Representation in: Number Sense, Mental Math and Subitizing

4. Enduring Understandings: What conceptual understanding are we seeking to help our students develop?

a. Math is figure-out-able. Too many of our students believe that math is a set of rules that they have to memorize. They don't believe that math should make sense, and that they can use their understanding of the world to figure it out. Equating our symbolic work with concrete actions helps them internalize the meaning of what they're learning and giving them time to develop non-standard approaches to solving problems, and seeing the wide variety of methods possible, helps them see themselves as mathematical thinkers.

b. The benchmarks of 5 and 10 are really important. Since our number system is based on the number 10, and 5 is half of 10, computations with 5 and 10 are typically pretty easy. We can extend that knowledge to other numbers by knowing how those numbers relate to 5 and 10. This is the basis for most strategies used in estimation and mental math. Learners should be familiar with the pairs of numbers that make 5 and that make 10, and they should know how far each single digit number is from 5 and from 10. Using ten-frames, number racks, and even fingers are great ways to develop this familiarity.  Students without this familiarity struggle to remember or quickly derive basic math facts, often having to fall back on counting by ones instead of using more efficient strategies.

c. We compose and decompose numbers when doing arithmetic and we need to understand why and how it works. It isn't just mental math that relies of composing and decomposing numbers: our standard algorithms are all built on the premise that we can decompose numbers by their place value, and so understanding why and how this works is critical to understanding those processes. For example, why do we only add within each place value (ones to ones, tens to tens, etc.) but we multiply every combination of digits?  Giving students a chance to work through the process of composing and decomposing small numbers using structures like the ten-frame and number rack help them to internalize when and how the process works, and this understanding will help them develop computational flexibility and understanding.

d. Solving problems mentally develops number sense. Giving students opportunities to solve problems mentally is one of the best ways to help them develop number sense. It helps them shift from blindly following algorithms to working with numbers based on their meaning, and it helps them move from counting by ones to chunking numbers in helpful ways, which a conceptual leap that is important for multiplicative and algebraic topics they will encounter later in their studies. Activites such as dot talks help students start to decompose numbers into recognizable parts, and number talks are a great way to develop flexibility in computation.

5. Let's build some collective understanding through discussion. We know that reflection and connection helps to deepen knowing. Please take a few minutes to add your thoughts to the google doc shown below.