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 Celia MacDougall Math Summative Assignment
Elementary Mathematics
Summative Assignment
Celia MacDougall
1 Celia MacDougall Math Summative Assignment
2 1) Annotated Bibliography Fractions are a concept that many students struggle to grasp and many teachers struggle to teach. Due to the high importance fractions play in our everyday life it is important to investigate and develop best practice so that students become proficient in the area. The ultimate goal is to have proficient students who can apply their knowledge in everyday situations and communicate their knowledge. This bibliography explores best practice in teaching fractions. D’Ambrosio, B.S., & Kastberg, S.E. (2012). Building understanding of decimal fractions using grids can help students overcome confusion about place value. ​Teaching Children Mathematics 18​(9): 559­564. The focus of this article is addressing the problem of how to order a set of decimals such as from smallest to largest. A typical problem would go as follows: Place the numbers in order from smallest to largest: 0.606, 0.0666, 0.6, 0, 0.66, 0.060. Illustrate and explain your thinking about this problem. Students struggle with this problem, they are unable to solve it correctly and their justifications are also skewed. In order to help these students, the researchers introduced a grid that was subdivided into tens, hundreds and thousands. Using a grid proves very useful in a grade 4 or 5 classroom. Since some numbers are difficult to represent on a grid it is important that students have an understanding of how tenths relate to hundredths, how hundredths relate to thousandths and so on. Based on the difficulties students experienced, the researchers suggest focusing instruction on one particular grid at a time to show different decimal quantities. There also needs to be an emphasis on the place­value system. It is important to teach students what a grid means and how to properly divide it into the appropriate pieces so that they are able to truly understand what decimals mean. Flores, A., Turner, E.E., & Bachman, R.C. (2005). Posing problems to develop conceptual understanding: “two teachers make sense of division of fractions.” ​Teaching Children Mathematics 12​(3): 117­121. Two teachers who each teacher a grade 4/5 split class collaborated to develop a lesson about the division of fractions. They looked to explain the connection to the algorithm of “invert the second fraction and multiply.” The teachers used to fraction kit heavily in their planning. They worked alongside each other but each teacher posed their own problems. The worked through a variety of problems with the same denominator and different denominator. They observed that when working with pieces of the same size, what matters is the numerator. By understanding the rule and the role of the numerator and denominator the teachers were able to effectively create a lesson that would allow their students to see the connection. This article Celia MacDougall Math Summative Assignment
3 highlights the importance of posing multiple problems to make sense of a concept. It is important to see concepts from a variety of perspectives. Goral, M. B., & Wiest, L.R. (2007), An arts­based approach to teaching fractions. ​Teaching Children Mathematics 14​(2): 74­80. The NCTM recommend that students represent fractions by using physical materials and number lines. Recent research highlights the importance of a sensory approach in the elementary classroom. This article focuses on how the arts­ musical, visual, and kinesthetic can be used as a means to teach fractions. The idea is to create lessons that are both developmentally appropriate and engaging. This article focuses on three ways to teach the material; having students recite a poem about fractions, having them jump fractional distances on a number line, and having them beat their fractional rhythms of a song (half notes, quarter notes, and eighth notes as well as whole notes) with rhythm sticks. The teacher experienced success with this lesson, the students were engaged and were able to meet the objectives targeted. Incorporating the arts into the classroom provides a different pedagogical approach for students and thus can be engaging and long lasting in their learning. Powell, C. A., & Hunting, R. P. (2003). Fractions in the early­years curriculum: more needed, not less. ​Teaching Children Mathematics 10​(1): 6­7. Powell and Hunting (2003) argue the importance of starting conversations with students at an early age as it establishes important ideas from which formal symbols later will flow naturally. Since children are capable of solving sharing problems which are developmentally appropriate it is an authentic opportunity to have it integrated. Children as young as three years old can divide a set of items into equal subsets. Since children can deal with sharing and division tasks in the earlier years it is important to introduce them to these concepts. Fractions are a difficult concept for students to grasp and thus it is important to begin introducing them in the primary grades. By simply asking students to divide in half a certain quantity students become aware of fraction equivalence while being exposed to multiplication by 2. Although not specifically in the primary curriculum there are still opportunities for teachers to integrate fraction concepts into the curriculum. By doing this, teachers lay the foundation for further concepts. Siebert, D. & Gaskin, N. (2006). Creating, naming, and justifying fractions. ​Teaching Children Mathematics 12​(8): 394­400. Based on the rules we were taught fractions it makes sense that we approach fractions as an arbitrary and confusing concept. It is important that we convey meaning and help ensure that Celia MacDougall Math Summative Assignment
4 students understand what fractions and their operations mean. Fractions should be taught as part­whole relationships, quotients, or ratios. This article focuses on two powerful images for thinking about fractions that move beyond whole­number reasoning. The two images are iterating and partitioning. ¼ is the amount we get by taking a whole, dividing it into 4 equal parts and taking 1 of those parts is an example of partitioning. ¼ is the amount such that 4 copies of the amount, put together, make a whole is an example of iterating. It is important not to describe a fraction as “out of”such as ⅝ as this implies that they are taking 5 from those 8 things, thus they are seeing the fraction as merely whole numbers. Words that imply partitioning include: cute evenly, split equally, or separate into equal parts. Iterating can be implied by using phrases such as: making copies, or repeating end to end. One of the main messages conveyed from this article is the language we use regarding fractions. The way we talk about fractions is important for students to develop appropriate meanings for the fractions. Wilkerson, T. L., Bryan T., & Curry, J. (2012). An appetite for fractions. ​Teaching Children Mathematics 19​(2): 90­99. A group of grade 6 students have gathered on a Saturday as part of a prep program for high school and undergraduate mathematics programs. The topic is fractions as it is one of the most crucial areas of mathematics to fully understand. By grade 6 students should be flexible in their thinking and able to efficiently work with fraction models. Wilkerson, Bryan and Curry (2012) felt that the students needed further work with a different fraction model. The lesson is a combination of small­group, partner, and whole­class activities. Students were offered a whole or a half of a Hershey’s chocolate bar or they could refuse. The chocolate bar was the basis of this fraction lesson. Students who wanted a whole were given a miniature chocolate bar and the ones who wanted a half were given a half of a normal sized bar. This activity emphasized the importance of knowing the size of the whole and that fractions have different meanings. The context of the Hershey’s chocolate bar was the foundation of the day’s activities. They used the chocolate bar to explore how to represent, and combine fractions. A conclusion drawn from this study was the importance of using a variety of models in order to develop a deep conceptual understanding of fractions. Celia MacDougall Math Summative Assignment
5 2. Critical In­Depth Review of 3 NCTM Articles: ALGEBRAIC THINKING Algebraic thinking has become an important theme across the curriculum in areas such as patterns, relations, number properties, variables, equations, and functions. In elementary school the focus for algebraic thinking is centred around the type of thinking and reasoning necessary for students to think mathematically. This type of thinking allows students to form generalizations, and formalizing these ideas with a symbol system. Since algebraic thinking is necessary for all forms of mathematics it is important that students make the connection to real life situations. Blanton, M.L., & Kaput, J, J. (2003). Developing elementary teachers’: “Algebra eyes and ears.” Teaching Children Mathematics, 10​(2): 70­77. Blanton, & Kaput (2003) propose three strategies to develop “algebra eyes and ears” that teachers can implement in their elementary math classroom. Although many elementary teachers have little experience describing what expressions and equations mean it is important that they apply best practice to develop these skills in their students. The main focus of this article is how to “Algebrafy” mathematics. This strategy allows students to make connections to their personal experiences. Blanton & Kaput (2003) highlight different ways teachers can create authentic opportunities to integrate algebraic thinking into everyday instruction. It is important that students generalize their thinking and are able to express and justify these generalizations. They propose three types of teacher­based classroom changes: algebrafying instructional materials, finding and supporting students’ algebraic thinking, and creating classroom culture and teaching practices that promote algebraic thinking. 1) Algebrafying Instructional Materials Simply put, this strategy is the process of including or adapting existing educational resources into algebraic activities. Teachers should take problems that have a single numerical answer and work on building and recognizing patterns, generalizing and justifying relationships. A classic example of this is the adapting the classic handshake problem. Normally the question is, “How many handshakes will there be if each person in your group shakes the hand of every person once?” This problem can be further algebrafyed by varying the number of people in the group, this allows students to recognize patterns in the data. It is also beneficial to include a number of people in the group that is too large to compute pictorially, thus they must identify the pattern. By creating the pattern and using it to solve further problems they are making generalizations that describe the general truth. Approaches that include algebraic thinking in normal instructional materials are beneficial as they integrate various mathematical topics such as: number sense, number facts, and recognizing and building patterns. Celia MacDougall Math Summative Assignment
6 2) Finding and Supporting Students’ Algebraic Thinking It is important to focus on the process and thinking that students use to come to their answer. By putting an importance on the process, students are prompted to justify, explain, and build arguments. This in turn deepens their understanding. The following questions deepen student understanding: Tell me what you were thinking? Did you solve this a different way? How do you know this is true? and Does this always work? By asking questions teachers can shift the focus from arithmetic to algebraic thinking and gain a better understanding of students’ strengths and areas needing improvement. 3) Creating a Classroom Culture and Practices That PRomote Algebraic Reasoning By fostering an environment that values algebraic thinking students incorporate conjecture, argumentation, and generalization in ways that build their knowledge. These activities that incorporate algebraic thinking need to part of the daily routine and not merely seen as enrichment. It is important that the environment created in the classroom values exploring, arguing, modelling, predicting, and testing their ideas. This will be reflected in students’ ability to reason algebraically. I have every intention of algebrafying my mathematics classroom. The benefits are clearly highlighted throughout this article, students perform better on standardized tests, they build deeper understanding of the content and become more engaged while taking ownership of their learning. Algebrafying the existing material will be routine and will encourage natural conversations that require generalized thinking within my classroom. Blanton and Kaput (2003) provide a compelling approach as to why we as teachers need to algebrafy our classrooms and how this can be achieved. Johanning, D., Weber, W.B., Heidt, C., Pearce, M., & Horner, K. (2010). The polar express to early algebraic thinking. ​Teaching Children Mathematics, 16​(5): 300­307. Johanning, Weber, Heidt, Pearce and Horner (2010) reiterate the importance of teaching arithmetic and algebra concurrently especially in the primary grades. A main learning target for algebra is to have students identify, think about, describe, justify, and make generalizations regarding mathematical situations. According to the NCTM standards, algebra is an emerging focus amongst the early grades (K­2). Teaching algebra can easily be integrated with other subject areas and taught thematically. This article presents using a story, The Polar Express, as a context for developing algebraic thinking. A team of Grade 1 teachers used The Polar Express as a theme that integrated reading, social studies and mathematics. The algebraic component of the lesson was to have students use an algebraic approach to skip counting rather than using a more Celia MacDougall Math Summative Assignment
7 arithmetic approach. The Grade 1 students identified patterns and looked for relationships among quantities. This developed their recursive thinking. The task was a functional­thinking task that involved two different number relationships: an odd number pattern and an “add three” pattern. This task allowed students to see and create models that represented the pattern. The students made predictions, collected data and checked the accuracy of their predictions. The activity was similar to a previous activity done within their class thus the students provided input into creating the question. They looked at an individual train car and observed that it had 3 wheels and 1 window. They created their own train and added additional cars to their train. All the students received enough materials to manipulate and create their own trains as well as a functional table. It was modelled how to complete the table and the teachers table was used to spark discussion and collect consensus. The students collected data and then were asked to make predictions based on the patterns they are able to see in their functional tables. The idea was not to learn to count independently but rather to count quantities that are inter­related. This is where the algebraic potential can be seen. By asking students to justify their answers they are able to make connections and develop a more algebraic approach to mathematical problems. Rather than telling students they have answered correctly they could build the train to support their answer. As a teacher it is important to maximize opportunities that develop algebraic thinking. Students may have difficulties recognizing and explaining patterns and should explore questions that involve a single parameter first. Johanning, Weber, et al., (2010) provide a simple problem that involves one parameter to work through as an introduction. If every student in the class brought two boxes of tissues, how many would there be? This could be solved by simply counting but they recommend that teachers ask questions that encourage students to see the patterns and thus draw conclusions. With continued exposure and practice students will be able to recognize and name patterns with more ease. Finally, the more experience students have generating patterns from situations, the more able they become to recognize patterns and describe the changes within them. As a teacher I believe strongly in integrating subject material as a strategy to engage and interest students. Integrating subject material is also a beneficial strategy for ensuring more outcomes are met. This article provided insight into how to create situations and problems that evoke algebraic thinking. The importance of developing the ability to use arithmetic in context was highlighted throughout the article. It is no longer important to solely teach students to skip count but rather teach them how to see this as a naturally occurring pattern and use that pattern to make predictions and inferences. This article was interesting and useful for developing algebraic thinking in the K­2 grades. Celia MacDougall Math Summative Assignment
8 Sores, J., Blanton, M.L., & Kaput, J.T. (2006). Thinking algebraically across the elementary school curriculum. ​Teaching Children Mathematics 12​(5): 228­235. Sores, Blanton and Kaput (2006) focus on effective ways to integrate algebraic thinking across the elementary school curriculum. They point out that many schools have a strong focus on literacy. By opening a mathematics lesson with a children’s book the students are more likely to be engaged, enthusiastic and will remember the investigation for weeks to come. Algebraic thinking is described as a process in which students build general mathematical relationships and express these relationships in increasingly sophisticated ways. In the elementary grades students can express their generalizations in words or symbols. Problems that are arithmetic in nature can easily be adapted to take an algebraic approach. For example have students expand the common handshake problem by asking what if there were 20 people in the group, or 100 people. By using high numbers such as 100 students must think about the problem algebraically rather than arithmetically. When working on problems that are algebraic it is important to give students time to process the question and think individually and then encourage them to work in small groups. By working in small groups, students are able to justify their answer, hear alternative solutions, reach conclusions and make conjectures with their peers. By introducing algebraic problems often, students are able to see the connections between them and then ultimately they can see connections to their own personal experiences. In the article, a grade 3 student provided the class with an algebraic problem about the wedding she attended on the weekend. She said she had to dance with all of her relatives. The class co­constructed a problem that aligned with her scenario. To track the number of dances some students drew a diagram of pairs, some added initials to their diagrams, and other students acted out the problem by dancing in the class. This is an excellent way to incorporate multiple learning styles. The students needed to come up with a statement that would hold true regardless of the number of people. It is important that in the elementary grades students use everyday language. A key learning outcome regarding algebraic thinking is to have students become more sophisticated in how they understand and express mathematical generalizations. Another problem presented in the article was developed based on the children’s book, Spaghetti and Meatballs for All ​(Burns 1997). The problem from this story was to think about the number of people who could be seated for any number of rectangular tables joined together. Manipulatives, the Unifix cubes were used to build a rectangular table that would seat four people. Then the teacher asked how many people could sit if there were two tables and the students immediately said eight. Since this was a common misconception, the teacher asked them to build the new table out of the cubes and then modelled it by pushing the desks in the classroom together. They then were able to understand why six people rather than eight could sit Celia MacDougall Math Summative Assignment
9 at the table. Students kept track of their data using a t­chart and then were asked to think of a number sentence that would show how many people could sit at ​n ​tables. Depending on the group of students and their previous exposure some students by grade 3 are able to represent unknowns with letters. By teaching algebrafyed questions there is no need to worry about the lack of arithmetic instruction as these questions embed arithmetic computations. Algebra questions also are much more engaging for students and an interesting way to practice their skills. By integrating algebra into other curriculum areas teachers are able to work with their strengths and provide a context for students. The more mathematics is integrated into other subject areas the more likely students are to see connections to their everyday lives. By algebrafying questions students talk and discuss their answers and predictions in a way that develops their algebraic thinking skills. It is necessary to ask questions such as: how did you arrive at you pattern? What representations did you use? How do you know your pattern will always work? I have every intention of integrating algebraic problems into my lesson and to challenge students. The most captivating part of the article to me was when the student came to class explaining that the problem at the wedding was similar to the problems they had been working on in class. By taking the time in class to work through the problem it was clear that the connection was made with that student. Most likely, the other students will be paying attention to problems that fit the model in their lives as well. This is an excellent time to be flexible as a teacher and work through these problems as we want students who recognize the importance that mathematics plays in their everyday lives. Celia MacDougall Math Summative Assignment
10 3. A Review of 5 Math Manipulatives Mathematical Manipulatives Review Manipulatives are an essential part of both teaching and learning mathematics. Students need to have a variety of materials to manipulate in order to construct their own mathematical knowledge from different perspectives. To understand mathematics at a deeper level students must first be given the opportunity to experience mathematical problems concretely before learning to represent problems symbolically. By implementing a constructivist teaching perspective and allowing students to use manipulatives they are able to actively construct their own meaning of mathematics. Mathematical manipulatives have been proven to strengthen the foundation in areas such as number sense, place value, fractions, ratios, problem solving, etc. Although mathematical manipulatives are suggested and necessary for elementary mathematics, they are also useful for students of all ages when used appropriately. Base­ten Blocks Base­ten blocks are one of the most comely used manipulatives in elementary school math classes. Base­ten blocks are useful when teaching number sense, place value, addition, subtraction, multiplication, division, fractions/ decimals, patterns and many other concepts. Base­ten blocks are effective in teaching number sense and place value (one, rod, flat) and can be used to visually introduce regrouping ideas with two­digit addition and subtraction. This can be done by asking students to represent numbers in various ways using the base­ten blocks. Base­ten blocks are an essential part of teaching mathematics and can help students visually develop a deeper understanding of a variety of concepts in mathematics. Multi­Link Cubes Multi­link cubes come in a variety of different colours and sizes. The traditional size is 1cm cube with a weight of 1g. The multi­link cubes are ideal in teaching measurement, mass, spatial sense, probability, graphing, algebraic thinking and patterns. The variety of colors allows for optimal use in creating, identifying and extending patterns. Students can create bar graphs using the cubes, can extend their understanding of volume and develop an understanding of central tendency. These manipulatives are often used in early elementary classes but certainly have their place in upper elementary and middle school. Pattern Blocks Pattern blocks are an effective way to have students explore concepts related to symmetry, geometry and patterns. Pattern blocks can also be used for fractions, area/ perimeter, tessellation, transformational geometry, spatial sense, angles and composite figures. The pattern blocks are a visual way in which students can manipulate patterns, extend them, and create Celia MacDougall Math Summative Assignment
11 patterns. Standard sets of pattern blocks include yellow hexagons, green triangles, blue rhombuses, beige rhombuses, orange squares, yellow hexagons and red trapezoids. Geoboards Geoboards are an effective mathematical manipulative when teaching concepts such as area/ perimeter, polygons, patterns, 2­D shapes/ angles, fractions/ decimals/ percents, spatial sense, symmetry, transformational geometry, and pythagorean relationships. By using rubber bands, students are able to create their own 2­D shapes and can then extend their understanding by finding the area and perimeter of their 2­D shape. The use of geoboards particularly promotes conceptual understanding of area and perimeter. The geoboards are a manipulative that is just as useful at a middle level as elementary. A geoboard can support understanding in middle school areas such as Pythagorean Theorem, transformations, tessellations and patterns. Pan Balance The pan balance offers multiple uses within a mathematics classroom. An obvious use of the pan balance is measurement in the form of mass. This gives students a concrete representation of the meaning of mass. The pan balance is also an invaluable tool when teaching the basics of algebraic equations. A common misconception for students is the meaning of the equal sign and this manipulative gives students that concrete meaning of equality. It is an effective tool to use when beginning to balance equations. With algebraic thinking being one of the most difficult concepts to teach this tool is essential in every elementary math classroom as students need to have a concrete understanding of the meaning of the equal sign before they are able to further develop their algebraic thinking. Celia MacDougall Math Summative Assignment
12 4) Technology Technology certainly plays an intangible role within a 21st century classroom especially when teaching mathematics. When teaching mathematics technology can be used to enhance student learning. Resources such as virtual manipulatives can accompany concrete manipulatives easily, websites, apps for iPads, Smartboard lessons and tutorial videos can easily reinforce lessons. The National Council of Teachers of Mathematics (NCTM) feels strongly about integrating technology into regular mathematic instruction. The NCTM feel strongly that both students and teachers have access to technologies as a way to support and advance mathematical sense making, reasoning, problem solving, and communication. Technology can be used to further develop students’ understanding, stimulate interest, and increase their proficiency in mathematics. With all the known benefits of technology I will undoubtedly integrate technology into my mathematics lessons. Review of Websites 1) Caribou Mathematics Competitions​ is a website that provides online math competitions 6 times throughout the school year. This website is targeted for students grade 3­12 and is comprised of mathematical puzzles. The older tests that have expired are uploaded on the website and can be used as practice. The contests and website content can be accessed in both French and English. In order to access this website, students must have a code that can be received from a school account. I’ve seen this site used as an enrichment tool in a grade 3 math class. Students are able to work through the older quizzes that challenge them to extend their thinking and apply concepts that they previously learned to a more abstract question. They also had the opportunity to participate on the contest days. These students were recognized within the school for their accomplishments and achievements. This website is an excellent enrichment tool and can also be used in collaboration with students who do not have an account. URL: ​https://www.cariboutests.com 2) Sumdog ​is a website that integrates reading, writing, and mathematics for students in grades 1­6. Students play games in which they play against other students worldwide while the teacher can still control what concepts/outcomes they are working on. Since each student has a personal account, Sumdog saves data on each students’ progress and asks individualized questions to meet their needs, thus it is an excellent differentiation site. The training tables allow students to see their progress and motivate them to improve. Another interesting feature of the site is the live tracking in which the teacher is able to see each students’ individual progress. This is an excellent formative assessment tool as trends can be identified amongst student performance or areas to be retaught. The reverse is also true, the teacher may identify an outcome that has not yet been taught but Celia MacDougall Math Summative Assignment
13 that all students have demonstrated competency on sumdog. A quick review of that outcome would thus suffice. This is a tool that students can use during free time, class time or at home as a way to master their skills. URL: ​http://www.sumdog.com 3) Front Row Website and iPad App This website is an excellent resource for teachers to use and is an effective differentiation tool. This resource is an individualized program that allows students to work independently and work towards mastery in their skills. The data it provides teachers allows the teacher to meet the needs of all students whether through small group or individual instruction. The program offers diagnostic tests to assess students’ current level and then uses these results to create a differentiated plan for students.The program creators suggest doing the pretests early in the academic year allowing the students to do them individually without support. This gives the program an accurate representation of the students level. If students get stuck while working they can refer to the tutorial video for extra support. This feature allows students to work independently without requiring much assistance from the teacher. The categories and questions are aligned with the common core standards. The teacher can create class lists and monitor student progress, the results can be displayed in graphs. The program is user friendly, especially for students and keeps the students engaged and active in their learning. As they continue to answer questions correct, they receive coins that can be used to purchase extras for their character. As a teacher I believe it is important to always find new ways to keep students engaged and invested in their learning. By introducing an online program that students are able to complete independently they are more likely to take ownership of their learning. I would use this program as a part of guided math since the students can work independently. It is compatible online as well as on an iPad. URL: ​https://www.frontrowed.com NoteBook Files for the SmartBoard The first attached file is a TIC TAC TOE game that students could play in small groups during a guided math lesson. This file was created for a grade 3 french immersion class practicing the expanded and symbolic form. The students would need to divide themselves into 2 groups. They would alternate answering the questions and filling in the Xs and Os when their team gets the question correct. The second attached file is a lesson I used to introduce fact families to a grade 3 french immersion class. On the first page there is a fish tank with 5 fish, 3 yellow and 2 orange. I prompted the students with questions such as how many fish are there and how can we represent them writing an addition sentence. They would say 3+2=5 and 2+3=5. One student suggested Celia MacDougall Math Summative Assignment
14 1+1+1+1+1=5. I told him that he was correct but that we only have 2 colors of fish. Next I removed the two orange fish and ask them how I would represent that in subtraction and so on. By having the interactive smartboard I was able to easily represent the equations to the students. We then moved into representing fact families using a triangle to visualize the equations. Students were able to come up to the board and express their answers. Having an interactive board absolutely helps students visualize new concepts. Celia MacDougall Math Summative Assignment
15 5. Areas of Interest Teaching and Practicing Mathematics Through Games Games provide a meaningful context in which mathematical skills can be mastered. Children see more connections to everyday situations and tend to express more motivation. When playing games students develop more self confidence and more positive attitudes towards mathematics as games reduce some of the fear of failure and error. Games provide an excellent medium for differentiation as students are working in small groups or pairs. Games also allows students to connect various outcomes and discover their link naturally. Finally, games are useful in the classroom as well as at home and thus provide consistency in their learning and practice. In the elementary grades it is important that students master their basic facts, addition, subtraction, multiplication and division. In my classroom I would expect students to practice these at home so that they can put them to use in the classroom activities. The following games are efficient to use at home as they only require a deck of cards. I would provide my students with new games to take home periodically so that they can continue to master their basic facts in fun and engaging ways. Before sending the games home I would model how to play them and give the students time to explore and play them in class. Addition Games: 1) ADDITION SNAP This game helps students master their recall of addition facts to 18. It requires 2 players and a deck of cards with the face cards and 10s removed. The ace=1. Players divide the cards evenly between themselves. next, each player turns over a card at the same time. Players add the two together as quickly as possible and say the sum out loud. The player who gives the correct answer first collects both cards. Play continues until one player collects all of the cards. In the event of a tie, players leave their cards down and let the pile build. Play resumes until one player gives a correct sum before the other and takes all of the accumulated cards. 2) ADDITION WAR This game helps students master their addition facts. It requires 2 players and a deck of cards with the face cards and 10s removed. The ace=1. Players divide cards evenly between themselves. Each player turns over two cards and adds them together. The highest sum gets all of the cards. In the event of a tie (ie: each player has the same sum), WAR is declared. Each player deals out three more cards face down and then turns over two more cards. These two cards are added together. The highest sum wins all of the cards. Play continues until one player has collected all of the cards. Celia MacDougall Math Summative Assignment
16 Subtraction Games: 1) SUBTRACTION SNAP This game helps students master their recall of immediate recall of subtraction facts. It requires 2 players and a deck of cards with the face cards removed. Ace=1. Players divide the cards evenly between themselves. Next, each player turns over one card at the same time. Players subtract the smaller number from the larger number of the two cards. The first player who says the correct difference out loud collects both of the cards. Play continues until one player has collected all of the cards. In the event of a tie, players leave their cards down and let the pile build. Play resumes until one player gives a correct difference before the other and takes all of the accumulated cards. 2) COUNT DOWN This game helps students master subtraction of two and three­digit numbers. It requires 2 players and a deck of cards, ace=1 and 10=0 with no face cards. Players turn over two cards each to make a two­digit number. Players may choose which card is the ten’s number and which is the one’s. This number is subtracted from the beginning number of one thousand. Players continue alternating turns and subtracting their two­digit number from the previous difference. A player may receive an extra turn if their difference ends in a zero or a double figure, ie. 466. The first player to reach zero is the winner. Multiplication Games: 1) MULTIPLICATION SNAP This game helps students master immediate recall of multiplication facts to 50. It requires 2 players and a deck of cards with the face cards removed (Ace­10). The ace=1. Players divide the cards into two piles. Cards A­5 are in one pile, and cards 6­10 are in another pile. Each player has one pile of cards. At the same time, each player turns over a card. Players multiply the two cards. The first player who says the correct product out loud collects both cards. In the event of a tie, players leave their cards face down and let the pile build. Play resumes until one player gives the correct answer before the other and collects all of the accumulated cards. Play continues until the common piles are finished. Players count up their cards to determine the winner. 2) IT’S A FACT SNAP “2” This game helps students master their immediate recall of multiplication fact families. It requires 2 players and a deck of cards without the face cards and without the 10s. The ace=1. The object of the game is to practice recall of basic facts. If practice is required for one fact in particular, this game allows for such repetition. Players decide (or teacher directs) the particular fact to practice (example: x5, x8, x3, etc.). Once the constant factor is determined, that card is placed between the two players. One player turns over a card from the common pile and places it beside Celia MacDougall Math Summative Assignment
17 the constant factor. The players multiply the two numbers together, and the first player to say the correct product out loud collects the card. Play continues until one player has all of the cards. In the event of a tie, players leave the card and let the pile build. Play continues until one player says a correct answer before the other and collects all of the accumulated cards. Parental Involvement It is important to keep parents involved in their child’s learning, especially in mathematics. Parental involvement has been shown to increase student achievement in math and also increase intrinsic motivation. By practicing at home and receiving support from their parents students are more likely to develop a growth mindset, meaning that they believe if they put the work in, they can learn! This is opposite of a fixed mindset where students believe that they are good at math or they can’t do math. As a teacher the praise and dialect we use in the classroom can foster certain mindsets and thus it is important to praise students on their effort rather than their intelligences. When working at home with parents they are more likely to develop a growth mindset. It is important to communicate to parents and students the importance of mathematics in everyday situations and why it is important to practice and further develop their skills at home. I do not believe in assigning nightly homework but believe it is important for students to practice their basic facts (addition, subtraction, multiplication and division) and work on any concept they may be experiencing difficulty with at home. Some parents are not overly confident in their mathematical skills and have a difficult time helping their students so as a teacher I would send home newsletters monthly that explain the concept that we are currently working on and how to solve the given problems. If students are experiencing difficulty in any area I would send home another note explaining the concepts in further detail with practice problems for the students to work on. These together with in school interventions should be the push students need to achieve at an appropriate level. See example of parent letter regarding estimation strategies and corresponding practice problems. By sending home letters such as these students do not need to get further behind and can then come to school and apply these concepts within a context. Celia MacDougall Math Summative Assignment
18 We've been working on estimation strategies in math class this past week. We will continue to work on estimation when we begin adding 2 digit numbers and it is important that students find an estimation strategy that works for them. The first strategy that we worked on was ​looking at the 10s place value​. With this method of estimation students look at the tens place value ​only ​and use that number as their estimation. Here are 3 examples of this estimation method: 1) 67 + 46 = 2) 43 + 79 = 3) 52 + 87 = 60 + 40 = 100 40 + 70 = 110
50 + 80 = 130 The other estimation method that was introduced was ​rounding to the nearest tens place. With this strategy, we look at the units and determine whether to round the number up to the next tens place or round it down. Students are encouraged to draw a number line to help them with this concept. Students are expected to explain why a number gets rounded up or down. For example, if the number is 27, they would say “I only have to add 3 to get to 30 instead of subtracting 7 to get back to 20.” Here are the same 3 examples but using the rounding strategy: 1) 67 + 46 = 2) 43 + 79 = 3) 52 + 87 = 70 + 50 = 120 40 + 80 = 120
50 + 90 = 140 Please see the attached practice problems. Your help at home working on these strategies with your child would be greatly appreciated! If you have any questions please feel free to contact me. Celia MacDougall Math Summative Assignment
19 Practice Problems for Estimation Strategies
Estimate by looking at the 10s place value.
15 + 96 =__________________________________ 35 + 67=__________________________________ 26 + 81 =__________________________________ 42 + 65 =__________________________________ 87 + 31 =​ ​__________________________________ 92 + 53 =​ ​__________________________________ Estimate by rounding to the nearest 10.
46 + 38 =__________________________________ 32 + 75 =__________________________________ 41 + 59 =__________________________________ 68 + 52 =__________________________________ 30 + 42 =__________________________________ 48 + 94 =__________________________________ Celia MacDougall Math Summative Assignment
20 Celia MacDougall Math Summative Assignment
21 Formative Assessment Strategies for Elementary Mathematics The assessment strategies used within the classroom have the power to motivate and facilitate learning. Appropriate assessments can provide the necessary information to put interventions or enrichments into place. In this section I focused on formative assessment strategies that I would use to guide instruction in my mathematics class. Traffic lights, snowball toss, ABCD cards, call sticks, individual white boards, and entrance/ exit slips are a few of the formative assessment strategies I would use regularly. Traffic Lights Each student will have three index cards connected on a key ring. They will have one red, one green and one yellow. These cards will be for students to make a judgment call based on their understanding. Green means I am working fine and can continue to work independently. Yellow means I need help but I can continue to work. Finally, red means I need help and I can’t keep working. This strategy will allow me to make judgments on which outcomes should be retaught to the entire class or small groups. Using traffic lights is also a way to have students self reflect and take ownership their own learning. Snowball Toss The snowball toss is a unique way to introduce or review a topic. Each student records on a scrap piece of paper something they know about a given topic. Once each student has their fact written, they crumple up their paper and throw it a predetermined target. Each student then goes and picks up a piece of paper and reads it aloud. This is a unique and authentic way to start conversation and further develop understanding. The results could be recorded on a KWL chart as an extension activity. I would use this activity in my classroom in a variety of ways. One way I would use it would be to put a number on the board, the answer, and have students create a question to go along with the answer. ABCD Cards Each student will have 4 cards with either A, B, C, or D written on each. These cards can be used when the teacher asks any multiple choice questions. This strategy eliminates the common trend of one student answering every question by keeping all students accountable. This would be a beneficial strategy to use during warm ups in math class. Call Sticks In my classroom there will be a container with call sticks. On each craft stick will be the name of each student. Whenever asking a question students will know that they are all expected to come up with an answer as any student could be chosen. This ensures engagement as students will be expected to answer at any random time. Celia MacDougall Math Summative Assignment
22 Individual White Boards If every student has a whiteboard they have the opportunity to work out their thinking and provide a quick answer in the class. The whiteboards are another tool that can be used to see what students know at a given time. Ask students a question, give them time to work through an answer and then give them a cue to show their answer all at the same time. Entrance/ Exit Slips A entrance/ exit slip is an excellent way to see if students have grasped the appropriate understanding. They should be designed so that students can answer them in roughly 2 minutes. When students hand these slips in they can be divided into two piles, they’ve got the learning target or they still need work. Based on the size of the piles the teacher can decide whether it is effective and necessary to reteach to the entire class, whether they can teach a small group in a guided math class or individual support during intervention times. Assessment plays a critical role in the classroom and should be the driving force that determines where and when the instruction moves. See attached a poster I have created that highlights three forms of formative assessment for a math class, traffic lights, snowball toss, and ABCD cards.