I find myself reflecting on what I believe is best for my students and best for my students’ beliefs about what mathematics is often. When I get the opportunity to take a look at my students’ work and time to determine next steps, I can’t help but reflect on how my beliefs inform what next steps I would take. However, I wonder, given the same students and the same results, if we would all give the same next steps? Let’s take a look at a few common beliefs about what our students need to be successful and discuss each.
My kids need to know their facts:
Often we see students who make careless mistakes and wonder why they could have gone wrong with something so simple. To some, the belief here is that if we could just memorize more facts, they would be able to transfer those facts to the problems in the assessment. While I agree that we want our students being comfortable with the numbers they are working with, I’m not convinced that memorizing is the answer here. Our Provincial test includes a few computation questions (for grade 3 only) and none of these are timed. Most of our questions involve students making sense of things (across 5 strands), some with contexts and some without.
Instead of spending more time worrying about memorizing facts, I wonder if other strategies have been thought of to help our students as well? For example, other than the 4 questions in grade 3, all other questions allow students to use manipulatives or calculators, and all questions have space for students to write in the margins any rough work or visual models might be using. The question below is one of the few computational questions a grade 3 student is expected to do. Many who might use the traditional algorithm might accidently pick 41. How might a number line help our students visualize the space between the two numbers here?
My kid aren’t reading the questions:
Often we notice that our students understand a concept, but the question itself requires several steps and students don’t end up answering what is asked. For many the solution is having students do some kind of strategy (whether they need to or not) like highlighting key words.
I wonder what answer students might get to the above question? Will they get the right answer here? What would you have liked them to do instead?
Highlighting specific words or filling out a standard graphic organizer isn’t the answer for all kids, nor for all questions!
Personally, I think the issue isn’t that our students can’t read the questions, it is that they are jumping to a solution strategy too quickly. Instead of believing the solution is to have our students highlight or fill out graphic organizers, what might be more appropriate is to help our students slow down and think deeper about the questions they are being asked. I wonder, however, about how our students’ prior experiences might be a big part of why they jump to solution strategies too quickly? If students typically receive questions that are simple and closed, and typically follow a lesson directly telling us how to answer those questions, then I wonder if the issue is that our students can’t read or if their experiences have actually been counter-productive? If students don’t experience mathematics in ways that help them make sense of a situation, and instead see math as answering a bunch of questions, then it is no wonder why they aren’t reading the whole question! They have been trained to believe math is about getting answers quickly, and that we get rewarded (less homework, better grades…) when we are fast.
Instead of more time practicing reading and highlighting questions, or filling out graphic organizers, we might want to spend more time building questions together, asking students to pose their own problems, or providing experiences for students to notice and wonder. What if we started by showing this:
What do you notice here? What do you wonder?
What might our students see? They might notice things we didn’t realize they were not even aware of (e.g., each row has 7 boxes, some rows are missing numbers, the numbers go in order, there are 2 Ss and 2 Ts at the top, the letters at the top probably mean days of the week…..).
What might our students be curious about? They might wonder why April 1st is on a Monday and not a Sunday. Or wonder about the “Chapter 1” part.
Then we could continue to show more of the question and again ask what students notice and wonder.
No matter the grade level or content, my students need to realize that mathematics is about the development of mathematical reasoning, not just quickly jumping to a solution strategy (especially not one that my teacher told me to use all the time). Taking the time to think deeply about our mathematics is what I want from my students! Numberless word problems, notice and wonder strategy, contemplate and calculate… any strategy that helps my students slow down, pay attention to visuals, and start to think about the situation more will probably help many of our students given enough opportunities.
They need more practice with questions like these:
Many believe that if students are doing poorly on something, that the best course of action is to continue practicing that thing. For example, if our students are doing poorly on Provincial testing questions, then giving students more questions like these will answer all of the issues.
To some, the answer to the problem is to give sample questions every week in a package or even more frequently. While there are times when practice is helpful, if our students are struggling with the content, giving more questions will not be helpful! Take a look at Daro’s quote:
More often than not, the quick fix solutions like this (noticing our students struggle with something, then providing the same instruction or same types of practice again) will not be successful. Developing our math knowledge for teaching is probably the most difficult aspect of teaching mathematics, and is definitely NOT a quick fix, but it is probably the answer here! If our students are struggling with concepts we believe they should be able to do, it is likely that they haven’t had the right experiences to help them learn! Have we provided experiences for our students to deeply explore a variety of representations? Have we provided experiences where our students are able to develop reasoning skills? Have we provided ample opportunities for our students to consolidate their learning?
Remember, questions that are designed to show evidence OF student learning is not necessarily the WAY students learn! Handing out these questions toward the end of the learning is far more reasonable.
My students need more stamina:
Often when giving young students extended time to sit and focus independently on an assessment task we have many that struggle to remain focussed. For some, the solution is to help students build stamina through quiet seat work regularly. While I do agree that we should have our students work independently at times, I’m not sure this is the answer to the stamina problem.
To me, I think the issue has more to do with how our students experience mathematics. Do they get lots of short closed questions where the right answer is apparent quickly? Or do they experience rich problems where they reason through and figure out their own way of making the question make sense? Do they learn math through independent think time and cooperative problem solving, or are they told material then asked to remember all of the steps and terms. Is their mathematics class structured in a way where students come to rely on themselves (individually or within their group) to make sense of challenging problems, or do they feel the need to access their teacher every time they don’t know what to do? When our students are working, are we monitoring all of our students’ thinking, or are we spending a lot of time guiding our students’ thinking?
If stamina is the issue, I wonder if we are allowing our students to productively struggle enough? If we see a bunch of hands raised around the classroom all wanting us to help, this is a huge red flag moment. Our students are asking US to think for them! If we find ourselves sitting beside our students helping out a small group all of the time, this might be another red flag. Our students are learning that they always have access to us right beside them when they learn, but the unintended problem is that we aren’t allowing our students to struggle enough!
Providing our students with a variety of manipulatives to learn and puzzle through their mathematics on a daily basis might be a big step in the direction of allowing our students to gain the confidence and stamina they need to do well every day. Notice how these students are using manipulatives to help them make sense of their work:
When our students learn their mathematics using manipulatives and have access to any manipulative at any time to solve new problems, we start to notice that our students come to realize their role is to slow down and make sense of things. When our students have had various experiences with manipulatives and can see their role as “thinking tools”, we start to notice fewer hands asking for help, less need to have to sit down with a group, and more time for us to really notice our students’ thinking going on in our classroom. When this starts to happen, we no longer see stamina as a big issue.
I want you to consider for a moment the differences between the beliefs I’ve mentioned. What messages are we sending to our students about what is important in mathematics? Strategies that get us to do better on the test, or strategies that help us slow down and think more? Is math about memorizing or figuring things out? Is math about removing the context to mathematize a situation, or about using the context to make sense of things?
Sure I want my students to do well on any assessment they are given, but I want them to do well every day! Quick fixes and silver bullets often don’t help our students in the long run though!!!
So I leave you with a few things to reflect on:
- What are some of the quick fixes you’ve heard about? Did you try any of them? Did they work?
- Is there a strategy that you see working for all of your students? Was it actually helpful for everyone, or just some? Do you expect everyone to use this strategy?
- Have you been asked or told to use specific strategies? Do you see it being successful for everyone? Do you have the autonomy to choose here based on the students you have in front of you? Do your students have any autonomy over the strategies they use?
- When looking at your student work, are you determining next steps for your students, or for yourself?
It is far easier to determine what your students can and can’t do well, than it is to figure out what to do next. While we absolutely need to help our students notice the things they could do to improve, we also need to do the hard work of reflecting on our own practices. Our beliefs about what is important and how we learn mathematics have a direct effect on how our students will do in our classrooms!
Thinking Mathematically 
In my four years of teaching math, the only fix that I’ve discovered is slow, sometimes painfully so. It is changing the understanding and the perception of the subject. No one would love reading if all they read were random lists of words in the dictionary, though some might have taken pride in their ability to do it fluently.
You describe this change throughout your post. If teachers and students see math as full of creativity, imagination, questions to ask, discussions to have, art to bring to life, everything starts shifting and changing. I love reaching this point in the year when every one of my kids participates in the number talks, builds models, constructs mathematical arguments… Some are moving to multiplication and division, decimals and integers operations. Some are still figuring out two digit subtraction and just barely got comfortable with their benchmark of ten. Regardless of their numeracy skills, they use any available tools (symbolic or physical) to engage in mathematical reasoning. This week, one of my students who worked really hard for half a year to figure out addition and subtraction of numbers to 20 (with rekenreks, and 10 frames, and subitizing cards) joined our Splat! warm ups (created by Steve Wyborney). He explained how he convinced his dad that square is, in fact, a rectangle. He noticed, wondered, estimated, measured, calculated, made diagrams to explain his thinking and was cheering to find if his estimate was close.
There are a lot of things that I don’t have any fixes for. Efficiently addressing numeracy gaps. Giving enough attention to every student. Providing timely feedback. Staying focused on specific learning intentions. You are right, a lot of it is about “what to do next”, and how to prioritize. But I think my slow fix has been working for me. I am learning to teach mathematics as an adventure of human mind that everyone can join.
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Thanks for sharing. Real change takes time! I think most of us that are on the learning journey are figuring this out as we keep learning. There is no quick fix! There is no silver bullet!
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Lots of good things to think about here. I recently gave a test on solving one-step equations to my sixth-graders. I basically gave them one way to show the work they are doing with inverse operations. This was not because they couldn’t solve any other way, simply because this is a method that works for all equations, even those that can’t be solved by inspection. However, more importantly, it is a method that will continue to work in seventh-grade when we are solving multi-step equations that cannot be solved by inspection. But, I struggled when I came across a student who told me that 2r=12 means that since 2*6=12 then r = 6. Is that sufficient work? I deemed that it was not, because if you cannot solve by inspection, then where does that 6 come from? Am I simply forcing them to use a system just because it works for me? I don’t know. I do know that the same student said on a different problem that the equation -3x=4 could not be solved. This reinforces my idea that students need a technique for how to solve when inspection doesn’t work.
In class we solved with Algebra tiles, we solved by inspection, we solved using cups and counters. We did card sorts. We did a variety of things, but the majority of them involved equations with integral solutions. I would argue that they very time we need a standard algorithm is when we have messy solutions. Also, a standard algorithm gives us a way to communicate with other mathematicians that is efficient and understandable. It gives us a technique to try when we don’t know what else to do.
Did it work for every student? No. I’ve never seen anything that worked for everyone. But it did work for 48 of my 51 sixth-graders, and I anticipate that the majority of them will continue to use it next year and beyond. That feels good enough to me.
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I agree with the first comment — most of your descriptions describe “fixes” which are based on math being this procedural ritual… and I revel in the description of the student who’s using all tools available to *build sense,* not to get answers. Yes, a square being a rectangle 🙂 🙂 🙂 🙂 THat’s a growing mind.
I think I”ll beg to differ just a tad with your idea of ‘not worrying’ about the facts and looking for other strategies. In my experience with older students (college age and older), they rely on counting and strategies (most are allowed a times tables chart) to find hte facts with basically no evidence of number sense attached. I think ‘learning the facts’ should be a path towards getting that number sense. (I’m thinking of designing some online exercises that will dress up reknrek as a game — like Candy Crush, maybe? — and seeing if some of my folks who seem to have skipped subitizing could grow those paths…)
I also struggle w/ the ‘by inspection’ to algorithm path… I think most materials, in their desire to get students past intuitive inspection, leap too quickly into examples to which students don’t assign sense (reducing 17/51 is one…) Using language and manipulatives — and more than once, for crying out loud! — to help students construct that understanding so that they can explore the ‘harder’ questions and make sense of them is … yea, it’s a slow fix. There’s also a *real* danger of giving students those “challenge” problems where we would like to *think* they are engaging in rich reasoning. I’ve had students come down to me anxious or angry about that and rightfully so. To them, it was as if the problems were delivered in a foreign language — the struggle is not always constructive.
Now, I work with adults in a tutoring center… lots of my strategies and “fixes” involve shifting student attention from “what does this look like?” to “what does it mean?” especially for things like “x/4 = 12.”
THanks for your tangible examples of places where we slip into “math is a procedural ritual.”
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Thanks for sharing your thoughts. Hopefully my statement of “not worrying” about facts wasn’t taken as not worrying about number sense. I completely believe we need to help our students develop their math fluency (flexibility), but I’m not sure memorizing is how this is done.
And your comment about “challenge problems” is right on! Handing out work that is not developmentally appropriate isn’t helpful at all. We need to make sure we are asking our students to do tasks / activities / problems that allow them to access their prior knowledge and use their reasoning skills to make sense of things. Again, we need to continue to learn more about what is developmentally appropriate to know what our students’ next steps are.
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