Ever since winter break, my students and I have gotten into such a great math groove. They’re engaged and excited in a new way. Ready to really tackle difficult problems and I get to witness their learning skyrocket. One of the things that really helped my classes make it to this point is when I made a “Rewarding the Process” bulletin board. This bulletin board is completely devoted to posting student work that shows multiple ways of solving the same problem. The answers aren’t goal, it’s focusing on the work and strategies being used.
At the beginning of the year I was really struggling with getting my students to care at all about showing their work. They wanted me to tell them the fastest way to solve a problem and begged for short cuts that they didn’t really understanding. Back in September I once overheard a student say “Yeah, 250 plus 10 is 2,500 because you just add a zero.” Yikes! The horror I felt when I realized that many of my students had no conceptual understanding of these short cuts that they had been taught before. And that the lack of conceptual understanding lead them to misuse these short cuts left and right. I spent most of the fall teaching and modeling different ways to solve a problem. We used bar diagrams, guess and check, pictures, formulas, and a lot of manipulative modeling. They had been exposed to so many ways to solve a problem, but the excitement still wasn’t there.
After winter break I set up this new bulletin board and titled it “Rewarding the Process”. I put up some motivational quotes, including my favorite from S. Gudder “The essence of math is not to make simple things complicated, but to make complicated things simple.” And we began working. Each class I would start with an anchor task. Students would have a colorful half sheet of paper and worked in partners to solve the problem. Once they solved it one way, they could get another paper and try a different method. Partners took turns explaining the method that made most sense to them and at the end I would pick a few groups to share their work. I would then pick as many different methods as they came up with to post on the bulletin board. If a partner group used manipulatives to solve the problem, I would use my phone to take a picture that I could post. I wanted to be sure they knew that the use of manipulatives is a valid way of solving a problem. Students were so encouraged to have their unique work up on the bulletin board. And what sixth grader doesn’t love the fact that they knew how to solve a problem in a way that was different from everybody else. They love thinking that they discovered a new tool or strategy. This bulletin board has been a critical part of building community of creative problem solvers in my classroom. If you don’t have space for a bulletin board, I would encourage you to find another meaningful way to reward the process. Let students know that you are interested in how they solved the problem and why their strategy worked, not just the answer.
After we returned from winter break my focus has really been on getting kids to reflect and speak about their thinking process. This is a huge part of Common Core in both Mathematics and English Language Arts. Traditionally, we can see how easily English Language Arts lends itself to metacognition, discussion and critical thinking. However, when we really look at our math instruction and how it aligns with the Common Core standards it is evident that there are just as many opportunities for students to examine, explain and share their thinking as well. I remember when I was in my teacher’s preparation program I had a professor that posed a math problem to us and asked us to solve it on our own. After a bit of time passed she asked for volunteers – not to share their answer, but to share how they solved the problem. To me this was revolutionary. Sure the answer, the correct answer, was important – but just as important was how you arrived at that answer. Your journey and your cognitive strategies were something of value to be shared with the class. As different people shared their process with the class there were many “aha!” moments heard throughout the room. I have to say it was pretty cool. I thought of myself as a student who always struggled in math and at times felt like I just didn’t “get it.” I thought about how if I had a teacher who taught math as something not totally black and white I may not have had such a hard time. Above all, it gave me hope for the future of math instruction! Now in my own classroom it is my mission that children see that there is not just one way to approach math, but many ways, and the more ways you know the more tools you have in your toolbox. When students talk about math and are challenged to explain how they arrived at their answer we are asking them to engage in higher-level thinking. Higher-level thinking leads to critical thinking and problem solving, and isn’t this what we want for all our students and in all content areas? When students share their strategies or their thinking they are showing that there are other perspectives and more than one way to arrive at an answer. In Kinder this can be as simple as using fingers to count, making a mental picture or utilizing manipulatives. We long ago abandoned the “one size fits all” approach to teaching and now it is truly starting to feel like math is catching up.
Easier said than done. That’s how I often feel about having students work effectively with partners in a middle school math classroom. At the beginning of the school year I had each of my students complete a “Math Interest Survey”. The survey asked students to rank how they liked to learn and work in math class – whether they preferred to work alone, in partners, or in groups. I was surprised, but pleased to find out that almost 70% of my students preferred working with a partner. We all know the great things that come out of partner work; more engagement, deeper learning, concept mastery. But we also know the obstacles that often stop us from using partners as often as we should.
One of the most helpful and time-saving things I have implemented this year is a “partner key”. To ensure that I have strategic and varied partner groups I gave my students a “partner key” to keep in their agenda. This key has three symbols and three names on it, so that each student has a “tree” partner, a “star” partner, and a “car” partner. The tree partners are students who are about the same level. The star and car partners are grouped strategically so that one student is inherently the teacher and the other is the learner. Depending on the task I will decide which partner I want them to work with. This took a little bit time upfront, but saves me from having to try and strategically choose partners on the spot. I wish I could take the credit for this idea, but like most of the successful things I use in my class I borrowed this idea from another teaching blog. Sharing is caring!
The other aspect of partner work that we teachers always worry about is keeping all students accountable for the work. I personally remember being the student who did all of the partner or group work and then just put everyone’s name on the top of the paper. Of course, I’m sure that I had amazing teachers who monitored the classroom diligently and somehow my group just got away with it. Ha! All joking aside, the point of partner work is not for the higher achieving student to do all of the work while the kids who really need the practice just skate by. First, I require my partner work to have both people’s handwriting on the paper. Whether it is going to be turned in or not I check that both partner have contributed. Students also know that they are responsible for knowing and being able to explain everything on their joint paper. I am not accepting the “Uhhh, what I didn’t write that. He got the wrong answer, not me!” Sixth graders are eager to push the blame on someone else, especially when working with partners. Asking students to present partner work gives them the opportunity to explain their thinking. If they don’t understand how their partner came to an answer it becomes a great time for that other partner to become the master teacher and prep their friend so they can then explain the work to someone else. And just like that we’ve circled back to the main intention of partner work – to allow students to talk through problems, to learn from a peer, and eventually to be able to teach their peers. I know that none of this is new and we’re all believers in productive partner work, but it’s always a good idea to reflect on how that’s going in our classes. I encourage you to think about one new way you can incorporate partner work this month!
After reading about Kapur’s theory on learning thorough productive failure I am both inspired and frustrated. I think that a lot of us teachers can resonate when I say that his theory makes a ton of sense, but I see some major obstacles when I look at fully implementing it in my own classroom. As a sixth grade teacher I am going to examine Kapur’s theory from the middle school perspective.
The “Preaching to the Choir” Part
Kapur believes that students, who are presented with unfamiliar math problems and then left to solve them, without teacher guidance, build a deeper understanding of the math concepts. I think that concept makes sense to anyone. In college I worked as a barista at the school’s main coffeehouse. I didn’t learn how to make creamy cappuccino foam from reading directions. I learned how to do it from making a lot of really bad cappuccinos and experimenting with techniques, until I finally realized where I was going wrong and how to fix it. We can all agree that learning from your mistakes leads to a deeper understanding.
The first obstacle I see is in the statement “learn from their mistakes.” A lot of people make mistakes, both in life and academia, but never learn from them. Many of our students are discouraged by their mistakes and too quickly jump into the “I’m not good at math” boat. There needs to be a complete shift in how we view mistakes. On one hand we encourage students to persist through the struggle and not to see their mistakes as roadblocks, but on the other hand most of our schools requires us to give them assessments that point out all of those mistakes and penalize students. The relationship a middle school student has with their mistakes is often one filled with embarrassment, frustration, or anger. There needs to be a complete shift in that relationship in order for students to be able to productively learn from their mistakes.
Motivation! With any lesson it can seem like a quite the task to motivate a room of middle school students, but especially when they are being asked to stay motivated through struggle and confusion. And while nobody wants to put it out there, we need to be honest and admit that middle school students can be a rather lazy bunch. Yes, I said it! They are lazy and almost always look for the easiest or quickest path to an answer. Please don’t get me wrong, I love working with this age group, but staying motivated is not typically on of their strengths. For this method of teaching to truly be successful we need to focus on fostering students’ intrinsic motivation and desire to problem solve in unique and creative ways.
When all is said and done, Kapur’s theory could lead to a deeper understanding and true mastery, but if it is poorly implemented it could lead to heightened levels of frustration and apathy.
The weather has finally cooled, fall break has come and gone and now with just a few days into December the talk around my classroom has turned to Christmas. In our class we build our calendar together each day and I was pointing out important days coming up in December; when our book order is due, Hanukkah, Christmas, Kwanzaa, students’ birthdays and our Winter Program performance. Of course I was immediately hit with, “How many days ‘till Christmas?” I explained to my students that we will not be in school during Christmas and that we will be on our winter break during that time. We counted the days until Christmas together. After counting I realized that the number of days at this point does not mean much to my students. The students need to SEE the amount of days. They need something tangible. We currently count the days we have been in school, and use straws to represent those days. With this in mind I decided to create a “Countdown to Winter Break” for my students (i.e our last day together before the vacation). I created a simple paper chain out of construction paper to represent the days left until winter break. As part of our morning routine I have a student remove a link from our chain each day as we count down the days until our break. Then we count the remaining links and I write the number up on the board. This way the students have a visual and tangible representation (the chain links) in addition to the number that represents the amount of days. This further reinforces the concept that numbers represent quantity. I really like using the chain links but there are many different ways you could represent the days; marbles, craft sticks, or any type of manipulative. Further, you can use this strategy for any time you want to count down to an important day, such as the 100th day of school, classroom celebrations, spring break and even the last day of school! This concrete representation allows the students to physically interact with an otherwise abstract concept like time. The students love taking away the days, which is a foundation for subtraction and seeing the chain get shorter each day!
One thing I have noticed is that common core has got some parents freaked out! Obviously the level of “freaked out” varies amongst individuals, grade level, curriculum familiarity, student, school, etc. However, in my experience I have heard multiple times from parents that “this is not the way I was taught” when it comes to math in particular. This can create uneasiness, anxiety and downright frustration amongst parents. One way I am addressing this in my classroom is by frontloading vocabulary. In my weekly communication via a classroom newsletter I let my parents know what is going on in the classroom and what we are working on. For math, I look at the chapter and assess what might be problematic or in need of a further explanation for parents. As teachers we often forget that our parents may not be so well versed in what a ten frame or number bond is. I have a little section in my newsletter with parent tips and this is where I include any math vocabulary, terms or strategies that I think they might need more information on. For example, when we were learning about composing and decomposing numbers I gave them a definition and an example of each. Further, I explained how it related to addition and subtraction (something I know they are familiar with). I think it is always important to relate new concepts to the bigger picture. Just like our students, parents also benefit from understanding why it is important to learn something. Visuals are also very helpful. I have found this is a great way to keep the lines of communication open between home and school. In addition, parents know that you are available to assist them with any concerns they might have when it comes to common core. I am continuously working to make the common core standards less “freaky” and frontloading vocabulary with parents has proven quite successful.
After three great years of teaching Kindergarten I have made the move to upper grades. This year I am teaching sixth grade at a K-6 elementary school and I couldn’t be happier. We will be following Robin through this school year as she makes the transition from Kindergarten to sixth grade.
We’re on our third month of school and I finally find myself identifying as a sixth grade teacher. If I had met you two months ago and you asked me what I did I would have replied, “well I’m teaching sixth graders this year, but I used to be a Kindergarten teacher.” As if I were standing there with one foot in the water, wondering when I would be ready to jump in. But we all know the thing with teaching is, you just have to jump in. The kids need you to be 100% and there is no time to waste.
The best part of this move to sixth grade (besides my commute being a tenth of what it used to be) is that I am teaching just math and art. While I enjoyed teaching other subjects, my heart has always sided with math and I love getting to pour all of my energy into it. I am also really excited about the sixth grade curriculum. It is so applicable to daily life and the real world links don’t seem like a stretch. Sixth graders, in the midst of some major emotional and physical changes, can be very self-involved. Being able to relate math to their worlds is key for engagement.
While I have absolutely loved the switch to teaching sixth grade math, there are definitely some new and unique challenges. When I was teaching Kindergarten, I could feel the gap between beginning and advanced students, but sixth grade is a whole new beast. The gap between low and high achieving students can be incredibly daunting, seeming to grow and widen with year. How to push my advanced students into deeper thinking while helping the many who don’t know their basic facts is a daily question. And honestly, what I find to be the more alarming problem is the gap between levels of student interest. Kindergarten was full of bouncy five year olds ready to take on the world. I felt like I always had so much joy and energy in the room during math time. The little ones loved problem solving and what they believed was just “playing” when we pulled out the manipulatives. I remember everyday when we would review the daily schedule the students would cheer when we got to math. By the time these kids reach sixth grade they have pretty much divided themselves into two groups, those who like math and those who abhor it. As we all know, this gap in interest and will only widen the gap in achievement. A goal that I have set for myself this year is to bring back the love of math. What are some ways that help create a classroom that is excited about math again?
We all know that fluency is more important than ever since new math standards have arrived. Each Eureka/Engage lesson in K – 5 provides 3 fluency activities for each lesson. I’ve been in classrooms where teachers are devoting 30 – 45 minutes to fluency because their students don’t have mental math skills and don’t know their multiplication/division facts. This is definitely not recommended! Fluency is important, but so are concept development and problem solving. How are you handling the issue with your students?
Many teachers utilize something called an exit ticket as part of their daily instruction. An exit ticket-basically a student’s “ticket” to exit the class-is a strategy that can provide feedback to a teacher about a lesson, require students to do some synthesis of the content being covered and challenge students with a question requiring some application of what was learned in the lesson. Exit tickets are used frequently and with great success in the older grades and in secondary. The teacher generally writes a question or poses a problem that pertains to the lesson and the students must answer or provide a solution in writing in order to exit the class. This is a great way to get instant feedback on your lesson and serves as an informal assessment for the students.
I have always loved the idea of the exit ticket but I needed to make it work for my kindergartners! Posing a question on the board and requiring a written response would clearly not be developmentally appropriate for my kids. I decided that a verbal exit ticket would work very well in our classroom. Recently after conducting a standard district math assessment I noticed that while many of my students were able to count all the way up to 100 (or close to it) many were struggling with number recognition, particularly the TEENS. They were either switching the numbers, for example changing 18 to 81 or saying “seventy” in instead of “seventeen.” While many children enter kindergarten with strong number recognition skills, many do not come will this prior knowledge and therefore review and repetition are crucial. This version of the exit ticket may be used for any content area but for the purpose of this article I am focusing on mathematics, specifically; number recognition. The beauty of the exit ticket is that it takes minimal prep, it is quick and it is a great way to interact with each student. I have my students line up and when they are exiting the classroom for recess, lunch, a library visit, physical education, etc. I will use this strategy. I put a number and dots to represent that number on a post-it. The dots allow the lower level students to be able to count the quantity and say the number. I place the post-it on the frame of our door at eye-level for the students. I stand in the doorway and point to the post-it and ask them “What is the number?” if they aren’t sure I tell them “Try counting the dots.” Many children take the initiative to count the dots on their own, others will need the guidance. The children get excited to show their knowledge and the children that need extra help with this are exposed to the content repeatedly while waiting for their turn. Not only is it a low-stakes activity for children who may be a bit shy, as opposed to coming up to the front of the room, but the repeated exposure is beneficial to all students. If a student is struggling I will have the students behind them in line help them out. The goal of this strategy is a quick check-in and for the students to feel successful. It is not meant to cause anxiety; so make it fun, show lots of enthusiasm and provide assistance if needed. When I want to challenge the students I might put two different numbers up on the door frame to identify and/or ask them which number is greater or less than? I have numbers 1-30 on cardstock, laminated with a magnet so I can easily take them on and off our magnetic door frame and reuse them. Post-its of course will always work too! As the year progresses and their math knowledge increases simple addition and subtraction problems will become exit tickets.
The possibilities are endless and your own exit tickets should be aligned with the content you want to assess, review and that is applicable to your students and their needs. You can also switch it up and make this strategy an entrance ticket that can be used when entering in the morning, or after recess/lunch, or whenever. When the children get more familiar with this strategy it would be possible to transition to written responses but I strongly suggest beginning with a verbal exit ticket and working up from there.
Good luck and Happy Teaching!!
Barkelonging down the trejinga, the skeppy flappit floppered the flippit.
You may be asking yourself, “What is that even supposed to mean?” Regardless of what it means – if it indeed means anything – almost 100% of people could answer the following question correctly:
What did the flappit do?
If you said that the flappit floppered the flippit, then you got the right answer. Is it meaningful or has it improved your understanding of the sentence? Not at all. This is exactly what our students experience when we focus solely on the procedure. While many are able to perform the process and arrive at the right answer, the path they have taken has not been meaningful. They are just blindly following the steps outlined for them. Conceptual understanding of the problem is essential for students to develop deeper levels of understanding and be successful with mathematics. Without understanding the WHY behind the HOW, they will traipse along until they stumble over something new. Once that happens, it is very difficult to undo.
Tips for developing conceptual understanding:
- Always provide a familiar context when introducing a new concept.
- Follow the CPA approach – start with a concrete item and build into the pictorial stage before introducing the algorithm. One such example of this can be seen while using the Place Value Disks to introduce the four operations.
- Give students a chance to explore a concept independently or in small groups before providing formal instruction.