# data[].attributes.body
1.  One of the toughest concepts for third graders, especially at the beginning of the school year, is subtracting with regrouping.  Students can memorize the algorithm, but it's not very helpful to them unless they understand why and how regrouping works.  We try and use a lot of different models, (both physical manipulatives and pencil-and-paper,) to show students how regrouping works and why we do it.  Models are not only important when introducing a topic, but they should also be used throughout the year to check for understanding.
2. Most third grade students struggle with taking what is in their head and turning it into written expression. While many of them can talk about and explain their thinking, putting it into sentences usually proves more challenging. I feel the best way to help students is for the teacher to be a model as often as possible. Teachers should try and take every opportunity to model their thinking in words, and give students an opportunity to practice informally as much as possible.

# data[].attributes.body
I love teaching adding and subtracting just for this reason! When learning this concept I always "show" them that there is different strategies that can use to come up with the sum or difference. I stress the importance how some friend strategy mat be different from their own and that is okay! That they need to pick the right one for them, not their friends, When I do problems for my math meeting, I have them use the strategy that is just right for them and they model it for the class and talk through working out the problem.
They maybe able to write more at the end of the year, but I would love for them to draw more pictures and explain their thinking!

# data[].attributes.body
1.  I love using Base ten blocks with my kiddos to help them understand what they are doing.  They first need to understand that 57 is the same as 4 tens and 17 ones as this deeper understanding of math will be immensely beneficial to them when regrouping three and four digit numbers with multiple regroupings.  They must first learn the "why" instead of just memorizing the algorithm.  We also draw hundreds, tens, and ones instead of using the Base Ten blocks.
2. I feel that the first step in getting kids to write more to explain their thinking in math is to first model how to think through what they are doing and why and then model the writing using the correct time order words and math vocabulary as the teacher writes out her thinking. Then having students pair up to discuss the what and why in various problems that they solve. Then taking the time to have them write out their thinking and sharing it first with a partner and then with others in the class.

# data[].attributes.body
1. I like to use Base Ten blocks with my students to support subtracting with regrouping. At the beginning of the school year, this is a tough concept to master. Using the Base Ten blocks, the students can see "why" there is a need to regroup. We also draw the Base Ten blocks, so the students can cross off and add to the smaller denomination. For example, cross off a ten (stick) and add 10 ones (blocks). Then the students show that same regrouping with the numbers. They can count how tens are left and change that number, and the same idea for the ones.
2. Modeling can support students with writing because it provides them an idea of how a drawing can be put into written expression. Teachers need to model written expression. I think a good strategy to strengthen student's written expression, is to have the students tell you what to write to explain a drawing from a whole group lesson. Another thought is that the students could work as partners to complete a written expression. Or even as one partner draws the model the other partner writes down how the example is being completed. Maybe students could demonstrate a solution to a math problem by drawing and recording it on FlipGrid and another student watches the FlipGrid and writes the solution - then share with each other. This might be a way to help with the thought of not having enough time. It could be done during centers. One more thought is to have a written expression and the students show the work - opposite way - maybe? For the students that have difficultly with reading, the written expression could be recorded on Google Classroom.

# data[].attributes.body
1. Base 10 blocks are some of the first manipulatives that we use with the students for place value. We use these to help show how the value of a number changes as we move left or right across the place value chart. We could also shift the place value blocks in order to use them for decimals, to help students see that one tenth is bigger than one hundredth and so on. Students are already familiar with base 10 blocks for whole numbers, so it could be an easy shift to teach the same concepts with decimals.
2. I think students need to have that model for writing so they know what is expected of a written response. They can write paragraphs in ELA in 5th grade, but can't seem to write a complete sentence in math. So I think it is important we model for them what a written response should include. I know many of the students know what they are doing and why, but the don't always write it. In order to strengthen their written expression they need to have models and practice doing so in concrete, representational, and abstract phases of learning.

# data[].attributes.body
1.  We use decimal tiles in fifth grade.  These tiles are helpful because the students learn that base ten blocks are ones, tens, and hundreds, but what do we do when we have decimal numbers.  Therefore, the students use the same idea as the base ten blocks with the decimal tiles to regroup and make numbers when adding, subtracting, and multiplying decimals.  I think that there are students who could use the base ten blocks and get the same understanding.  I also think there are students who could draw these tiles instead of actually manipulating them.  The decimal tiles evens the playing field for all students to participate and use the manipulatives.
2. Writing in math needs to happen more often. I totally agree! However, students need an understanding of what they are doing to be able to write about their strategies. I think our number sense routines are helping students to describe more of the strategies they are using and will help us move the students to complete more writing in math. Hopefully, as we continue down our road of building consistency at ACES, we will incorporate more writing throughout math and the concept will build from year to year.

# data[].attributes.body
1. Models that I have used to make math more concrete: base ten blocks for place value, addition and subtraction. fraction tiles, number lines, and bar models for fraction units. Open number lines are my best friend for elapsed time, but maybe there are other ways students would show it if given the opportunity. Time: judy clocks. Multiplication: counters to use create equal groups with before moving to drawing those counters, but I should I provide more options. I am really interested in thinking through how I could be providing my students toolboxes to pull from at each grade level that I teach.

2. A big idea from this chapter was that not only do representations help our students make sense of the math by making it more concrete, it helps make it easier for students to talk about their math understanding. So if modeled at least once a week during whole group where students are given an open ended question that prompts students to represent an answer using models, students could share with partners their ideas, and discuss the similarities and differences. And then to connect it to writing, students could have a math journal where they share their model and explain with words how it shows the answer to the problem, and then also if time allows show their partner's model, and explain that one as well. I think you move to students independently writing their math thinking by first showing a lot of think alouds, where you are taking them through your thought process of explaining their ideas. Then you work together as a class to create scaffolded writing frames or sentence starters to help students get started/ and be more organized when sharing their thinking when trying it more independently. It can move from being modeled in whole group, to guided practice in small group with the teacher, to an independent activity during centers for students to complete and build on throughout the week. And then maybe students are able to share that out during the launch or closure of a lesson.

# data[].attributes.body
Our students connect very well with Base Ten Blocks and it helps to lead into the beginning stages of 4th grade addition and subtraction. It becomes a challenge when our numbers grow into 4, 5 or 6 digit sized numbers and then representing them with models. Even when we are using the multi-digit numbers to add and/or subtract the base 10 blocks help especially with the mental process of regrouping/borrowing numbers during computation. If you ever visit my classroom you will see us routinely using drawing and diagrams on our white boards to create pictures, charts, models to help and assist us with problem solving. Our goal is to make abstract ideas/concepts more concrete and visible. Writing in math is also important because it enhances our understanding more by requiring the students to explain their thoughts and tell why in writing. I encourage students to use sequencing words such as first, next, last, finally, etc. to help develop the multistep understanding of problem solving.

# data[].attributes.body
I love using AngLegs. I used my PTO money last year to buy a complete class set:) They are great for angles and 2D geometric figures and their properties. The students are completely engaged and enjoy using them. My other go to are fraction bars. With these, they can compare, order, add and subtract. An idea would be to give each student a maniplulative and have them write what they think it is used for, or purpose, in class.

# data[].attributes.body
Using models and manipulatives in 5th grade is an area I feel we have worked very hard over the last few years to explore different possibilities and incorporate both previously used and new options.  We use base 10 blocks early in the year for reviewing and introducing place value and computation skills.  We then familiarize the students with using decimal tiles as an alternative to base ten blocks, as it can sometimes be difficult for some of the students to understand how the same blocks can denote two different place value representations of numbers.  Number lines are another modeling strategy that we use frequently, especially during our Number Sense routines.  By the time you get to 5th grade skills that often use larger value amounts, it starts to become difficult to use some of the manipulatives that can be incorporated into math lessons at the lower grades.  To continue to build consistency in materials and modeling strategies, we have created many different skill mats that we frequently use with our students.  We designed them on the computer, printed them on cardstock, and laminated them so that they can be used repeatedly with Expo markers during all components of our math block, and they are huge paper and time savers. 😊

An area I feel we could certainly make improvements to is incorporating more writing into our math block.  Having seen the many shifts and changes that have occurred at our intermediate level in math over the last ten plus years, I think this is probably the one component where I can say we previously spent more time modeling and practicing.  As a result, I would have to rate the overall level of our students’ written responses as being less well developed now than they were a few years ago.  Truthfully, I would have to say that time is the biggest factor that has worked against continuing to make improvements to this area.  We have worked very hard to incorporate verbal discussion and explanation, and this can be done with a quicker turn and talk with a partner, small group, or even in a whole group setting, that it is often the more time-efficient and “easier” option than doing it through a written response, which we all know can be a VERY long process for some of our students.  I would love to see this be a primary focus area for our math team to put our wonderful minds together to make improvements to.

# data[].attributes.body
I agree with the book that "through representations, students make their understanding visible and it helps them gain a deeper understanding." In fourth grade we use a variety of models throughout the year. When thinking about our fraction unit we use actual plastic fraction bars and pattern blocks, drawings of fraction bars and number lines to model what a fraction looks like, to compare fractions, to find equivalent fractions, to discover what a unit fraction or a mixed number is, and to add and subtract fractions. I think it is crucial for the students to actually see what the numbers mean. An example of this is why when you add and subtract fractions with like denominators do you not add the denominator too. The kids get to see it not just have memorize and go with it because we said so. We can show them why it does not look right or make sense if we do add the denominators too. At the small group table you will see packs of fraction bars laying on the table, in whole group you see pictures of the fraction bars to move on the screen for them to see during our slide shows. We have also found websites with virtual manipulatives so kids can discover on their own using various manipulatives. It can be difficult for students to explain a topic in math so using models can help them be able to describe in their writing what they are seeing. They can see how adding the denominators does not make since and then write about it easier than without models. I think the two biggest strategies we can use to help students strengthen their written expression using models is to do more of it and to have teachers model this writing. It reminds me of a retired teacher who taught reading. One of his beliefs for increasing a students reading ability was to read, read, and read.

# data[].attributes.body
One of the tougher concepts I have taught was the concept of integers.  I have primarily taught this concept using number lines.  Incorporating the use of 2 different colored counters to represent the negative and positive numbers would be useful when teaching how to add and subtract integers.  I could also use diagrams to help the students see patterns and discover the rules for adding and subtracting integers.

A lot of times, students have a difficult time explaining their thinking in math.  When I ask my gifted student how they solved a problem,  I often hear, "I did it in my head."  An important part of getting our students to explain their thinking in math is to model our own thinking.  Another way we could help our students with writing is to work through problems as a class; model the problem together and write down how we solved the problem, step by step. We also need to give our students lots of opportunities to work with other students to help practice and become more confident in modeling and explaining their thinking.