I told her I had a problem and asked her to solve it. (This intervention student shall remain nameless due to the fact that the internet is an infinite digital footprint, and I believe some day she will be a famous scientist and doesn’t need to read this story about herself online.) My problem? I have 45 books that I want on three shelves, equally please. I like when books are evenly spaced out. It’s my math brain I guess.
She sat there, solving this division problem right in front of me for a few minutes. She wrote the number sentence and acted it out, even got the correct answer. Conceptually she was SOLID. But the second I asked her what the 45 stood for, I got the deer in headlights look. I waited…nothing. Then, I asked her what the 3 stands for in her number sentence…no idea.
The very next hour I was in a kindergarten room. This problem was up next. Students were solving with drawings. They wrote number sentences to match their drawings. When I asked “What does the 3 mean?” No idea. Their understanding of what the 3 could mean became incredibly fuzzy.
I was not shocked to see this. This is a daily occurrence at every grade level with students working at every level. This is a fundamental problem with Math Practice Standard #2 (Reason abstractly and quantitatively.). I’ve realized that once the students pull the numbers OUT of the problem, they aren’t thinking about what they mean as you try to connect that back IN to the problem. Remember all of your teachers saying “Label your answer!”? I believe they were onto something. It is not enough to say “Label your answer!”. We won’t get over this barrier until we start asking what ALL the numbers in the number sentence that you wrote represent.
3 ways to deepen this understanding, and practice it over and over:
- Ask: “What does that number stand for? How do you know?” The simple act of going back INTO the problem after solving it will deepen their understanding.
- Write words along with the number sentences for a while, until they begin to see how they can move back and forth between the words in the problem and their number sentences.
- Give a number sentence all alone. Ask them to write a story. It’ll be pretty hilarious at first. Apparently my student had potatoes on the brain. Notice, I’m not asking for a story problem, but a story. That means the story will not have a question or an unknown at the end of it. This helps them make sense of ALL the numbers in the number sentence.
This is something that we must all commit to to help students make sense of mathematics, make sense of problems and to make math less abstract. I would love to hear any other suggestions you might have to strengthen our little mathematicians in this area.
OK, I believe strongly in feedback and using it for my own personal improvement. I’m really looking for ways to find out more about my audience, and to improve my practice. So along with getting good, honest feedback, I decided to give something away. I mean, who doesn’t love to win stuff???
Would you please fill out my short survey by clicking HERE? It’ll be open until midnight central time on January 31, 2017. From the responses I’ll randomly choose one awesome winner.
Or you can click on this great photo of my cat to take you to the survey. Cause she’s awesome. She really wants you to win.
I read a statistic somewhere that English Language Learners in the United States talk only 4% of the time during their day. That shocked me, but when I thought about it I realized that for most students there is not a lot of talk in their day. Classrooms that are looked upon as “in control” or have “good management” tend to be quiet in our own minds as teachers.
But talking can be done with good management and can be powerfully done. I’m hearing over and over that research says that talk is so important for both language development and understanding new concepts.
So I thought I’d try out some mini debates with my intervention groups. I was thinking that these are my students who are talking the least because they are the most unsure in the classroom. What I did was present them with 4 solutions to a problem.
At first it was painful trying to get them to see anything, or even say a word. They are so used to waiting out the teacher, or waiting for another student to see what they notice. And then, they began to see things…I started to write what they saw, and it was exactly the things that I’ve been showing them for weeks. When it was their time to talk about it, the light bulb clicked on. What an eye opener this was for me! I’ve been talking about tens and ones and addition strategies…showing them one at a time. I WAS DOING ALL THE TALKING. Ugh. I don’t know how many times I need to learn this hard lesson. Check out what they noticed:
So I whipped up a few this weekend for you to try. I’d love for you to try it with small groups, with partners or with your whole classroom. Just click on the image below to download and let me know what you think!
Pro-tip: Give them tools. Give them tools! Really…give them the tools!
I was told this over and over by a math coach when I was a third grade teacher, and for some reason it really did not sink in at all. I made them “available”, which means I had buckets of them in a cabinet and they could go and get them if they needed them. No one ever got them unless I brought them out. I don’t know what my problem was, were they too messy? Too loud? Too much of a pain to have available? Did I not know how to use them properly?
It wasn’t until I was an interventionist and coach that I have gotten to see why the tools are not only valuable, but essential.
Without tools or imagining what the tools can stand for, students are really not connecting the numbers to anything whatsoever. We can tell them what to connect it to, but you know they aren’t all with you all the time. The tools are something they have to manipulate and understand.
Right now I’m using The Cupcake Shop for my intervention group. Just because it’s cupcakes it’s a highly motivating projects, but also the stones we are using make it so REAL. They are building orders and figuring out that in this case, the multiplication symbol actually stands for the words “boxes of”.
If I were to tell them that the multiplication symbol stands for “rows of, groups of or boxes of” without concrete materials, what could they possibly connect it to but something in their mind? And with students of all different backgrounds, we really can’t rely on them connecting it to something that we are thinking about.
Let them touch it and try it! In the preview of The Cupcake Shop you can try this project out for free to see if it’s right for you.
Tonight I watched my beautiful daughter at her archery class. It’s rare that I ever talk about anything personal, but I was so moved by her tonight that I felt compelled to write something. It was only her third time attempting to shoot a bow and arrow and tonight she hit the target over and over again. Why is this so special to me?
Well because the first time was awful. The first time was absolute chaos for her brain. She dropped the arrows over and over, never once hit the target and looked like she might cry the whole time. She was always the last one at the line struggling to get all 5 arrows off her bow. But she never stopped. The master archers teaching the class gently guided her, adjusting her arm, her stance and making small tweaks to her form.
The second class was a bit better, but not much. She was hitting the target this time, but the arrows wouldn’t stick. She didn’t give up, and as she listened to the instructors you could see her confidence growing.
And tonight? BAM. She nailed it. Over and over again, she hit that target, and the arrows stuck. Her smile when she turned to me was ear to ear. She was SO PROUD.
This reminds me not to be discouraged when I’m working with my students at school. Every day, that small amount of feedback and gentle guidance is moving them forward. They really are learning, and sometimes it’s so small that I’m frustrated and wanting more. But I need to trust the process, and not stress about the data, the protocols, the progress monitoring, and the pressure from above. I know they are learning, and they are PROUD of their learning. I know that I won’t stop finding new ways to teach them because they deserve to feel the same success that my daughter got to feel tonight.
I think I’ve discovered the number one problem in just about every math classroom. It’s a problem that I see over and over in every math classroom. Students that struggle and students that excel have this problem. They tend to look only at numbers, not thinking about what they represent. They try to memorize vocabulary without connecting those words to anything in their life. We race through math curriculum teaching quickly because we have to get through SO MUCH CONTENT. With standards and interventions and assessments to give, we don’t have time to think about what the numbers and vocabulary mean.
So we have to sneak it in.
We need to make math real all day. Not in some silly contrived way, but to make our thinking known as we notice math throughout the day. Take this example of real life geometry:
When we connect these terms to their real life, they begin to sink in, and you don’t have to write some fancy lesson for it. When they open their pencil box, their desk, their books we can mention these words and bring the vocabulary into the front of their minds.
When we have two packs of gum, let’s connect it to numbers. Do we have enough for the whole class? Is there another way I can write that number sentence?
If you think about it, we read all day long, we write all day long…let’s make an effort to bring mathematics forward all day long!