For math so far this year, all concepts and lessons directly use or build off of place value knowledge. The beginning of the year involved whole and small group instruction on place value, moving quickly into centers with leveled math groups. Lessons focused on place value included learning the various number forms (drawing, word, expanded, and standard), visual representations of place value (ones, tens, hundreds, and thousandths cubes), and identification of numbers’ place and value.
My understanding of teaching place value increased significantly as I taught math groups within the centers. I am also able to work towards a personal goal of developing skills in working with leveled groups and how to modify or accommodate lessons/problems to support students in the lower math levels and students performing at a higher level. To support student learning and target misconceptions, I focused on representations of numbers when there was a zero in one of the places. My goal was to teach students conceptually that you must place a zero in the number when there is no representation because it fills a place and impacts the digits’ and numbers’ value (ex: write 4,026 in expanded form, write in standard form a number represented by hundreds and one cubes with no tens sticks, etc). Another goal was to build understanding directly between expanded form and standard form – where I taught the strategy of stacking the numbers so that the zeros (places) decreased as you went down. Through teaching these various small group lessons, I learned the importance of repetition in teaching the same math concept but using different representations and strategies so that every student and type of learner can identify a strategy or model that works best or most makes sense to them. Additionally, the practice deepened their understanding to a conceptual level and not just memorization.
After students started to develop proficiency with place value skills through the hundreds and thousandths place, Mrs. Martinez introduced the concept of rounding. Rounding uses place value to help them first identify where we were rounding (i.e. rounding to the tens or hundreds) and then how to determine what number to round up or down to for the final answer. To teach this concept, we used a number line method where students first circled the place they were rounding to (ex: circle 4 in 344 for tens), then placed the two numbers right above and below the number with no numbers in the ones place (ex: 340 and 350) on either end of the number line, next they marked the center of the number line with the magic middle number (ex: 345), and lastly they identified on which side the number fell and then circled the rounded number they were closer to (ex: 340).
While in rounding the use of place value was evident, in the most recent lessons on regrouping it has become even more evident. With regrouping, we are focusing more on the concepts of regrouping numbers using the ones, tens, hundreds, and thousandths cubes than we are on the algorithm for regrouping and carrying. These lessons connect to what we have read about and learned in class about place value and base ten knowledge because the lessons are based on the concept of unitizing and being able to work flexibly with units of ten. For example, when adding 45 and 67 students are learning that the regrouping algorithm they use (5 + 7 = 12, carry the one above the 6 and 4) they are learning that 5 and 7 ones cubes makes 12 cubes but also 1 tens stick and 2 ones cubes. They then trade in for the tens stick, and eventually also trade in 10 tens sticks for 1 hundreds square. These specific small group lessons deepened my personal understanding and pedagogical knowledge with regrouping because understanding the concept prepares me to assist students with mistakes that reflect misconceptions and not just careless errors.
My overall work with math and place value has also given me the chance to both observe and implement strategies for place value/base ten and developing a growth mindset and using mistakes as a teaching tool. The success of these strategies has been very evident! I have done some examples where I give them incorrect answers with common mistakes and asked them to explain the errors (ex: 4,000 + 30 + 4 = 434) as well as had students come to my board and explain the process instead of me just ‘going over the answers.’ Now, students are better checking their work for mistakes with place value because they know what mistakes are more common. Moving forward, these lessons have shown me the benefits of differentiated small group instruction and using manipulatives.