Building Thinking Classroom-It is not just hype!

This past school year was a banner year. Students in our district were back in person 100%, well, unless they had Covid. If students had Covid, they were permitted to join the class via webcam and participate along with the live class. Regardless, most students were physically in school and I spent the previous summer bringing out all of the stops to get ready for our return to in person learning. During this time, I was fortunate to be part of a small group (forgive my Scrubs reference here, but the arrangement could be best referred to as a brain trust!) of math teachers from around the country. We exchanged ideas with one another and the book Building Thinking Classrooms (BTC) was introduced to my life. My virtual colleagues inspired me to purchase the book and I was hooked immediately. 

Peter Liljedahl performed years of research before determining 14 practices to implement a thinking classroom. Although encouraged to adopt only a few practices at a time, I attempted to implement aspects of all 14 practices and I don’t regret diving in this way. Rather than listing the 14 practices all here, I would encourage everyone to purchase and read the book! If you have been hearing about Building Thinking Classroom or Peter Liljedahl, there is a very good reason. 

My first admission-I still have a lot to develop in many of these practices. Regardless, is there a teacher out there still growing and learning that does not feel that drive for improvement every year or day for that matter? I cannot express how excited I am to revisit and refine my own spin on BTC practices again after having a year’s attempt under my belt. The learning curve was steep, but worth every strained brain muscle and synapse in my head. In future posts, I will focus on different aspects of the 14 practices, but this first entry is just to encourage everyone to jump on board. If anyone wants to join our brain trust, please reach out to me. As always, teachers are better together!

For those of you out there new to the BTC philosophy, don’t be afraid to take liberties with the suggestions in the book. In my interpretation, BTC is suggestive, not prescriptive! We, as teachers, do our best when we make practices our own, so long as we don’t lose the heart of the idea. Keep the passion, hold on to the heart, but be your own BTC advocate and adventure seeker. You will not regret taking this leap!

In response to the question…

What it’s been like teaching during the pandemic. All these opinions posted below are primarily mine from my own experiences. I sort of speak on behalf of others, but don’t really have the right to do so.

We teachers are not experiencing the very worst as far as situations during this pandemic. I consider my friends who are in healthcare and those who work in supermarkets or have had to close restaurants or even those who have lost their jobs. But…like so many other people, teaching has been an exhausting challenge in a way that has changed my perspective on it forever.

1. We worry our students who may or may not be vaccinated are going to get Covid or have seen our students come down with Covid. It is terrifying since so much is still unknown about the short or long term effects from the disease. Some students have described a very difficult illness and their reported suffering has been heartbreaking. We just want everyone to be ok.
2. Many students will not keep their masks above their nose (sometimes mouths) and we are constantly having to function as “The Mask Police.” We hate this role, but understand we are trying to do all we can to keep everyone safe. I teach 7th graders, but I have had to chastise my first grade daughter while out in public for the exact same transgression multiple times. My heart goes out to all elementary teachers!
3. Many teachers also hate wearing masks (I know I do). I am not saying we should not have to wear them, I completely respect any information or efforts to protect me or others, but this does not mean I have to like it! Think about the nature of a teaching…Teachers must be heard clearly, which means we have to project our voices with a mask on. Our strained vocal cords hurt our throats every single day, but I don’t want to lower the mask to drink water because, well, it kind of defeats the purpose of wearing a mask.
4. Hybrid teaching can be summed up in one word-nightmare. I challenge anyone to try to interact with people physically in the same room and over a screen simultaneously with a changing number of students on a daily basis in different home environments with cameras on and off. Creating lessons that will be effective for both in class students and at home students have limitations. This seems especially true with my teaching style as a math teacher-but I have heard from many other teachers in other subjects that it is difficult for them as well.
5. And about that hybrid situation-wifi strength, technology, sound, cameras all have different consistencies-essentially, the unpredictability of teaching rose to unimaginable levels when the at-home learning became part of the in-person teaching responsibilities. Sometimes, lessons lose instructional time in attempts to troubleshoot connection and/or other issues for our quarantined or Covid-stricken students.
6. Standardized tests are still part of the equation-we don’t love this decision.
7. SEL-that is on teachers to manage as well, even on those days we are barely hanging on ourselves.
8. Our colleagues have gotten Covid-we don’t always have staff to cover all the absences, so we are covering their classes too. We are all expecting to contract Covid sooner rather than later, even with all of the mitigation strategies in place. The sub shortage is a crisis.
9. The boundaries that might have existed between home and school have blurred. or completely disappeared. Personally, I have made countless phone calls on my private phone to parents during off hours, because there are just not enough hours in the day to get it in or parents are unavailable to speak when we can do so. My weekends are often entirely dedicated to creating new lessons and/or grading. This has not been entirely because of the pandemic, but it has been exasperated by it. If parents truly understood the amount of hours that have been poured into the creation of each lesson, I do not believe an unkind word would be said about education.
10. We are all entirely saddened that the goalpost keeps getting moved on if/when “normal” school will return and wonder if such a life exists out there again. We crave teaching without masks, temperature checks, quarantining, social distancing and remote education just as much as everyone else. I dare say I might wish for this more than most…

#Findyourmathpeeps

On the precipice of a new school year, I have found myself at a familiar crossroads. Lest not forget we are approaching the third solid school year with Covid calling the shots…But I digress.

Once again, like so many other teachers, I spent the summer immersed in research, webinars, books, and as a result, feel inspired to integrate a multitude of new instructional techniques. As I get closer to the start of school, I find myself overwhelmed by all of the shiny new teaching strategies. I WANT TO USE ALL OF THEM. 

This happens each year. I know if I try to adopt all of these strategies simultaneously, none will be incorporated with fidelity or impactful for my students. So, once again, I find myself whittling down all of the new ideas and trying to choose the most effective options. When I attend a Q&A from a presenter, I tend to ask a version of the question, “What would a year roll out plan look like using method A,B,C.” The response is usually vague, as it would have to be since each attendee has different grade level and school expectations.  It’s really a crazy expectation for the wonderful presenters to truly answer such a question.

The difference this year is I have been lucky enough to find like-minded teachers across the country to work with in planning. We represent California, Maryland, Connecticut, Texas, and Iowa! How is that for a national movement?

Our amazing group does not have the exact same curriculum, but we have the same passion, dedication, and fall into the category of “Math instruction is also my fun hobby.” Don’t get me wrong, I still find myself going down all of the instructional rabbit holes, but now I have people trying to point me towards better paths when I would normally continue to get lost. 

I cannot pretend that this professional circle of professionals has cured me of wanting to try everything at once, but having a group rally around each other to determine what might serve our students best has given me something that there is never enough of-unwavering support. There is no better gift for a teacher.

My wish is for every teacher to find what my group and I have found. So with that,  I would like propose a new movement for math teachers #findyourmathpeeps. 

As we’ve been saying since Covid hit, we are all in this together, not only that, we are better together. So go, before you dot another I, cross another t, pass go, collect $200, etc., find your math peeps!!! Your students will thank you.

Dear Students of 2021

Dear Students, 

Happy New Year! Yes, we have exited the dreaded hashtag of 2020, but although the number we use for our year has changed, the hybrid and/or virtual school setting has not for most of us. So for those of you who are learning from home, we the teachers have a very important request:

Please turn your cameras on and keep them on your whole face for every class.

We know there are reasons you don’t want to turn you cameras on. We hear you and we understand, but we are still going to insist that you turn them on.

When we ask to see your mini whiteboard, your work, or a show of thumbs, we actually need to SEE it in order to serve you best. Virtual teaching can be very effective and provide the interaction necessary to produce amazing results for everyone. We teachers can see nodding of heads, we can notice confused looking faces, even on a small screen. In some situations, we have even learned to read lips! There are great technological teaching platforms out there like Demos or Pear Deck where we can literally view your work in real time. There are so many powerful tech options out there and we use them with extreme gratitude, but nothing can replace the live interaction between you, your friends, and a teacher during a lesson.

Let’s be frank…We teachers are not naive, we know that some of your friends do not actually attend class, even though their names are visible on our Google or Zoom meetings. We are aware that these same students may watch youtube videos, visit Tik Tok, play video games, and text friends while they are supposed to be participating in their class. When a camera is not turned on, it is a bit trickier to manage these distracting behaviors, so you may not know that we understand that these situations are happening. 

If your friends, these students, were physically with us, these issues would rarely occur. If they did occur, we could see it with our own two eyes and redirect them on the spot towards a more effective environment. In the virtual world, we need to call their name out to a black screen or strange icon with their name. If that fails, we have to track their attendance and participation, and then we need to reach out to their parents/guardians to let them know what is happening. We can’t just ask your friends to hang back for a second at the end of class or quietly talk to them about their distractions during class because even if we put them alone in a breakout room, the students physically in the classroom would overhear a conversation meant to be private.

Some of your friends keeping their cameras off have opted for a 100% virtual school experience this year, which is completely understandable, so our chance to physically speak with them privately is a challenge. Although we can and do send emails about concerns to these friends, these messages often go unread unless we cc their parents. In pre-Covid times, teachers would not have to bother parents with such learning issues at all. We also know your friend’s parents have to participate in their children’s education far more than is fair during this pandemic. Regardless of the distracted behavior or the possible consequences for it, the real issue is that there is a lack of learning when these alternate activities are siphoning your friend’s attention from their lessons.

We also know that for every friend that has been distracted during class, you have other friends who have been thriving in an online environment more so than they did in physical school. Perhaps there were awkward social issues that were disrupting their learning opportunities when they had to sit in a classroom that they no longer have to deal with. Maybe these students are introverts who have discovered that they learn best in the comfort of their own room. Please know, success for these students doesn’t just magically happen. These are your friends who ask questions when needed and participate in the chat box or during live conversations. These students are the same friends who complete all of their assignments, reach out to their teachers with questions, and sign up for virtual extra help sessions if necessary. Guess what else these students all have in common?

They keep their cameras turned on. 

Which brings us back to the original plea that is not just for your friends, it is to you as well as all students. Please show your teachers your full faces and use your camera in every virtual class meeting (if this is what your teacher expects). If you were unaware, please know that we teachers are putting in everything we have in order to make this long distance learning situation productive, and dare we say it, even fun! The truth is, this situation can only work optimally if you the students come along for the ride. We pledge, there are enough seats for everyone.

With love, support, and dedication to your development,

Your teacher

Teaching in a Pandemic World

Two days before our school shuttered their brick and mortar doors we were told at an instructional team meeting, “The schools will likely close for 2 weeks. We will assign student work for those two weeks and then continue on from where we left off. These two weeks may not include new learning.”

Whatever horrors Covid-19 was causing around the globe, none of us really anticipated the long-term impact it would have on teaching and learning. Perhaps I misspeak, maybe the more accurate truth is that I absolutely did not imagine a world where my classroom and my students would be physically off limits for the foreseeable future. 

During the first week into Quarantine, I sent messages to my students to say hello and checked in on their families via email and through our LSM, Google Classroom. In addition, I served as both a mom and  a teacher to my 5 and 8 year old girls. I jumped on Google Meets to chat with students who had questions about the mandatory (but not idea) two weeks worth of work.  To all of you with toddlers or babies who had to do this, my hat is off to you!

In piecemeal fashion it came to our attention that we might not be able to return to our buildings in another week as it had first been anticipated. As we learned more about the virus and its serious impact on the health of our world, the severity of the pandemic made its mark, especially on my New York neighbors. As a school district, we collectively recognized that an end date could not be anticipated to this school quarantine situation. I can only speak for me, but I went from, “oh my gosh 2 whole weeks away from my students to, oh my gosh,  how do I teach effectively in an online environment for a few more weeks to,  oh my gosh, how will I keep my connection to my students and their connections to each other in this new and unexpected situation that has no end in sight?”

Teacher to student and student to student connection in education is vital.  Like so many of my dedicated educator friends and colleagues, I began submerging myself into webinars on the many free (sadly, but understandably, for a limited time) technology options for educators. If there was a tool brought to my attention and it could be used in a math context, I investigated. This meant a dive into the following platforms: Loom, Kami, Flipgrid, Google Forms, Classkick, Nearpod, Peardeck, Educreations, Desmos (a long-time user already), whiteboard.fi, Jamboard, all things Google Meet, Bimoji, Quizziz, Assistments, and others I am certainly forgetting as I type this reflection. Working 17 hour days to prepare lessons was the average, and I couldn’t wait to get back to the classroom where I could be with my math peeps again.

Kyle Pearce and Jon Orr encourage us to be better curators. I certainly was not decisive enough at this time, sending one platform out to my students after another to see what worked and what did not. Thank goodness my students were amazing sports and game to try whatever I sent their way. Eventually, I gravitated to a few favorite tools, one of them being Loom. As great as it is to have some consistency, this too came with challenges. Creating a Loom voice over became an all encompassing event, and hundreds were created during the quarantine.

Allow me to provide a sampling of anecdotes to the numerous challenges of recording a voice over component in a small home with a family. In one memorable recording session, I was almost done and my youngest daughter loudly announced, “I have to use the potty!”  In a separate Loom attempt, my dog, who rarely makes a noise, howled randomly, and in another, my husband did not realize I was recording, and entered the house with our two girls singing loudly about sharks. When something like this happened, that particular recording had to get scrapped. It would often take me 9 or more attempts to get one Loom presentation done because of issues like this and also, because my internet connection would fail, but the internet connection is a whole other issue…

As I searched for tips from brilliant educators around the world, I studied their work spaces. I glanced at elaborate classroom-like backgrounds that educators were expertly forming in their homes. The images I studies were downright inspiring, but the only space I had to create a workstation was at my daughter’s toddler table in the basement. My students got a kick out of the Barbie Dream House in the background.

Regardless of what tools I used, my main goal (other than keeping the math thinking and investigating in check) was my student’s social and emotional well being. We were all suffering in our own ways, but collectively, we were a society grieving over the loss of shared time and space. Arguably, when schools moved to a virtual set up, what we all grieved over the  most was that loss of socialization in the classroom. I tried purposeful activities to try to maintain a connection with my students and to each other in any way possible, including number talks, three act tasks, and Desmos activities, but they just weren’t the same.

During synchronous sessions,  several of my students would enhance our experience with their humor and perpetuated our classroom dynamic in the Google Chat. The jokes and witty responses continued, even when students’ microphones were muted. My intervention groups were small, so we were able to participate in discourse as if we were together in many instances during our time together, and it was wonderful.  Was it as  effective as it was when we were physically together? No. Many students elected to turn their cameras off, which I really disliked. I often felt like I was talking to myself when staring at inanimate icons instead of familiar faces. Because every household had different noise in the background, the rule for the school was to keep the microphones muted unless asked to speak. That situation is absolutely not the status quo of my in-classroom culture. Yeap Ban Har jokes that a quiet classroom is suspicious. Imagine the addition of faceless students plus those muted microphones and you can envision that suspicious situation amp itself up.

I am not going to pretend that my live sessions were perfect and being of a growth mindset know I can and must do better for all students who choose a 100% virtual situation for this upcoming school year. In addition, regardless of the model of school we begin with this year, there is no guarantee of where we will end up. We can all end up back 100% virtual in a moment’s notice, it happened before and we now know it can and likely will happen again. I want us all to feel like we (teachers, students, parents) have some control in education in a world where overall control is sorely lacking.

This entry is really an invitation, a plea to all educators, parents, students, etc. to share what platforms and techniques worked to keep that connection alive during the virtual experience. Please reach out to me and share out any tips, tricks, lessons, methods, etc. that keep the student-student-teacher connection alive and thriving during this time. We have been and continue to all be in this less than ideal situation together. With that in mind, how can we all be in this together for the better?

 

Comparing Fractions-Reference Sheet

After taking Graham Fletcher’s fraction course, attending both Greg Tang’s Math Institute and intervention workshop,, I have completely transformed my own understanding of fractions. The following are the strategies taught to me by extraordinary professional math educators as best summed up as I can offer for anyone who has been on a fraction journey too. 

Strategies for Comparing Fractions, Estimating their value, etc. Please keep in mind that when comparing any fractions, we are either told or assuming that the pieces we are comparing have the same size whole.

Strategy 1) Common Numerator* and sometimes specifically Unit Fractions. To use this strategy (possibly obvious) the numerators have to be the same. You can find equivalent fractions to make this happen or the fractions might start out as having common numerators.  Let’s say you are comparing ⅜ and 3/12: You can see that ⅜ of the pizza is larger than 3/12 of the pizza. This is because you are slicing the pizza pie into smaller sections or partitions when you slice it into 12 slices rather than 8 slices.  This strategy works for comparing any fractions that share a common numerator because the concept in this regard never changes. Unit fractions strategy is this exact strategy, and more specifically, the numerators are both 1. This would be used in comparing ⅕ and ⅛. ⅕ of a pizza would be greater than ⅛ of a pizza. The larger the denominator, the smaller the partition (when the numerators are the same).

Strategy 2) Benchmark numbers -Benchmark numbers are those numbers that we can visually picture with ease. In fractions, imagining what ½ looks like or 1 whole is likely easiest for most of us. To use the benchmark of ½, all one needs to do is to look at the relationship between the numerator and the denominator. If you multiply the numerator times 2 and your result is a value equal to the denominator or greater than the denominator, than the fraction has a value greater than one half. If you multiply the numerator by 2 and the product is less than the value of the denominator, the fraction has a value less than a half. If you have two fractions and one is greater than a half and the other is less than a half, then you have figured out which fraction value is greater. The inverse operation holds this example as well. When the numerator is exactly ½ the value of the denominator (found by dividing the denominator by 2), you can quickly determine whether the fraction value is greater or less than a half.  If you are looking at comparing fractions and one of them is greater than one half and the other is less than a half, you can conclude that the fraction value less than a half must be less than the fraction value greater than a half.

Here is an example:  Comparing 7/16 and 30/59

If you multiply the numerator of 7 x 2, the result is 14. 14 is less than the denominator of 16, so 7/16 has to have a value less than one half. For this fraction, the numerator would need to be 8 or greater (16 divided by 2) to be greater than one half.

For 30/59, if you multiply the numerator of 30 x 2, the result is 60. 60 is greater than the value of the denominator of 59, so 30/59 has to be greater than a half. Exactly half of the value of 59 would yield 29.5, of which 30 is greater.

Therefore, 30/59 is greater than 7/16.

You can also use the benchmark of 1 when the opportunity presents itself. 

If you have a fraction value greater than 1 whole (13/12)  and you are asked to compare its value with a fraction that is less than 1 whole (11/12), then you can see that 13/12 has to be greater.  Just be on the lookout for a fraction that has a numerator greater than the value of the denominator compared with one that does not!

Strategy 3)  Common Denominator*-This strategy is the inverse of the Common Numerator strategy. Let’s say you are comparing 5/14 versus 9/14. The denominators are the same, meaning, the fraction is cut into the same pieces. In this situation, 5 out of those 14 is decidedly fewer than the 9 out of 14, so the 9/14 must be greater. 

Strategy 4) Missing Piece-This is another example where pie, cake, brownies, etc. provides an excellent visual. If you are comparing fractions such as ¾ and ⅘ , students tend to think they are equal because the numerator is one digit away from the denominator in both fractions. However, anyone who has an understanding of the value of fractions will immediately know that they are absolutely not equal. If you have two pies and one is sliced into exactly 4 slices, while the other is sliced into 5 pieces, you can think about the unit fraction and recognize that each individual slice in the pie that is cut into 4 pieces (or ¼) is larger than the unit fraction/one slice in the pie that is divided into 5 pieces (or ⅕). Once you remove one piece, (the missing piece) the remaining pie is left. Since the unit fraction of ¼ is a larger piece, removing it from the pie means that the amount of pie remaining must be less than the pie that only lost a ⅕ slice of pie. Think about it, someone took a larger piece of the first pie, so there is less remaining! So, going back to the example of ¾ and ⅘, ⅘ has to be larger than ¾ because the missing piece is smaller than the missing piece from ¾ of a pie. The larger slice of pie removed, the greater amount that is remaining behind. 

Once students master these strategies, their conceptual understanding of the meaning of fractions increase, which will make working with them in advanced concepts much easier. Increasing number sense through incorporating these strategies is time well invested, no matter which way you slice it!

 

*Often the fractions will need to be renamed/converted to equivalent fractions to use this strategy.

The Importance of Play in Math

Dan Finkel is famous for saying something along the lines as “Books are to reading as play is to math.” I cannot express how much that viewpoint has transformed my thinking and how often this year I have spread his message. However, if you are like me, you used to introduce math games all of the time, watch the students play, interact with a few, and think, “Well, that was fun!” It wasn’t until I took Graham Fletcher’s course and watched how he played a few games with his daughter that I realized just how powerful games could be in a classroom.

With my intervention groups, I tried to follow the lesson structure of launching with an open ended task, direct instruction, and then follow up with practice through games. It was this year that I realized that yes, you can have formative and summative assessments through math games and it has been…well…a game changer!  The small change I made was taking turns sitting in with groups for longer periods of time to make sure the students were playing in a way that would provide them the practice they needed, and also, allowing me to check in on their understanding. Previously, I would circulate around the room-check for students to be on task, maybe answer a question here and there, and call it a day. Now I see how much my participation mattered. I make sure I stay with each group long enough to be able to formatively assess or summatively assess their understanding.

Here are some games I recommend for this:

Conceptual Bingo. This is a game that was ordered from a supplier. Here is a link if you are interested in purchasing:  http://www.cmmstore.com/category.aspx?categoryID=3  I had tried this game in the past without impressive results. This year, I teamed students up and gave them multiple boards. The competition element made it supercharged. With having multiple game boards in front of them, they did all sorts of extra math practice to see if they could find a match with each card. We were studying percents, so I used the game about percents to check for understanding.

Sales Discount. Desperate to have students practice the ever-useful math strategy of finding how much a discount would change an original retail price, I reached out to Graham Fletcher and asked him if he had any suggestions. Sure enough, he came through! He sent me a photo of a game he had not yet tried, so I recreated it. The link is below.

https://docs.google.com/document/d/1PyuXX8HTkFBUlpvjX8hwHxL5jreOscTFMNj5FGwf4d4/edit?usp=sharing

Anything four in a row. Think Connect Four, use different colored counters for different players. You can use this for absolutely any concept that needs practice. I have used it for all fraction operations, decimal operations (including the discount game above), and I am about to create my own in  other concepts..This idea really came from Graham Fletcher. I cannot recommend his Foundation of Fraction course with the enthusiasm it deserves! https://gfletchy.com/blog-2/

Closer to Zero. Who needs integer practice? Look no further than this game. Materials wise-it is a dream because all you need is a deck of cards. Red is negative, black is positive. It is very similar to Blackjack, except the object of the game is get a total sum that is closest to zero. Students can request to get “hit” and have up to a total of 7 cards per round.  This is a quick and intensely competitive game that propels students to hunt for zero pairs and use their number sense. If you are looking for a five minute filler-this will make your and your students’ day. I may create an equivalent version of this with rational numbers soon.

Anything by Greg Tang. I have to admit, I haven’t purchased his games yet because I have spent my quota on books and supplies for the year. However, Kakooma is the next item I plan to purchase when I can do so. I was unbelievably blessed to learn how to play Kakooma with Greg Tang himself (at a conference I attended this summer) and saw first hand how much mathematical thinking was involved during the game. What can I say, the man is a genius. https://tangmath.com/

I have used a few other games as well, but I have found the ones I mentioned in this post provide the most opportunities for student engagement, success, and productive practice. Obviously, I cannot take any credit for creating these games, but I am happy to try anything out and report on it. I always say that the best ideas are likely out there, we  have the good fortune of having the ability to discover them for our own classroom from our friends who are willing to share. And on that note, if you have any additional games that are easy to recreate (or better yet, can share a free online template) and would like to share them here, please do so via a direct message or in the comment section.

 

Game on!

Dan Meyer has Done it Again: Nana’s Chocolate Milk

You might crave a glass of chocolate milk while teaching this lesson!

I am not sure who to thank first, Robert Kaplinsky or Dan Meyer… If it were not for Robert Kaplinsky, I would not have come across a tweet of excited anticipation to try Dan Meyer’s ratio lesson entitled Nana’s Chocolate Milk. If it weren’t for Dan Meyer, there would be no such lesson. What can I say, both mathematically inclined gentlemen have my gratitude.

I have been talking about ratios with students for as long as I have been in the middle school grades. In my, oh my gosh, I can’t believe I used to teach math that way years, I would simply teach students how to find an equivalent fraction by scaling up or down by the same factor. Perhaps, as an aside, I would mention real-world tie ins like, recipes, but I really could have done so much more. In Dan’s lesson, he delivers the conceptual gift of ratios with ease through the power of chocolate milk.

In his three act task, Dan “accidentally” puts in an extra scoop of chocolate into a glass of milk, even though he knows that Nana’s preferred chocolate milk beverage has a 4:1 ratio of chocolate powder scoops to cups of milk. He asks the simple question, how can he fix the situation?

The students were invested as soon as they saw the video clip. They could all relate to accidentally putting too much of an ingredient in something. For some, it was to much milk in cereal, for others, it was accidentally over-measuring a tablespoon of vanilla in a batch of cookies. The point is, the Mathematical Practice Standard of abstraction and quantitative reasoning was in sharp focus here.

When students were challenged with how to fix it, some students immediately suggested to spill it out and start again.  Although, definitely a solution, that option was not deemed an economical option, and the double number line was presented. As I walked around the room and heard students debating each other about how to change the milk and/or chocolate amounts with keeping the Nana-preferred ratio in tact, I noticed the mathematical conversations were appropriately everywhere. Some students immediately thought to double the amount of milk and add three additional scoops, and then put the remaining chocolate milk aside in the fridge for the next day. Others didn’t think that another cup of milk would fit in the glass (part of act 2) and asked if fractions were okay. Some had not quite understood the ratio concepts by their responses, which was addressed in the closing thanks to the anticipation model touted by both Graham Fletcher and Robert Kaplinsky. The point is, there was rich, robust conversation about ratios through the relatable chocolate milk scenario.

When students were told that the glass could not fit 2 cups of liquid, students wondered how much liquid displacement occurred with the powder to see if adding an additional ¼ or ½ cup would be a more appropriate answer. I wasn’t sure the first class had understood the idea of ratios, so I used Dan’s sequel on Nana’s eggs. It was clear that the lesson was powerfully effective as they all came up with correct solutions keeping the egg to flour ratio intact immediately. Every, single, student… I have always been a huge fan of cooking, baking, and math. How fun it was to watch the concept of food make a beautiful day in the math classroom!

The lesson link is here:  Nana’s Chocolate Milk

 

What Should Math Intervention Accomplish?

What is math intervention?

Perhaps it depends on who you ask. When I have been asked to explain what I do in my current role, I have found it difficult to respond with a quick phrase, which is likely the type of retort people are looking for when asking such a question. On several occasions, before I have had a chance to formulate a thought, I have heard, “Oh, so you work on math facts and torture kids who hate math already with more math.”

Math intervention is not just about math facts and algorithmic procedures. Although, for the record, I do believe that all students do need to know their basic facts. Not because it was something I learned and knew as a kid, but because not knowing them is too taxing on a student’s working memory. Students who struggle with their math facts are realistically struggling with something much more vital, which is number sense. That, in my opinion, is a math interventionist’s number one focus.

The follow up question I am often asked is, what is number sense? The best answer I have come up with, thus far, is that number sense incorporates all of the aspects of common sense, but with numbers. Let’s say, for example, you are driving to a restaurant. Typically, you take a specific route in your car from your home to the restaurant. One day, on your way to the restaurant, a pole has crashed on a street you usually drive through to get to the restaurant. If you do not have common sense, you just wait in your car for the pole to be fixed and the road to be cleared. This could take hours, days even, so obviously, not the most efficient strategy at your fingertips. Instead, you could try another route, decide to travel to a different restaurant, or even turn your car around and go home and cook a meal. The point is, you have flexible thinking and options of your choosing.

The same is true in math. If you are solving a division problem and there is no calculator in sight, you can of course, use long division. But, if you make a mistake in the procedure, you will just be thoughtlessly following steps and generate an incorrect solution. (Like waiting endlessly for that pole in the road to be fixed).  If instead, you decided to first use estimation, you might calculate a precise answer using long division, but in the event of an error, you would be able to stop yourself and at least know to go back over the problem and try to fix the error.

Another focus of intervention in developing student number sense is training them to contextualize math problems.  If a textbook problem requires students to solve 312 divided by 56, there is little sense making involved. Think about it, numbers are adjectives. Outside of Textbook Land, You are never really dividing 312 by 56, you are dividing 312 somethings into groups of 56 other things. If a student is trained to visualize a scenario to bring meaning to the problem, sense making is happening and the solution will reflect this to be the case. It could be as simple as there are 312 sandwiches to be shared by 56 people, or $312 to be distributed to 56 charities. Visualization is a powerful tool in any content area. The point is, once visualization and estimation are options, any algorithmic procedure at that point is fine. It won’t matter if a student prefers partial quotients, long division, even exploding dots. The important component is that students aren’t blindly following steps like Math Zombies or Math Robots (Dan Meyer and Robert Kaplinsky terms that I use all of the time), they are instead, actively making sense of the math in their lives.

So, to answer the question, math intervention, in my humble opinion, is helping to bridge student thinking and procedures. Some days, the students achieve this through a 3-act task, other days, it might be through lengthy number talks, and still others, it is through math games to promote automaticity. What intervention isn’t, is an additional burden on students who already struggle in math. My hope for these students is that their time with me builds their appreciation and understanding for math. Anyone with common sense can achieve number sense, and anyone with number sense can learn to love math.  That is really, in my humble opinion what intervention is, teaching everyone that they can love math, or at the very least, hate it a little less.

Are Procedures a Dirty Word in our Current Math Classrooms?

I accidentally created an online argument between myself and other passionate educators on a Jo Boaler dedicated Facebook page. After countless hours of endless research and sending messages to math educators I admire, I still had a question that could not seem to be answered. In my perpetual attempt to improve my craft, I innocently asked how to make those “procedure teaching days” meaningful for students who had prior knowledge. So, for example, if we have students who go to Kumon, Russian Math, or even have access to a private tutor and are taught lots of procedures, how can we challenge them on days we are having the rest of the class engage with them? Many of these students feel the discovery piece is a waste of their time since they already know the end result. As often as I remind students that procedures without a solid conceptual foundation are weak, there is still push back and I was looking for a way to engage them, not for my ego, but for their betterment. Furthermore,  if a teaching goal is still for students to be able to perform algorithms to help them “Look for and make use of structure,” what is the best way to make those days of instruction inspirational for everyone?

My post got a lot of comments, and not all were kind. Disgusted responses such as, “You should never teach your students procedures, what are you even doing as a member on this Facebook page” were peppered in along with, “Following, if you find out, please tell me.” Ok, I am exaggerating the first statement, but only a little. The message was underlying and clear. Although my post was not about the straight teaching of procedures, it got me wondering:

Is it true, are procedures a dirty word in our current math classrooms?

Now listen, I am all about trying to have students discover and conceptualize before utilizing any procedure. A circle unit my students recently completed as a review is a great example. Students  measured the distance around a circular object and then across the diameter of it to provide them an opportunity to discover the pi ratio. Before the lesson, I asked them what they knew about pi. They responded that it was 3.145…and used in circles. Many commented on “pi day” celebrations of the past. None knew it was a ratio. This was a small example of procedure without substance.

For the area of the circle, students studied images of a circle being sliced up into wedges and slowly transforming into a rectangle, which they were then able to compare to the area formula for a rectangle, hence, discovering the formula for the area of a circle.

For circumference, students engaged in Rolling Tires 3-Act Lesson by Andrew Stadel.

However, after these periods of discovery are over, students still apply the formulas in and out of context with the procedure. The conceptual piece is enhanced by the algorithm, and vice versa. My critics questioned my inclusion of this piece of teaching, so I have to question it too. However, how will isolated discovery without practice allow concepts to enter the long-term storage area of the brain? Aren’t both needed? I think yes.

I have researched (through the printed word, professional development courses, and on-line videos of educators I admire). The same message delivered in different ways is that the algorithms of the past are not meant to magically disappear. “There is no new math” is a popular phrase. In my current state of teaching, I agree. Algorithms are wonderful, important, and revolutionary methods. It is not wrong to know how to use algorithms, but it is not productive to use them and have no idea why they work. Shouldn’t we have both?

I really want to know.