Teaching & Twitter (it’s worth it)

Five to ten minutes. I get in the habit of being involved with Twitter but time passes and it I get “too busy”. This post is that reminder to take the little bit of time in a day and check in. It is so worth it.

On Wednesday, I saw this post:

Screen Shot 2018-05-11 at 11.48.12 AM.png

What a cool idea. Get students quickly working with area and perimeter in an interesting problem solving idea.

IMG_1705On Thursday, a few students were finishing up early on their area/volume project so I pitched this idea and a stack of old newspaper at them. It was fun watching their process. They decided to start with getting the amount of paper to make a square meter.

A few moments later, they realized that their perimeter was going to be quite on the short side. At that point, a student said what could be the quote that makes my year worth it:



Should we just model it first? 

They sketched out a model of a few different ideas and included a scale.


Once their model was done, they quickly put the newspaper together.


Oops, it can’t be a rectangle so they made a quick adjustment:


A big thanks to @S_ODwyer for sharing her idea and to Twitter for a platform that makes sharing ideas easy. Take a few moments each day to touch base. Hidden gems are everywhere!


Let the students show the way (spreadsheets to Desmos)

Recently, I posted about the CER structure that I was starting to use more with my 7th graders in science. There, time was spent on representing evidence from multiple trials as well as different measures of center. I wondered about the use of Desmos in creating graphs. The goal is for students to be able to quickly create graphical displays so that the bulk of their time is spent discussing patterns within their evidence. Honestly, I hadn’t poked around in Desmos enough recently to know if it would be challenging or not. Graphing in Excel/gSheets often opens up the destructive side of me as it is quite cumbersome to make and modify graphs (at least at my skill level).

Today, students were finishing up assessments and and I tasked them with getting into Desmos and graphing the data they had recently done by hand. The goal was to explore, learn skills, reproduce the graph and then share out if it is possible.


Students got to cracking! I had them work individually to both allow for quiet in order for others to finish up assessments and to also allow for each student’s creativity in finding a solution. They were quite resourceful. Some began by putting in individual points. Others typed in entire tables. I nudged to find a way to copy and paste from the data in the spreadsheet. (I’m fully onboard in the power of spreadsheets to analyze data – I just can’t get the displays I want without frustration.) They reported back that they had. Entire tables can be copied from a spreadsheet and pasted into Desmos! Yeah! That’s quick! However, we had to do some restructuring of tables to make better meaning.

Screen Shot 2018-05-09 at 3.54.02 PM

gSpreadsheet Link

The trials and averages went into two different Desmos tables so that the formatting could be differentiated.

Screen Shot 2018-05-09 at 3.57.31 PM

Desmos Graph

Desmos Tip from students: Holding down on the color in the table provides extra options such as adding in a line between points if that is desired.

By the end of class, students were telling me that they were quite comfortable bringing tables into Desmos and then creating a graphical display. Desmos Tip: The wrench in the top right corner allows labels to be put on each axis as well as adjusting the scale.

So, I’m pretty excited both in having a potential system to make graphical displays and with the reward of putting my trust in the students. They could and did dive into the Desmos and finished by increasing the understanding of our community.

CER – Developing Structures

Evidence. It’s challenging for my 7th graders to keep coming back to evidence. Personally, I’ve struggled finding the right scaffolds for students to have them put together investigations and share their thinking. The following is my latest installment based upon some past work with Paul Andersen (@paulandersen) and continual shifts in thinking towards NGSS practices. In the end, student goals are to have strong conversations about their investigations, clearly model design, display key evidence and conceptually model their ideas.

Screen Shot 2018-05-07 at 4.22.09 PM

(Link to above document)

The structure above is used at the end of an investigation though students need to show their experimental design. (A topic for a different time – how much emphasis are people putting on the physical writing of a procedure versus setting up the experimental design and then moving forward?)

Experimental Design: In the experimental design, students create a model that shows their set-up with the independent, dependent and controlled variables clearly indicated. Below are a couple of student examples for an investigation where the mass of a dynamic cart was varied as students examined change in velocity with constant force or an applied force was varied.

What I like:

  • The visual nature of setting up an investigation through a model.
  • Highlighting the variables. My students are not always 100% confident on independent, dependent and controlled variables. This color connection makes them again talk about the variables. They also quickly realize if parts are missing from their model.
  • There is a focus on the experimental set-up. Students are talking about the parts of the investigation they are working with.

What I wonder:

  • For seventh graders, is this good enough? In the past, I’ve spent a lot of time word smithing on procedures and I’m not sure that that is necessarily time well spent. What do I really want? I would like students to have a strong foundation in designing investigations that focus on a relationship between an independent and dependent variable. I would like students to truly grasp the need to keep all other items controlled.
  • How can I mix in a strong foundation of procedure-type vocabulary that students then show on the model?
  • What can be done to improve on the current model? Does more need to be done regarding the independent / dependent variable relationship? What about the controlled variables – how do the key ones really stand out?
  • Multiple Trials – One group wrote to repeat for every trial though they did not indicate how many trials would be necessary. Maybe I should make this more specific.

Data: Once the set-up is ironed out, students move to collecting data. Organized data tables are challenging to set-up! We eventually get data tables constructed in a spreadsheet and students begin collecting. This post is more related to the workup of ideas so let’s move to the display of evidence.

What I like:

  • Paul Andersen was the one that led me to showing all of the data collected in trials on a graph. In the past, students showed the average only but a lot of information was lost. We now have conversations about the range of data and the confidence level of groups in whether or not they have a strong data set.
  • The graph is another place where we check in on independent and dependent variables.
  • Physics! We got reproducible data almost across the board and it really does make a difference.
  • Working up the data into a graphical display and then having students write a claim.

What I wonder:

  • Students are doing more with data in spreadsheets though get really turned around when it gets to graphing with the software. I like the investment of time to go into discussing the data instead of physically graphing but I don’t have a good solution yet for graphical displays. (I’m toying with Excel/gSheets, Desmos and CODAPany others?)

Conceptual Model of Claim: In support of the claim that student groups generate based upon their data, I’m working with the idea of a model to explain ideas. Evidence is shown on the graph, so the focus here is more in showing the big idea of the claim. This is definitely a work in progress and in need of refinement.

What I like:

  • Students are returning to the relationship of the independent and dependent variables.
  • By having this piece not focus on numbers, students talk more about representing their ideas.
  • A focus is placed upon the claim being a representation of the data. Our conversations were around what patterns are in the data and how can this be shown.

What I wonder:

  • Dot diagrams were provided as a way for students to model their thoughts. In other areas, is it good for groups to have similar models for better comparison or better to have groups represent in any way they like?
  • What additional description should accompany the conceptual model. Is the claim and the model enough?
  • Many more thoughts running around…

Reasoning – With my 7th graders, I’m finding that more structure is still a good idea when writing a claim. We return to the relationship of the independent to dependent variable and I want to see students including specific information from their trials of what was changed and what was measured. My focus has been on developing a solid reliance on evidence that I hope can then be used to eventually link to strong reasoning.

Working towards relevance (Project Launch)

Recently, I’ve cracked open a few great books about teaching and learning  that have inspired me. Katie Martin’s Learner-Centered Innovation and George Couros’ The Innovator’s Mindset got me thinking and then I hopped into John Spencer’s Design Thinking Master Course

Screen Shot 2018-04-11 at 2.12.11 PM

I really like teaching mathematics and feel that my classes are engaging and that students actively work on problems nonstop. Estimation 180, visual patterns, WODB and Graphing Stories are some of the starting points for class that then moves to 3-act tasks and other problem-based scenarios. Students continually work together on whiteboard tabletops to discuss, model and justify their solutions. It’s often loud, it has a lot of student talk and I like the dynamic atmosphere. But…

The “but” is that I feel that while my students are highly engaged and are actively solving problems, they miss the big idea that mathematics is a powerful system to help explain and understand this world that we live in. The question from the books and courses is am I moving beyond engagement to empowerment? Am I allowing students the opportunity to grapple with issues and problems that surround us? I’m not sure. Many of my problems are based on events happening around the world but we wrap them up in a class. And what are my students not experiencing? Are they having to define problems, develop solutions and then use mathematics as a tool or for justification of their ideas?

Extended projects in the classroom have more often happened in my science classes instead of mathematics but as I look forward to only teaching mathematics next year, I feel that it will be nice to bring a project or two into the rotation. As such, with two feet pointing in (I hope) the right direction, I’ve dived in to pilot a project. My goal with the next few posts is to keep track of my process through this project and reflect on how it’s going.

Screen Shot 2018-04-11 at 2.13.31 PMAs part of The Guardian’s City feature, a series of articles and photographs were put together regarding urbanization and the future. Being located in Beijing adds a bit of extra significance – we’re already over 20 million! This was the launching pad for the project. I took an idea of John Spencer – Maker Challenge: Design the Ultimate Tiny House and began modifying. Cities build up, so let’s look at an apartment. Keeping the theme of “tiny” and wanting students to have a physical point of reference, the footprint of our design challenge is the classroom. Our room is a long shoebox that measures approximately 5 x 15 m.

Day 1:Project Kick-off

We began with a silent gallery walk. Students came in to see a variety of images on vertical whiteboards around the room. With a whiteboard marker, they walked around noticing and wondering.


This was followed by an imagine – Imagine that you are a designer of apartments. You have an apartment to design with the same footprint as this classroom. What would it look like? What would be in it? Students individually began sketching out this potential apartment. Some started to realize issues posed by the size constraint but others sketched away without this consideration.


From large water-beds to basketball hoops to blank rooms, students had lots or little ideas as they began work. After a bit, students paused on their work to bring together a list of items and features they thought important to have in designs.



With students invested, we began to dig a bit deeper into the problem and students were asked to generate as many questions as possible. This is hard! Students really struggled to come up with questions and made me realize that they have not had enough opportunity to develop this skill over the year.

The types of questions asked also made me think about the timing of information that I provide to students. With the first class, I gave them a “sneak preview” of the upcoming project prior to the questioning phase and found that the nature of most questions were through the lens of how will I complete the school project? So, I switched it up for the second class and did not give the sneak preview until later. The result? Most of the questions were oriented around the design of the apartment. Students then each took a “burning question” to research and share out with the class.


Sketch Video Project – Expressions & Equations

Skills-based portions of the year are often a struggle for me as the vast majority of the class is spend on active problem solving. Is there a place for learning strategies and practicing “naked number” skills? Some students who have tutors or take additional math classes walk in with a strong foundation in these areas before we start and others who are a bit fearful cringe. At the moment, I believe that time does need to be given to work on skills such as simplifying expressions and solving equations as this is an expectation of students moving on. This year, I tried something different after spending some time on John Spencer’s blog. Among many great resources and ideas, the one that captured my attention was his Sketch Video Project. A few examples are included and this is a technique for students to quickly put together a video of their ideas. When the making of the a project takes substantially more time than the learning or creating of a concept, I often have trouble following through with the project. In this case, I felt that students would have lots of great conversations in how to explain expressions and equations and not spend much time pulling together their ideas for the video.

I gave a few parameters:

  • explain the difference between expressions and equations
  • show an understanding of key vocabulary (variable, constant, like terms)
  • provide examples of and show how to use the distributive property
  • solve 2-step equations

The video nature of the project lent itself to showing models and I nudged students towards including appropriate visuals. Everything else was their decision.

The project spanned three class periods. The first was a kick-off, understand the project and get-to-work kind of day and the other two were work sessions. During the two work sessions, I pulled students to individually take a formative skills assessment. By only pulling one student from each video group at a time, groups were able to continue moving projects forward while I could ask questions of students. Were they getting the big ideas? Were they able to simplify expressions and solve equations? In general, the formative data was quite positive. Many students demonstrated a positive ability and those that needed extra help were able to get some one-on-one attention. In addition, they returned to groups where other students helped talk them through scenarios.

On the day that the projects were due, we shared projects. The videos were 3-5 minutes long so groups rotated through each video station to watch and give positive feedback. This process also served as a review / reinforcement of skills. I hope the linked student video can be accessed as it is on a school server…Expressions & Equations Video

At the end, the same assessment used previously was used to gauge student skill levels. I was quite pleased to find that students did as well if not better than in the past. In addition, the level of organization and communication in student work was much higher. Was this due to spending the last few class sessions in a format where communication of ideas was at a premium?

A quick survey asked students for some feedback on the project.

In general, the feedback was quite positive and students both enjoyed the process of working together to make a video and learned the math. One student indicated that the video did not help but when I had a conversation with him he indicated that he put a “1” because he already knew all the math. Ok, point made. But, when I asked him if he learned something about making videos and explaining his ideas he quickly responded that he had.

Again, a big thanks to John Spencer for sharing his ideas. I enjoyed the process and will try to incorporate Sketch Videos another time.

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