Teachers' Hub

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The Importance of Considering the Mathematical Tasks We Pose to

Young Children

September/2018

Importance of Mathematical Tasks

 

Recent research from the international PISA study indicated students’ mathematics understanding influences opportunities for future schooling as well as better paying and more rewarding jobs (OECD, 2013). Mathematics experts have written that students who are proficient in mathematics are able to demonstrate conceptual understanding, procedural fluency and problem solving skills (National Council for Teachers of Mathematics, 2014). In multiple countries, mathematics standards and recommendations call for teachers to pose more complex mathematical tasks embedded in real-world contexts (European Commission, 2011; Ministry of Education, 2011; Mullis, Martin, & Loveless, 2016;  National Council for Teachers of Mathematics, 2014). Mathematics education experts (NCTM, 2014; Smith & Stein, 1998) in the United States categorize tasks into two broad categories based on their rigor or level of cognitive demand. Within each category there are two more specific types of tasks. These are described in detail in the table below.

 

Table 1: Types of Mathematical Tasks

Examples of Mathematical Tasks

 

 

 

Counting and Cardinality Example

 

One of the most foundational concepts for young children involves counting and cardinality skills. Tasks which integrate attributes of number by embedding real life tasks and measurement and geometry concepts with counting and cardinality help raise the cognitive demand of tasks. One such task for counting and cardinality exposes students to using mathematical reasoning to justify how many dinosaurs are in a hidden picture (see Figure 1).

Figure 1: Example of Digging Dinosaurs Task

Teachers can launch the task by quickly showing the dinosaur image (Figure 1). The teacher might ask students to predict how many dinosaurs are in the photo based on looking at the picture of the 8 feet provided.  In addition to the image of the dinosaur feet, the teacher could also have students build the picture with toy dinosaurs, wiki sticks or cubes. After students predict the amount of dinosaurs in the picture allow students to talk about their prediction before letting them explore the task.

 

As students explore the task, the teacher should engage the students in encouraging them to work with other students and discuss strategies to determine the number of dinosaur legs. Students’ own ways that they explore and represent the picture is the springboard for mathematics discussion after time to explore. Discussing students’ strategies and solutions is a critical element of learning and helps teachers determine if they have developed a deeper understanding of the mathematics concepts. During the discussion of this task, the teacher should pose questions about how students started the task, how they counted the number of legs, and how they know their answer makes sense. After possibly grouping the objects in twos as pairs of legs or counting them one by one, the teacher may opt to give the students a second picture with more than ten legs or a word problem such as a farmer saw 10 legs. He had 1 duck and some pigs.  How many pigs did he have?   Teachers should continue to nurture students to explore numbers so subitizing becomes a more natural process as well as use open questioning so  higher levels of thinking such as creation and connection are applied.

 

After two or three students have shared how they predicted the feet in the image, the teacher can pose questions to the entire class such as: “Can you use words and mathematical language to explain what you did?” and “How do you know your answer makes sense?” When the teacher asks students about the number of legs, this is when counting and cardinality becomes the focus.  They may begin to count by individual legs or groups of two. The question, “Explain how you know you are correct” encourages young learners to think deeply about the number of objects and the context.

 

Joining and Breaking Apart Groups of Objects

 

 Young learners need experiences exploring how to join (compose) or break apart (decompose) numbers and groups of objects. One example of a task involves working with similar (e.g., cubes, blocks, popsicle sticks) with the focus of adding the objects in groups. To begin, the teacher gives the directions to the students: add the objects using groups (2, 5, or 10, for example) to reach the total sum. While adding, the students should pay close attention to how they are grouping the objects. After students have counted their objects by grouping, the next key part of the task requires students to draw a visual representation of their number-composing process on a piece of paper using pictures and symbols. The different representations students draw, supported by explanations of their thought processes, can lead to a rich mathematical discussion as a class about the different ways to compose numbers. To prompt the students while sharing, the teacher can ask questions such as why did you choose this grouping strategy? What does this (pointing to the representation) mean? After two or three students have shared, the teacher can pose follow-up questions to the entire class that may deepen students’ understanding, including how are these strategies (representations) alike? How are they different? or Who can explain (name’s) strategy?

 

A variation of this same task can be used to focus on breaking apart or decomposing numbers, too. After they decompose the original pile into groups, ask the students to arrange the containers in a way that makes sense to them (some students may line them up or put them in groups of two, for example). Students’ arrangements of containers and explanations of sense-making can lead to a rich discussion similar to the first activity. During the discussion, the teacher should be mindful of capturing these ideas and translating them into mathematics language and symbols (for older children). Encouraging students to explore ways to join and break apart numbers in an open-ended, hands-on activity such as this is one way to increase access to understanding while promoting higher levels of thinking.

 

Reasoning about Geometric Shapes

 

Early in life, children begin exploring with shapes– watching a colorful ball roll down a slide or selecting different blocks to build a tall tower. It is important to recognize these experiences and build upon them in elementary school. Higher-level cognitive demand tasks, “require students to access relevant knowledge and experiences and make appropriate use of them in working through the task” (Smith & Stein, 1998, p. 348). One such task that can encourage students to apply their prior experiences with shapes and engage them in cognitively-demanding thinking is called “What Shape Am I?”.

 

The purpose of this task is to engage students in identifying various shapes using defining attributes. This task can be used when students are first exploring shapes or later when they have some experiences with the formal names and attributes of two- and three-dimensional shapes. Depending on your students experience with two- and three-dimensional shapes, they may use more informal or formal language to describe the different shapes.

The materials for this task include two- and three-dimensional shapes, paper bags, and a two-column chart for each student. Place one shape in each paper bag, and label the bag with a number. Some shapes to include are circles, quadrilaterals, triangles, cubes, rectangular prisms, cylinders, and triangular prisms.

 

Individually or in pairs, students will travel to each of the paper bags and find out what shape is inside the bag without peeking! In the first column of their chart, students will write the number on the bag. In the second column, students should describe the shape and if they know, write the name of the shape. For example, in a bag with a sphere, students may notice that the shape rolls like a ball and that there aren’t any edges or faces. Students may also draw a picture of the shape. This will provide students with an alternate way to describe the shape that’s in the bag and engage them in creating a representation of the shape. As you walk around observing your students, you will be able to see how they are progressing in their use of formal language describe the different shapes. You can also formatively assess your students’ understanding of the various shapes by asking questions such as, “How did you know that the shape in the bag has four vertices?” or “Why do you think that the shape in the bag is a triangular prism?”

 

After students have explored the different shapes, bring them together for the “Big Reveal.” Hold up the first bag and have all the students look at their papers. Ask some students to share what they think the shape is and why. To encourage student to student discourse, ask them to add-on to their classmates’ ideas, ask clarifying questions, and agree or disagree with the ideas presented. Once your students have discussed what the shape might be and why, have a student reach into the bag and reveal the shape for the class. If the students are correct review what led them to successfully identifying the shape. If the students are incorrect, discuss what might have been challenging and what they could do next time to correctly identify the shape. The students can then try this strategy out on the next bag. Engaging your students in exploring and discussing their reasoning for identifying each shape can help deepen their understanding of the defining attributes for various shapes.

 

Teaching with Cognitively-Demanding Tasks

 

 The launch-explore-discuss approach has started to gain momentum as an effective way to organize the process of teaching with cognitively-demanding tasks. The process involves launching or getting students started on a task, allowing students time to collaboratively explore a task with support, and then discussing the strategies and mathematics concepts. The idea of directly teaching concepts or a mini lesson about strategies does not exist in this format. Any direct teaching would occur as students share strategies during the discussion or after the discussion in targeted small group time.

 

Drew Polly

Professor

University of North Carolina at

Charlotte, United States

Madelyn Colonnese

Assistant Professor

University of North Carolina at

Charlotte, United States

Amanda Casto

Doctoral Student

University of North Carolina at

Charlotte, United States

Wendy Lewis

Doctoral Student

University of North Carolina at

Charlotte, United States

References

 

European Commission (2011). Mathematics Education in Europe: Common Challenges and National Policies. Retrieved from: http://eacea.ec.europa.eu/education/eurydice/documents/thematic_reports/132EN.pdf.

Keqiang, X. (2011). Examining changes between China’s 2001 and 2011 mathematics curriculum standards for basic education from 21st century key competencies perspective. Higher Education of Social Science, 9(6), 79-85.

Ministry of Education. (2011). National curriculum standards of mathematics for basic education (2011 edition). Beijing, PRC: Beijing Normal University Publishing Group.

Mullis, I.V.S., Martin, M.O., & Loveless, T. (2016). 20 Years of TIMSS: International Trends in Mathematics and Science Achievement, Curriculum, and Instruction. Retrieved from: http://timssandpirls.bc.edu/timss2015/international-results/timss2015/wp-content/uploads/2016/T15-20-years-of-TIMSS.pdf.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

Noyce Foundation.  (2014).  Problem of the Month:  Digging for Dinosaurs:  Creative Commons.   (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).

OECD (2013). PISA 2012 Results: What Students Know and Can Do (Volume I): Student Performance in Mathematics, Reading and Science. Retrieved from: http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

Smith, M. S. & Stein, M. K. (1998). Selecting mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.