This paper elucidates the Common Core’s vision of sound Mathematical Practices for young children, including reasoning abstractly and making sense of problems.
The Common Core Mathematical Practices were designed to guide teachers in helping children to develop a deep understanding of mathematical concepts. At first glance, they might seem like lofty goals for preschoolers. But a look "under the hood” can provide insights into why these eight practices can be so valuable and how they naturally fit into the preschool classroom. You’ll be surprised by how many traditional preschool skills they support as well!
Why are the Mathematical Practices important for preschool?
The mathematical practices:
- represent a way of thinking about mathematics that all children can benefit from.
are based on years of research by the National Council of Teachers of Mathematics and the National Research Council.
provide teachers with a way to engage children in thinking about mathematics instead of just doing mathematics.
provide children with tools that will be useful all of their lives.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
MP1. Make sense of problems and persevere in solving them
Wow—developing problem solving skills and perseverance! What terrific goals for preschoolers! For young children to make sense of problems and persevere in solving them in the math world means that they have to think about a variety of ways to solve the problem (such as counting on their fingers, using teddy bears, asking friends to help, or measuring) and then try out the solution that seems to be most promising. If the first solution doesn’t work, they can go back to the drawing board (that’s where we see their perseverance!). Making sense of problems also means making sense of the mathematics of the problem. What quantities are involved? Are we putting together sets? Taking them apart? Are we balancing more than one block? Do the two triangles fit in the square? If not, why not?
MP2. Reason abstractly and quantitatively
We often think, mistakenly, that children do not have the ability to think abstractly. It is true that initially children reason with quantities in a very concrete way: “If I have four trains and I have to share fairly with Simon, I’ll only have two left!” Later, children learn that these sorts of quantity questions can relate to numbers themselves in a more abstract way, for example, that one plus any whole number equals the next number. Likewise, children can use number words and the terms more or less to describe differences between unequal sets of train cars.
MP3. Construct viable arguments and critique the reasoning of others
Anyone who has ever argued with a four-year-old over whether bedtime is reasonable in the light of a midsummer evening, or exactly how many pieces of Halloween candy are enough, understands that children can develop negotiating skills that involve sophisticated arguments full of astute critiques of an adult’s reasoning. Preschool children can develop these very same skills in mathematics (and use them in a more healthy and constructive manner!). Why isn’t that a square? Well, because all the sides aren’t the same. Why isn’t a split of six tunnel blocks and four tunnel blocks fair? Well, because you have more. These are the rudiments of later thoughtful and proficient reasoning skills.
MP4. Model with mathematics
Preschoolers begin by counting with concrete objects like teddy bear counters, play dishes, and cars, and by joining and taking away from sets of objects. Later they develop more abstract mathematical thinking, for example, knowing that being “five years old” is older than being “four years old.” However, for both types of thinking, modeling with mathematics is a perfect fit in preschool. Ask children to “Show me how you know that.” Even early in preschool, children can become very adept at using objects and drawings to illustrate their thinking and problem-solving activities.
MP5. Use appropriate tools strategically
Preschoolers love tools! Teachers can introduce children to appropriate tools for solving a variety of mathematical problems. Classroom, small group, or one-on-one conversations about what sorts of tools are good for measuring weight, height, length, and temperature can also introduce vocabulary useful in multiple settings (heavier, lighter, smaller, longer, infinitesimal). Initially, young children use non-standard tools to measure and do comparisons. For instance, they will measure the length of a block road by counting how many same-sized rectangular blocks were used to make the road, or they will hold two pumpkins to estimate which is heavier.
MP6. Attend to precision
You may have noticed that as children go through their last year of preschool, rules become increasingly important to them. Just watch them play a game out on the playground. Negotiating rules and requiring adherence to those rules can take almost as much time as playing the game! In these activities, preciseness is common, and obviously within their realm of ability. While some mathematical activities require estimation, a lot of mathematical problem-solving requires precision: in most cases, numbers are not interchangeable. Ask a five-year-old if he is four years old and you’ll get an earful or laughter at your silly question. So you can have conversations about when estimation is useful and when precision is really required. When engaging in mathematical problem-solving, encourage your children to communicate precisely.
MP7. Look for and make use of structure
Children notice structure early in life. They use structure to assign animals to categories: “Has a tail, four legs, fur, and barks… must be a dog.” And to pick out clothes that they like: “I want the striped one, not the polka-dot one.” They can sort shapes and create a table setting so that there is a plate, fork, and cup for each person at the table. Before long, they begin to investigate the patterns in number; for instance, anyone who is one year younger than them can be described as their previous age in the number sequence (“I’m one year older than you! I used to be four, too”). Our job is to provide opportunities in which we can support children as they notice increasingly sophisticated structures in mathematics (“You have three crackers and Samuel has two, if I switched your plates, will there still be five all together?”).
MP8. Look for and express regularity in repeated reasoning
Can young children really do this? Yes! One-plus-one-is-two-plus-one-is-three! Anytime we add one, it is just the next number in the counting sequence! If we match identical square pattern blocks side to side, every square fits up directly against four other square pattern blocks. If we split each square into two triangles, we will always have two triangles for every square. Count the squares and then count the triangles, and you will always have twice as many squares as triangles. Young children are very perceptive and have a lot of fun looking for patterns in their environments.