This quote is well known and has been adapted to include many other groups over the years. (You can read more about the original quote HERE) Though I have loved the message of this quote for years, it is this week, yesterday and today at the start of a new presidency that I find myself so distracted wondering how I need to change where I invest my time, resources and energies moving forward. I find myself distracted from wanting to think about math, math teaching, and so much more and instead worrying about my students, my neighbors and my friends.

I post this post today publicly asking you to hold me accountable in how I stand up on behalf of those around me. There are so many others that have and will write things about what is going on in the world more eloquently than myself. I thank them. You move me to act. I am simply stating to any of you reading this that I commit to stand up and speak for those around me who are being treated unjustly. I am not sure what will be next in my journey. I plan to spend time daily in prayer and meditation seeking truth and guidance as I move forward. I am though also committed to action. It may be quiet. It may be loud. It may be public or private – but all I know for sure is my silence and inaction is wrong and a sin.

I do not support the bans put in place in the last days. I will not stay silent. I will reach out to those in my community with love. I will stand with them. Thank you for listening to my public commitment.

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**UPDATE (2.12.17)**: I created a valentine postcard for all my students on Tuesday. If you want to create one similar to mine, all you need is 1 sheet of card-stock, 8 pixie sticks and 8 adhesive dots, & a colored pencil to create 4 nerdy valentines for your students. Here is the Word Doc if you want to edit and make your own: valentine-card We love it because the front is a bit angsty and the back is a bit sappy. Enjoy.

Update 2.13.17: My former student teacher who now lives in the middle of Superior National Forest near the Canadian border could not find pixie sticks and adhesive dots to make her cards – but made equally lovely ones with smarties and a hot glue gun. Check them out.

Desmos is where it is at for Valentine’s Day. Enjoy using the Math-o-Gram site to send a Desmos Valentine via facebook, twitter, email or to post a link anywhere. Share the site with your students so they can send 2.14 desmos grams to one another. You have 5 options to select from.

Many people have created equations that make hearts. Again, DESMOS is where to use them. If you’ve taken 3 minutes to sign up for a FREE desmos account you can log in and just hit ‘save’ for each of thee links and then you can find them always in the future by clicking on the 3 horizontal parallel lines in the upper left corner. Here are just a couple I have saved. Sierpenski Triangle in Hearts 1 red heart Flower to Heart Click on link and then click ‘save’ once you get to desmos.

Also, check out this amazing new video from @DesmosDude and his youtube channel (bonus – @DesmosDude is from MN! we do great stuff in this state) It’s a Desmos love story set to 80’s themed music. Love. This is a must watch!

Check this card out – if you got a bit of time you could make a Pop-Up Sierpinski Valentine Card. Amazing.

Check out this youtube video on how to make a Two-Mobius-Hearts. Who doesn’t want to do arts and crafts in math class.

Last year I shared told my students that if they had someone they were secretly in love with they could tell them secretly using one of these 2 math equations.

- The first equation requires you to see only half of the story. Check out other versions of this equation below.
- The 2nd equation, I tell my students, tells you if your true love is good enough for you. If they can’t solve it then they will never know the truth. See below.

For some of us Valentines Day is lonely. For those students I shared this sad math story with them.

or in GIF form….

Here are just a few fun things I found connecting mathematics to Valentine’s day or Love or Hearts or Cupids or…. Enjoy.

First a Which One Doesn’t Belong with HEARTs from the amazing (and Minnesota’s own) Christopher Danielson. (Seriously, have you bought his book yet? No, click here and do it NOW – so good)

If you are teaching ‘**ratios**‘, this video entitled ‘Bad Date’ is perfect this week.

If you are teaching (or have taught) ‘geometric mean’, this video will teach you how it is used in identifying good matches for your valentine. (video is called ‘inside OK Cupid’)

Brian Marks, over at Yummy Math, posted a bunch of resources related to Valentines last year. **Check out his resources HERE.**

Check out this Valentine Brain Teaser at NCTM’s site.

If you are wanting to send a math valentine to your loved one, your mom or your child – here are a few pictures for inspiration.

Click to view slideshow.Check out this post from Jennifer Cook who had her students make math valentine cards. She got the idea from Matt Vaudrey’s post **HERE.**

Another youtube video Valentine for Math Nerds

Or try these 15 math pick-up lines out on someone….

Or throw this GIF in your slides:

After writing this post, I search twitter before hitting ‘publish’ and found this… Last year the amazing Annie Forest posted some resources. Some of which overlap with mine. **Check it out HERE** (and if you are not following her on twitter, do it now, go, run, click follow)

Do you have a math valentine/love resource to share? Send it my way and I will add it to my list. Twitter @saravdwerf or email saravdw@gmail.com or post below.

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Here are some questions I have after watching this…

- Is there truth in what Simon is saying? My gut is there is. My sister works as a Dean of Students at a college. We often find ourselves talking about the changes we’ve seen in students over the last 25 years of our careers.
- If there is truth, what is my responsibility as a math teacher?
- I know my job as a math teacher is not just about teaching math. I am called to make sure my students are prepared for the world they will enter after HS. How does what is in this video change what I focus on in my classroom?
- What is the role of technology, social media and phones in my classroom?
- What can I do to build my student’s self esteem?
- Should we be talking to students about what Simon talks about in this video?
- How do I help my students build deep and meaningful relationships and practice this skill set? Should I invest more time in my math classroom on this?
- What is the role of impatience and instant gratification in my math classroom?
- Honestly, after watching this I am personally committed to using my cell phone less around the adults in my life.

I’d love to talk to you about your thoughts. Let me know what you think after watching this. Happy Martin Luther King Jr. Day.

Note: Simon followed up the video above with this video response on New Years Day 2017.

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I get asked pretty frequently for help finding math activities for all levels of math. I know why I get these emails. I led K-12 math for 5 years in my district of 200 math teachers and answered these emails all the time. I also sent frequent emails with all kinds of cool things I found for the classroom. The problem is though for the last 4 years I’ve been back in the classroom full time and I don’t have the same time to research for others the cool stuff out there. That said, I will often send something on to those that ask if I have the time. I’m sure that Chris and Colleen (the people who currently hold my old job) get these emails too.

In the last couple of years my response to these emails has slowly changed. Instead of sending something that I find, I send a response like the one I sent today…

I have changed from giving links to activities I find to sending quick emails telling them how to find what they are looking for. Really, it only takes a few minutes to find great resources for the mathematics classroom. Last summer I blogged about some tips for finding resources **HERE**. I am finding I need to add to this blog post with a bit more information. So this post is part 1 of 3 I will do in the near future.

I sent the advice above without testing if it would work. So after I sent the email I made sure by taking my own advice. I timed myself to see how long it would take to find something great on ‘piecewise functions’. Here is what I did..

- I went to my favorite search engine, Google, and entered the following into the search engine: What came up were pages of possible activities to use. ( By the way, I prefer google’s search engine to the search engine at teacher.desmos because I find activities and also things I can use on the online graphing calculator & videos &…). The key to using google is to add the word ‘DESMOS’ to your searches if that is what you are looking for. The favorite thing I found was the the first thing on page 2 that came up on my search (I am not going to give it to you, you need to go find it yourself, I promise you, you will learn more if you do it yourself and it would only take you 2-3 minutes – go now).
- If I don’t find anything at google (and sometimes even if I do), I go to
**teacher.desmos.com**and sign-in (If you don’t have a FREE account yet, what are you waiting for, SIGN-UP now. It only takes a few minutes). I then search Desmos site for what they offer. Not only do Desmos employees write cool stuff here, but they also look at all the stuff we all make and ask for permission to tweak it and add it to their search features. Here is what I searched: Note: at teacher.desmos I search first using just the one math vocab word I am most interested in. My search brought up 4 things. The introductory POLYGRAPH activity being the best. - Go to twitter (create an account if you need to, it only takes a few minutes) and search ‘piecewise desmos’ on twitter’s search engine. After I did this
**my favorite things I found came up**(by the way, this is my #1 thing I do usually, but I made it #3 since many the people who ask for my help are twitter phobic). Do this and you will find a link to ‘Harry and the purple dot’ and ‘Piecewise functions’ by Nick Corley in the first few tweets that came up. Great stuff. Go find it for yourself. Once found you can click the 3 vertical dots on the right and ‘Copy and edit’ to make these activities your own. - If #1, #2 and #3 bring up nothing, then I tweet the following: I tweeted this at 5:12 pm today. I always hear something in 24 hours or less. (usually way less). It is really important in your tweets to tag the math education on twitter using the hashtag #MTBoS and also tag @Desmos on twitter. I promise, even if you don’t have a ton of followers like I do, someone will answer you. The online math community is where 80% of everything I use in my classroom comes from these days. They are welcoming and want to help. Here are the responses I received after tweeting this out:
- At 5:30pm I received this tweet: with this picture.
- & an hour later I received this tweet from Nick (who I mentioned above…

After spending less than 6 minutes (remember I timed myself) online doing #1, 2, 3 and 4. (note: this 6 minutes included opening up 20+ links I found – I could have opened more…) I found lots and lots of Desmos piecewise activities I could use if I were teaching this topic right now. If you want to find what I found, invest 5-10 minutes yourself. Or if you have a different topic you want to find something on, do #1, 2, 3 and 4 with a different topic.

**Note #1:** I still share activities I find interesting with my coworkers (both at my school and around MN) pretty often. I am not done doing this. If I can help I will. What I do want is others to help me find great things.

**Note #2:** The 3 things I do above do not include my 2 favorite ways to find activities. To do what I do, you need to invest a bit more time into twitter and the #MTBoS community. II will share these things in a future post.

Note #3: An email I love getting from people is the following….”Sara, Can I buy you a coffee sometime soon and you help me learn about twitter and the #MTBoS community?”. I will always make time for you (If you can find your way to Minneapolis) and will happily get you involved in the best place to be now days for math PD and resources.

Peace & love Until, next time…..

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In the early 90’s I taught a curriculum (IMP) that changed me & defined my core values as a teacher for the rest of my career. (I am working on a ‘Love letter to IMP’ post sometime). At the start of IMP year 3 (11th grade), the first unit problem was one I am using as an introductory task with my Advanced Algebra students tomorrow (20+ years later). Here is the task, titled ‘Fireworks’: fireworks-task-sheet

My Goal with this task is to review what students should remember from their 9th grade Quadratics Unit (vertex, intercepts, max/min….) within a contextual problem. In the 90’s I did not have Desmos to help make this task more visual. That changed last year & this is why I LOVE Desmos.

**THIS** is the Desmos graph we used with our class after students struggled with the task. When you open the graph it is pretty basic (only labels on the axes).

To tell the ‘Fireworks’ story we (my colleagues & I) click open the equations in the graph one at a time, saying something like the following….

Resulting in this Desmos graph:

Then we click more equations in Desmos and say…

The animation looks like this (but animated!):

Then we click some more Desmos equations and add mathematics vocabulary connected with quadratic functions (note: for this we turned off the background images first).

The graph then looked like this:

Thank you Desmos for helping me and my students visualize the ‘Fireworks’ task. Love you!

**Note:** Before the ‘Firework’s’ task we taught a 3-day arc of lessons on Fawn Nguyen’s Duck Pattern that I highlighted in my **Number Talks Post**. Following the 3-days we used **THIS DESMOS GRAPH**, to determine if the 5 quadratic rules students wrote were equivalent. (later we will prove equivalence algebraically). We also added quadratics vocabulary to this graph as well. (again, click through the equations-scroll down-& see what happens)

Well, I did it. This post is only 416 words. (way less than my normal posts which are always 1000-3000 words). To be fair, I cheated and incuded words within the images I used. I am working on someday doing posts that are 100-200 words with only 1-2 images/links. (this may take me a while though – I have a lot to say and feel like I need to include it all in every post). LOL.

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I teach Advanced Algebra (Alg 2) and last year (2016) I started with this visual:

I then reminded them of the story of Karl Friedrich Gauss we had looked at the first week of school where as a child he had quickly found the sum of the numbers 1 to 100. . Students went on to write rule for the expression by noticing the patterns Gauss noticed.

In the fall I followed this up with a number talk about a ‘football’ pattern.

Together we found a rule for the extended pattern and introduced the tern ‘triangular number’. We talked about this equation being equivalent to the one Gauss had found. On the first day back in 2016 we reviewed all of this and I told them 2016 was also a triangular number. I asked them how many rows of dots were in this picture – instead of counting them we replicated the equations we had built in the fall using a smaller pattern.

We annotated the pattern and wrote the rule.

Students then threw the quadratic rule (it was so great that this rule was quadratic as that is what we were currently studying….Love it) into Desmos to find the number of rows of dots in the number 2016.

It was a great start to 2016. Now I want to do the same in 2017.

So tomorrow is the start of 2017. This is what I am doing tomorrow….

- First I am going to ask them what they know about the number 2017. Most likely the fact that it is a ‘prime number’ will come up. I am going to use this time to deepen students understanding of ‘prime numbers’. I may use these GIF’s.
- I will then share a bunch of other geeky facts about 2017. I tell students that math teachers are geeky and at midnight on twitter & facebook this is what I found people tweeting about 2017. For example look at all of these……Click to view slideshow.
- One of the tweets I plan to use tomorrow is this one…..I will be teaching my students about ‘i’ and complex numbers soon and I want to create a ‘subaru effect’ for my students so they are looking for ‘i’ and/or the term ‘complex numbers’ in the next month or 2 in my classroom.
- My favorite tweet was this one from David Radcliffe (from St. Paul, MN – love my MN math geeks). His tweet caused me to pull out a piece of paper and figure this one out. I was pleasantly surprised that this pattern was quadratic. Love that the unit I am teaching tomorrow is on quadratic functions. I will be using his puzzle this month in my classroom.

I pulled out Desmos to check the work I did on paper and to solve his challenge. The answer is 2017! - This is the 8.5×11″ sign I will post outside me doorway tomorrow. Yes, it’s intentional that there are 17 boxes of 2017. You can print it out using this word doc. Enjoy. happy-2017

- Check out David Radcliff’s amazing blog about the number 2017. He has tons of geeky number facts you can use tomorrow.
- From Matt Parker

- From Alex Bello’s – a math challenge.
- From Brian Marks & Yummy Math. Is 2017 prime? task.
- Do the
**2017 Challenge**with your students with either The Math Forum or Denise Gaskins! - Here is what Wikipedia says about 2017!

Lastly, here is an image I will be using in my classroom this week for sure. In 2017 I am excited about the opening of ‘Hidden Figures‘. You should be too.

I will be walkin’ into 2017 like…

**Happy New Year to all my friends and family. Love you all.**

p.s. My Christmas card blog post (and videos highlighting a decade of my homemade xmas cards) is coming, but is a bit late. Hopefully later this week. Until then, Which One Doesn’t belong?

Kind of like the credits at the end of the Ferris Bueller movie…I have one more 2017 thing for you. Here are other ‘mathy’ dates in 2017.

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This was one of my first posts and has 19 times the views of my average post this year. It taught me a lot! #1, I learned that my most popular posts (not always my goal) will be ones that are ones that you can easily do in your classroom tomorrow. #2, I learned about the power of Dan Meyer. He tweeted a sassy tweet with a link to this post and my numbers of views & followers skyrocketed. #3, I learned about the power of Pinterest. This post has been ‘pinned’ thousands of times. Crazy. All that said, this game is GOOD. You should play it with your students. It is always a win.

This post is one of the reasons I started a blog. I kept having people ask me for this task (I reference it often when I speak). One of those people, **Megan Schmidt (@veganmathbeagle**) lives near me and kept bugging me to stat a blog. So I did. I’ve used this task week 1 every year for the last 10 years. It is great for setting group work norms. I think I had 100 of you tweet me photos using this task in your own classrooms. I loved everyone of them.

I am convinced that every math teacher should be doing Number Talks regularly. They have changed me as a teacher. I have also learned that Secondary teachers are scared of using these and that there are very few of us talking about using them at the secondary level, particularly HS. I have received speaking requests from this post and also numerous requests for me to write more on this topic. If you live in MN, I’ll be doing some PD on this topic summer 2017. Until then, you can read about my journey above. I am still learning a lot on this topic.

This post was a favorite of the Ed-Tech community. Somehow they found this post and shared it widely. I heard from many non-math teachers thanking me for writing about the importance of modeling googling in your math classroom.

This post was not about math really, but it is something I think all of you should be doing week 1 of school every year with me. It works in any subject area. Again, I loved all the tweeted photos from those of you who used the Name Tents. What I know for sure is we need to all be doing more in our classrooms to solicit and give feedback to our students. The time we invest in this is always well spent.

I have become a bit of a minimalist in my classroom. That said, I do have an entire wall dedicated to 8.5×11 inch laminated Math Fail Photos. This post has 80 you can download and use. This post is the closest I’ve come to creating a hashtag, #mathfail or @mathwallofshame. About once a week I get a tweet with a new math fail sent my way. I received so many I posted ‘Math Wall of Shame, part 2’ with 70 more downloadable fails you can use. Part 3 will come out in 2017 sometime. Keep sending them my way.

The response to this post taught me that everyone is looking for ideas of how to support students who struggle. It was another post that generated emails asking me for more information on this topic. This post is about my favorite thing I did in my support math classes, but I also bluntly speak about all the other things I believe about support math classes.

If you are looking for a good post to share with a newer teacher (or a teacher struggling with engagement) this is a great first post from my blog to share with them. It has 7 ideas for engaging students that I talked about when I did a **Global Math Department presentation** in June on the same topic.

This post brought out all of your creativity. Many of you created your own badges after reading this post. Many of you asked for me to share my documents where I created the letters – I’ve not done this yet, but it is on my 2017 to do list. This post also taught me the power of Sarah Carter. She tagged me in several of her posts this year including this one where she was inspired by my math badges. So many people come to my blog through her blog almost everyday. She has an amazing platform in math ed.

This post is a simple task that I’ve used for 25 years. So great. I’ve been amazed at how many of you had the old school version I had turned digital. Love it.

Everyone should be blogging. Really. I learned a ton this year. I learned things that I never would have learned if I had not been blogging. During Twitter Math Camp I tweeted this image. Blogging had widened my world in a really great way. Above you can see what I have learned so far – I highlighted my learnings in** yellow**. Here are a few more…

- I learned I love feedback on my posts. I have loved the tweets with photos from your classrooms showing how you used what I wrote about. I love that when I meet some new math teacher now I often have them tell me something they have used or loved from my blog. Thank you.
- I learned that the numbers/metrics I can read about views and things are fun and a bit addictive. I am not blogging so that I get views – really I am not – but I am continually interested in how you find my blog and what resonates with you.
- I learned this year that my favorite posts are not your favorite posts. I have lots of posts that represent what made me a better teacher for all my students this year that were not as widely read as others. Perhaps I’ll share my favorites in another post. Let me know if you are interested.
- I learned that if I am open and vulnerable in my blog it opens me up to more intimate conversations with people I don’t know well. So many of you thanked me for being real in my blog (not sure what this means entirely) and thanked me for speaking about things like Grief.
- I have learned that I can be brave and speak about issues that are often elephants in the educational professional world. I got lots of feedback from non-math teachers about these posts. Here are 2 that did not make my top 10, but that resulted in the most satisfying conversations after I posted. I hope to do more like these in 2017. Check out The Subaru Effect and The Number 61 & read about when I get a bit political.
- I learned this year that writing is an amazing way to reflect and change in your classroom. I have never written a ton and frankly I have never thought about myself as a writer, but I have loved what I have learned this year through the writing I’ve done. Seriously, you all need to join me in this. Writing is where it is at.
- I learned this year that math teachers are frighted to blog because they all say to me when I ask them to write something…”Sara, I am not a writer.” Seriously, almost everyone I’ve asked said this. And I’ve asked 100 people to write something for me (or our state organization this year) and us math teachers are seemingly scared of writing. Here is what I have to say to you if you are thinking this….Have a growth mind-set about writing. You get better at it when you do it. You all need to do more writing. Join me.

**Some things I hope to do with my blog in 2017:**

- My goal is 52 posts again in 2017. I can do it! 1 per week. (most likely it will mean a bunch over the summer and fewer during the school year). I have 55 posts started on the back side of my blog – this should be a no brainer.
- I am going to do a redesign of my blog soon. I know a lot more of what I want from my blog. My brother-in-law, Brian, has offered to help me.
- I want to post more guest blogs from the amazing math people I know who do not have blogs of their own. There is so much great stuff out there.
- I want to post at least 10 blogs that are short….just a couple of paragraphs. My blogs tend to be LONG. I can’t stop the long, but I can post some short blogs once in a while.
- What ideas do you have for me? Let me know. I love hearing from all of you. If you have ideas, suggestions, feedback… TWEET me
**@saravdwerf**or EMAIL me at saravdw@gmail.com

**Thank you** to everyone who came to my blog in 2016. I am humbled by how many of you come to my blog. I love how many of you actually used things from my blog. I feel like such a bigger part of the MTBoS community now. I look forward to meeting more of you in 2017 and sharing more thoughts and ideas from my classroom in Minneapolis, Minnesota.

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I am always surprised a bit about what my former students remember from their time in my classroom. It is almost never what I would hope they say. But I smiled when she told me and thought, I should blog about this.

**So, What is the ‘Subaru Effect’? **

Years ago – like 20-30 years ago – I saw a news story where someone was talking about what they called the ‘Subaru effect’. Essentially it works this way. One day you go and buy a new car. You decide to buy a Subaru for the first time. The next week as you are driving your new Subaru around town you see Subaru’s everywhere you go. You wonder why you are seeing so many Subaru’s when you would have sworn last week there were very few Subaru’s on the road. The ‘Subaru Effect’ is your brain being awoken to seeing something around you that was always there, but you just had never paid attention to it before.

I have never owned a Subaru, but my most recent car purchase was a Dark Blue VW Passat. When I bought the Dark Blue car I thought, ‘I never see cars this color’. As I drove the car off the lot I started noticing dark blue cars everywhere. My conscious mind was awoken to noticing the cars. For years I had have mentioned this effect in my classrooms.

When I had Jahlie as as student I had to teach a Social Studies elective to 7/8th graders. I chose Economics because it seemed like something kind of mathy. One day I challenged them to name the most powerful person in the world. I told them it was someone who was on the news all the time. Students named the president and other leaders of other countries. They named sport and music stars and all kinds of other people. Each time I shook my head, ‘no’. I reminded them we were in an ‘economics’ course so this person must have something to do with money and I introduced them to the Federal Reserve Chairman at the time, Ben Bernanke, and told them about his role in controlling the markets world wide. All my students swore they had never heard of him despite me saying he was in the news all the time. The next day all kinds of students came to class and exclaimed “Ben Bernanke was on the news last night. The news said he raised interest rates.” For the rest of the year these 7th/8th grade students would tell me every time they saw Bernanke on the news. I had awoken their eyes to the financial world that had always surrounded them.

When Jahlie reminded me of this and the ‘Subaru Effect’, I went home and googled it and nothing came up. Nothing. I thought I had perhaps made it up. After a bit more googling I realized it was a real thing it just had other names. (though I prefer my name, the ‘Subaru effect’). This phenomenon goes by the names: **‘*** Baader-Meinhof Phenomenon‘ *OR

I love these quotes from the video…

** “our brains are pattern recognizing superheros, able to find meaning in countless pieces of data” **

**YES! Isn’t that really why all of us are mathematicians. **

**I also love…**

** “ what is interesting to me are all the patterns that are zipping by in your day to day life that you don’t register at all. The Baader-Meinhof principle proves that we only see the things we are looking out for.” **

**He goes on to say in the video…**

* “It shows how ignorance can be maintained through a lack of active learning.”*

The ‘Subaru Effect’ has been a lens that I bring to my planning for my classroom since I learned about it. I think about it so often that I usually end up telling my students about it sometime each school year. I don’t set out to teach them about Subaru’s, it just happens because it is always prevelent in my brain as I plan for making them all successful in math.

**Here are a few examples from my classroom.**

If the Baadof-Meinhof Phenomenon is correct then we only see things we are looking out for. If you’ve ever felt like your students should know something in your class because they’ve seen it so many times, perhaps the problem is you have not given their brain the lens or structure for it to register that what you are teaching them is important. Once you help them see what you can see they will start to see it all the time.

Here is a recent example of where my focus on trying to help students see the structure/relationships in graphs that I see worked for the first time in my career. I teach Advanced Algebra (Alg II or Alg Trig). I always have

students that do not see the infinite number of invisible points on the functions we graph. They see the line

or curve, but don’t see it as points. For example, in a linear equation pictured many just see the blue line. Many students struggle to write an equation of this line is slope-intercept form because they do not see the points on the line that will help them write the equation. Many students will see the point (0, 2), the y-intercept, but may struggle to see the points that intersect grid-lines that will help them calculate slope.

In a recent unit on ‘function transformations’, we wanted them to see a set of important points (vertex, inflection points….) in each of the parent functions we were studying. In the past we had created tables of parent functions and graphed 5-7 points before drawing the curve. It looked something like this:

We would then talk about the relationship of the points on the parabola to the vertex, wanting students to see that in the parent graph there are symmetric points one unit over and one unit up and another set of symmetric points 2 units over and 4 units up. Students who can see this relationship generally have an easier time with transformations of the parent functions when we change the values of the parameters a, h and k. No matter what we did to encourage students to see these points on the curve, many students could not make the connection between the table and the graph of the curve.

So here is what we did differently this year. I created this visual to start the unit and we did some noticing and wondering about the points on this visual.

This visual was void of axes, lines or curves. It just had important points of the parent functions we would study in this unit. I have no idea why this worked, but at the end of the unit – students begged to have this visual as part of the reference sheet they could use on the test for 10 minutes. Many students found this graphic useful and we saw students ability to apply parameters to parent functions correctly increase. Their ability to explain what happened to the shape and location of a graph when looking at an equation increase. It seemed like students ccould see structure in graphs that had been invisible to them before. Some students drew in the curves on the visual in their notebooks & some did not, but all annotated the visual noticing the relationship & location of points in each of the functions we were studying. The points for these parent functions were my students ‘Subaru’ in math. Once they saw it, they started noticing the points in many other functions.

**Side Note:** If you want a copy of the visual I used with my students you can download it here: what-do-you-notice-function-dots and what-do-you-notice-function-dots-for-notebook

Understanding the Baadoff-Meinhoff Pehnomenon (‘Subaru Effect’ for short) gives us teachers an interesting lens to look at curriculum planning. What can we do in our classrooms to help students see the invisible in math. How do we help them notice the patterns right in front of their eyes. How do we empower them to see things that will cause them to see these things around them all over the place. This power to see things is empowering. for students. I challenge you to think about how the ‘Subaru Effect’ can play out in our classroom.

15-20 years ago my district started training all of the staff in an equity framework. As part of this training (Courageous Conversations from Pacific Educational Group) named for myself for the first time ‘white-privilege’. I defined it. I talked about. I read about it. I noticed its role in my life. More importantly all my colleagues were with me on the same journey. Over the last many years there have been many forms of this training in my urban district. It is frustrating at times, because it seems like we are stuck in a rut just talking about equity issues (we do lots and lots of this) but it is void of action.

My larger district construct may be frustrating, but on a more individual level, this shared language with colleagues has allowed us to look at the role race plays in our lives. When ‘white-privilege’ was given a name for me I started noticing what had previously been invisible to me – the fact that as a middle class white woman I walk through life never thinking about things my peers of color and students of color think about all the time. I was humbled. I also started changing what I did in my classroom.

**White privilege was my ‘Subaru Effect’ in my classroom & life.** I started noticing it in the classroom, when I was shopping, in conversations with family and friends. I could not unsee it. When I saw its role in my life I had to say something I had to act. I had to change my personal actions in my classroom and in my life. I had to learn more. I had to read. I had to listen.

(**side note:** I am sitting right now hesitating to type my next words because I know what a politically charged world I live in and have prior experience with internet trolls lashing out- Ugh. I am not sure why I get nervous as my white privilege affords me the privilege of having little repercussions for my next words).

This last summer a police shooting happened in my neighborhood. It made national headlines due to it happening days after another high profile shooting in New Orleans and due to its aftermath being viewed by millions on Facebook Live. Philando Castile was pulled over by police. At the time they told Philando that it was because of a broken tail light. Within an hour of his death I had viewed the facebook live video shot by his girlfriend. When I saw it I thought several things – 2 I’ll share quickly here:

#1: I drive the same road Philando was killed on. At the time of this shooting the tail light on my car had been out for 4 months. Not once had I been pulled over. (I know. I should have gotten it fixed long ago, it is fixed now)

#2: If I had been pulled over, I am pretty sure I would have left that situation alive – even if I had told the officer I had a gun permit and/or gun on me and acted in a similar manner as Philando.

It is my belief race played a large role in the events of that evening. In the days that followed I participated in a couple of marches and loudly shouted ‘Black Lives Matter’ surrounded by thousands in my community and many of my friends and colleagues.

In my life I have friends and family who hear ‘Black Lives Matter’ and hear it as only a cry against police. They share on social media phrases like ‘all lives matter’ and ‘blue lives matter’. They see my actions in support of BLM as terrorist threats against their way of life. I’ve been frustrated that they don’t see what I see. When I say Black Lives Matter, I am not saying that other lives don’t matter as much, I am saying that there is *demonstrable evidence* that black lives matter *less *than white lives to the criminal justice system. I am also saying that this unfortunately happens in classrooms. I am saying this to wake both myself and others up to looking at how we can change this.

As a teacher, when my unconscious biases cause me to make decisions that harm students, this harm is often invisible. I do feel for Police. When their unconscious biases get it wrong, death is a possible outcome. As teachers we are afforded the privilege of being in a profession that our mistakes do not result in death and make the news. That said, my unconscious biases can result in just as much harm in students. When I assume that students of color can’t learn the same level of math as my other students and teach them half of Algebra in a year versus the entire curriculum, my actions are screaming to my students that I don’t believe you can go to college. My actions are causing them to enter the post-secondary world unprepared to be successful. When I am not educating myself about issues of race and equity, I can cause harm to my students that will have lasting effects in their life.

I need to learn more and so do all of us. 98% of Minnesota’s teaching staff is white. We owe it to our students to educate ourselves on matters of race.

For more information on white privilege or issues of equity I highly recommend the following:

- Teaching Tolerance website.
- Educolor Resource Page (seriously – so much great stuff here. Enter 2017 well by reading 1 book & 1 article from their list or watch 1 movie & 1 video link)
- Read Lisa Delpit’s book ‘Multiplication is for White People.’
- Read ‘Blindspot (hidden biases, good people)’ and take the Implicit Association Test from Harvard.

**The Baadof-Meinhof Phenomenon says we only see the things we are looking out for. I humbly ask that you make a deep understanding of ‘white privilege’ your personal ‘Subaru Effect’ as you enter 2017. If you have good resources for this, please send them my way. I have lots left to learn.**

Start by listening to Royce Mann, age 14. “White Boy Privilege”.

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This year I gifted my students a copy of a Math Magic Trick I taught them. Well, I did not so much as teach it to them…they had to figure out HOW I was doing the trick and tell me WHY it worked before I would give them a copy of the trick.

Before reading further – check out the trick in action (thanks to my friend Abir – a great math teacher – for her help).

**Do you know how this trick works? **

Here is how I presented it to students:

- On day 1, after completing our unit on Exponential and Logarithmic

Functions, I told my students that I had a Math Magic Trick that they could figure out using one of the number patterns we used in our most recent units. - I showed the students 7 cards and said I could ‘read their minds’. I then asked one student to write down a number from 1 to 127 large enough for the class to read it. I turned my back so the student could show the number to the rest of the class without my knowledge. I then told the student all they needed to do was tell the truth. I told the class to watch carefully and try and figure out what I was doing.Click to view slideshow.
I then proceeded to show them each of the 7 cards one at a time asking,

**“Is your number on this card?”**and noting if they said “Yes” or “No”. After showing all 7 cards I announced the number to the class – to their amazement I was always correct. It is not easy to impress my Advanced Algebra students, but this amazed them. I told them that I would do the trick every day this week and if they could figure it out I would give them their personal copy of the trick to take home and amaze their friends and family. - On day 2, I did the trick again. This time I reminded them it had to do with one of our number patterns from the exponential unit and it had something to do with how computers were programmed way back in the day (I used the visual below.)
- Day 3 I did the trick again and sorted the ‘yes’ and ‘no’ cards into 2 piles as I announced the number they had secretly chosen. (Note: In this picture the secret number was 109. Can you figure out what I am doing?) The arrangement of cards was done to help you a bit.
- Day 4 I made a simpler problem/trick with fewer numbers and cards. After doing this smaller trick a few times, students started to catch on. Can you see what I am doing in the 2 sets of cards below? The answer to the first is 10. The answer to the 2nd set is 7.
- On day 5 many students had started to figure out what I was doing. I had students do the trick for their peers. Finally I had them explain
**HOW do do the trick**(Are you ready? I am about to give away the secret to my Math Magic Trick……..The secret number is the sum of all the numbers located on the upper left of all the ‘yes’ cards). - My next goal was to help them figure out WHY the trick works. I showed them the cards arranged in order and asked them to look at the number patterns we know and look for a pattern……
- We then talked quite a bit about the Binary Counting system and that all natural numbers can be written as the sum of powers of 2. This would be a great place if I had time to do some ‘Exploding Dots’ work (such cool stuff) from James Tanton. Check his stuff out
**HERE**and**HERE**and his video on dividing polynomials**HERE**blew my mind. - I then handed out a copy of the trick (on card-stock 1/4 of a sheet of paper size) to every student. This was their end of 2016 gift from me (with a free homework pass attached).Click to view slideshow.

Here are some resources if you want to print out the trick and do it yourself. Enjoy.

- A pdf of my slides: power-of-2-puzzle-slides
- My gift (recommend printing on cardstock) math-magic-trick-cards-for-gift
- A word doc with the 7 cards (1 per page) powers-of-2-card-trick
- A word doc with 4 cards (2 per page) small-power-of-2-puzzle
- counting-in-binary

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The K-8 school had a phenomenal Kindergarten teacher named Kate. I loved watching her in action. My favorite thing I saw her do most days was stand outside of the boys bathroom making sure her little darlings were learning to negotiate bathroom etiquette. I would hear say things like “Austin, don’t come out here until you’ve washed your hands” or “Samuel, don’t put your hand in the toilet” or even “Zip up your pants, we don’t touch each others private parts”. I thought ‘WOW, I am so glad I do not have to talk about these things as a middle school teacher.

Today, the teacher I co-plan with daily reminded me of Kate when she used the sign above (& below) at the start of her Advanced Algebra lesson, asking students to notice & wonder.

We are currently teaching students how to solve equations that involve exponents & logarithms and several students were struggling to remember the order of operations. To help, Morgan did a quick review of the Order of Operations with them. She used this visual to talk about the need for a standard order of doing things. (note: the students had a great conversation ’cause talking bathroom behaviors seems to have magical ways of increasing engagement – even with HS students)

I loved her use of the bathroom sign because I think bathrooms and PEMDAS have a lot in common. My wish is for you to **STOP using PEMDAS** and begin doing something better. Let me continue….

A few days earlier at our team meeting I told Morgan that (like the K-5 teachers avoiding teaching Kindergarten) I now avoid teaching PEMDAS like the plague. This led to a great conversation with our team as to why this is and what to do instead. I thought I’d share a bit of or conversation and share what we use instead. I also encourage you to work to eliminate the tricks in your curriculum and consider how you teach concepts.

I started by pointing my team to Tina Cardone’s (FREE download) book ‘Nix the Tricks‘. In her book Tina gives reasons or avoiding 50+ common ‘tricks’ math teachers have used forever to teach math, including asking us to Nix (quit) using PEMDAS. What I love about Tina’s book is that not only does she explain why to eliminate a trick but provides a fix to replace it with. Go now and download her book. She talks about PEMDAS starting on page 9.

I personally started moving away from using PEMDAS 15+ years ago when I last taught HS. My students would continually apply the order incorrectly when simplifying expressions like this:

Many students would first multiply 2 and 5 resulting in an incorrect solution of 3. Their mistake was believing multiplication always proceeded division because M is before ‘D’ in the PEMDAS acronym.

As a high school teacher I knew I had to disrupt their use of PEMDAS. It is kind of like the backwards bike that I am trying to learn to ride. I can ride a normal bike so unconsciously that any change to these skills require such effort to rewire my brain. My students first understanding of PEMDAS has several misconceptions but they apply them almost unconsciously. To change their bad habits I had to change things up and stop using PEMDAS altogether to disrupt the pathways in their brain.

One way that I disrupt PEMDAS is to use this simple warm-up with secondary students at some point at the beginning of the year.

All 4 of these expressions lead to potential misconceptions often with the misuse of PEMDAS. These expressions have led to great discussions with my students about why we should review the standard order of doing math and learn a better list of the order of operations that works best for HS. #1 is the classic doing multiplication before division misconception. #2 is a super famous Facebook expression that keeps floating around every few months with millions of comments with incorrect solutions. Students will often do #3 correct without a calculator, but then I tell them to type it into their calculator (often they type in 24+8/4+12 resulting in an incorrect solution) which leads to great conversations about the term ‘grouping’. #4 includes a great discussion about whether 4(3) induces parentheses or multiplication.

With Advanced Algebra we then gave out a 1/2 pager to glue into their notebooks with a ‘new & improved’ order of operations.

Morgan and I made this and we spend time talking about what ‘grouping’ symbols mean. We also make sure students see radicals and exponents as related. We intentionally wrote Division before Multiplication (and Subtraction before Addition) on our 1/2 pager in an effort to further disrupt incorrect bad pathways related to OOO in students brains.

We are not 100% in love with this 1/2 page of notes yet, if you have suggestions for improving it, please let us know. Here is the word doc of our current 1/2 pager: 3-4a-order-of-operations When I teach middle school or 9th grade I don’t just hand this 1/2 page out. I do a fuller lesson on order of operations and we develop this list together. In my Advanced Algebra courses, I speed this up a bit and have the conversation of this concept through a warm-up followed by this 1/2 pager.

**A few extras related to Order of Operations for your classroom:**

#1 I love using this GIF. Good stuff!

#2: Read this great NCTM article titled “**12 rules that expire in middle grades**“. Check out #5 – more great information.

nctm-12-math-rules-that-expire-in-the-middle-grades

#3: Use **Kate Nowak’s great ROW GAME** for Order of Operations. This is great for partners to self check their work motivating them to rely on each other to find their mistakes.

row-game-order-of-operations-1

#4: My favorite Order of Operations Lesson that I used when I taught Middle School was taken from an episode of Survivor (Spinning Propellerheads) that that got the math wrong. I first have students work in pairs to simply this expression (Here is the word doc: day-3-survivor-order-of-operations-answer:

Students (at least those in middle school) really struggle with this. After 10-15 minutes I tell students that this expression lost someone $1 million dollars and I have the video to prove it.

I then give a bit of background (what an immunity challenge is, the prize for winning the show…) on the TV reality show Survivor and proceed to show them **this video**:

Before showing the video I tell my students that there are at least 3 math errors in this video and their job is to find them. I then make exaggerated sounds of disgust at 0:53 seconds when Jeff Probst (the host) calls operations ‘math symbols’ and continue my exaggerated disgust when I see J.T. write

‘PEMDAS’ (he does not win) on his chalkboard at timestamp 3:46 as the host says ‘JT is doing the math’ (UGH!). I scream at the video when the winning answer is worked left to right completely ignoring the order of operations all together. This is wrong.

The students and I then have a great conversation about why we have a standard order for operations in mathematics. I follow this conversation up with some practice (here is a HW I used with my middle schoolers: u2-order-of-operations-homework or this partner task in class: u2-order-of-operations-partner-task) for students and **a challenge** to add ‘grouping symbols’ to the expression above to make it equivalent to one (the incorrect solution preferred by Survivor’s host).

Note: Here is the solution the TV show Survivor should have accepted if the winning solution had followed the order of operations: day-3-survivor-order-of-operations-answer

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