In Fathoms once a month between now and the end of the year we’ll dig deeply into a particular area of the curriculum to discuss why it’s important and what it looks like across the ages and grades here at The Little School. For each of these editions we’ll use content from teachers’ recent Friday newsletters to provide an authentic window into what this area of learning looks like, in action, during any given week of the school year. In this Fathoms we’re looking at math. In our Annual Report, we outlined our process for adopting a new math curriculum, Bridges. This curriculum offers us rich mathematical experiences to share with your children and ensures continuity and depth as your child engages increasingly complex mathematical concepts and skills.
Math has many facets to mastery and for this article, we will focus specifically on how children develop number sense. What is number sense? It is how we make sense of numbers and use this understanding in increasingly complex ways. A few characteristics of number sense include:
- Understanding that numbers are tools used to measure things in the world.
- Developing a fluency and flexibility with numbers and their relationship to one another.
- Understanding of place value and the base-10 number system.
- Developing useful strategies for solving complex problems.
- Processing with a mental number line, on which numerical quantities can be understood, compared and manipulated.
- Generating, inventing and applying strategies to conduct number operations.
- Using numbers and quantitative methods to communicate, process and interpret information.
- Having an awareness of levels of accuracy, error and the reasonableness of calculations.
- Understanding quantity, sets and groups of numbers and how quantities can be composed and decomposed.
Building number sense happens in a learning environment that is abundant with math talk, where curiosity and inquiry drive learning, and where hands-on experience, games and visual models allow children to construct their learning – along with lots of practice. Let’s take a look at how the curriculum in the older grades challenges children to think critically and flexibly, using their number sense to achieve mastery and understanding of skills in experiential ways and how recent experiences in classrooms are building a strong foundation in number sense in the early childhood and primary years.
One hallmark of our approach to math at The Little Schools is having students work with multiple strategies to solve problems. Learning, understanding, working with and discussing multiple strategies enriches students’ number sense, strengthens their conceptual reasoning and encourages their critical thinking in math. Traditionally, students learned an algorithm to solve each operation, but rarely did students understand how that algorithm works. Research shows that learning a conceptual strategy at the beginning supports students in understanding an algorithm later on, and therefore internalizing it with greater accuracy. Our fourth graders have been working with a wide variety of multiplication strategies. Deeply understanding the concept of multiplication is vital in fourth grade as so much of the math work that follows builds on this essential skill set. Our fifth-grade students have been working on adding and subtracting decimals. The strategy of lining up the decimals and stacking the numbers is effective here, but we have been working with a variety of strategies that provide students with the foundation they need to understand decimals to the thousandth place. When students are ready to fully understand, we do teach them the traditional algorithms, which tend to be more efficient. Here, we illustrate and explain some of the strategies and conceptual models that our math groups are currently working with.
Use Money as a Benchmark: Think of quarters, since 26 is very close to 25.
Decompose the Numbers: Break 26 into 20 and 6 and then multiply by 6, applying the distributive property. Add the two products to get the total.
Skip Count: Make 6 jumps of 26 or 6 jumps of 25 plus one of 6 using a number line.
Reducing: Students learn this strategy on a number line but it can be done without as well, as demonstrated here. In this strategy, the student subtracts (reduces) enough from the minuend (first number) to reach a whole number (here it is 7.0). They then figure out how much more they need to subtract to have subtracted the value of the subtrahend (second number). Not only does this make subtracting more manageable, but there is a necessary understanding of place value required to execute the strategy accurately.
Differencing: As opposed to Reducing, in Differencing students begin with the subtrahend (second number) and add to the minuend (first number). Again, finding a landmark whole number supports students’ understanding of place value and simplifies the expression.
Constant difference: Here students add or subtract the same value to both the subtrahend and the minuend to simplify the problem. The concept of constant difference means that if the same amount is added or subtracted to both values, the difference between them remains the same. There is significant critical thinking and problem solving necessary to figure out what value should be added or subtracted from both numbers.
In 2nd grade, students have been exploring story problems with scenarios involving “parcels” (representing 10’s) and “presents” (representing 1’s). While studying and solving a variety of problems as a whole group, they learned to recognize the key information within a problem. By organizing what they knew and what the question was asking, they were able to demystify an otherwise complex question. Using familiar templates and an artistic touch, students created story problems of their own, and they eagerly moved about the room, solving each other’s story problems. As students worked through the problems, they noted and discussed the strategies they were using, such as counting by 10’s or other landmark numbers and using their understanding of place value to more quickly solve multi-step word problems. Lastly, as they checked the accuracy of one another’s solutions, they realized the importance of how to present an answer to a story problem. Many students reported back:
“They didn’t even answer the question, they just told me how many in all!”
“They just wrote an answer. I don’t know how they solved it, what if they just guessed!” Together we discussed how showing the strategies they use, in an organized fashion, with the answer clearly shown and labeled, was truly important. As we continue to solve story problems, reminding ourselves of these strategies we generated to show our work and explain our answers will be helpful.
In 3rd grade, students have been working to understand measurement concepts related to time and volume. We began our unit by focusing on reading analog clocks and measuring elapsed time, considering questions like: How is measuring time different than adding or subtracting numbers in our base 10 system? Students had fun enacting time on giant clock models with partners. As the unit has progressed, we have shifted our focus to understanding metric and customary units, particularly as they relate to mass and liquid volume. This week, students used pan-balance scales to measure the mass of various items around the classroom. As they measured each item, they made an initial estimate of its mass and then found the difference between their estimate and the actual. They worked to understand how different something with a mass of 1 gram feels than something with a mass of 1 kilogram. Later in the week, students were introduced to terms for different units of liquid measurement. There is so much new vocabulary associated with different areas of measurement that we encourage you to discuss amounts that you see with your child, in places such as your home or in the grocery store. Over the next few weeks we will continue to explore mass and liquid volume including the ways in which the metric system is different than the customary system we use in the United States.
Number Trees and Facts to 20!
January is the time to ensure that students increase their knowledge of number facts to 20 and beyond. Some strategies we practice in class include counting on, using the nearest double and using doubles plus or minus one. The best way to get used to number trees or families is to play games that reinforce these facts in a fun way! Ask your child about Spill the Beans and Race to 100!
This week we taught the class a new game, chip-trading! We introduce kindergarten children to our base number system and explore how we can trade (regroup) to represent larger quantities more efficiently. With a role of the die, children add red chips to the board and work toward the goal of getting a white chip. We started by trading in base 5: five reds for a blue, and five blues for a white. Chip trading is fun and highly motivational and helps kindergarteners intuitively understand place value. For children who are ready, we begin to discuss the values of different arrangements of the chips on the board, using chips to represent ones, fives and tens to add or subtract. This week we’ve also been working with doubles. A double is a number added to itself. Learning the doubles is a powerful addition strategy that helps students solve tricky math problems, using the facts they already know and applying them as they approach new problems. It also helps them understand that there are many different ways to solve a problem.
Regardless of how high a preschooler can rote count, a child’s sense of what those numbers actually mean develops gradually. We call this understanding number sense, and it requires relating numbers to real quantities. This week we explored numbers and counting with a dot card game. A variety of dot arrangements helps children develop different mental images of quantities. The dot cards are small quantities (up to four or maybe five) that children can perceive without needing to count. The dot cards were flashed for a short time, and players were asked to show the same amount of gems as dots on a given card.
Way up high in the apple tree, five little apples smiled at me. I shook that tree as hard as I could, and down came an apple. Mmmm that was good.”
This week, we used this rhyme to explore subtraction, subitizing, and composition of numbers. Here are some examples of how we start to explain these concepts to the children:
Subtraction: “We had three apples and we took one away. Now we have 2.”
Subitizing (instantly recognizing how many): “We can see that there are four apples on the tree without even counting them.”
Composition of Numbers: “There are different ways to make four. You can use two and two (show on fingers) or three and one more. “
Another popular activity this week was a math game which uses brightly painted wooden fish each labeled with a numeral 1-10. First, we read the book Splash! A Penguin Counting Book, then, using a miniature fishing rod with a magnet on the end, the children took turns fishing, saying aloud the number of the fish they caught, and placing it inside the boat. Controlling the fishing rod with enough precision to pick up a fish also proved to be an excellent fine motor challenge for the children. Lots of fun was had while identifying numerals and honing their hand-eye coordination.