# Becoming a Better Storyteller

## Graham Fletcher

**Video of My Talk**

**ShadowCon Session: **Stories have been around since the beginning of time. They’re passed down from generation to generation and shared through countless children’s books. No matter how good the book, the portrayal of the story falls back in the hands of the storyteller. Some people are good storytellers; others are GREAT. How is your storytelling skill-set in math class?

**When:** Friday, April 15, 2016 | 5:00PM – 6:30PM

**Slides: **Becoming a Better Storyteller

**What Twitter said: **https://storify.com/Zakchamp/graham-fletcher-shadowcon16 and my livechat.

**Resources**

Common Core State Standards: http://www.corestandards.org/Math/

The CCSS Progressions: http://math.arizona.edu/~ime/progressions/

Graham, I really enjoyed your talk. In particular, the discussion on standards first and curriculum or textbooks second was powerful. As a district math coordinator, I know that we all can get caught up in having the textbook guide our daily, weekly, and yearly plans. At our next middle school department meeting, I am going bring up your talk and allocate time for teachers to review the standards and discover something new. I will report back after.

Awesome Matthew and thanks for hanging out with us at ShadowCon. It’s only a launching point and obviously the magic happens back at home. Can’t wait to see what you all uncover. The fact that you’re already looking for ways to give teachers the time they need to look at the standards is huge! Keep in touch.

A standard I taught and discovered something new is 8.EE.B.5:

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Found at http://www.corestandards.org/Math/Content/8/EE/B/5/

The last part of this standard intrigues me: compare a distance-time GRAPH to a distance-time EQUATION to determine which of the two moving objects has greater speed. I taught my students to identify and group different representations of the different stories: table, graph, and equation. However, I don’t recall ever explicitly giving my students two stories, each one with a different representation, and asking students to compare just those two.

More importantly, I love that this standard expects students to be able to understand one representation enough in order to compare it to a different representation. For example, a student should be able to look at a proportional relationship in a graph and be able to at least find the rate of change in order to compare it to an equation with a rate of change, meaning the students must also identify the rate of change in the equation.

Does that sound about right?

It does sound right Andrew and thanks for sharing.

Looking back I can only recall times where I’ve had students represent one context as multiple representations, or had students compare rate of change within the same representation. I think there is some really great potential here for designing a task that centers around this NEED to compare to representations. Wondering if there’s already one that exists.

Yes…we need to provide experiences where students are able to move flexibly between the representations. Check out this task that is listed as a 6th grade task when companies were first trying to make sense of the standards: (I think it hits more towards the 8th grade standard) http://ccsstoolbox.agilemind.com/parcc/middle_3788_2.html

And Illustrative Math does a nice job providing tasks that allow for students to make these comparisons: https://www.illustrativemathematics.org/content-standards/8/EE/B/5/tasks/184

Perfect Bridget and thanks for sharing this. It’s great to know that as soon as we find gaps in our storytelling others are on standby with the missing pieces. Nice find!

Working on writing a geometry training for 3rd-5th grade teachers, and having some crazy a-ha moments about 4.MD.5a: Understand concepts of angle measurement: an angle is measured in reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

I can’t help but compare it to its counterpart from the 1997 CA Math Standards: “Understand

that 90°, 180°, 270°, and 360° are associated, respectively, with 1⁄4, 1⁄2, 3⁄4, and full turns.”

I don’t know if I EVER made the connection between angle measures and turns. Of COURSE our angle measures are in the context of a circle. Why have I never picked up a circular protractor before? I’m in the beginning stages of my exploration with this standard, but wanted to share so far.

*Christine

Sweet Christine and such a huge connection to make for us as teachers… and for students!

This seems like a wonderful opportunity to explore angle measures but there’s a limited number of tasks or activities make the understanding of this standard accessible. If you find some please attach it to this thread so we can give it a whirl. I’ll start by adding this one to the list: http://nzmaths.co.nz/angle-units-work. It’s a rabbit hole so be careful.

We look forward to you reporting back.

Great discovery! Just FYI this is an important HS standard as well, but students have to compare not only across different representations, but also different functions. After 8th grade in Algebra 1 students are supposed to compare and contrast linear, exponential and quadratic models, including Arithmetic and Geometric sequences, represented in different forms. This is a tough standard to find questions for and so my team has been creating and sharing these over the past couple years. (And reminding ourselves to pull them in because it is easy to forget when you are focused on a particular function.)

A cool way to do this is also to integrate in your 8.SP.1&2, having students analyze data about a real world situation and then present them with a model for another similar situation and make comparisons. For example, download real data off tuvalabs.com (like the data on Athletic Training Certification in the US) and then present them with the model for another career. This is also a good place to do a “Which One Doesn’t Belong” or a “check all that apply” where students have to think about scenarios presented in different forms. It certainly raises the rigor and depth of their understanding. Good luck!

I appreciate you connecting the vertical progression of where this standard is going Phillip. A lot of the times we look past standards because they might not be “focus” in our specific grade level. What’s important here is that as we begin to look at standards first, we begin to make connections to the progression of of strands and concepts. And when we do that, we enable ourselves to tell a more complete story….from the beginning, to the end.

Graham, I’m very excited for the release of your video next week…the slides just don’t do it justice from what I remember!

Just a bit of background, I had taken a couple years off teaching to raise my kids (right at the start of common core implementation) and when I finally came back to the classroom last year, I was lucky enough to land a job at a school that had no “textbook”. I never did use textbooks much…just for the occasional homework assignments and for lessons at times when I was feeling particularly overwhelmed…but not having one forced me to begin my lesson planning from the standards. Because of that, I feel like I had a pretty good handle on common core. So, I was thinking about what I could do for you CTA.

I think I’d like to spend a bit more time on being familiar with the standards of the grade level below me. I’ve looked on occasion when I was wondering “what exactly are they supposed to come to me knowing about fractions???” but I haven’t delved as deeply into them as the grade I teach. I’m also going to post the standard in which I feel the weakest in the hopes someone can give me amazing strategies for teaching it to my students:

Draw informal comparative inferences about two populations.

CCSS.MATH.CONTENT.7.SP.B.3

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Statistics and I have a love/hate relationship. 🙂

Can’t wait to re-watch your talk and continue the conversation here and on twitter!

Found my “simplifying fractions”…I’ve been working on sprucing up my 7th grade ratio and proportions unit and realized that nowhere in the standards does it say students need to create a proportion. I even checked 6th grade and it’s not there, either! Everything talks about using “proportional thinking.” It specifically talks about graphing and equations, but no proportions. What better excuse to get rid of cross multiplication???

Funny, isn’t it. Out there in the real world the word “proportion” is in common use, as in “the proportion of cars failing the test was 1 in 10”, or more daringly “.. 1/10”

Such a huge thing to notice Elizabeth. I think “setting up a proportion” is purposefully left out of the standards AND the progressions because it quickly becomes forced on students as a procedure. When this happens the focus shifts from the relationship to the computation.

Page 10 in the progressions (https://goo.gl/UVYUeW) offers some alternative examples where the relationship remains the focus. They provide some nice counter examples.

Thanks for sharing and please keep sharing what you find.

I loved the talk Graham. We are storytellers, but I also strongly believe that our students have a math story to tell. As teachers, parents, and coaches we need to take the time to listen to their story, without any preconceived ideas. Listening to a student share how they arrived at an answer can be a beautiful thing. Sometimes, just like when we hear a story we want to ask questions, such as “what happened next?” or “will it happen again?” – these are the questions we can then ask our math storytellers.

Great point Molly and I absolutely agree! Interesting and compelling stories make us want to ask questions. Intuitively students are going to ask questions but we don’t necessarily put them in a place where they can ask meaningful questions that add to the story their trying to tell.

Thanks for pointing this out.

We secondary folk can learn much from your talk Graham. In HS, we often ask kids to write lines in “standard form”, or express a quadratic in “vertex” form – only because that’s what we feel like asking. Instead, let’s get kids thinking about why certain forms have advantages and cause kids to think about them and communicate them – move away from the one-way streets.

For many years, I have taught 9th graders the binomial theorem and related probabilities. In my earlier (less ripe) years, this could often be done through mastering procedure, and hoping to make some connections. Now, much of the unit is approached through discovery and story-telling. Instead of computing a distribution, we discuss plausible characteristics of what we would expect to see, then let the math verify our feel.

Inspiring talk Graham. Always a pleasure to hear your thoughts.

Thanks for chiming in here Bob! Perfect example of how we all learn and grow from one another no made the grade levels in between. I can’t help but think how we’re all guilty of pigeonholing our student thinking in our early years.

Funny how you mention “letting go” by providing opportunities to explore and discover. Seems like best practice is best practice…whether it’s in 1st grade or AP Stats.

Thank you – I enjoyed listening to this presentation.

Thanks for watching Troy and please report back when you find some nuggets worth sharing.

So I planned to share this video with my middle school math team in our meeting this afternoon. At the beginning of the meeting we were recording which standards we had worked on with our students this week. I was rereading the standards and noticed for the first time the word “numerical” in 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. I have been having students work with both numerical and algebraic expressions with exponents and this makes me realize that we should only be doing work with algebraic expressions if it is in service of finding equivalent numerical expressions. This was a great moment to share with the rest of my team after we watched the video.

Awesome Nicole.

This is a perfect example of why we need to go back and “unpack” our standards. I can’t help but think about how selective the authors were when choosing the words to describe a standard. Sure “numerical” has a meaning, but now we understand the meaning it has relating to the standard.

I appreciate you sharing and please add any other hidden gems you find.

Thank you so much for another great talk. I am an elementary teacher from California, and I must say that our Mathematics Framework revised in 2013 has been instrumental in helping me to better understand each of our standards. My “simplifying fractions” was 1.OA.1 which asks first graders to solve various word problems within 20. The component that I had originally missed was “by using OBJECTS, DRAWINGS, and equations…” Thanks to the framework I was reminded of how important direct modeling is for our budding mathematicians… Rather than have first graders jump straight to working with abstract number strategies to solve word problems, the standard itself tells students to model with objects and drawings. I would have missed that had I not dug deeper. Thanks, as always for your passion for math education. It’s so inspiring.

This is HUGE Meghan and thanks so much for sharing!

Far too often we look to move students away from models (objects and/or drawings) which is super scary. As students begin to develop their understanding, models allow them the opportunity to explore and anchor their understanding on a visual. I have to keep reminding myself that the turtle won the race. You’ve definitely identified a big piece I think we all forget or overlook.

Thanks!!!

I’ve watched this many times here and shared with many others to generate some great conversations. I just wrote about a standard here in Ontario that many of us read with different perspectives. Annie F reminded me about your call to action. Take a look:

Concept vs Procedure: An anecdote about what it means to be good at math https://buildingmathematicians.wordpress.com/2016/07/01/concept-vs-procedure-an-anecdote-about-what-it-means-to-be-good-at-math … …

The standard I picked last year was 2 way frequency tables for 8th grade statistics. I didn’t like the lesson in CPM after teaching it for 2 years as the kids couldn’t relate and get past all the structure of the representation.

I adapted a task prompt from Illustrative Mathematics after searching out the standard “CCSS.MATH.CONTENT.8.SP.A.4

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?”

The key vocabulary of the standard I see is relative frequencies. These are the inside, guts, or area of the square or rectangle that is the 2 way frequency table. The total on the right and bottom are called marginal frequencies, and allow for a denominator to be used in relative frequencies.

My lesson hook was my hypothesis: students who play a musical instrument are more likely to also play a sport. I conducted a survey with each period of students asking if they played a musical instrument and if they played a sport. I did so on a copy of the class roster with 4 spreadsheet columns with headings. I asked if we could tell. They said it was hard. Here is where the 2 way frequency table has a need to display the data.

Interestingly, the data could also be represented with a Venn diagram which is how CPM’s Core Connections Algebra 1 curriculum first introduces the idea. I wonder if it would be useful to include this as well.

My hypothesis was correct most of the time. I then empowered students to come up with their own hypothesis that could be answered using two questions with yes or no answers.

Anyway, the point is, I am writing a blog post that will hopefully have a scoring rubric for the task that will help students have all the parts necessary to answer their hypothesis using their data.

I am writing a blog post about this standard’s lesson and I’ll be sure to link you and your talk.

Oh and my original comment was totally unrelated: “.@gfletchy I thought of you when we taught surface area of cylinder in 8. Checked CCSM. volume only. Once again, can’t trust textbook..”

This was regarding a lesson in Core connections course 3 on construction gym bags that were rectangular prisms and cylinders to see which one used the least amount of cloth and had the most volume. After looking closely at the standards, colleagues and I saw there was no mention of surface area in the standard, so the time would be better spent on the suggested shapes in the standard as far as volume.

We have used the great task on the map.mathshell.org for 8th grade called Making Matchsticks which gives a need for volume of different types of 3 dimensional figures.

http://map.mathshell.org/lessons.php?unit=8300&collection=8

And of course trashketball by Mr Stadel is a once a year lesson for spheres.

http://www.101qs.com/2008