Cluster 1 has a small, tight group of participants in the middle with a cloud of mostly remote participants (RP) on the outside. There are a few blue lines radiating outward, which show attending participants (AP) tweeting with other APs. The red lines show one of three possible communication patterns, either AP to RP, RP to AP, or RP to RP. [I am not sure how to show all 4 on a graph. I tried, it looks so busy as to be almost unreadable.]

The thickness of the line is proportional to the quantity of tweets between the nodes, so not every node is a one and done communication. In fact, it would be wrong to assume that a single line indicates a single tweet, for at least two reasons. First, the software has a stepwise function for determining thickness of the line. This keeps the graph from being a solid blob of color. Second, it is common practice in tweeting to NOT include the hashtag in any replies. **This means that that any graphs or counts from the dataset should be understood as an undercount and underestimating the behavior.** This cannot be stressed often enough.

What was said in this cluster? What were the word pairs? Who was most mentioned?

Top Hashtags | Count | Top Words | Count | Top Word Pairs | Count |

tmc17 | 4336 | math | 401 | math,teachers | 40 |

mtbos | 428 | graceachen | 302 | morning,session | 33 |

pushsend | 128 | session | 192 | blog,post | 32 |

tmcequity | 96 | thanks | 182 | math,camp | 29 |

talklessam | 74 | more | 180 | jgough,sharing | 28 |

1tmcthing | 73 | jreulbach | 163 | sharing,sketchnote | 28 |

clotheslinemath | 60 | want | 158 | flex,session | 21 |

tmcplans | 56 | love | 156 | looking,forward | 20 |

descon17 | 51 | one | 146 | nice,things | 20 |

iteachmath | 50 | 1 | 142 | graceachen’s,keynote | 19 |

I was honestly puzzled by the “1” that was the 142nd top word, until I connected it with the “one” and the “1tmcthing”. Maybe the use of “1” as a standalone word had to do with 1TMCThing? I won’t know until I do the qualitative analysis, but that is a question I have already.

Already, the idea of equity is coming through strongly, with TMCEquity, Grace Chen showing up in all three columns. IN addition, other sessions also showed up, with TalkLessAM and ClotheslineMath. Gratitude is a surprise word, with “thanks” having a strong showing, along with “love”. The context of those words being used is unknown as of yet, but the fact they are there indicates something positive.

The people who were most replied to or mentioned in Cluster 1 were:

Top Replied To | Count | Top Mentioned | Count |

ryansethjones | 29 | graceachen | 310 |

gwaddellnvhs | 26 | jreulbach | 141 |

graceachen | 21 | mathprojects | 128 |

davidkbutleruoa | 21 | davidkbutleruoa | 118 |

jreulbach | 21 | anniekperkins | 111 |

rawrdimus | 20 | gfletchy | 104 |

misscalcul8 | 18 | carloliwitter | 102 |

cheesemonkeysf | 18 | cheesemonkeysf | 93 |

tracyzager | 17 | rawrdimus | 84 |

overzellis | 16 | gwaddellnvhs | 80 |

Comparing this list with the list above, it is clear that “pushsend” had a lot of traction, even if the participants didn’t connect it with CarlOlitwitter every time. In fact, it is clear from the counts that it is much more common to name someone than to reply to someone using the hashtag (see my bolded comment above). Also, that Grace Chen was mentioned 310 times, which is almost 3 times as much as the next closest person is telling. It suggests that “thanks” and “love” show up in the word count above because the three Keynote speakers also show up in the mentioned column, Grace Chen (graceachen), Graham Fletcher (gfletchy), and Carl Oliver (carlolitwitter).

It appears this cluster is composed of individuals who were having conversations about the keynotes and sessions, typically. Compare this with Cluster 2.

Top Hashtags | Count | Top Words | Count | Top Word Pairs | Count |

tmc17 | 1703 | math | 188 | sharing,sketchnote | 91 |

mtbos | 201 | jreulbach | 139 | jgough,sharing | 77 |

descon17 | 96 | sharing | 111 | want,use | 33 |

sketchnote | 93 | love | 110 | blog,post | 31 |

pushsend | 42 | session | 103 | use,incorrect | 31 |

coachtmc | 42 | desmos | 97 | incorrect,work | 31 |

tmcequity | 24 | descon17 | 96 | work,more | 31 |

mathandell | 24 | more | 94 | more,person | 31 |

iteachmath | 21 | sketchnote | 93 | person,confused | 31 |

tmcjealousycamp | 20 | jgough | 92 | confused,led | 31 |

Just looking at the counts, there is something different with the cluster. Looking at the top word pairs, in cluster 1 there was a small range compared with the range of counts in cluster 2. The counts of word pairs in cluster 2 was also higher, across the board. The smallest of the top 10 was 31, whereas the third largest in cluster 1 was 32. This may indicate there was more unanimity in the cluster. More people saying the same thing.

Top Replied-To | Count | Top Mentioned | Count |

samjshah | 13 | jreulbach | 128 |

stoodle | 10 | jgough | 104 |

gfletchy | 10 | desmos | 91 |

jreulbach | 9 | graceachen | 83 |

mrvaudrey | 7 | heather_kohn | 56 |

druinok | 7 | illustratemath | 53 |

jfinneyfrock | 7 | gfletchy | 51 |

mrschz | 6 | jstevens009 | 49 |

mrsjenisesexton | 5 | crstn85 | 49 |

dingleteach | 5 | mrsjenisesexton | 42 |

There was also a lot of math talk in this cluster, with Desmos and IllustrateMath being among the top mentioned. Grace is mentioned, but the other two keynotes are not. Jenise Sexton is is mentioned, and she had one of the top 10 links for the entire data set. This indicates that equity is playing a role in the cluster, which connects with the TMCEquity hashtag being in the top 10 as well. The surprise is the hashtag TMCJealousyCamp being used 20 times. That requires further analysis as to why that was used in cluster 2.

I am still exploring how to compare clusters with this quantitative data. If you have any suggestions, please let me know. If you have any questions, please ask!

]]>For example, who are the most ‘influential’ people in the data set? There are several ways to indicate influence, and one way is ‘betweenness centrality.’ This measure calculates the shortest (weighted) path between every pair of nodes in a connected graph. So, for my data set of 1319 unique nodes, it calculates the shortest path between each of those nodes. The node with the largest number of shortest paths, ends up with the largest value of betweenness centrality. A node with only 1 connection would have a value of 1.

So who are the most influential nodes of the data set? In order they are:

jreulbach | 215883.14 |

mfannie | 139677.23 |

heather_kohn | 86266.32 |

graceachen | 82643.39 |

gfletchy | 77538.27 |

mathhombre | 76284.53 |

approx_normal | 70797.31 |

davidkbutleruoa | 70564.82 |

jgough | 70180.90 |

marybourassa | 67656.08 |

It is interesting how much larger Julie Ruelbach’s betweenness centrality value is compared to everyone else. Annie Fetter has a solid middle number in between Julie’s and the rest of the bunch. If these were placed on a numberline, we would have a cluster of values for 10 – 8, a gap to Annie, and then another larger gap to Julie. What creates that huge value for Julie? She is in the center of group 1, and she pulled many people to her in tweeting and retweeting. She also had a very large number of self-tweets as well. Self-tweets are a ‘circle’ dyad, where no one replies, retweets, or likes her tweet, and includes no other person in her tweet. Julie had both the largest number of connections AND a large number of self-tweets.

NodeXL also gives the top 10 elements of the entire data set, and each of the top 10 clusters. In the data set, the top two shared links were the google form for archive submission (#1 at 44 times) and the single page with reminder links for the conference (#2 at 35 times). That is unsurprising. Numbers 3 and 4 are more interesting. Jenise Sexton’s blog post on “Being Black at TMC was shared 33 times, while Annie Perkin’s blog post on “The Mathematicians Project: Mathematicians Are Not Just White Dudes” was shared 27 times. This shows a strong element of equity attention by the community. The 5th most shared website (23 times) was CreativeCommons.org, which is surprising to me. Perhaps one of the morning sessions had a great discussion about creative commons licences?

Finally, the last 5 sites were both the first and last 20 minutes of Grace Chen’s keynote on equity (22 and 16 times respectively) [links not included on purpose], Julie Ruelbach’s post (21 times) on using Desmos for assessment, how-old.net (19 times) and a google document entitled “Learning about Contemplate then Calculate“.

What can be learned from this list of top 10? There was strong interest in equity. Of this top 10 list, four of them are directly tied to an equity discussion. Three of the top 10 are connected to sessions and mathematics, and the last three are connected to conference management somehow.

Notice he difference between the Top URL’s and top domains, however.

twitter.com | 976 |

google.com | 224 |

wordpress.com | 139 |

youtube.com | 120 |

blogspot.com | 70 |

blogspot.ca | 41 |

ispeakmath.org | 38 |

arbitrarilyclose.com | 35 |

pbworks.com | 33 |

desmos.com | 24 |

Twitter.com comes from sharing someone else’s tweet, and tweets were shared a lot. Five of the top 10 are blogging platforms, with three of them individual’s blogs, and pbworks.com is where the conference stores documents from the sessions. No one session sessions shared a pbworks link enough for the link to be in the top 10, but in total the sessions shared links enough to make it to the top 10 domains. Desmos is, well, just awesome. Still, only 24 times was a Desmos link shared with the hashtag TMC17.

When the top 10 hashtags are listed, the similar connections as above are seen.

tmc17 | 7783 |

mtbos | 861 |

pushsend | 195 |

tmcequity | 173 |

descon17 | 173 |

sketchnote | 142 |

iteachmath | 119 |

1tmcthing | 107 |

coachtmc | 100 |

talklessam | 99 |

Ignoring the top two as pro-forma hashtags, Carl Oliver’s “PushSend” was the top hashtag used at the conference. In fact, there is essentially one use for every single attendee. Similarly with TMCEquity and Descon. ITeachMath makes a showing during the time that TMC17 was occuring as well. That Descon was the day before the conference, with a smaller group of people shows its strength in the discussion. The Sketchnote hashtag showing up at number six is interesting, because if word pairs are examined, the top word pair has to do with sketchnotes.

sharing,sketchnote | 137 |

jgough,sharing | 123 |

math,teachers | 81 |

relationships,kids | 79 |

kids,think | 79 |

numbers,mfannie | 79 |

math,camp | 77 |

blog,post | 76 |

think,math | 75 |

math,relationships | 73 |

In fact, the top two word pairs are connected with Jill Gough’s sharing of the sketch notes created during conference sessions. In addition, the pairs “relationships, kids,” “kids,think,” “numbers,mfannie,” “think,math,” and “math,relationships” show a that a good deal of the conversations on Twitter had to do with math content and how to bring math learning to kids.

This is the entire data set, however. The different clusters may show differences in discussions (which hopefully they do, otherwise what is the point of different clusters?) That is another post, however.

]]>RQ1: What are the network behaviors of participants in TMathC in 2017?

RQ2: What is the human activity of conference participation of TMathC in 2017?

RQ3: How are the network behaviors and professional development activity of the participants interrelated in TMathC in 2017?

This blog post is a first attempt at answering the first question only. The second question is a qualitative question that will be answered by actually reading each of the tweets and blog posts from TMC17. The third question will be answered by putting the results of the first and second together.

The software I am using to do this analysis is NodeXL. NodeXL is created by the Social Media Research Foundation (https://www.smrfoundation.org/nodexl/) and is available in both free and pro versions. I am using the pro version to do the analysis you will find in this and later posts.

To collect the data for this analysis, I every day from 4 days before TMC17 through the end of August, I downloaded the Twitter activity using a search on the hashtag #TMC17. This only collected public tweets that contained the hashtag. That means that followup tweets between people may not be collected unless they used the hashtag in their replies. **This is typically not frequent behavior, so all of the analysis below should be understood as underestimating the actual patterns of communication.** I bolded this statement, because it cannot be stressed enough.

First off, is the pattern of communication different from a “typical” math teacher conference (whatever that may be?) The easy answer is yes. Working from the public list of TMC attendees (https://twitter.com/TmathC/lists/tmc17) the number of attendees was 189. The data set I downloaded and compiled has 1348 unique accounts in it. Comparing the two lists, 169 of the 189 listed show up among the 1348. This means that 89.4% of the participants at the conference had some kind of Twitter activity. That is a huge percentage of attendees in the attending participants (AP) in the data set. A little arithmetic shows the number of remote participants is 1159. This means the ratio of AP to RP is 0.1458, or 14.6%.

These 1348 nodes (they are not all people, some are accounts like NCTM) created the following network map.

The G1-G25 reference the Clusters the software segments the nodes into based upon their patterns of communication. So, for example, the G1 cluster has a tight group of nodes in the center, with a radiating pattern of nodes around it. The small central cluster communicated with each other frequently, and the radiating nodes had less communication. The words are the top words used by the nodes within each cluster. I set the software to ignore the TMC17, MTBoS, and ITeachMath hashtags. Otherwise they showed up in every cluster.

This graph tells me that there is a great deal of communication between the different clusters. In fact, almost every single cluster has strong communication ties with cluster 1, 2, 3, and 4. Cluster 8 has none, but given that TMCJealousyCamp shows up in that cluster it makes sense.

But which of these nodes are AP, and which are RP? When that question is asked, I realized there are actually four different communication patterns which must be explored.

AP to AP; AP to RP; RP to AP, and RP to RP.

It is very difficult to see all four one map. The tightness of the communication patterns means that the colors overlap, and wash each other out. What if, I only look for RP to RP communication in the data, and I hide all other communication? Is it reasonable to think that remote participants talk to each other about TMC17?

Surprisingly, the answer is YES! In fact, in some clusters (notice I changed the default “G1” notation to “C1” to align with the vocabulary I am using) there is a tremendous amount of RP to RP communication. In addition, each of the clusters has RP to RP communication. C9 has strong grouping of communication, which aligns with the fact that is a cluster of GlobalMathDepartment communications. I also limited this graph to only the top 10 clusters, and added the count of the number of unique nodes into the labels. This helps give further context to the graph (I hope).

Do the RPs engage with the APs?

Again, yes. The RPs engage with not just other RPs, but in large amount with the APs. This graph shows RP to RP and RP to AP directed activity. The graph shows that the RPs engage with the APs more than with RPs, which is to be expected. But do the APs engage with the RPs?

Adding in the AP to RP directed activity creates more edges where there already existed edges, resulting in more lines between the participants, and a darker red. The RP AP communication definitely worked both ways. When the AP to AP communication is added back in the result is:

This graph shows in blue the AP to AP communication along with any RP communication in red. At this point, it is clear that TMC17 was not just a conference for the people in attendance, but it was a conference for a much larger number of people who were participating remotely. The numbers also show this:

Total number of APAP dyads: 7592

Total number of APRP dyads: 1620

Total number of RPRP dyads: 1559

Total number of RPAP dyads: 2778

APAP/(all others) = 7592/5957 = 1.274

A dyad is a two part communication pattern. For example if I were to tweet (not a real tweet, mind you):

Hey @TMathC, you should check out the blog post on the initial analysis of TMC17! @cheesemonkeySF, @druinok, @lmhenry, you should too!

This tweet creates the following four dyads:

gwaddellnvhs to TMathC

gwaddellnvhs to cheesemonkeysf

gwaddellnvhs to druinok

gwaddellnvhs to lmhenry

Should one of those individuals reply, it would create an additional four dyads. This is how the directionality of the tweets is maintained, and how I managed to show the directionality in the graphs above. It also means that while the largest number of dyads is between AP and AP, the ratio of APAP dyads to any dyad which contained an RP is 1.3. This Remote Conference Participation Ratio (I think I just created a new metric) is interesting. A ratio of 1 would mean that there is equal participation between attending and remote attendees. It isn’t a ‘clean’ ratio, because AP shows up in RPAP and APRP categories, but it does suggest a level of RP participation.

That is enough for today. Hopefully this gives the mtbos and iteachmath community something to challenge and think about.

Please, give me feedback. If there are questions here I have not addressed, ask. I welcome the pushback and opportunity to answer your questions. After all, that will just make my dissertation better.

]]>I had an alarm set to go off 5 minutes before class ended, every single day. When the alarm went off, the learners in my Knowing and Learning class knew there were 5 minutes left, and they reached for their devices. It worked so well.

The learners were honest with the questions as well. I had some learners say, “Agree” to the fact that they were not challenged in class. They told me why. I had an agree with “treated kindly by my peers” one day when a learner was sarcastic and mean.

The real payoff was in the “Answer the question on the board.” I had a day by day reflection on what the class learning was for them. I will do this again, definitely. I encourage others to as well. It worked.

I have some other things to say about things I tried this semester. I created an asset/ deficit thinking exercise that I want to share. It was a good semester, but crazy busy. I was in a writing rut, and was not making progress on my dissertation, and the entire thought of writing anything outside of that scared me to death.

Good news is, I passed the proposal by the end of the semester, and am furiously doing my data analysis. I will have much, much more to say about this, because I am going to use my blog as a rough draft writer. Stay tuned!

]]>I could never really figure out how to close the lesson. No, scratch that. I knew how, I never really could REMEMBER to do a good closure! With that in mind, I have done a couple of things this semester to help me.

- I have set a repeating alarm on my cellphone. It goes off 5 minutes before class ends. I have selected kind of a soothing, interesting music to sound, as well as vibrate. This is my reminder to stop, and DO the closure.
- To make the closure process easy, efficient, and solid, so that I WILL do it every day, I have spent some time this summer collecting ideas for solid closures. That link will take you to a google doc where you will see many of the ideas and questions I have collected.
**[Please feel free to add more!]**One thing that is going to make it easier, is my use of google forms to do the collection. (thank you Mari Venturino!) I think her form is amazing, and I have adapted it for my own use. This takes my 50 minute class down to 45, but I think it will be worth it. - I really like the 6 questions she has embedded in the middle of her form.

I have challenged my apprentice teacher to ask question 2. I think he will be surprised by the answer. I really wonder why we don’t ask that question every single day, every single class! I think we are afraid of the answers.

So, my first goal this semester. Every single class, an exit ticket of some form. Non-negotiable. I must close the class by modeling exit tickets, and USE that information in the next class planning.

I am excited to begin the new semester! It is going to be a rough one. 5 different classes to teach, AND writing a dissertation. No big deal. I got this!

]]>But, they are the 800 lb gorilla of PBL. There are not a lot of resources for teaching mathematics through PBL. Heck, we can’t even agree what PBL means in math. I have seen the acronym used for Project Based Learning (this is how BIE uses it). I have seen it used for Problem Based Learning. I am helping teach a class for preservice teachers this semester called PBI, which is short for Project Based Instruction.

Way too many acronyms here. So in the interest of keeping things straight and organized, I will consistently use the following acronyms:

PBL: Project Based Learning

prBL: Problem Based Learning

PBI: Project Based Instruction

prBI: Problem Based Instruction

I think each of these has merit, and a use.

That is the goal of this post. To tease out the differences, and justify the need for both.

First off. I am not going to focus on the difference between Learning and Instruction. I don’t think it is that important. The difference is the difference between whether we are talking about the teachers (Instruction) or the Learners (um, learning :)).

So the issue is the difference between Project and Problem. Where does that fall? What does it mean for the classroom?

Projects, according to BIE has 8 pieces.

- Key knowledge, understanding, and success skills
- Challenging problem or question
- Sustained inquiry
- Authenticity
- Learner voice and choice
- Reflection
- Critique and revision
- Public product

An amazing new book by NCTM called Rigor, Relevance, and Relationships: Making Mathematics Come Alive with Project Based Learning mirrors these elements.

But, there needs to be some explanation. When they say “Sustained Inquiry” in the third element, and “Learner Voice and Choice” in the 5th, they are not talking a free for all. If we put the inquiry into the context Trevor MacKenzie uses in his inquiry books (see the image below).

No, we are not talking about Free Inquiry when doing either PBL or prBL. We ARE talking about Guided Inquiry. This was never made explicit at PBLWorld, but they hinted at it strongly. In fact, it must be. The whole point of a teacher coming up with a guiding question (second point), and making sure that standards are being hit (first point) means we cannot be doing Free Inquiry.

And the standards to be addressed are not just the CCSSM or CCSSELA standards. The standards are also the 21st Century Practices standards. Some districts are requiring teachers to show how they are incorporating these standards into your teaching. When doing PBL, it is simple. They are put into the project right from the start, and clearly practiced at the end in step 8: Public Product.

Problem Based Teaching or prBL is very similar. If you want to read an expert on the topic, Carmel Schettino has a wonderful series of posts on the NCTM’s Blog. There are 4:

- Aspects of Problem Based Teaching
- Aspects of Problem Based Teaching – The need for community
- Aspects of Problem Based Teaching – Assessment
- Aspects of Problem Based Teaching – Student Conversations

Carmel also runs a conference in the summer of prBL, called the PBL Math Summit, and she has many resources available on her blog.

If you examine what she writes, she has essentially the following steps:

- Key knowledge, understanding, and success skills
- Challenging problem or question
- Sustained inquiry
- Authenticity
- Learner voice and choice
- Reflection
- Critique and revision

I think Carmel may actually merge a couple of these. I think it is silly to have step 3 “Sustained Inquiry” separate from “Learner Voice and Choice” but that is a Buck Institute thing. I left it, because Carmel talks about inquiry and learner choice as well. (Carmel can correct me if I am mistaken in that, I don’t want to put incorrect words in her mouth.)

The big differences are the public product, and the nature of the ‘guiding question’. The guiding question (step 2) for PBL should be open ended and allow for a great variety of learner choice. That isn’t to say that prBL questions can’t, but types of Open Middle questions or even closed ended, but interesting questions avaiable to prBL would be excluded in PBL.

The biggest difference is the Public Product. In prBL, there is not a demand for a public product as a concluding phase to the prBL. This means that many more prBLs can be done in the course of a school year than a PBL. It also means that prBLs can be done as practice elements for a full PBL. Imagine doing a couple of interesting prBLs and then opening up the question more on the next one and turning it into a PBL.

There are other people who work on the topic of PBL. John Spencer is an author of materials on PBL. I definitely get the feeling he is very ELA based. He writes in that link how he failed at a math PBL, but writes at length about how to incorporate ELA standards into his classroom.

If you are looking for math, specifically, there are people to follow and engage with.

Carmel Schettino for starters. If you are interested in expanding your horizons on this, follow her. Talk to her. She is amazing. Her publications include the following:

Schettino, C. (2003).

Transition to a Problem-Solving Curriculum. Mathematics Teacher, 96(8), 534-537.Schettino, C. (2011).

Teaching Geometry through Problem-Based Learning. Mathematics Teacher, 105(5), 346-351.Schettino, C. c. (2016).

A Framework for Problem-Based Learning: Teaching Mathematics with a Relational Problem-Based Pedagogy. Interdisciplinary Journal Of Problem-Based Learning, 10(2), 42-67.

Another person is Tia Vandermeer. She works in Canada, and does PBL in mathematics.

Also, follow the hashtag #PBLChat. Many, many teachers using PBL in their classroom (and some of them are doing prBL too!)

Get the book mentioned above. It is FABULOUS! If you want the link again, here it is: Rigor, Relevance, and Relationships: Making Mathematics Come Alive with Project Based Learning. Buy it. I promise the full retail of $36 is a much, much greater value than the $1100 of PBLWorld.

]]>Instead, I want to draw your attention to this image. You can click to embiggen it.

Source: https://arxiv.org/pdf/1806.03163.pdf

The beauty of the proof is immediate and powerful. No mathematics beyond the image is necessary to understand the idea, but it helps. The 1 page paper also has the following math to support the proof.

That is it. The author gives only those two lines. Which really, are three lines, because the author abuses the equal sign notation in line 1. I spent some time deconstructing this, and came up with the following, more easily followed, version of the proof.

Brilliant. It is just beautiful in its presentation. Such an amazingly approachable proof. But notice that red square. There is some magic going on in that square.

Could this proof be done with a function f(x)=-x? Or how about f(x)=(x=pi)^2? The visual element appears to yield the same information.

The area of the rectangle is greater than the area under the curve. But does this yield the same proof? I worked it out, and I ended up with some cool math, but nothing which approached e^pi or pi^e. Visually good, mathematically on the wrong track.

So what is going on in the red box in the hand drawn proof above? What is so different?

The difference is that in the original paper, the author recognizes that out of infinite number of functions that are decreasing from the interval { e, pi } the one function that will yield the area under the curve as log(x) + constant is f(x)=1/x. Recognizing that and realizing that one must work in base e, and from there we can create the necessary exponential.

That is very creative.

Amazingly creative.

I have been enamored with this problem ever since Taylor Belcher posted it. It is obvious at one level, and so not obvious at other levels. The graph and areas are obvious, but it is not clear how that translates to the idea that pi^e < e^pi. That leap, that jump from any decreasing function, to a specific decreasing function which creates the necessary area is the creative leap that can be difficult.

I see that with the college learners as the move from the calc series to the proofs based courses like groups, fields, and rings. The approach to the proofs classes must be different because instead of procedures, the learners must push creativity into their work. They must think of the options they have for an approach and pick and choose with intentionality.

The balancing of the goal with the multiple paths which can approach the goal, and choosing made me think of Bob Ross and his painting show on PBS. “I need a happy tree here, so I will use this brush and …” He talked about what outcome you wanted, and decisions that had to be made along the way. He spoke of the joy of selecting color, making mistakes, fixing mistakes, and not becoming trapped in decision, but pushing forward to obtain the outcome you wanted.

That is the beauty and joy of math. The art that can be found in math.

I really need to stop procrastinating, but I found so much joy in this proof that I wanted to share it.

]]>Here it is in it’s entirety:

Engaging the Mathematics Community; Miscellaneous Assignment 2: 10 points

The goal of this assignment is to encourage you to engage with the existing larger mathematics education community. You must do one Global Math Department video presentation and one Twitter Chat.

To get full credit, you must provide a transcript of your interactions (copy and paste) to verify the Twitter chat. For the Global Math Department, if you participate in a live chat, copy and paste the transcript. If you watch a video of a past presentation, reflect on the material. For the reflections, you should answer the following questions at a minimum. What did you learn from the activity? Who was the person or persons who had the largest interaction with you? Describe them based on your interaction. What is the value, in your opinion, of the activity? How would you use this in classroom practice?

Global Math Department:http://globalmathdepartment.org/ These weekly video chats occur weekly on Tuesday evenings, 6:00 pm Pacific time. They are an hour long, on a variety of topics relevant to mathematics K-12 educators. For example, recent presentations have been on the topics of ‘Improving student discourse through debate in the math classroom’ or ‘Interactive notebooks: Tapping into left & right brain thinking.’ Go to the link and hit the big blue “Conference Info” button for more information. All videos are archived at the end of the session.

Twitter Chats:https://twitter.com or even better, https://tweetdeck.twitter.com Twitter chats are hashtags used to collect people who have a similar interest. There are ‘live’ chats at certain times (usually an hour, sometimes 30 minutes) and there are ‘slow chats’ where individuals post a question using the hashtag and others reply when they can. Every hashtag is used throughout the week in the ‘slow chat’ model even if there is a ‘live chat’ time. When engaging in a chat of either type, always introduce yourself, and use the hashtags in every tweet.

- #alg1chat: “slow chat” style, no set time or evening
- #alg2chat: “slow chat” style, no set time or evening
- #geomchat: “slow chat” style, no set time or evening
- #msmathchat: Monday evenings, 6:00 pm Pacific time
- #mathchat: “slow chat” style, no set time or evening
- #statschat: “slow chat” style, no set time or evening
- #teachNVchat: Thursday evenings at 7:30pm Pacific time
- #swdmathchat every other Thurs 6pm Pacific
- #elemmathchat every week, 6pm Pacific
- #makeitreal every Wednesday, 6:30 pm Pacific
These last two hashtags are not chats, but are general hashtags used to include the larger community of mathematics teachers. Make it a practice to include both of these tags in each tweet. #mtbos #iteachmath

My college students laughed at me. They said, “I don’t use Twitter, it is just for celebrities and stupid politics.”

Ha! I had the last laugh.

Three brave souls said, “Okay, Waddell, I will try it.” I retweeted their first post (because they had zero followers), and the Math Community stepped up! Within 1 minute, all three had responses. And then more responses. And they had a discussion. And within 5 minutes they were so excited that other math teachers were actually sharing and talking with them.

They were so loud, that within 5 minutes, three additional PSTs joined Twitter and tweeted. The same thing happened. One PST said, “I can’t wait to be out in the classroom, get stuck, and realize I have a whole community of people to help me out. This is awesome.”

I was proud. I am proud.

This is a heartfelt Thank You! Thank you to all who helped my learners see there is a larger community of math teachers who want to help. Thank you to all those that didn’t help last night, but help others every day, once a week, once a month, or whenever they can. Thank you to all the teachers who are lurking, and thinking they are gaining something from reading others posts. Thank you to all those who will be joining the community in the future.

We all grow when we learn from others.

Thank you.

(edited to add additional chats, 1 July)

]]>I made something! I haven’t been very ‘makey’ lately. Focusing a lot on teaching college students, observing college students in the classroom, and then observing more. But, we got a new classroom on campus at the beginning of the semester, and we needed decorations. I looked all over for some good, high quality posters of the Standards for Mathematical Practice, as well as the Mathematical Teaching Practices (from the Principles to Actions) but couldn’t find anything .

Seriously, nothing. I looked on publishers sites and couldn’t find any good looking posters that matched.

So, I made my own. Much thanks to Meg Craig who helped me get out of my own head and improve the design. I used the golden ratio in the design elements a couple of different ways. Below I attached files that are standard sized (good for printing on regular paper) and a high resolution version for poster printing. I am having them printed 2ft wide by 3ft tall to hang in the classroom.

I got permission from Trevor Makenzie for his “types of inquiry” graphic. I think it adds something to the poster to have that additional information, especially since it relates strongly to MTP 4 and 5. The other images are public domain, no attribution necessary, or taken by me personally. The diagram of the types of representation is Figure 9 out of the Principles to Actions, and the book is cited at the bottom.

As I was making the two, however, I realized that I needed a third. My learners were asking for questions to ask and how to develop better questioning strategies. I had just read Cathery Yeh’s book and she listed a whole series of questioning strategies. So, I made third poster.

Not all of these are Cathery’s. I modified them, deleted some that wouldn’t work well for the secondary school arena, and added some from Steven Leinwand’s presentation at the Breakfast and Math we held in Reno a couple of weeks ago.

All in all, I think they turned out well, they represent what we are trying to do well, and they will be something that can be printed off and given to future math teachers as they graduate.

Let me know what you think!

Math Posters for classroom (standard sized for viewing or printing on 8.5×11 paper)

Math Posters for classroom – high resolution (larger file size for printing poster sized)

Update: Half Tau Day 2018

Here are pics of them in the classroom.

Right wall (facing students) Left wall

]]>One take away (and I had more than one) was from Samuel Otten, (twitter: @ottensam) host of the MathEd Podcast and Zandra de Araujo, both are assistant professors of math education at the University of Missouri. The team gave a fabulous presentation on why we need to nudge teachers, and not expect radical transformation of teachers. They are right on the money, in my opinion. Radical transformation is hard, but incremental steps are easy. Think about changing my diet. If I say, “tomorrow I will become a vegan” that will take huge amounts of energy and effort. However, if I say, “tomorrow I will start eating an apple every day” that is easy, and a great first step. Day 2 I swap out regular milk for soy milk, day 3 I swap out breakfast, etc. Small, incremental steps are easy to adopt and lead to the same results.

Think about Steven Leinwand’s 10% rule. No teacher should be expected to change more than 10% of their practices in a year. However, every teacher should be expected to change 10% of their practices in a year! Small, incremental changes.

During their presentation, they put up this image.

Don’t get hung up on the percentiles or quality of instruction axis. There is no scientific basis for this graph, just a gut check, a collective grasp of what do we think is the distribution of teachers at each level of quality. But, we believe there is a distribution of teachers who are at the low end of quality of instruction, and some at the high end, and some in the middle.

And we have been talking about radical change in the quality of instruction since 1989 when NCTM published the Curriculum and Evaluation Standards. That is 30 years of conversation about changing math instruction.

30 years.

At that point, we really have to admit, that the real leaders of mathematics education are those teachers who are on the low end of the Quality of Instruction scale, whether they are 25% or 75% of the teacher population.

That group has been driving and leading the discussion on Quality of Instruction for 30 years.

We don’t discuss the gains we have made in changing mathematics instruction, we constantly talk about and research the lack of gains. They are the real leaders.

That is really sad!

They also showed what happens when we stop talking about radical change, which only moves the right side of the graph higher, and does little to move the left.

Moving the left side of the curve requires nudging teachers. Introducing small changes, that build over time. If we really want to change the shape of the curve, we need to stop expecting radical change, which only a small number of teachers adopt. We need to gently add to the skills and abilities of all teachers.

This was a powerful presentation. I only cherry picked 2 slides out of the entire thing. He gave a much stronger argument, using a variety of approaches.

Thank you Samuel and Zandra!

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