About Frank McGowan

I teach middle school students.

MSIS #5 Rational Numbers & Ratios

The final institute in Shanghai. Ramblings and note takings

Opening with convince us that 1/10 of 450 is 45.

  • Equivalence
  • “Who did it differently?”
    • chop of the zero
    • move the decimal to the left
    • percent
    • multiply by a tenth
    • divide by 10
  • Record – why?

“This is unquestionably the hardest piece of mathematics to teach and the second most important to learn (after operations).”

Rates / ratios – do they “live” on the number line?

  • 6/10 + 6 /10 = 12/10, but…
  • In the first half, someone made 6 out of 10 free throws. In the second half, an additional 6 shots out of 10 were made. 12 out of 20 were made
  • Examine the relationship between the values (double number line versus single)


Bridge to HS -shift from additive thinking to multiplicative thinking

Model – what happens between the problem and the solution

Convince us in two different ways. How do the two different ways talk to each other?


A blurry miserable image but…writing ratios as fractions. Should a part:part ratio be written as a fraction as it does not represent a part to whole?


Student Talk

  • Who is struggling? Let us know what you are thinking.

MSIS #4 – Geometry

Back in Shanghai again for the fourth institute! Moving past the vast, unmanageable resources such as Engage & adapting to our students and classes. General and random notes and thoughts.

Geo-metry: measurement of the Earth

A balanced brain –> numerical, verbal, spatial

What is Geometry

  • study of shapes
  • relationships between objects
  • connection of mathematics to the physical world
  • dimensions (point, line, plane,
  • space
  • logic / reasoning
  • making / defending conjectures

Connection of geometry to number & algebra

  • bar models – multiplication area models
  • number lines – Cartesian plane
  • distribute property
  • tables to graphs
  • composing & decomposing (shapes like numbers & expressions)

Data – clumps, bumps and holes

Van Hiele Levels

  1. Identify – name
  2. Draw – create
  3. Describe – characteristics
  4. Analyze – properties, examples & counterexamples
  5. Classify / categorize

Language: informal to formal + Picture


  • benchmarks or referents are essential
  • Number is discrete, measurement is continuous
  • Estimation is essential
  • an approximation with tools (i.e. nearest kilogram)
  • measurement conversion similar to common denominator (must have same to compare / work with)

Resource Reminder – spend 5 minutes on LearnZillion to see how they unpack the standard.


Day 2

Justification – Give problems more ambiguity. Justification involves explaining why certain mathematical decisions were made.

PreK-2 Progressions of Measurement

  • Describe and compare measurable attributes
  • Measure lengths indirectly and by iterating length units
    • Order first (G1) then measure (G2)
    • G1: express length as whole number of length units spanning with no gaps or overlaps. End to end
    • G2: measure with different units, estimate lengths, how much longer is one than another
  • Relate addition and subtraction to length
    • PreK – directly compare, “more of / less of”
    • G2 – using number lines
  • Tell & write time (G1-G2) Note: G2 begins with money as well (not quite sure what that has to do with time)
  • Classify objects and count the number of objects in each
    • PreK – sort
    • K – classify & count
  • Represent and interpret data
    • PreK: Compare categories (greater than / less than…)
    • G1 – Organize, represent and interpret data with up to 3 categories
    • G2 – generate measurement data. Make line plots of whole number units


  • Identify & describe shapes
    • PreK – match like shapes, group by attributes, correctly name
    • K – Describe objects using their name s and relative positions, identify shapes as 2-dimensional (flat) or 3-dimensional (solid)
  • Analyze, compare, create & compose shapes
    • PreK – describe 3d objects
    • K – analyze & compare 2D / 3D shapes, model shapes by building, compose simple shapes to form larger shapes
    • G1-distinguish between defining attributes (e.g. triangles are closed & 3D) versus non-defining attributes (color, orientation, size…)
      • build shapes
      • draw shapes
      • compose 2D shapes or 3D shapes to make composite shapes
      • Partition circles & rectangles (halves, fourths, quarters)
    • G2-Recognize & draw shapes with specified attributes
      • partition rectangles into rows & columns. Count to find total number
      • partition circles & rectangles into shares. Describe, and recognize part:whole

Notes looking from 3 to 7

  • Rectangles are hammered in the lower grades. 2D, 3D. Area. Volume.
  • Circles pop up in 7th but have not been discussed since 2nd grade. Wow! Students might not know these key vocabulary terms.
  • Angles have not been discussed since 4th grade
  • Triangles are worked on in 6th
  • Scale drawings are new – build upon ratios

Big Middle School Ideas

  • Circles (pi)
    • Pizza is 78% of the pizza box
  • Pythagorean Theorem (G8)
  • Using formulas (generalization of relationships)
    • Why do they work?
  • Surface Area
  • Transformations – Scale
  • Congruence & Similarities (G8)
  • Lines & Angles
    • Begin becoming more precise of justification

Reworking Sugar Packets

Seeing the look on student faces as the video plays makes the problem. Seriously! What is that guy doing? (This is a Dan Meyer 3Act Problem – please see link for original problem and video.)

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I ran two versions of the problem in class. Due to a testing schedule and late arrival of many students, my first class’ time was whittled down to 30 minutes. We’re getting into a rates & proportions unit and a goal was to connect back to work done in sixth grade to continue getting a feel for where my students are meeting me in the class. So, the problem unfolded generally as given in the 3 Act flow. There was high engagement at first but then the problem was solved immediately. Woops – good thing that the class was a short one! However my next math class of the day was a full length class (80 minutes) so reworking was needed. Here is the new flow and it provided for a better lesson where students were more engaged and actively discussing mathematics for an extended time.

The hook – got ’em.

Immediately after playing the video, student tables were given a red and an orange card tent and asked to come up with an estimate that was “too high” and one that was “too low”. We stretched a clothesline across the room and members of each group came up to begin placing their tents on the line. The clothesline is a great tool for these problems as students work together to sort their estimates along the line. Students then gave a quick explanation for their team’s estimate. The best was the estimate of 50 packets of sugar due to someone’s little brother who once put lots of sugar into a drink and a “stickiness” scale was developed…

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So far, the problem is unfolding as the 3 Act is laid out. The next step is the shift. Instead of providing the given nutrition level, I grabbed multiple nutrition labels from the Coke site. The intent here was to have students work with more data. Does Coke keep the same proportion of sugar in each size beverage?

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Multiple labels were printed and placed about in the room. For the smaller sizes, the serving size changed but for 1L and up, the serving size was based upon 12 oz (360 mL). This led to great small discussions as many students did not understand how to read a nutrition label and what a serving size meant.

Compared to the first pass of the problem, students were now up and out of their seats collecting data from the different nutrition labels. Their target had shifted as students now needed to determine whether or not all Cokes are created equally. Yay – we now had a need for determining the unit rate. (Interesting side note – the 7.5 oz bottle does seem to have a higher sugar content if you are looking for a sweeter Coke.)

For me, the shift in the problem opened up the class to a higher level of activity and discussion. I appreciate the window this gives me to have more conversations with students. Working through the nutrition labels resulted in a good stretch for some students. My weakness in wrapping up and really pulling out key understanding through class discourse showed itself again and I’m back to reading the book below. Processes that others have for wrapping up a 3Act task to pull out the learning for students would be much appreciated.

5 Practices for Orchestrating Productive Mathematics Discussions


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 @ tagul.com


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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