I was out all day today at our Board’s Junior Empowerment Conference, and one of the activities that I left for my supply teacher was a diagnostic activity on multiplication. Our next big math unit is multiplication, and this skill overlaps with our current focus on measurement. I wanted to see what the students knew and where we needed to go next. Last year’s EQAO results in math indicated that our junior students need to continue to focus on explaining their thinking, and this is an area that I continue to focus on in the classroom. While I knew that many students knew their basic times tables, I wondered if they could explain their thinking and make connections to real world reasons to use this skill. I basically kept the questions very open-ended in an attempt to truly see what the students knew. And this is where a student challenged my thinking with her very interesting response.
Here’s a look at the tweet that I sent out today of this Grade 5 student’s work:
As a school, we’re really focusing on “student voice,” and this is a voice I couldn’t ignore. I have to admit that my initial thought was, “Well students need to explain their thinking because we assess thinking and communication.” And while this is true, I don’t want students to only do this because “they’re getting a mark.” Real learning needs to be about more than the mark!
Then I started to question why she thought this way. (Truly honouring “student voice,” I’m going to need to ask her more about her thinking tomorrow, but for now, I’m going to do some inferring.) I started to wonder what I could have done differently to have a different response. The truth is, I think that I made a mistake. My intentions were good. I wanted an open-ended, diagnostic assessment that could show me what students knew, what they didn’t know, and where I needed to go next. But I didn’t give the students a meaningful context.
The truth is, given the activity that I created, I don’t disagree with what this student said. Why should she have to explain her thinking? She knows that 12 X 2 = 24. This is a basic fact, and one that she’s committed to memory. Isn’t it enough for her to just say that she knows it? If I want her to explain her thinking, I need to give her a better question. I need her to see the value in multiplication, and I need to provide her with something to really explain. While not intentional in the least, I think that I ended up creating a make work project for her, and one that was really no different than a giant worksheet, but without the photocopying.
You see, the one thing that I learned looking through the student work today is that almost all students don’t know any real, meaningful reason to use multiplication. They know reasons that they’ve seen on worksheets or in math textbooks before. They gave me lots of examples of six students have four cookies. How many cookies to they have altogether? But then I start to wonder, why do the students have the cookies? Why do I care how many there are altogether? No student should be eating that many cookies!:-) Given this information, it’s no wonder that this child does not want to explain her thinking. She can tell you that there are 24 cookies … and so what?
How am I going to change this? I’m going to create the distinction between the need to memorize basic facts and the meaningful application of these facts. Strangely enough, this takes me back to a conversation I had about an hour before seeing this student’s work. Today, I was working with a Grade 4 teacher, Jennifer Beattie, and as we were cleaning up, we were talking about math. Jennifer mentioned that her students are working on multiplication right now, and she said, wouldn’t it be great to have our students FaceTime to practice their skills? From there, we looked at how we could have a small group of students FaceTime each other to first practice their facts (writing down questions for each other on whiteboards and then each showing their answers), followed by us presenting a problem to both groups of students, giving them the week to solve it, and then coming back together on Friday to share their thinking. Today showed me that this needs to be a rich problem. It needs to be one worth explaining. And maybe as the students talk about their solutions together, the need to “explain their thinking” will matter more.
What do you think? What meaningful problems would you suggest that would work for both grade levels and would inspire students to want to “showing their thinking?” I’d love to know your thoughts.
If you want a truly open Q, try “how many ways can you make 24” instead.
I totally agree with your entire line of thinking. I say kids should be taught operations deeply, but once they know algorithms, they shouldn’t need to resort to things like arrays or diagrams or repeated multiplying. I call that getting your Algorithm License.
Look at it this way: she has already passed into a more efficient and proficient use of multiplying. Challenge her to come up with her own problems that require multiplying. Or let her work with bigger numbers, to show her skills.
Now some in your class may still need to show you how they understand multiplying, and that’s okay. Communication looks different for everyone!
You continue to be the most reflective practitioner out there!
Thank you so much for your comment, Matthew! I really like your focus on the fact that “communication looks different for everyone.” That’s so true! This open-ended activity worked well for some, but not for all. I also saw that coming up with meaningful ways to use multiplication was a struggle for everyone. Maybe I need to show them some of these meaningful ways in the questions that I ask, and maybe I need some provocations to get the students to think about more of these ways on their own. Hmmm … 🙂 Challenging her to work with bigger numbers might help as well. I definitely saw that there are students in the class ready for this challenge, and maybe this would also inspire greater thinking. There’s always so much to think about!
P.S. Thanks for the kind words too! Much appreciated!
Matthew, I couldn’t agree with you more. Once students understand algorithms they should be using them and not need to explain. They also have to be able to use them interchangeably, sometimes and algorithm is not the most efficient strategy. That’s why I tell my students it’s important to know how to explain their thinking and be able to understand the math. However, I don’t need long explanations just be able to see their thinking.
I also like the idea of opening up questions to a point, I think that students still need a good context to grasp and really understand the concepts. This is even if they can show or tell you the answers. However, Marion small work has some great talking points.
Good comments, your also a great reflecting educator.
Jonathan, the line in your comment here about “not needing long explanations just to see their thinking” really had me pause. I wonder if we tend to attribute “showing your thinking” to great big written pieces. I know that in the past, this was how I thought. Some of the best thinking that I’ve seen though from students came with few words and lots of pictures. Sometimes it was an audio comment or a short video that really helped capture this thinking. I wonder what students think about when they hear “show me your thinking,” and I wonder if all teachers have the same impressions of this as well.
So much to consider …
You know I can’t help talk about math. Research shows that we can hinder students when we make them explain their known facts. We can actually train them to explain only because we told them too. If kids get it they get it, and you know when they do and when they don’t.
As for contexts that I like in multiplication:
1) I love asking students ratio questions: the school 6 times smaller then the can tower, if the school is 23m how tall is the tower? (I would make the context realistic, just trying to give an example)
2) also like talking about GDP and the ratios between different countries
3) good old area, in fact have you seen the Dutch algorithm for multiplication? Works really well for area and then double or higher digit. It actually works for students when they start learning factoring and algebra. Students can plan houses, floor plans, etc.
4) the fosnot units are good but they are meant for more 3-4 then five. However, they have great string lessons and the landscape is also helpful.
Great reflection and thanks for sharing.
Thank you Jonathan for chiming in and with such great suggestions! I don’t have the multiplication Fosnot unit, but I’ll check with the Grade 3’s and 4’s. Maybe there’s something I can adapt. You’ve given me lots of other options to look into as well, and all very meaningful ones at that!
The only thing that worries me with not explaining their thinking when it comes to the algorithms is the number of students that I noticed (up until now) that had the memorized facts, but didn’t seem to understand what they meant. The meaning piece is important — right? Maybe this meaning best comes though through meaningful problems instead of almost “rote thinking” on “rote facts.”
Again, so much more to contemplate …
I do find a difference between fact and mathematical understanding of concepts. I think that is where the context comes into play. With the more open ended questions you will get more factual response, which is still important but with contexts they should be able to use those facts to solve the problems. This is were the explanation part is crucial. I also said in matthew’s comment that to me it’s all about the versatility of their strategy and understanding what to use and why.
I also find that asking ratio questions really tests their understanding because most textbooks and questions as part active questions (how many things are in a group of…). They really have to see where the multiplication is in the problem. It also allows great talk.
Also teaching the Dutch algorithm, which is an array model. For double digit think of four boxes and aphave students break numbers into place values. Each box is multiplies by the length and width. Sorry hard to explain in words and can’t draw in word press.
Don’t know if that answers your question.
Thanks for sharing more here, Jonathan! I really love the idea of ratio questions. As a Board, we’re spending a lot of time looking at proportional reasoning, and this would tie in wonderfully. I also need to do some investigating about the Dutch algorithm. I think that I know what you’re talking about, but taking these written words and creating a visual is definitely not my strength. 🙂 I must be able to Google it. 🙂
Thanks again for all of your help! I’m sure to have some more questions for you!
What about turning that very question you have asked your readers back on to the students? What if we asked our students “In what situations do you think you would use multiplication? or have you used multiplication?” and create some sort of chart that they can reflect on as we work through the unit. This way they can reflect on this question and use some serious thinking skills (as I have to actually think about what answers I would anticipate them saying), and perhaps even provide reasons for their thinking. Right off the bat, we have touched on 3 of the 7 process expectations.
Now, with this question, we also bring to the table, the real-world connections that we strive for our students to make all the time and we are providing them with the opportunity to make powerful connections because they will have to think deeply about this question and give reasons why.
What do you think?
Thanks for the comment, Jenny! I really like the idea of turning this question back on the students. I wanted them today to think of ways that they used multiplication in real life, but the only ways that they seemed to come up with were ones that they saw on worksheets or in textbooks from before. None of these ways were really meaningful ones or inspired deep thinking. Maybe these ways are something that they could consider at home too, and even talk to their parents about. I wonder if parents and students could look closely at “meaningful math” together, and then we could further support and push thinking forward in our classrooms and through our FaceTime calls. What do you think?
Aviva, great post. I think you are spot on about students needing the math to be purposeful- tasks in general. When they connect yes they will demonstrate understanding through richer representations. I still wonder where her head was at with the cake analogy… That in itself is worth investigating. She was using a metaphor to EXPLAIN HER THINKING all the whole refusing to explain her thinking….oh the irony! I wish I could interview this our student for student work study inquiry! I love the use of the word provocation in your title. For that is exactly what her voice has done for you and me and others. Now Aviva you may have just provoked me to start blogging myself!! Love this co learning. Thank you.
Thanks for the comment, Mona! This student definitely got lots of us thinking today. I’m so interested to hear more about her cake analogy, and what was going through her head at the time. It really is quite incredible!
And on a side note, I do hope that you start blogging. It’s a great way to learn together!
I really don’t see the point of explaining your thinking, and I don’t understand why it is so important. What I said about cake is true. Why does everyone else think otherwise?
Layla, thank you for commenting here! I loved your blurb about the cake, and it really got me thinking. The adults that read your cake comment (both on Twitter and on the blog) absolutely agree with what you said. You’ve made me question if I’d want to explain my thinking (given this question) either. Why do you think people don’t agree with you?
Some other teachers here have helped me think of better questions. I’m hoping that we can have a good math talk tomorrow and discuss some of these questions as well. I’d love to know more about why you don’t like to explain your thinking. What does “explaining your thinking” look like to you? I wonder if there’s ever a good reason to explain thinking. So much to think about!
See you tomorrow!
To me, explaining your thinking means a lot of extra time taken up for nothing when you could be learning other stuff, or doing something else. If you know the answer, you know the answer!
Okay, I understand that … but how do I know that you know the answer and what this answer means? What if I get a different answer than you? Who’s right? Who’s wrong? Does it matter? So much to think about, Layla! Thank you for such a great conversation in class today too.
I love your cake analogy, it really made me reflect on why it is that, as teachers, we ask students to explain their thinking in math. So, going along with the analogy: what if I enjoyed the cake so much that I’d like the recipe? Not only the ingredients, but the step by step of how you made it. Being a visual learner, I’d really like to see you make it. How does that apply to math? Well, the ‘ingredients’ tell me if you got your facts straight (that you’re not using anything others may have an allergy to ;), and when you give me the step-by-step I (or someone else, a peer or younger student) can follow along your reasoning and duplicate it. It’s like you’re teaching me your way of solving a problem, because even in math there’s usually more than one way to get the right answer. If you get the answer wrong (or I hate the cake) maybe it’s so that by seeing the process, we can discover where you made a mistake and help fix it (what if your cake could have been great if you only used 1 teaspoon of vanilla extract instead of the 1cup you mistakenly poured in). Maybe we’re hoping that if you have to show your thinking, you will be forced to examine each step closer and correct your own errors.
I think it’s great that you posed that question and that you posted a comment on your teacher’s blog – you made your ‘student voice’ heard and have proven that it really matters. I love the question that your teacher posed to you in reply, what a great challenge you have ahead of you. So exciting to have a real reason to do math. Good luck, I’m looking forward to reading what you come up with if you’re willing to share.
Magdelena, I love your response to Layla! I’m going to have her look at it this morning, and see what she thinks. I have a few initial thoughts as well — and they really coincide with my issues based on my question itself — but I don’t want to influence Layla’s thinking. I’ll let her reply first, and then maybe, depending on what she says, share more later.
Thanks again for pushing my thinking and Layla’s too!
Aviva, what a thought provoking student reflection and post. Made me stay up past my bedtime 😉
It is so amazing that your students read your blog and feel confident enough to comment. It’s a testament to what an amazing teacher you are!
As I was reading, I was reminded of the first time I did Scratch with my junior students. It was incredible to watch them do math that was beyond what was required for their grade level in the curriculum. They were so excited to program that they did not even realize they were doing math. It really does make a huge difference when learners are tackling real problems and tasks that are meaningful and motivating to them. Thanks for the reminder that as teachers we should always start with the ‘WHY’.
Magdalena, thank you so much for your comments to both of us (and for staying up late to do so). Your Scratch example got me thinking. What did you do with your students? What was the math connection you made? Programming certainly provides a real world application for math (depending on the activity), and it may just increase the interest in students “showing their thinking.” I’m wondering if posing a real world problem and giving options about how to solve it (programming being one way) would help. This almost sounds like the beginning of a meaningful design challenge. I don’t know if it connects to multiplication, but it might. Hmmm … Lots to think about on this Thursday morning!
No, no multiplication involved unfortunately. Just plotting coordinates of +/- numbers.
Thanks Magdalena! I wonder if there are any programming options that do involve multiplication. I need to think about this one.
I think it is fantastic that “student voice” created this deep thinking on the parts of teachers. It makes me wonder how many times we neglect to hear and understand the voices of students. I am so proud of Aviva and Layla for creating a safe classroom where voices are heard and respected!
Thanks for the comment, Paul! I completely agree with you. I love the fact that “student voice” is such a focus for us as a school this year. Without this focus, I’m not so sure that I would have heard Layla’s voice and made some important changes to my instruction. Equally as amazing is the fact that her voice inspired so many educators from around the world to look closely at their practice. Our students are the BEST!
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