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Take It Away – CGI National Conference 2017

At the end of June, I attended (and presented at!) my first CGI National Conference. I also visited the Pacific Northwest for the first time in my life. Seattle was beautiful and the learning was great. I know there are folks out there who aren’t able to attend many conferences, so hearing from attendees is one way they learn from afar. So, in case you weren’t there, let me tell you what resonated with me from the conference.

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One thing I especially liked about the conference was the essential questions. Speakers weren’t required to connect with them directly. Rather they were designed for participants to personally consider and reconsider as they attended keynotes and sessions:

  1. In what ways are your students allowed to bring “their whole selves” to the learning of mathematics in your classroom and school?
  2. What do you know about the cultural and lived experiences of the students in your mathematics classroom? (How can you broaden your knowledge?)
  3. How does your mathematics classroom interrupt and/or reinforce narratives of who is and who is not capable mathematically? (How could your classroom become more interruptive vs. reinforcing of these narratives?)

Not what you’d normally expect at a math conference, right? The focus on culturally responsive pedagogy was a breath of fresh air.

I also appreciated the emphasis on making connections Рboth in person and virtually.  A special thanks to Tracy Zager for giving folks a nudge as well as support. There were quite a few #MTBoS members in the audience, and I hope by the end of the conference that number increased.

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The Opening Keynote was a panel discussion called “Talking Math With Kids.” The panel included Christopher Danielson who blogs at the aptly named talkingmathwithkids.com; Allison Hintz and Tony Smith from the University of Washington; and Megan Franke, Angela Turrou, and Nick Johnson from UCLA. They told stories of their experiences working with young children around mathematics. The (extremely important) theme of their talk is that young children have mathematical ideas. We should listen to, value, and encourage them.

Then we moved into our first of six sessions. I happened to present during the first session. It was a little stressful, especially since the projector was not cooperating at first, but I was happy to get it out of the way right up front. ūüôā My talk was called “Numberless Word Problems in the Elementary Grades.”

In the talk we solved a numberless word problem together to create a shared experience. Then I shared the story of Jessica Cheyney using numberless word problems in her classroom to help students connect the act of separating to the concept of subtraction. Next I shared the story of Casey Koester, an instructional coach who used intentional planning and numberless word problems to help 2nd grade students make better sense of word problems. I closed by sharing resources teachers can use to implement numberless word problems in their classrooms.

Since we started in the afternoon, the opening keynote and session #1 were all we did on day 1. Day 2 opened with another keynote called “Equal Math Partners: Families, Communities, and Schools.” The keynote included Erin Turner, Julie Aguirre, and Corey Drake from the TEACH Math Project; and Carolee Hurtado from the UCLA Parent Project.

I loved this keynote! We often talk about what teachers and students are doing in schools and gloss over or ignore the role parents can and should take in their children’s mathematical development. We also ignore the role that students’ family, community, and culture play in their learning of mathematics. The two projects shared in this keynote were inspiring to listen to and so important for us to hear.

The first story was about the UCLA Parent Project, a multi-year project that invites parents in to become partners in their children’s math learning. It also builds up the parents into leaders.

The second project was the TEACH Math Project. Pre-service teachers were required ¬†to take a community walk to interview people and learn more about the community in which their students lived. We often ask teachers to create tasks and problems based around student interests, but this often leads to generic problems around what we assume the students’ interests are. In this project the pre-service teachers had to get to know their students, their lives, and their interests for real. Then they had to use what they learned to create relevant tasks and problems. I loved it.

After the keynote we attend session #2. I went to Megan Franke’s “No More Mastery: Leveraging Partial Understanding.” This resonated so much with me because it matches my current thinking about how we should be analyzing and interpreting student work.

According to Megan Franke, mastery learning “breaks subject matter and learning content into clearly specified objectives which are pursued until they are achieved. Learners work through each block of content in a series of sequential steps.” The trouble with mastery learning, however, is that actual learning isn’t that clean. Further, it sorts students into two groups – those who’ve got it and those who don’t – which contributes to inequality.

A partial understanding approach, on the other hand, looks at understanding as something we can have varying amounts of. What’s important is finding out what students’ current understanding and capabilities are and build from there. Megan shared an example of a preschool counting task where students had to count 31 pennies. According to the mastery approach – they either counted to 31 correctly or they didn’t – only 2.5% of the students demonstrated mastery of counting. However, when they scored the students on a range of numeracy criteria – knowledge of the counting sequence, 1-to-1 correspondence, cardinality, counting the whole collection, and organization – the picture changed completely. Only 12% of the students demonstrated little to no number knowledge while 64% of them demonstrated understanding of multiple criteria.

For session #3 I got to attend Christopher Danielson’s “The Power of Multiple Right Answers: Ambiguity in Math Class.”

I especially love the power of the phrase, “Well, it depends…” and hope to help teachers in my district see the power in crafting questions and tasks that lend themselves to some ambiguity. I also love this thought by Allison Hintz retweeted by Christine Newell:

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If you haven’t seen Christine Newell’s Ignite Talk from NCSM 2017, “Precision Over Perfection,” check it out because it touches on this very idea.

During session #4 I went to lunch, and I’m going to skip talking about session #5 because it didn’t really resonate or push my thinking very much.

Session #6 was fantastic though! I saw Jennifer Kolb and Jennifer Lawyer’s talk “The Importance of Counting in Grades 4 & 5 to Support Complex Ideas in Mathematics.” I noticed that counting in general and counting collections specifically appeared across the conference program. I have made the counting collections routine a mainstay in my primary grade curriculum materials. I was especially intrigued to hear stories of how intermediate grade teachers are using the routine. The two Jennifers did not disappoint!

In the example above, counting groups and then groups of groups helped nudge these 5th grade students into an understanding of the Associative Property of Multiplication.

This same idea of “groups of groups” led students to explore groups of 10 in a way that led to deeper understandings of place value and helped introduce exponents:

Counting is a skill we naively think students “master” in the early grades, but taking a partial understanding perspective, we can open up the concept to see that there’s so much more to learn from counting in later elementary grades and beyond!

On day 3 of the conference we opened with another enlightening keynote “Anticipatory Thinking: Supporting Students’ Understanding of How Subtraction Works.” This keynote was led by Linda Levi from the Teachers Development Group and Virginia Bastable from Mount Holyoke College.

Linda Levi’s portion of the talk reflected on the meaning of computational fluency. She reminded us that while many people think of fluent as being fast, the definition is much broader and more nuanced than that.

“Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands.” (Principles and Standards for School Mathematics, 2000, p. 152)

We started with a video example of a student solving 5,000 – 4,998 using the standard algorithm. Is this an example of computational fluency? According to the above definition, no, it’s not. Producing an accurate answer like a calculator is not the same as demonstrating computational fluency. In this example the student did not demonstrate flexibility in the methods he chose, he didn’t understand and couldn’t explain his method, and his method is not based on mathematical ideas that the student understands.

We then watched videos of two other students who used subtraction strategies they invented. Were these students demonstrating computational fluency? The students clearly understood their strategies and they were based on mathematical ideas the students understood. However, we then watched these same students solve another problem and realized that these students were not flexible in their thinking. They used the same strategies for subtracting even though other strategies would have been more efficient for the new problem. It’s really important to remember how multi-faceted computational fluency is and attend to all facets as we work with students.

One of Linda Levi’s main messages was that understanding how operations work is the foundation for computational fluency. She shared with us how we can use equations that represent students’ strategies as objects of reflection for discussing why a strategy works and to help make explicit important mathematical ideas.

Virginia Bastable followed up with a talk about mathematical argument which was along the same theme of helping students understand how the operations work.

One thing that resonated with me from her talk was the important work of opening up mathematics learning beyond the narrow focus of answer getting. Rather, mathematics is a landscape that also involves sense making, exploring, wondering, and even arguing.

After the keynote I attended Kendra Lomax’s session “Learning from Children’s Thinking: A CGI Approach to Formative Assessment.” This session dovetailed nicely with Megan Franke’s session on partial understandings because the whole point of the CGI assessment is to get a sense of where the child is at in a variety of ways rather than a binary “yes, they have it” or “no, they don’t.”

If you’re interested in this assessment approach, then I have good news for you! A slew of assessment resources are available at Kendra’s website, Learning From Children. Look at the resources under “Listening to Children’s Thinking” in the menu at the top of the page.

For my final two sessions I went to hear more from Linda Levi and Virginia Bastable. Linda’s talk “Understanding is Essential in Developing Computational Fluency” gave us practice recording student strategies using equations as a way to make explicit the properties and big ideas embedded within the strategies.

Virginia’s talk “Support Math Reasoning by Linking Arithmetic to Algebra” dove more deeply into the role mathematical argument can play in helping students develop a deeper understanding of the operations. When I think back to the skill-based worksheets of my youth, I’m jealous of the deep thinking elementary students are given the opportunity to do in classrooms today.

We came back together for a closing session and that was the end of the conference. Spending three days with like-minded educators who care so deeply about mathematics education and nurturing children’s mathematical ideas helped recharge my batteries before coming back to work for the 2017-18 school year. It will be another two years before the next CGI conference – this time in Minneapolis – and I can’t wait to attend!

Inspiration – Summer Edition

As you may or may not know, I have a tendency to roam¬†the seasonal aisle at Target, looking for mathematical inspiration. So far I’ve shared photos I’ve taken at Halloween, Valentine’s Day, and Easter. You can find them all here.

Today I was stopping by Target for some bug spray which just so happens to be next to the summer seasonal aisle. I couldn’t resist the urge to take a stroll and take some pictures. Here’s what I’ve got for you today.

How many large wooden dice are in the package?

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It’s totally obvious, right? For younger students, maybe not so much. But even after everyone is in agreement that it’s 6, what do you think they’re going to say once you reveal the answer?

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Not what you were expecting, is it? You probably thought I was wasting your time starting with such a simple image. So now you get to wonder, “Why/How are there only 5 dice in this package?” Perhaps this will help:

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That burlap bag has to fit somewhere!

Let’s move on to another large wooden product. How many dominoes are in this pack?

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It might be a little hard to tell from this perspective. Let’s look at it another way.

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Barring any more burlap sacks, you might just have the answer. Before we find out, stop and think, what answers are reasonable? What answers are not reasonable?

Ok, time to check if you’re right.

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No surprises here. Although after the first image, I probably had you second guessing yourself. There’s something to be said about the importance¬†of how we sequence tasks.

Speaking of sequencing tasks, let’s move on to another one. How many light bulbs on this string of lights?

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I really like this box because you get this tiny 2 by 3 window, and yet it’s such a perfect amount to be able to figure out the rest. This would be one I’d love to give students a copy of the picture and let them try to show their thinking by pointing or drawing circles on it.

Again, this is a great time to ask, what answers are reasonable? What answers are not reasonable? Assuming the light bulbs do create a rectangular array, there are definitely some answers that are more reasonable than others.

After some fun discussion about arrays, it’s time to check the actual amount.

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So fun! Like I said, I love this image. Let’s look at another package that caught my eye.

How many pieces of sidewalk chalk in this box?

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I was pleasantly surprised to find that Crayola put arrays on top of all their summer art supplies.¬†It’s like they were designed to inspire mathematical conversation! Granted, the box doesn’t give it away that the dots represent the pieces of chalk, I wouldn’t point it out to students. I’d let them wonder and make assumptions about it. It’ll turn out that their assumptions are completely right, and how satisfying that will be for them!

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Since we’re talking about arrays, which means we’re talking about multiplication, let’s shift gears a bit to look at some equal groups.

How many plastic chairs in this stack?

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And to throw a wrench into what looks to be a simple counting exercise, how much would it cost to buy the whole stack?

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Now students have got some interesting choices about how they calculate the cost. The fact that half the stack is blue and half the stack is red is just icing on the mathematical discussion cake.

My final image from the summer seasonal aisle has been a real head scratcher for me.

How many water balloons do you estimate are in this package?

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What is an estimate that is too low?

What is an estimate that is too high?

What is your estimate? How did you come up with that?

Take a look at the box from another angle, and see if you want to revise your estimate at all.

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We clearly have groups – eight of them to be precise – but the question I’m not entirely sure about is whether there are eight¬†equal groups. Maybe? And if there are equal groups, then there are certain answers that are more reasonable than others.

So how do you wrap your head around this?

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I’ll give you a moment to think about why this is confusing me a bit.

Assuming there is an equal amount of each color, this doesn’t make any sense! But then I noticed the small white tag on the set of purple balloons.

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Oh! That explains it. There’re only 260 balloons in here so…no, that still doesn’t work if these are eight equal groups.

Oh, then maybe it’s 5 more than 265 so it’s actually 270 so…no, that doesn’t work either. So I’m left to conclude that either this is not a pack with eight equal groups or there is some funny math going on! Sadly, $25 is a bit steep to satisfy my curiosity. If any of you purchase this pack and want to count balloons, I’d love to get the full story.

And with that, my tour of the summer seasonal aisle comes to an end. If you’re just finishing the school year, bookmark this post to revisit when school gets back in session. What a fun way to start the year! If you’re still going strong, then I hope you’re able to use these to spark some fun, mathematical discussions in your classrooms.

 

 

My First Ignite Talk

I’m a fan of Ignite talks. They’re short – which makes sharing them and using them in PD very easy – and they’re to the point – which means a focused message to get others thinking and talking.

Here’s the catch. To ensure each talk is short and to the point there’s a format each one follows:

  • Each talk is 5 minutes long.
  • You have exactly 20 slides.
  • The slides auto-advance every 15 seconds.

Sounds stressful, right? And up until a couple of months ago I was always happy someone else was in the hot seat. That is until I got an email from Suzanne Alejandre from The Math Forum inviting me to be one of 10 speakers at the NCSM 2017 Ignite event in San Antonio in April. My emotions upon reading her email were a mix of feeling honored and terrified.

The event has come and gone, so now I get to look back on it with relief and a sense of accomplishment that I took the challenge and saw it through. I’m proud of the final result. I chose to talk about doing less of 3 key things in our math classrooms, which sounds counter-intuitive, but it actually results in getting more of a few things we want. Check it out:

My Other Blog

As you may know, I’m the district curriculum coordinator for elementary mathematics in Round Rock ISD, a school district just north of Austin, Texas.¬†I recently created an RRISD Elementary Mathematics Department blog to share posts on specific topics with teachers in my district. If you like what I write here, you might want to check out the posts over there as well. It’s only recently started, so you’ve only missed three posts so far:

You might notice a theme to all three posts. Maybe.

Here’s links to the three posts:

I’ve been on a bit of a geometry kick lately. In my work, I’ve focused a lot on computation generally and number talks specifically the past couple of years. Geometry is an area I haven’t explored much recently so I’ve been making a more focused effort to do so.

The blog also has a great resources page with lots of stuff for teachers and parents. Take a peek at that if you go by for a visit.

I’ll still be writing here. I just wanted to make sure people knew I was writing¬†there, too.

Mathematically Correct

Last night I posted a short survey on Twitter. I asked participants to analyze the following question.

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Then I asked them to answer the question. I purposefully set up the survey so participants could select more than one answer, though I gave no encouragement to do this. All the question said was, “Your answer.” Finally, I left a section for comments.

When I tweeted out the survey, I didn’t provide any details about the problem or my intent.¬†I was trying really hard to see if other people saw the same thing I did without leading them to it.

Turns out many people did. I feel validated.

So here’s the story.¬†This is question #28 from the 2016 Grade 3 Texas state math test, called STAAR, that students took last spring. Back in February, one of our district interventionists emailed me to say that he thought both choice G and J are correct answers. I opened up the test, analyzed¬†the question, and realized he was right. I immediately drafted an email to the Texas Education Agency to ask about it.

Good morning,

I have a question about item 28 on the grade 3 STAAR from spring 2016.
The correct answer is listed as J. This makes sense because the number line directly models a starting amount of 25 people and then taking some away to end at 13, the number of people still in the library.
However, the question isn’t asking for the model that most closely represents the story. Rather, it asks which model can be used to determine the number of people who left the library.
In that case, answer choice G is also correct. Our students understand that addition and subtraction are inverse operations. Rather than thinking about this as 25 – __ = 13, answer choice G represents it as 13 + __ = 25, which is a completely valid way of determining the number of people still in the library.
I look forward to hearing TEA’s thoughts about this question. You can reach me at this email address or by phone.
Have a great day!

About a month later I still hadn’t received a response so I emailed again and got a call the next day. It turns out I wasn’t the only person who had submitted feedback about this question. Unfortunately, according to the person on the phone, after internal review TEA has decided not to take any action. However, they do acknowledge that the wording of this question could be better so they will do their best to ensure this doesn’t happen again.

I told her I wasn’t happy with that answer and that I would like to protest that decision. She didn’t think that’s possible, but she offered to pass my email along to her supervisor or ask the supervisor to call me. I asked for her supervisor to call me.

Surprisingly, my phone rang about two minutes later.

The supervisor asked me to go over my concern with her so I explained pretty much what I said in my email. She said she understood, but if we looked at G that way then all of the answer choices could potentially be right answers. This was confusing to me because I don’t think F would help you determine the answer at all. If anything it shows 25 + 13, which will not give you an answer of 12.

I stressed that my concern is that answer choice G is mathematically correct with regards to answering the question asked. I get that J is a closer match to representing the situation, and if the question had asked, “Which number line best represents the situation?” then I probably wouldn’t be emailing and calling.

But it doesn’t.

The question asks, ‘Which number line represents one way to determine the number of people who left the library?” If you know how to use addition to solve a subtraction problem, then answer choice G is totally a way to find the number of people who left the library.

She said that is a strategy, not a way of representing the problem.

“That’s exactly what a way is. How you would do something, your strategy,” I replied.

She decided to redirect the conversation, “Let’s look at the data on this question. 68% of students chose J. 9% chose F, 12% chose G, and 10% chose H. The data shows students weren’t drawn to choice G. It’s not a distractor that drew them from choosing J.”

“I don’t care about that. The number of students who selected G doesn’t change the fact that it’s mathematically correct. If anything we should give those students the benefit of the doubt because we don’t know why they picked it.”

“Exactly,” she replied. “We don’t know why they picked it, so we can’t assume they were adding.”

“That’s not okay. Since we don’t know why they picked it, we’re potentially punishing students who chose to use a perfectly appropriate strategy of addition to solve this problem. There are a lot of 3rd graders in Texas, and 12% of them is a large number of kids. Who knows if this is the one question they missed that could have raised their score to passing?”

From this point she steered the conversation back to the question and how J is still the best choice because this is a subtraction problem.

“But you aren’t required to subtract to solve it! We work really hard in our district to ensure our students have the depth of understanding necessary of addition and subtraction to know that they can add to find the answer to a subtraction problem. We want them to be flexible in how they choose to solve problems. And again, the question isn’t asking students which number line best matches the situation. It just asks for one way to find the number of people who left, and both G and J do that.”

She went back to her original argument that if I’m correct then all of the answer choices could be used to find the answer to the question. She talked about how choice F shows both parts of the problem, 25 and 13, so you could technically find the answer. I disagreed because you end up with a total distance of 38. There’s nothing that makes me see or think of the number 12.

We went round and round a few more times. She wasn’t budging, and I was having a hard time listening to her justifications. She assured me they were going to be much more diligent about how number lines are used in future questions, but this question was going to remain as-is because she believes J is the best answer.

The whole exchange left me livid. In some small way, TEA is acknowledging that this question is flawed, but they aren’t willing to do the right thing by either throwing it out or making it so either G or J could be counted as correct.

They’re just going to do better next time.

But we’re talking about a high stakes test! Our students, teachers, principals, and schools don’t get to just “do better next time.” They are held accountable for their scores now. They can be punished for their scores. People can be moved out of their jobs because of students’ scores. So much is at stake that if a question is this flawed, TEA should show compassion to our students, not stubbornness. They should admit that both answers are mathematically correct and update each students’ score.

Because we’re not talking about a small handful of kids.

12% may not sound like much, but when 327,905 students took this test, that means nearly 40,000(!) of them chose answer choice G. That’s 40,000 students who are being punished because of a poorly worded item that has two answers.

That’s not correct.

Play With Me

On Wednesday I had the chance to visit my first classroom this school year. Sadly, in my role as curriculum coordinator, I don’t get to do this nearly enough. So I relish opportunities like this. Even better than visiting, the teacher allowed me to play a math game with her class.

I had so much fun!

I wanted something simple and quick to get the kids engaged before moving on to another activity. I also wanted it to involve adding 3-digit numbers because her class is in the middle of a unit on that very topic. I also wanted to bring in some place value understanding and reasoning, which are very much related to adding multi-digit numbers.

Basically I brought two decks of cards – one had Care Bears on the back and the other had Spider-Man on the back. I wanted different backs to the cards so it would be easier to tell which cards were mine and which were my opponent’s in case we needed to reference them during or after the game. I also pulled out all of the 10s and face cards, with the exception of the aces. I kept those and we decided to use them as zeroes. I tell you this because if you ever want to play a game that involves digit cards, here is a great way to get some without having to painstakingly cut out cards to make your own sets. Decks of cards are cheap enough. Just use those.

The game was me vs. the class. The goal is to make two 3-digit numbers. Whoever has the greater sum wins. On my turn, I drew a card, and I had a choice of putting it blank spots that I used to create two 3-digit numbers. Once a digit was placed it couldn’t be moved. On the class’ turn, I drew the card for them, but I let them tell me where to place the digit.

My favorite part of the game was at the end when the kids started shouting out that they’d won without even finding the sum. Take a look and see why they got excited: (Just pretend I hadn’t written the sums yet. I took the picture after the game was over.)

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“You have a 9 and a 4 in the hundreds place. We have a 5 and a 9.”

“Interesting, and how does that tell you you’ve won?”

“Because the 9s are the same. And we have a 5 which is greater than 4. You should have put your 5 in the hundreds place.”

“I was hedging my bets and I lost.”

Such wonderful thinking from a 3rd grader! How often do students rush to calculate and find an answer to a problem? How amazing that these students were paying attention to the place value that matters most in these numbers – the hundreds – and then comparing the digits to determine who had a greater sum?

Since I was just the lead-in to the day’s activities we only got to play once, but I would have loved to play again. I would have liked to change it up a bit. I would still construct my number on the board, but then I would have allowed everyone to create their own number at their desk using the cards that I drew on their turn. At the end we would discuss who thinks they have the greatest sum and talk about their placement of digits.

Even though I didn’t get to play again, I’ll take the time I did have. It was the highlight of my week!

More Than Words

Yesterday Tracy Zager shared a heartbreaking post that every teacher should take a few minutes to read.

The gist of it is that teachers need to be mindful about the messages they send students and parents about learning and doing mathematics. Sometimes damaging messages come across in the form of words – “You may not talk to anyone as you work.” – but they also come across in our choices of lessons and activities we do in our classrooms – such as a long pre-assessment that most students will “fail” because they unsurprisingly don’t yet know the content from their new grade level.

But there’s hope! This Tweet sums it up nicely:

I’ve been especially encouraged while reading the latest blog posts from the members of my Math Rocks cohort. Back in July we watched Tracy’s Shadow Con talk. Afterward everyone took Tracy’s call to action to choose a word to guide their math planning at the start of the year.

Flash forward a month and the school year is finally getting underway. Our latest Math Rocks mission was to re-watch Tracy’s talk and to watch my own Shadow Con talk¬†since the two are very much related. Then they had to choose one of our calls to action to follow and write a blog post reflecting on their experiences as they kicked off the school year.

The results have been so inspiring! I’ve collected all of their posts in this document. Take a look. Just reading the titles of their posts makes me happy, and if you go on to read them, I hope you’ll finish with as big of a smile on your face as I have.