Opening with Estimation180 on a clothesline

I’m a big fan of Andrew Stadel. If you haven’t checked out his awesome contributions to teaching math, please head on over to his blog, Divisible by 3. For years, I used resources from Estimation180 to get my students estimating and talking math. Last year, I decided to institute a more regular warm-up and cycled Estimation180 into my weekly rotation. Then, Mr. Stadel began stringing clotheslines all over the place as dynamic number lines. For no good reason, I did not string one up in my classroom last year – no clotheslines in China? (yea right) – but came back to the start of this year with a clothesline in my bag.

Talks surrounding Estimation180 have often been rich as students explain their reasoning. I’m a fan of the “too low” and “too high” bounds. I want my students to be able to set these limits. I want them to get the feeling of a range and that they do not have to zero in on a specific number. So, airtime in discussions leans towards setting these bounds, discussing the reasoning involved, and working as a class to develop (or recognize the levels of) confidence in our estimations.

To start this year, I decided to use a clothesline as a way to make the math talk a bit more dynamic.


Here is the flow from my first go – any suggestions / tweaks greatly appreciated!

  1. Start with the image – students Notice / Wonder to get their minds into the math class. We then bounce a few ideas around as to what we could possibly be getting ready to estimate.
  2. Individual – students work on a “too high” and a “too low” estimate. A focus is on paying attention to the thoughts and processes involved in setting the bounds.
  3. Table talk – after each student has made some individual progress, get them talking.
  4. Group talk – this is where the clothesline comes in
    1. Too Low – orange cards used to place values on the clothesline. Several students came up and shared.
    2. Too High – red cards used and again several students placed a value and shared reasoning.
    3. Table groups then had a few moments to agree upon one “just right” estimate. We just finished a lot of processing so they take it one step farther and decide upon a value that answers the estimate question.
    4. A group member comes up to place the group’s value on the clothesline. Yellow card tents used this time. No discussion at this point unless it is making observations about the values.
    5. Check out the actual value and move on…

That was the process yesterday. What did I like about using the clothesline?

  • In the past, we had the conversations regarding low and high values but the lack of organization was clear. Students would write a value on the board and talk and it was challenging to visualize the range. Now, we begin to see the distribution of values.
  • Developing more number sense – When final values were placed by a class, the value 185 cm was used on multiple occasions. However, these values were not stacked to represent the quantity but lined up to represent each group’s idea. A nice conversation ensued about value.
  • Developing a sense of placement and spacing – As values are placed on the clothesline, we can begin talking about keeping a general sense of equal spacing. 10 cm on one end of the number line should represent the same value distribution as on another place on the clothesline.
  • Visual Representation – Wow! My lack of organization from last year’s conversations was quite apparent. The clothesline immediately organized and the colors of the number tents allowed us to easily talk of the highs/lows. Maximum and minimum values were easy to discuss. Students can walk the range.
  • Shift to students – I felt that more of the focus and attention was placed on students as they walked the number line and explained their reasoning. It is helping me fade to the background!

To improve upon? Time. I am looking for ways to tighten the process up a bit. I like the different colors of number tents to place onto the clothesline and wonder how I can better distribute them to students. Possible idea – provide each table group with a tent of each color (representing too low, too high and actual estimate). The too low and too high values are written first and at the same time a member from each group comes up for “too low”. They have to work together for placement purposes and then give an explanation. Repeat for “too high”. Have a number talk regarding patterns seen and explanations. Final estimates are placed. Other ideas?

Students review Math & Science

It’s that time of year again – as educators, we look back and wonder where the year went, get excited about the upcoming break and reflect on the past year. Listening to the voices of students is key. This year, I continued with using a gForm to obtain feedback but a few days before asking students for their thoughts, I came across this post from Julie at I Speak Math. I took two awesome ideas from the post:

  1. Keep, Change, Stop, Start
  2. Putting results in Word Clouds

The simplicity of keep, change, stop and start is great – all students have access to providing a response. The results in word clouds illustrates the big ideas as the most often expressed words pop out.

2015-16 Math

(shown as Start – Math is… – Stop – Keep – Change)

2015-16 Science

(shown as  Science is… – Change – Start -Stop – Keep)

I plan to share these with students prior to the end so they can see the results of their feedback. I’m sure that they will make the connections that there are some of the same items on each of the images. However, the difference is that for most of the items (for example Estimation180) there were many more for keeping than stopping. Our challenge in keeping class varied enough so that all students are able to be engaged and access the material.

Update: Word Clouds made @

MSIS #3: Assessment

ramblings from the 3rd institute at the Shanghai American School

Kick-off: What’s worth celebrating & sharing?

Themes for the Weekend

  • “Assessment is a process of gathering and using evidence of learning to improve teaching & learning.”
  • Formative assessment is a critical part of learning. If it’s “graded”, it ain’t formative.
  • Effective assessment balances DOK (Depth of Knowledge) levels 1-3. [Level 4 = projects]
  • What & how we assess drives what & how we teach.

What is our primary focus? Teaching Mathematics – therefore, where is the curriculum & assessment coming from. Our job should not be to create curriculum & assessment. So, where should we find these great items?

How are assessment questions aligned with instructional practice?

Characteristics of high quality assessment

  • Justification / explanations
  • Multiple strategies
  • Models can be used to support
  • Reasoning / critiquing
  • Fair
  • Aligned to standards
  • Limits complexity of language

DOK – what is the cognitive complexity?

  • Content is assessed @ DOK 1 & 2
  • Problem Solving – DOK 2 & 3
  • Communicating & Reasoning – DOK 2 & 3 (with some 4)
  • Modeling & Data Analysis – DOK 1-4

DOK does not equal level of achievement of student.

DOK Levels (What kind of thinking is needed to respond?)

  1. Recall & Reproduction
  2. Basic Skills & Concepts
    1. Mental processing beyond recall is necessary.
  3. Strategic Thinking & Reasoning
  4. Extended Thinking

Excel vs Exceed – does a shift to “excel” have more meaning

  • Evidence of complete understanding
  • Evidence of reasonable understanding
  • Evidence of inadequate understanding
  • No Evidence

Rigor – the pursuit of

  • conceptual understanding
  • procedural skill & fluency
  • application

with equal intensity

Standards Based drivers

  • What should my students be able to do?
  • How will we know when my students are successful?
  • What will I do if they “got it”?
  • What will I do if they did not “get it”?

Assessment –

  • something that we do with (not to) a student.
  • integrated with the learning.
  • DOK level of instruction should be above the level of assessment


  • What you are teaching – the standards
  • When you are teaching – scope & sequence
  • How you are teaching – teacher instruction


Students must benefit from formative assessment.

Comparing Tasks – how do we improve existing tasks / assessments ?

Justification & the Frayer Model – how do the mathematics and model justify each other?



Comment Codes

From Shannon Andrews(@andrewsshannon2) “I remember reading on Fawn Nguyen’s (@fawnpnguyen) website Finding Ways that she grades using a highlighter.  During the Principles to Actions math chat, Frank McGowan (@frankmcgowa) talked about using comment codes.  Instead of writing the same questions and comments on EVERY SINGLE PAPER, Frank attaches a code to each comment or question. I believe he collaborated with his English Department on this.  Then when the assignment is returned, he gives them a reflection sheet which includes the codes.

So here is how I applied the same idea in my class this weekend as I was grading.  Frank, maybe you can offer your insights as well.”

This post is an attempt to provide my beginning use of comment codes. Any ideas are appreciated! Comment Codes are something that I’ve begun working with this year. My hope behind the trial was:

  1. avoid writing the same comment on many different papers,
  2. reduce time in providing comments to students, and
  3. (most importantly) shift the burden of analyzing work and finding errors to students.

Credit for this idea comes from the post on Pragmatic Education titled: What if you marked every book, every lesson? In this post, Joe Kirby (@joe__kirby), describes making with icons or numbers.

As I begin looking over student work, I begin writing comment codes. On a student’s page, the code is placed inside a hexagon (bad move on my part – all assessments have been returned and I’ve no copies. Sorry!). Student work is returned with a reflection sheet that asks them to process their work. I’m still working on the format of this and any ideas are much appreciated.

Types of comments (In general, I tend to pose questions in the attempt to guide students):

  1. Error check – In many cases, students are doing great mathematics/science but need to look back over their work.
  2. Expansion of ideas / connection to evidence: A big theme is communication. Are students explaining their ideas? When available, is evidence being used?
  3. Basic understanding: sometimes a student might need a quick reminder or prompt. Is it notation (i.e. using absolute value symbols)?

Using comment codes to plan future lessons: I think there is a lot of potential in recording these comments linked to assessments/topics. On one hand, it’s easy to note the numbers that are being regularly written down. Hey! That’s a class issue. On the other hand, in a following year, the comment codes can be looked at when planning the unit and asking what were the typical areas of challenge for students.

Below are examples of comment codes used from a math and a science assessment.


Comment Codes in Math

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Comment Codes in Science

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MSIS Session 2, Day 2

more disconnected thoughts from the MSIS at the Shanghai American School

Opening: Formative Assessment is challenging to implement because it necessitates us to adjust our teaching. It’s so much easier to believe that students walked out of the door understanding what happened than to find out that they didn’t and need to adjust.

Why MSIS – to help shift mathematical practices in schools. How do we return and improve mathematical instruction at our schools?

Start with the Problem! How else will students begin to realize what they know, what methods are efficient / inefficient, and how these problems relate.

Algebra – create the expressions from scenarios. Relate independent and dependent variables. How can a simple scenario be reworked to focus sometimes on the dependent and other times on the independent? – keep using them!

MS Progression of EE

6.EE.1 Write and evaluate numerical expressions involving whole number exponents

6.EE.2 Write, read and evaluate expressions in which letters stand for numbers.

6.EE.3 Write expressions that record operations with numbers and with letters standing for numbers (i.e. Express the calc. “Subtract y from 5 as 5-y”)

3a – Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity (i.e. 2(8 + 7) as a product of two factors; (8 + 7) as a single entity and a sum of tw terms.

3b – Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order t when there are no parentheses to specify a particular order (Order of Operations)

7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related (i.e. a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05”

3c – Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+x) to produce 6 + 3x;

apply the distributive property to the expression 24x + 18y to produce 6(4x + 3y);

apply properties of operations to y + y + y to produce the equivalent expression 3y

7.EE.1 Apply properties of operations as strategies to add, subtract, factor and expand linear expressions with rational coefficients.

6.EE.4 Identify when two expressions are equivalent.

6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q (nonnegative rational numbers)

7.EE.3 Solve multi-step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals)

Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies (i.e. If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary, or $2.50 for a new salary of $27.50)

6.EE.8 Write an inequality of the form x>c or x<c to represent a constraint or condition in a problem. Recognize that inequalities of the form x>c or x<c have infinitely many solutions; represent solutions of such inequalities on number line diagrams

6.EE.9 Use variables to represent two quantities in a problem that change in relationship to one another; write an equation to express one quantity (dependent variable in terms of independent variable)

Analyze the relationship between the dep. and ind. variables using graphs and tables and relate these to equation. (example – in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d=65t to represent the relationship between distance and time.)

7.EE.4 Use variables to represent quantities in a real-world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities

a) Solve word problems leading to the equations of the form px + q = r and p(x+q)=r where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used. (ex. the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?)

b) Solve word problems leading to inequalities of the form px + q > r or px + q <r. Graph the solution set of the inequality and interpret it in the context of the problem. (ex As a salesperson, you are paid $50 / week plus $3 per sale. This week you want your paty to be at least $100. Write an inequality for the number of sales you need to make and describe the solutions.

Grade 6: x + 6 = 12 or 6x=12

Grade 7: 4(x+6) = 12

Grade 8: 9 – 4(x+6) = 12 + x

Better Questioning with Gradual Release


I believed myself set. My bookmarks linked up to plenty of 3 Act tasks. I worked them regularly into my classroom. But, how were those tasks getting launched? This question was not one that I thought about much. Students were engaged and exploring mathematics. Then, I watched Steve Leinwand roll out a task and I was sent back to the drawing board. Where? In the questions.

Take time getting to that big question that students are going to chew on. What do you notice? Get them involved! What do you wonder? Get students looking for questions and thinking about possibilities. Can you let us know why you are thinking …? Throughout this process students support their statements. “I notice that the objects are 3-dimensional.” what do you mean by that? These probing questions help students clarify their thinking while also allowing me to gain a better feel for background information. Looking back, I can think of numerous instances where the “word” that I was “searching” for was provided and I never questioned if the student had any idea what that term actually meant.

Gradually releasing a task requires a bit more of a set-up. I’m still trying to work out how much to provide at once as the task is unveiled. At the same time, it’s always possible that we will take a left turn at some point based upon what the students come up with and that’s ok. It takes more time to set students off on the task but an incredible amount of information can be collected and when students finally get going they are primed and seem to have a stronger understanding of the context and that big, meaty question.

MSIS #2 – Day 1

a random assortment of items through the day…

What are people doing from last time that is working

  • I notice, I wonder…
  • Problem solving
  • Using models
  • Students talking
  • Slowly opening up problems

Big ideas of mathematics: language of quantity, change, shape, dimension, chance

Goal: build students depth of understanding and comfort in these areas through discourse

Through: multiple representations

The 10% perspective

Screen Shot 2016-01-30 at 8.49.16 AM

Pedagogical decision – when a student provides a response, how much do you want to push? How far will you continue to going along the path of their thoughts?

Book Title: The Problem with Math is English

Number Talks: 2 + 2 + 2 + 10 + 10 +10

  • How did you work with the 10s? Did you add or group? Then, what did you do with the 2s?


Domain: Operations & Algebraic Thinking

Cluster: Understanding addition and subtraction 

?How does this relate to number sense?

**Progression is by age, not grade.

What are the representations? How are we putting together and taking apart?

  • preK – exploring addition and subtraction with fingers and objects. Decomposing quantity (less than or equal to 5, then to 10) into pairs in more than one way (using objects / drawings)
  • Grade 1 – using objects, drawings and equations with a symbol for the unknown number
  • add to / take from / put together

Focus on word problems – explaining! Use the practices.

1.OA.6 First Grade!
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as

  • counting on;
  • making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);
  • decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9);
  • using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and
  • creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Progresses to 2.OA.2

Fluently add and subtract within 20 using mental strategies (what’s the picture that you see?). By end of Grade 2, know from memory all sums of two one-digit numbers.

Where has the math been? fragmented, skill-driven, incoherent

Where are we going? fewer, deeper, examples & specificity (focus, rigor & clarity)

Why would I subtract when I can add-on?

Take home quote of the day – don’t stop pushing for student thought.

“If a student answers in the way that you were thinking, it should not keep you from asking did anyone do it differently?”