between the lakes. And, the first time on any route, I skip the pedal clips rather than trust that I’ll remember to unclip as needed. Often, great planners struggle to be spontaneous. And, people who flex easily when plans aren’t working seldom love to plan.

Self-understanding lets us capitalize on these kinds of insights. For example, a veteran teacher excelled at setting schedules or estimating how long activities would take. She could easily motivate students with reality-based promises such as, “I know all of you can finish this within 10 minutes. Let’s work hard and that will give us extra time to finish a read-aloud chapter.” A new teacher on her team tried the same technique but found that her estimates were off.

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New Teacher Goals

In my last blog, I discussed how my experiences with coaching myself as a cyclist parallel my most effective strategies for coaching teachers.

Setting individualized goals is a key component. When I first tried out my biking shoes, I aimed for my husband’s goal: clip in as much as possible. After the second fall, I changed my goal: clip in wherever it is safe. I unclip on any stretch of road or trail that might bring surprises.

Often, new teachers—or their mentors—assume that a strategy needs to be implemented just as other teachers use it. They end up either biting off more than they can chew at first, or struggling inordinately with something that looked so easy when modeled by a teacher with very different strengths. Think for example about ways to develop student responsibility for materials. A new middle school teacher wanted to use a colleague’s chart for tracking which class had the highest percentage of students who came each day with pencil, notebook, and homework.

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Differentiated Cycling AND Coaching

Have you noticed that “who you are is how you teach”? Colleagues who are a bit more reserved tend to run quieter classrooms. Teachers who love to read (in what little spare time teachers have!) share that love with students. Our classrooms mirror our strengths and interests in countless ways. (You can download Chapter 2: Who You Are is How You Teach from one of my books which explains a framework for working with normal differences among teachers–and students)

Who We Are is How We Bike–And Teach

I recently realized that who I am is also how I bike. As I did some thorough self-coaching to master new equipment, I found I was taking myself through the same process I use with teachers. Hopefully, my biking experience will help provide an image of key strategies that can help new teachers use their strengths to master their biggest needs in the classroom.!<

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Chocolate Chip Cookies and Instructional Coaching

How is baking chocolate chip cookies similar to implementing teaching strategies? It has a lot to do with “implementation with fidelity”–instructional coaches are often taught to look for key elements necessary for a given strategy to work. YES you need to identify those elements BUT when you’re observing a classroom, beware. They may be harder to recognize than you think. Just like a good chocolate chip cookie.

The Chef Effect

My mother taught my four big brothers and me to bake chocolate chip cookies. In education terms, we all had the same “instructional coach.” !

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Of Course Teachers Can Become More Effective!

A recent article proclaims that “The findings of several recent studies by psychologists, economists, and educators show that—despite many reformers’ claims to the contrary—it may be possible to make low-performing teachers better, instead of firing them.” (“How To Build a Better Teacher http://tinyurl.com/cnnl2pn)

Why would we think otherwise? How has the debate over teacher quality lost track of how excellence is developed in any field? It takes time, targeted practice, and the right support!

First, it takes 10,000 hours to reach excellence, as Malcolm Gladwell so eloquently documents in Outliers. That’s 10,000 hours–about 8 school years’ worth–with the right sort of coaching and support. Let’s be clear, too, that only actual teaching, lesson planning, assessment construction and analysis, and other core responsibilities count toward those hours, not preservice theory work (although that may guide practice).

A compilation of over 100 studies from many disciplines revealed, “When most people practice, they focus on the things they already know how to do. Deliberate practice is different. It entails considerable, specific, and sustained efforts to do something you can’t do well—or even at all.” [1] 

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