Teaching Strategies
Integrate language arts skills into mathematical instruction
Today most of the problems students solve are in the context of word problems (or an even friendlier term we use in second grade is "story" problems). It helps students see equations in real world situations which make the process of solving problems more meaningful to students, and it adds an element of students not just solving an equation, but engaging in reading and skills related to reading - understanding the main idea and details of the problem. Students have to analyze what is important information and what is not. By implicitly teaching students and acknowledging for students they are integrating reading into their math curriculum then they can begin to see connections between the two subjects and hopefully apply the skills they use to answer questions in reading to answering questions in math. Story problems allow students to "attach meaning to operations" and "extend their understanding of situations involving operations." 8 "Mathematics occurs in forms much more like word problems. It is by doing word problems that students realize the importance and applicability of mathematics. Word problems are also a very effective way to reinforce both concepts and calculational skills." 9 Word problems can be written by teachers to get students to start thinking about numbers and operations in context. They can also be written by students. When I challenge my students to write their own problems to match equations, they are thinking at a higher level to demonstrate they know what is being asked of a situation. I make sure to read word problems to or with students so that students who are struggling readers do not get hung up on the words, specifically the words they do not know or understand, and can spend time focusing on understanding the math behind the words or situations.
Another language arts and math integration is to teach students the vocabulary associated with the different structures that problems can be identified with having and have students categorize what type of problem they are solving. The structure vocabulary includes: Change plus or minus with initial amount unknown, change unknown, or result unknown; Comparison greater or fewer with greater amount unknown, smaller amount unknown, or difference unknown; and, part-part whole with one part unknown or whole unknown. This helps students understand the relationship between the number(s) they know in a problem and the number(s) that are unknown in order to understand how to solve the problem. Examples of all these types of problems for different grade levels can be found with the Common Core support documents. 10
Language arts is also integrated with math when I teach my students to use labels or units to describe numbers. Sometimes problems ask for students to add or subtract common objects (red balls and red balls) and sometimes they do not (pennies and nickels). When students are solving problems they need to label parts of the problem and especially the solution so others can know what each number is referring to (i.e. the units of the number). Students use this skill so they know when they have common units or to recognize when they need to find common units. This will later be applied when working in measurement, geometry, fractions, algebra and other topics in math. If my students can develop a habit of recognizing and labeling numbers as beginning learners, then they will apply it when working in more challenging math situations as older learners.
Use models to illustrate addition and subtraction and the relationship of how many more.
I try to scaffold instruction from concrete to pictorial to abstract. Some of my students will need to use manipulatives longer than other students and that's ok. Manipulatives help students reason and work with numbers in a concrete hands-on way. Numbers can be represented with any material. The number four can be represented with 4 crayons, 4 cheerios, 4 paper clips, 4 snap/pop/unifix cubes, 4 teddy bear counters, 4 base-ten-block units, or many, many other materials. In my classroom I call manipulatives "math tools" and they are referenced in the Practice Standards 4 and 5. When students engage in the use of manipulatives they are representing numbers in a very concrete way and supporting their reasoning for the operations they are conducting in equations. I always have students represent their models using pictorial representation on paper. When students are working with numbers to 10 or 20 then it's ok for them to use manipulatives to show one-on-one correspondence, however, for numbers past 20 I encourage students to start building number sense by making numbers represented as tens and ones and hundreds tens and ones so for students using math tools in the classroom it is a good time to transition to using Base-Ten-Blocks.
To prepare students for measurement and integrate the idea of measurement of length, and help them understand the relationship of size, students can line up cubes in a train to have a visual of the difference of sizes as a number increases or as a number is compared to another number. This is especially important in helping students understand the difference between one and ten, ten and one-hundred, and even one and one-hundred. Size models can look different for different problems. By lining up the trains on a number line I can reinforce the idea that length combination amounts to addition. (See Appendix II for Length Model).
Number Line
A number line is a perfect tool to use when modeling addition and subtraction situations. Number lines are used to build an understanding of size and distance – foundations for measurement of length. Before even introducing the number line I have students practice modeling equations using snap cubes or tiles which align to using an inch as the unit length for the number line or unit cubes which align to using a centimeter the unit length. My students line up the cubes and place them with the zero edge of a ruler (either inch or centimeter ruler or side depending on what math tool your students are using). My students line up their cubes for an addition problem and they can see the magnitude of the numbers they're working with. When they line them up along a ruler they are attending to the Practice Standards 4, 5, and 6. After students practice pairing using cubes along a ruler they can begin to conceptualize how to create a visual of a number line on their paper and iterate or evenly segment numbers across the number line and model the equation using the number line. Chris got three books. Then he received five more books. How many books does Chris have now? (See Appendix II for Number Line model).
Games
Students love to play games. Games are a great way to help them both gain an understanding of what they are learning and also repeatedly practice new knowledge and information. There are three games that I know and use to help students manipulate equations to build ten. It is helpful to play the two card games with number cards. I use cards that have both the number written in standard form and also include a pictorial representation. I always make sure students record the equations they create when playing math game.
"Tens Go Fish" is played like "Go Fish," but instead of asking a partner for a pair, a student asks for a number that would help him/her make ten. Each player gets five cards and places down any two cards that when added together make ten. Then students take turns asking each other for a number that when added to a number in their hand will equal ten. Students replace each card they place down with a card from the "Go Fish" pile until all the cards are used up. Students also record the equations while they get them in order to practice all the different ways to represent ten. 11
Another game is called "Make Ten." During this game students place number cards in an array of 5 x 4 and take turns finding combinations of ten. (A more challenging version is placing the cards face down and playing it like a memory game trying to find tens while remembering locations of cards). With both of these games students will record the equations they make with their hands in order to help them better commit to memory what they're doing and learning. 12
"Cover up" is a third game that can be played with tiles or counters. Students count out ten counters and place them on a surface. Students take turns covering some of them. (You can use a book or piece of construction paper to cover them). Then they write an equation using a square in place of the missing number, make a prediction, guess, or reason how many counters are covered and record it in the square. Then students should uncover the counters to check and see if they are correct. I always tell my students to validate themselves if they got the right answer with a check, star, or smiley face on their paper next to a correct equation.
Class Discussions
One of the Practice Standards requires students to "Construct viable arguments and critique the reasoning of others." The best way for students to apply in this practice is to engage in class discussions. Students need to share the work they are doing with others and defend both the process they use and the product they achieve. Class discussions can look like whole group, small group or partner. Sometimes it is hard to engage all students in a discussion if it is whole group, but class discussion is not only a great way for a teacher to assess student understanding, but also for students to share their strategies and thinking with others. At the beginning of the school year I establish the routine/procedure for Class Discussions. We brainstorm why class discussions in math are important and students tell me the rules for discussion time. I record the rules and make them visible as a review for every discussion. In addition to brainstorming rules and expectations for discussions, we discuss the role of the listener. Once my students have developed ownership on why class discussion is important to a mathematical practice I can begin facilitating discussion conversations. This is a great and safe time in the day for my students to ask questions, offer and illustrate different strategies and ways they think about and solve problems, share feelings, review and use vocabulary, explain their process and steps for solving in their own words, and much more. I can anticipate student participation or lack thereof by meeting and having a small group discussion with students who may be struggling and give them the opportunity to practice both the problem solving that they're doing, and their thinking and discussion about what they're doing. As an adult learner I value the time I have to discuss ideas and concepts with my colleagues to learn new strategies and work through struggles I'm having as an educator. Usually a light bulb goes off when I hear how another teacher solves something or figures out how to get his/her students to understand something. Students can benefit from classroom discussions as well.
Ten Frames
Ten Frames are a great way for beginning learners to see ten in a very visual way. As my students progress through their understanding of ten they can be used to see groups of tens and singles. Ten Frames are an array of two rows and five columns that illustrate a quantity. Students use what they can see to figure out how many they need to make a set of ten. An example of a Ten Frame is below.
Ten Frames help students visualize addition or subtraction. Students can see this Ten Frame as 5+2+x=10, 7+x=10, 10-x=7, or 10-7=x. Second grade students can use the variable x or some other symbol to represent the missing information, or they will probably express it in a sentence saying or writing 10 minus something equals 7. After students become proficient with understanding one Ten Frames they can be given multiple Ten Frames to figure out a quantity and/or missing amount. (See Appendix II).
Fact family houses
Operations have relationships. Subtraction is the undoing of addition just like division is the undoing of multiplication. Students can understand numbers better when they begin to see numbers in terms of their relationships, and when students understand their relationships they can choose which way of solving problems works best for them. Some students are more comfortable with solving subtraction problems through counting back while others may choose to count up. When students understand the relationships between numbers they can engage in making these choices. As students are working toward knowing and understanding their math facts and mental strategies it is sometimes helpful for them to visualize the relationship. One of the ways I help students to do this is to teach them how to build number houses. Visualize the house having a rectangular base and triangular roof. In each corner of the triangle students record three numbers that have a relationship. In each corner of the square base students record an equation that illustrates the relationship they have. Fact family houses can be built for any three numbers that have a relationship. Fact family houses can also be used to help students understand that numbers in a word problem have a relationship and students just need to figure out what the relationship is – what part of the family the missing number belongs in. Jan used 65GB of data on her phone. Her plan only allows her to use 100GB of data each month. How much more data can Jan use this month? (See Appendix II)
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