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In the newly adopted Texas Essential Knowledge and Skills (TEKS) standard 1c for each grade level states, “The student is expected to select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.” And this standard is stated in the Common Core State Standards under MP5 saying, “Use appropriate tools strategically.”
The term “tools” is used loosely here and doesn’t necessarily mean just those tools that you can manipulate with your hands but also encompasses the tools that are in your mind and built from strategies for problem solving.
In my classroom I always started by teaching my students the problem solving strategies that I wanted them to use. These strategies of course changed over time but in the end they were pretty much the same overall with a different focus on things that I wanted to see out of them in their work. Was I helping them by doing this? Was I hindering their thinking by doing this? The point could be made both ways but I also was open to letting students solve their problems in a different manner as long as they explained their thinking behind it. Communication was KEY!
Many of the abstract tools that we need to teach our students are DANG HARD! While it may take time to teach perseverance through a problem situation it is possible. Exploration through mathematical thinking and building conjectures- again totally possible although they take time. Now how about teaching students to develop strategies based on prior learning to help them solve current problem situations. WHOA, back up the truck there! You want me to teach my students to bring something out of the depths of their brains and use that to build on what we are currently learning? Yep, I sure do! Example time…
Around fourth grade students begin to learn how to add fractions with the same denominator. During this process they are introduced to fractions strips, part/whole relationships, composing and decomposing fractions, unit fractions and more. This is all part of building a solid fraction foundation. Now, look back at what they have been introduced to… fractions strips are a manipulative that allows them to “connect” their fractional pieces. We have been teaching students to build groups using manipulatives since Kindergarten. Part and Whole Relationships start when are working with Composing and Decomposing numbers back in Kindgergarten as well. Students learn the number bonds that create a larger number. Unit fractions… now this stuff is newer… students are to learn that anytime you have ONE PART of a fraction, it is a unit fraction and therefore teaches you what the smallest form of that fraction can be.
Now how can we build upon that? Well the next thing that students need to be able to do is add fractions with unlike denominators. To get to this step they must have an understanding of what we have already taught them as well as common mulitples of numbers. You can see in the examples provided that because the student was understanding of how to use bar models to represent fractions they were able to easily connect what the equivalent fractions were.
Not only has the student built on prior knowledge (a tool), they have also made conjectures (a tool), used bar models (a tool), and communicated their thinking (a tool). All of these TOOLS were used to solve one problem. Teaching your students different ways to fill their toolbox with strategies that will help them break down a problem is more than just a standard, it is a critical attribute to fulfill their understanding.
I can’t wait until next time
where I will be talking about choosing the right strategies to solve problems. I will discuss the different methods that I have used and taught with over the years and pros and cons of each so that you can decide what is best for your classroom.
Within the Common Core State Standards (CCSS) and Texas Essential Knowledge and Skills (TEKS) new standards have arisen to promote the communication of math skills. Common Core Math Practice Standard 3 states, “Students should construct viable arguments and critique the reasoning of others,” and Standard 8 states, “Students should look for and express regularity in repeated reasoning,” while the Texas Essential Knowledge and Skills Math Practice Standard 1f states, “The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas.”
Last week I talked about the importance of building communication in mathematics while focusing on problem solving and gave some ideas on how to incorporate those in your classroom. If you haven’t had a chance to read that post, make sure to check it out! Communication is the FOUNDATION for building proficient problem solvers and we must make sure that we aren’t the only ones that are communicating but our students are doing their fair share as well!
So what does it mean to analyze relationships in math? “When a student analyzes a mathematical relationship between two or more quantities, he or she looks for a pattern or a structure and uses it to solve a problem. He or she can see how two quantities are alike or different mathematically based on their attributes or properties.” (Strategies for Mathematics Instruction and Intervention, Weber/Crane-2015). Analyzing relationships is one of the highest levels on Bloom’s Taxonomy and therefore a very difficult area of understanding for students due to the complexity of situations and problems at this level.
When we are working with students we need to make sure that we are not only communicating effectively so that they understand what we are wanting from them but we also need to use words that will spark them to think clearly. Analyze may be a difficult word for many as it is so abstract so teach students other words that are similar so that they can think for themselves when working on an assignment. Here are some useful power words that you may want to use in your classroom.
A common way to use analyzing in the classroom is to choose two related concepts (or even numbers) and compare them. As you can see in this activity from an 8th grade classroom students compared the formulas for surface area and volume of a rectangular prism. While most students wouldn’t think twice about how they were related it is defintiely something worth making the connection over.
As I just stated, you can do this with numbers at the lower levels. Think of doing your Number of the Day program (or Number Talks) and how you can incorporate comparing numbers like 937 and 97. Students should see that 97 is approximately 10 times less than 937, the 7 is in the same place in both numbers therefore both numbers show the value of having 7 ones, and so on. There are so many connections that students can make with numbers and this allows analyzing in math to start at the early grades as well.
Building the deductive reasoning skills that are needed when analyzing items is important as that is what allows our problem solvers to think further and become that student that continues to ask WHY! What can you do this week to promote the analyzing in your classroom?
Within the Common Core State Standards (CCSS) and Texas Essential Knowledge and Skills (TEKS) new standards have arisen to promote the communication of math skills. Common Core Math Practice Standard 3 states, “Students should construct viable arguments and critique the reasoning of others,” while the Texas Essential Knowledge and Skills Math Practice Standard 1d states, “The student is expected to communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate,” and Standard 1f states, “The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas.”
Why is there so much focus on communicating mathematical ideas when focusin g on problem solving? Communication increases students’ self reliance and belief that they themselves can find answers on their own. We as teachers see it every day where a student will just want you to spoon feed them the information they are supposed to know so they can get to the next step. This is not teaching them to problem solve, this is teaching them to rely on others and not think for themselves. We have to change this mindset in the classroom.
Communciation is more than just talking about individual problems but is also about connecting topics from day to day. In my classroom I did great with allowing my students to talk about what they were currently learning and creating relationships with their peers on what they were learning. What I struggled with is making my students think to what they had already learned and make connections. So when I am teaching about ratios and proportions, have students connect their learning to what they know about percentages and fractions. How do they relate together? How are they similar? How does your knowlege of fractions help you develop your understanding of proportions and being able to solve them correctly? See how easy it is and can spark creative thinking with your students?
Learning to ask questions is a communication habit. Don’t get upset with the kid who CONSTANTLY asks questions, they are learning! I know it is hard when Trevor (totally made up student) is asking questions every 5 minutes and you just want to get through your lesson. Accommodate his need for asking questions by giving him some post it notes and allowing him to write them down when he has a question and stick them on his desk. You may end up sparking some really good conversations in your classroom with these questions. You may want to even create a Parking Lot for questions in your classroom and allow any of your students to do the same with questions they may have about what is being taught. This will limit the interruptions in your lesson especially when the questions may not completely align with the topic at hand. Build in time in your weekly lessons for classroom discussion/debate about some of these questions to help students develop their communication skills with each other.
“I know it but I don’t know how I know it.” How many times have you had this answer given in class? Many students struggle with the notion of being able to talk about their thinking and although they may know an answer but aren’t sure how they got there. This shows an inability to put math ideas into words on their own. Buidling in the time for classroom discussions/debates will allow students to start inputting ideas over time as they are comfortable. Challenge your students to contribue one mathematical idea daily in class whether it be on classroom brain dump done via Post It Note or outloud in class.
Ever student enters our classroom at a different stage on the communication scale (shown above). As teachers we are often overzealous in wanting to hear every student’s ideas that we don’t tend to think about what stage of mathematical communcation each of them may be coming to us at. We can’t expect a beginning problem solver to explain in detail their thinking and what they are doing. We have to allow them to communicate at their ability level and then foster that level to continue and travel down the path to the next level of communication. Our students all come into our classroom with a wide variety of skills, abilities and preferences when it comes to communicating and we must seek out new ideas to recognize that thinking and promote their growth.
Communication should ALWAYS have a purpose. As a learner we always prefer authentic tasks and communcation tasks that are motivated behind the lesson at hand. NO ONE wants to do busy work! Communcation comes in many forms from discussing problems within a group, writing out their solutions, oral explanations, etc. If a student struggles with one area of communication find what they excel in and focus on that! Just like the Post-It Note Parking Lot example earlier being great for written expression, allow students to communicate through technology. Submit questions via email, a Facebook group or Twitter. Heck, what about taking a picture and posting it on Instagram and tagging the classroom Instagram page. Students can them collaborate together and discuss the thinking that comes behind the problem.
In our classrooms we must create a culture of communication. Creating a classroom culture is more than just setting guidelines, it is also building up students to know how to talk about mathematical ides, know what it means to be respectful of others ideas and having tasks that require them to write for real communication. These three steps are easy to facilitate but also very crucial. Setting guidelines as a class not only allows students to take ownership but also helps to develop the family aspect. Once these guidelines are set, don’t just post them and be done but take time to model them regularly.
Later this week I plan to come back and discuss more about communication and give direct ideas on fostering communciation in your math classroom today beyond those given already. If you have any questions or ideas feel free to submit them in the comments so that we can all develop our mathematical communication together!
Don’t forget to check out Week 1’s post on Why Practical Problem Solving is Important. Next week will be all about Analyzing Relationships which you won’t want to miss!
What a whirlwind of a weekend around here. Friday morning I left for Phoenix bright and early (seriously before the sun even thought of rising) and when I got there I had the whole day ahead of me, what was I to do? My room at the hotel wasn’t ready yet so I decided that breakfast was on the menu. Off to Yelp! to find what to eat that might be interesting to try. Well, I found this AMAZING little restaurant called NCounter and oh my heck was it FABULOUS! Not only was my Chorizo and Eggs flavorful but so was the Passion Fruit Tea! Definitely recommend this place and will be back when in the area!
After I got a little nap at my hotel I had to do some driving around to grab some last minute supplies for my booth (candy of course). After a trip to 3 different Walmarts to get what I needed I was set for the next day. But before any of that could occur I got to meet up with my amazing teacher buddy Marie Cote (or as some of you may know her Sweet Tea and Squats… hehe). Marie is an awesome mom of 2 kiddos and a teacher at a charter school in
the area. We grabbed dinner (and drinks) at Pita Jungle and had a grand old gab session. Too bad we live so far apart! A great night was had by all!
The next morning was the morning of the Arizona Association of Teachers of Mathematics Conference and boy was I ready! I presented and exhibited at this conference last year and knew it would be great to come back once again. Bright and early (again… seriously was up at 5:30am) I awoke to head to Arizona State University (via Starbucks of course) to get my day started. Got the booth set up just fine although I forgot a tablecloth and I totally forgot to take any pictures.. DOH!
My session was during the first breakout and I got to talk to a ROOM FULL (we had to bring in extra chairs) of teachers about how to get students to go beyond just “notes” in their interactive notebooks. I told every teacher in there, “If we do not get students to communicate their thinking based on the lesson provided then we as a teacher can not be sure they have mastered the skill at hand.” I know this sunk in wtih many of them because I was seeing many of them write this down as well as getting questions about how to encourage students to communicate their thinking. This has honestly become my favorite workshop to teach despite it being something that the teachers have a lot of reflecting themselves to do afterwards and it takes extra planning, it is what makes Interactive Notebooks WORK!
And this conference wasn’t complete without a FANTASTIC conference put on by Mr. Que and Mr. D of Music Notes Online. Seriously I was on my chair dancing along with the I Did My Homework Song and if you aren’t already using this in your classroom then what’s up with that? Grab the Motivation CD today and you need to get to using it!
I personally used this in my classroom on Friday’s when my students turned in their homework and if they had at least attempted the homework for the week they got to stand on their chair and dance along to the song. The one rule was that both feet had to stay on the chair at all times! That means they could do the Stanky Leg but couldn’t do the Superman.
Whew! Such a great day overall and I loved every minute of it! Glad to be back home and getting ready for my next conference coming up in Oklahoma City in October.
Each school year we start with a new group of students and with that new crew we start fresh with working on problem solving in our math classes. I don’t know about you but over the summer it seems as if most of my students always forgot what they had learned about solving problems from the prior years so it was like starting with a clean slate.
The importance of teaching problem solving to our students stems from the fact that to truly learn mathematics we must teach more than how to memorize facts and algorithms. We as a teacher have to teach our students to think mathematically. Beyond the mathematical thinking students are able to develop mathematical reasoning and then apply that reasoning to computation and concepts to solve math problems. Taking time to make our students proficient in problem solving will allow them to stick with it when the problems seem hard.
With the new Mathematical Practice Standards in the TEKS and Common Core we are specifically given skills to make sure our students understand. These skills should be integrated daily into our math lessons and that is where I want to take time to help you. Over the next few weeks I will go through each of the standards and help you break them down for your classroom.
When it comes down to it, our students become problem solvers by solving problems. How are you implementing problem solving every day in your classroom? How are you providing feedback to your students? My goal is to help provide you tips and trick to implement problem solving in different ways as well as helping you give immediate feedback to your students to provide an effective learning environment for you problem solvers.