Yesterday, I started a new unit in math on addition and subtraction. Before beginning the teaching part of this unit, I gave the students a diagnostic assessment from Leaps and Bounds. This morning, I marked the assessment, and I was surprised to find out that nobody got perfect and that almost every student struggled with subtraction with regrouping.
Looking at the answers, it appears that most students know what they need to do to regroup and when they need to regroup, but with a focus on place value, many students are not just moving over the one, but moving over a 10 or 100. Then they’re struggling with then completing the question correctly. This is actually an error that I haven’t seen before, but one that I saw with much of the class this year. Now I’m figuring out the best way to address this.
I started with posting this Khan Academy video on my blog, so that parents can watch it at home with their children. This may be a good way to show students various ways to regroup, and allow for some additional practice at home, as well as the practice that they get at school.
Subtraction 3: Introduction to Borrowing or Regrouping: Introduction to borrowing and regrouping
I also think that I need to model how to solve subtraction with regrouping questions, and then give students a chance to practice this skill with solving various questions: regrouping once and regrouping twice. Many students may just need a quick reminder of what to do, and a short lesson followed by some practice time could assist them. Others may need more support, and this is where I’m going to need some guided math groups, where we can do more questions together and talk through the steps in the process.
For the few students that can subtract with regrouping correctly, they can work on showing and explaining their work in different ways: using both the traditional algorithm as well as some mental math strategies. I noticed that most students only solve the problem in one way and do not check to see if they’re correct. Focusing on the benefits of various solutions may help students see beyond just the “one right answer.”
Marking these diagnostic assessments this weekend reminded me that we cannot forget about the value of computations in math. Students need to know how to solve problems correctly and they need to understand that their answer is reasonable. I always pay attention to the four levels of the Achievement Chart when planning activities for any unit, and I will continue to be cognizant of this, but I’m going to ensure that students have a strong knowledge and understanding of math facts, so that they can think, communicate, and apply their ideas well.
How do you develop computation skills in students without making it just about rote learning? How do you balance knowing the math facts with thinking, communicating, and applying math skills? I’d love to know your thoughts on this!
Subtraction is a hard topic for many students. Many students find it hard because they have not developed a) a good place value system and b) a good idea of magnitude of numbers. I teach only through problem solving and contextual problems, so I tend to avoid the rote learning. My facts and practise come in games. I found Vandewalle’s book elementary and middle school mathematics is a great book to look for answers.
Have you tried using an open number line, using negative numbers, adding up, constant difference, or using a decomposition of numbers? Many of these strategies my students will automatically think about some version of in a problem. We then discuss the pros and cons. Very rarely will they even contemplate borrowing. In fact when in real life have you taken out a pad and paper to borrow. I know our curriculum says they have to learn standard algorithms and such but I find that to be the last resort.
You can also read Julia Anghileri(2001), intuitive approaches, mental strategies and standard algorithms. she has some great research on algorithms. Also Karen fusion (2003), developing mathematical power in whole number operations.
Sorry for the long comment. I love math.
Jonathan, I absolutely love your long comment and all of the resources! Thank you! I’ve used Vandewalle before, but have not seen the other books. I’m going to check them out. Your open number line idea is a great one, and my plan was to move into “adding up” and “decomposition of numbers.” Now you have me thinking about so many options … and at the perfect time too, as I’m just planning math for next week. 🙂
You can also try Fosnot. She is more primary for addition and subtraction but love her landscapes of learning and her other units in multiplication, and fractions.
Thank you so much! I loved Fosnot when I taught Grades 1 and 2. I recommend that resource all of the time to primary. I’ll check out anything that there is for Junior.
Has every student memorized the 0-9 addition tables at flash card speed? It’s the basis for all mathematics plus and minus.
This doesn’t appear to be the problem, but thanks for adding to the discussion. This is certainly something I can check.
Have you used the mental math strings? It only takes a few minutes each day, involves good thinking (beyond just rote memorization) & students usually like them as a quick minds on challenge.
Here’s a decent (ie better than mine!) description: http://mathcoachondemand.blogspot.ca/2011/03/mental-math-strings.html
Would so recommend mental math strings. Fosnot has a whole book of them in her primary section but I still use them for my juniors.
Thanks Kristi and Jonathan! I knew I saw these math strings somewhere before, but I couldn’t figure out where. I bet I used that Fosnot book with my 1’s and 2’s. 🙂 I hadn’t even thought of doing this, but this would be perfect for my students! In the games/activities I created to address “mental math” skills, I actually have students doing something similar to these math strings, but I just hadn’t thought of it that way. Too funny!
How would you do this with the class? Would you pose the question to the whole class and then have some students share their answers and strategies, or is this something you would just do in a small group? I can see how even a few minutes of this a day could help address some of my student needs.
I take ten to fifteen minutes at the beginning of the else’s on to do it. I then try to pick a problem that forces (as much as u can force a context problem) to work with the strategies in the strings. I really try to allow exploration but it is guided by the numbers I chose or the context. You can also do it in small groups and have a more direct discussion for those that don’t get it to quickly.
The best plan is the one that works the best with your class, but I would probably start day 1 whole class so everyone gets how it works, then have students do it in small groups after that – either self selected/teacher selected groups based on a particular strategy need, or whatever else works. Eventually, having groups set up strings for other groups to solve works too. After the first day, you’ll really only need a few minutes each time. This sets a good routine too, for math concepts you teach later in the year that may also need strings to reinforce learning.
Thank you both, Kristi and Jonathan! This really helps. I think that I will start with the full group, and then reinforce it in small groups after that. I know that I put together some activities/games/explorations that really focus on “strings,” even if that wasn’t what I thought at the time. This should allow for some good small group practice!
Thanks for the great conversation!
Your welcome. A lot comes down to the discussions that you have about the strategies. Let the students talk but bring out the math as they talk. Good luck!
Such a good reminder! Thank you, Jonathan! I agree with you about the importance of this talking time, but with keeping a focus on the math.