Last week when we were transitioning from models to formulas for adding and subtracting mixed numbers one of my struggling students showed me some absolutely amazing thinking and reasoning. My intervention group was doing such a great job with this topic that they started to get a little bit bored.

My instruction included the use of manipulatives and drawings. We did not go over any procedures except to combine the whole numbers with whole numbers and fractions with fractions, all had common units denominators, and then combine the fractional parts and make wholes as needed. We did not change from mixed number to improper numbers and then add the numerators… etc. These problems were fairly simple.

My student who has struggled all year is doing a great job with fractions. Like I said above, they were getting a little bit bored of the problems. So, I gave them a subtraction problem where you would need to break apart the whole number to regroup in fractional portion of the number in the minuend. I did not help them along the way on this problem. I was going to have them draw it out or use manipulatives to model the situation. This student looked at it, and said, “you can’t do that”. The other students subtracted incorrectly and then said oh year it won’t work. They were all curious about solving this problem. Side note, these students have continuously score under the tenth percentile on standardized assessments and don’t like math.

It did not take long for the student, who did not solve the problem on her paper but knew something wouldn’t work, to figure it out. she said, “Oh, it’s just like when you subtract regular numbers. You have to regroup!”

Now, they did not know how to regroup in this situation. Their experiences with subtracting with regrouping are in base ten, they wanted to add 10 to the numerator. The denominator was 4. It will be interesting next week to see how else they are understanding this. I was really excited when this conversation happened. Math Interventions work!

One thing I do not understand is why so many kids struggle with this concept. Is it the way we are teaching it? Does anyone ask the students if they really understand that exponents are repeated multiplication? While tutoring this week, I showed my student what it would look like in an expanded form rather than using the exponential notation and she started to understand what she needed to do for simplifying.

I know students in the US are taught that before Algebra 1. But, if they are not ready for the abstractness that comes with repeated multiplication they might not be ready for exponents at that time.

So far this year I have noticed several students making the same errors when subtracting. For example, with 5 – 3 student kept saying 5 was the answer. And for 2 – 2 they answered 2. So, I watched what they were doing. They were using their fingers as if the problem were an addition problem and then they subtracted the subtrahend. For 5 – 3 the students put up 5 fingers, then 3 more fingers, and then they subtracted the 3. When they are doing this, they think they are doing it correctly. To me this means they have a procedural understanding of addition. They are overgeneralizing that procedure with subtraction.

Additionally, this was observed with more students earlier in the year and is now only observed with students who have not transitioned to a better understanding of addition and subtraction. So, this makes me wonder if all students go through this or if it is just my tier 2 and 3 students.

Another common error I keep seeing is when subtracting 0 students answer 0. It is unclear what they are thinking on this right now. I will be asking them how they know that in the near future. We have mostly worked on addition and counting/number recognition than with subtraction.

12/14/22

Today I saw the moment a child noticed the repeating pattern of the tens and ones when he got to 109. He couldn’t count past 109 verbally but when he was writing his numbers in a 120 chart, he saw that it repeats. He completed his 120 chart (and accurately) for the first time. His face lit up and he said, “oh, it’s just the same as the top”. He finished writing and jumped up to say he did it! That’s what it’s all about folks. That sweet spot right there.

This past week I spent time with a friend and we talked about math teaching and leaning. She talked about how her child struggled learning division the way it is taught (she specifically stated-partial quotients and using an area model). She expressed how her daughter never struggled before and has not since. She ended up teaching her a long division standard algorithm, which I have heard teachers call it the “McDonald’s cheeseburger” way. My friends daughter must not have had a robust understanding of place value, multiplication, or using manipulatives. If a child understands multiplication and using base-ten blocks or area models, they should be able to divide. I always start out with base ten blocks when students are learning division because they need to understand what division is. Division is sharing a certain amount equally and is tied closely to fractions and decimals. It is really important for us to help students make the connections between and among different parts of the mathematical domains. Using the base ten blocks allows students to physically move and exchange pieces when splitting them up into equal groups. Also, with this method, students see what a remainder means. There can be a lot of great mathematical discourse by students using base ten blocks for division.

This is my very first blog, and I do not know if there are certain rules to blogging. So, I am going to just share a short snippet of my thoughts. I have seen several students struggling with basic number sense my entire teaching career. However, my experiences most recently have me questioning what I am doing (or we are doing) and wondering how I can help students. This year students have struggled with simple addition and subtraction facts. The most boggling, 6 – 6. Students have given answers such as 1, 3, 6, “there is none,” and “you cannot do it.” There is no answer for 6 – 6 for some students. It is worth noting that this was not an isolated case. There were several students who struggled with the concept of subtracting everything and having zero left. The students spanned two grade levels and came from different classes. Is this something that has always been there and I missed it? Is this typical for students in 3rd and 4th grade to struggle with basic foundations such as zero? Or, is this because of the lack of schooling students have dealt with the previous two years? The most important question, how can I help younger students so they don’t have these same struggles?