Over the course of my own math journey in a variety of roles from elementary teacher to currently an independent math consultant, there is one quotation that resonates with me the most: “There is nothing elementary about teaching elementary math.” Truly the more I learn, the more I realize I need to learn. Over the past 10 years as I have immersed myself in the research on how children learn math concepts, the most impactful understanding for me as an educator has been surrounding the progressions of not only the various math concepts as they develop over the course of a child’s school journey but also the developmental thinking progressions within the various concepts of the students. Once we are aware of these progressions, we can most effectively meet our students where they are and propel them forward by using strategies and models that build number sense and are naturally applied to other number sets from whole numbers to decimals and fractions down the road. One of the keys to this exploration of math concepts is manipulatives, since they allow us to see the underlying structure of place value and story problem structures as well as making our flexible thinking visible to others.

In a typical school year, each classroom has students at various places in their learning journeys, which may or may not match up to their grade level. This year particularly, fulfilling the expectation of exploring current grade-level content when so many of our students have unfinished learning from the previous grade level has proved to be an unprecedented challenge. I want to share with you a few examples of how we can use manipulatives to explore progressions of math concepts that will set a foundation of thought that will apply throughout the grade levels and accelerate our students’ learning.

One obvious thought might be where do we even begin? How do we know what each student knows so that we can connect what they already know to what they need to learn? In my consulting work, I have found that beginning with math fact fluency has the power to change the math climate in districts. My favorite assessments are Dr Nicki Newton’s (free) Math Running Records (www.mathrunningrecords.com). By sitting one-on-one with our students using her research-based protocols for each operation, we can see exactly which sets of facts on the progression are causing super slowdowns, inaccuracies, or inefficient strategies. In addition we can see where our students are in their thinking from using counting all, counting on, derived facts, or mastery. Armed with this information, we can then provide just-in-time scaffolds and supports to meet the students exactly where they are and propel them forward.

Two of my favorite addition expressions to ask students are 5 + 6 and then 6 + 7. The vast majority of students will tell me that 5 + 6 =11 because they know 5 + 5 = 10 and one more makes 11. Yet, when I ask them 6 + 7 they will count on the 6 from the 7. So I know immediately that they haven’t yet built their understanding of number relationships to help them figure out facts that they don’t yet know. Instead, they go back to what is familiar and count on. For the students who count on, I would show them an image like this:

I would then ask them how many beads are there and how do they know. By using the visual of the rekenrek, students can see the quantities and use groupings of objects they know to determine the total amount. Some students may mention that they see (5 + 5) + (1 + 2) = 13, others may see 6 + 6 + 1 = 13, and yet others use a Bridge 10 strategy as they imagine taking 3 beads from the top row and moving them to the bottom row to rename 6 + 7 as 10 + 3. These are the derived strategies that are the key to developing flexibility with numbers and building number sense. These very same strategies can then also apply to multidigit numbers, decimals, and fractions. Think for a moment how you would solve 56+ ^{3}/_{6}. Would you follow a procedure of adding numerators, keeping denominators the same, and then feeling a little weird that the sum has a larger numerator than denominator? What if we use the Bridge 10 strategy by decomposing the ^{3}/_{6 }and giving 1⁄6 to the 5⁄6 to rename the expression to 1 + ^{2}/_{6}? We can model this using Fraction Tiles.

In my experience, subtraction is the Achilles’ heel of all our students. Too often students think of subtraction only as take away. I’ve seen too many upper elementary students counting back basic subtraction facts within the context of subtraction in a division problem. We need to be sure to explore the relationship between addition and subtraction because sometimes, given the numbers we are given, adding up may be a more efficient strategy. Have you ever witnessed a student solve 104 - 98 by regrouping and perhaps even making a mistake during the process? What if we take a moment and think about the numbers first and think addition? It is 2 more to 100 and 4 more to 104, so the difference is 6. Let’s take a double-digit example of 52 - 38 and model this thinking using my favorite math manipulative, Cuisenaire^{®} Rods.

We can lay out the minuend of 52 and then place the subtrahend amount on top of the minuend and look at the difference. We can then use addition to solve it. 38 plus 2 more is 40 and then there is 12 more to 52. So, the total distance is 14. This is just one of many strategies for dealing flexibly with numbers when we are free from the constraints of algorithms which too often keep our students in the counting phase of reasoning.

Let’s take one more example with multiplication. The area model for multiplication is one of the most powerful models because the very same model for single-digit multiplication can apply to multidigit numbers, decimals, fractions, and polynomials way down the road. Too often our students get caught in additive reasoning when they are skip-counting basic multiplication facts usually even within a context of multidigit multiplication when using the algorithm. We want to facilitate our students’ movement from skip-counting and into multiplicative reasoning which again uses number relationships. Here are a few examples of how we can determine 4 x 6.

Very often when I administer the multiplication Math Running Record, I meet students who have memorized their math facts. If a student correctly answers all the facts, then I always ask them 4 x 17. For those who have rote memorized the facts, they tell me “I haven’t learned that one yet.” But for those who have explored number relationships, they will tell me that 17 doubled is 34 and 34 doubled is 68. They didn’t immediately know the answer, but they knew they could use something they knew to determine what they don’t yet know. When we explore x4 as double a double, then we can then multiply anything times 4—multidigit numbers, fractions, and decimals! Exploring math concepts using strategies and models like this is the key to accelerating our students’ understandings of mathematics. Not only will we be exploring the current math content, but we will be setting a foundation of understanding that will unlock future math concepts while at the same time having classroom communities of learners that promote positive attitudes towards math and the development of excited lifelong mathematicians!