Zandra de Araujo: All right, I’m gonna get started because I know many of you are on your lunch break, and I’m so thankful that you chose to spend some time in the middle of the day with us today. This is the first of five webinars for the Lastinger Center this spring, so we’re glad you could make it.

I’m joined today by Dr. Jose De Leon, who is the math manager at the Lassinger Center. Many of you might have met him previously, and Amy Strickland, who is our math partnership person, and there’s a couple of folks, Hope and Maxine, helping with tech today as well. 

So thank you for joining us. It’s always a team effort here at the Lastinger Center. So today, I get to talk about something that I am super excited about, and something that I always enjoy talking about, which is math proficiency.

And many of you, I know, have been participants in the Math Matrix. I made sure to make this different and new than things that you might have seen, but also to have a little bit of familiarity with the topic. So hopefully you get something out of today.

The chat is open. There’s also a Q&A box, so if you want to ask a question at the end, there will be time for Q&A. There is a chat, and that is, open, so you should be able to put in there. 

So, I’m going to start, and if you’ve seen me talk before, I like to start talking about the math that we care about.

So, my first question for you in the chat that you can answer is, in looking at the world’s top mathematicians, including Fields Medal winners, which is kind of like the Nobel Prize of Mathematics, what do you think are the top five characteristics that distinguish a brilliant mathematician in their own words? So these are kind of the top of their fields of mathematics.

How do you think they describe a brilliant mathematician? And so, that’s what the prompt is for you. If you could take just a second or two to think about it, and then put some ideas in the chat. You don’t have to get all five, you could just put the first one in.

Thank you, Madeline, for getting us started. An outside-the-box thinker, creative thinkers, analytical, flexible, great communication, critical thinking, love that.

Grit, risk takers, critical thinking, curious. Love it. Innovative.

Love for puzzles. I definitely have a love for puzzles.

Okay, so there’s a lot of different ideas in there. I noticed nobody said crushes time tests very quickly, or just knocks out a lot of procedures quickly, and so we’ll talk more about that in a minute. 

To get started, I just wanted to share, so I used AI to collaborate with me on this, for some answers. And the first thing that came up was a deep curiosity and passion for understanding. So, great mathematicians aren’t motivated by grades or rewards, they are intrinsically driven by the joy of discovery.

Many describe math as creative, which a number of you said in the chat. It’s an evolving process, and understanding itself is the reward, which I thought was a really nice way to talk about it.

So, another way to think about it is, brilliant mathematicians are driven less by answers and more by wonder.

And I’ve included a quote from mathematicians on each of these sections, just so you can hear from them in their own words. So, George Cantor probably didn’t say that correctly, but that’s how I’m reading it. “In mathematics, the art of asking questions is more valuable than solving problems.”

So, if you think about that, like, how do we foster that with our students? How do we support the art of asking questions, or do we do that, and should we if we don’t?

Number two is persistence, which, again, a lot of you said, grit, persistence, things like that, willingness to struggle. So top mathematicians consistently talk about grit and tenacity. Andrew Wiles is very famous. He solved Fermat’s last theorem, which was a long, long, long open problem that he spent decades working on, and he has a nice quote about it. 

He said, the problem with working on Fermat was that you spend years getting nowhere, and it’s fine to work on any problem as long as it generates interesting math along the way, even if you don’t solve it at the end of the day. The definition of a good mathematical problem is the math it generates, rather than the problem itself. So you can imagine what kind of math you would give your students to foster that kind of thinking.

Miriam Mirzahkani is one of my favorite mathematicians. She unfortunately passed away at the young age of 40, but she has beautiful quotes about doing mathematics. One of them is, “doing math for me is like being on a long hike with no trail, and no end in sight.” 

But she said it in a very positive way, right? Not like I’m lost in the woods and I’m not having fun, but she enjoyed the journey. And that’s what we want to make with students.

Creativity, a lot of you said creativity. Creativity is super important to being a brilliant mathematician. It’s… one of my favorite characteristics of a mathematician. So they don’t just follow rules, they make imaginative leaps and they see structure that others may miss.

Again, Mirzahkani had a nice quote about this. It is, she said, “math is like being lost in a jungle.”

She tries, you try using all the knowledge that you can to come up with some new tricks. And with some luck, you might find a way out. It’s such a beautiful way to think about mathematics in thinking about it like, how do you get through a jungle? You’re creating interesting pathways along the way, which is probably not how most of our students see mathematics.

Four is active engagement and deep thinking. So, I think Polya had a nice quote about this. He said, “math, you see, is not a spectator sport.”

So it’s not a spectator sport. You have to engage in doing mathematics to learn mathematics. And I think, unfortunately, for many of our students, sometimes it comes across as a spectator sport, where they are looking at what’s happening, but they’re not actively engaged, and we want to make sure not to do that.

And finally, patience and sustained focus. This is one that’s really important. Math is not a sprint. It takes a long time to think deeply and reason through mathematics problems that are interesting.

And I’ll highlight this second quote from, Dr. Mirzahkani. “The beauty of math only shows itself to more patient followers.” And I thought that was a beautiful way to summarize this.

Okay, so I just shared the five top traits for brilliant mathematicians in their own words.

I want you now to share in the chat. Are any surprising to you? Was there anything in there that, like, resonated deeply with you? Or I want you to think about, how does your classroom align with this, or are there opportunities to align with it? 

So take a minute in the chat and share some thoughts on this with me. In the chat, what thoughts did this bring up for you?

All right, I see, there is a hand up. I will just encourage you to, if you have a question, use the Q&A box at the bottom, and you can put your questions in there. You can also use the chat, we’re monitoring it.

Creating environments that support this. Awesome.

Seeing connections that others miss, for sure, that was great. A development of a growth mindset, we think that that’s really important. Willingness to struggle, which we tell students all the time, but it’s one of those things that we need to continue to impress upon.

Students need to learn to stick with it. Yeah, math is, like, hard, but it’s interesting because it’s hard, so that’s one of the things we need to think about.

So, I want to share this with you all. This is proof that I have been good at math for a long time. You can see this is from January 26th, 1990, which I didn’t realize we were so close to that date, but it’s almost my,

I don’t want to say how many years since then. It’s been 36 years since then. And you can see I got it for being a math whiz.

So, I’ve always had a really strong identity as a math person, and I’ve always felt really good about math.

However, that changed for me at two different points over my life. The first one was when I got to college at the University of Florida. I had done really well throughout K-12 schooling, got to college and I really struggled with math for the first time ever. Most of it was because I tried to memorize proofs.

I thought you just memorized things, I was a really good memorizer, and so you can’t memorize all the proofs in Calc 2. It’s not gonna go well. I’m pretty good evidence of that.

Then, the second time that I realized, ‘oh, I really don’t know a lot about mathematics’ was actually when I was a graduate student at the University of Georgia.

And I knew about division, so, I assume some of you teach elementary school, so this might look familiar to you.

So I knew about division. I thought it was pretty cool that there were actually two ways to think about division. That was the first time I had learned about that in grad school, where there’s a measurement or quotative type of division, where you think about six divided by two. How many groups of two can I make from six?

But there’s this other type of division, the sharing division, and that’s saying if there are six across two groups, how many are in one group?

So there are actually two different ways of thinking about division, and I didn’t realize that until grad school. But that wasn’t the point at which I knew I didn’t know a lot. 

I mean, I learned, ‘oh, hey, there’s something I didn’t know’, of which there’s many, but here’s what really got me is, how does that apply to fractions? So, fraction division is a really hard topic for our students.

And I actually figured out in grad school that fraction division was really hard for not just students, but for me, somebody who majored in mathematics and took a lot of mathematics. So I knew there were these two models of division, it made sense to me, but I didn’t know exactly how to connect those models to a problem like one and three quarters divided by one half.

That was a harder question. So I’m gonna give you all a chance to think about this.

If you have paper near you, and you want to write it down, you can. But think about it. Could you make sense of a quotative meaning for this? Like, a ‘how many groups’ meaning for this expression.

Could you make sense of a sharing or partitive model of division?

How would you do so? So, here’s the question.

And this is kind of how you think about it. So I had a lot of trouble with this. This is something that highlighted to me that elementary mathematics is actually quite complicated and difficult in a way that I hadn’t appreciated previously.

So, this same problem, if you’re thinking about it in terms of measurement division, one way to think about it would be how many one-halves fit into one and three quarters? How many groups of one half can I make out of one and three quarters?

The second way to think about it, which is less common, I would say, but also very valuable is if one and three quarters is half of a group, how many are in one full group?

That blew my mind, and it took me a long time to think about that. And in case you’re curious, I’ll give you my models of how I drew it out.

So, how many one-halves fit into one and three quarters.

One, two, three and then it’s really hard to think about what this last one means.

It’s half of another one, so it’s three and a half.

 Three full, and half of another. Which is hard to think about, actually.

And if you’re like, ‘whoa, this is a lot, I didn’t know that we were gonna do so much fraction division today,’ it’s okay, we’re gonna go all over the curriculum, in this hour.

Here’s the other way to think about it, though, which is one that I hadn’t really considered previously. If one and three quarters is half of a group, how many or how much is in one group?

So, here’s my model. Note, I always define the whole. So, this is one and three quarters of a whole.

So if that’s half a group, how much is one group? So I just iterated that twice, and so now I see that one whole group is composed of three and a half.

Very different ways of modeling this situation, but also both valid and driven by the context of a problem, if there is a context in the problem, which is a pretty cool way to think about things. Again, I didn’t learn this until grad school. I’d never thought about this. I had a hard time modeling this.

Until I was in grad school, and I was somebody that had a master’s degree in math education, and a lot of graduate math, and an undergrad degree in mathematics. So you can think about the kind of experiences that led me to a situation where, with all that math, I still didn’t deeply understand some basic operations.

Now, there’s also a nice story, and I love fractions, so I tend to go on about fractions, but there’s a nice research study that Li Ping Ma did in a book, I think it’s from 1999 or somewhere around there, maybe a little earlier.

She surveyed how many U.S. adults, I believe it was teachers, could make a story problem for a fraction expression such as this.

It was a very low amount, and I think that just goes to further evidence that sometimes, even though we can all keep, change and flip, or whatever you want to say, invert and multiply, convert this to an improper fraction, then invert and multiply.

It doesn’t mean that we understand it deeply enough to draw models in multiple ways, or to make story problems.

And I think that if we are all striving to become brilliant mathematicians, or at least competent mathematicians at the level that we attain, we need to think about mathematics a little differently.

What I would say happened to me is, like I said, I was a very good memorizer. I am still a very good memorizer of facts and information and numbers.

So, if you think about math understanding as a nice rope made up of various different strands, I had a really good strand around procedures and memorization and recall of facts. I’ve always known my times tables very quickly. I always loved time tests because I was really fast at calculating, but I was missing all the other stuff.

And if you’ve taken the math matrix, you probably think that this looks familiar.

I was really strong in procedural fluency. I think the other strands of mathematics were far less strong, or maybe not even present.

And that’s what happened to me, and that’s what happens to many students, or maybe many students don’t even have one strand that’s really strong. It’s tenuous at best. And our job as teachers and educators is to really help them develop mathematical proficiency that is more balanced and more complete.

So I am going to go through this rope. The focus of today is mathematical proficiency, and thinking about it as more comprehensive than just one little strand of procedures, which is what a lot of people seem to think that math is, is just procedures.

So, I’ll start with procedural fluency, because that’s the most common thought around mathematics. So, procedural fluency. It’s skill in carrying out procedures flexibly, accurately, efficiently and appropriately. So it’s not just correct calculation, it’s also being able to be flexible with your procedures.

And to be efficient. Like, one strategy that might be efficient for me may not be for another person. That’s okay. At least I can think through different options.

And knowing when to use what procedure. Those are all parts of procedural fluency.

So, what does it look like? It looks like accurate calculations.

It looks like an efficient strategy choice. Like, I can choose a strategy, and it’s going to be an efficient strategy. So that might mean, if I’m solving for the square root of 25 and I know I could do the quadratic formula, I don’t use it because I can use number sense to know, I know five squared is 25, and negative five squared is 25, so I’m just going to use my number sense. It’s more efficient in this instance than the quadratic formula.

You want to use flexible approaches, and you also know when a method applies. So you want to know, oh, it’s good for this situation, but not good for this situation.

I feel like most educators have procedural fluency pretty much down. It’s the one that gets the most attention, so we’re going to spend some time on the other strands as well.

Conceptual understanding has gotten a lot of spotlights in recent years, for good reason. It’s comprehending what you’re learning. You’re understanding why something works. So, it’s not just knowing the procedure, it’s knowing ‘I know why the procedure works and where it came from.’

You understand the relationship between ideas, which is really important. Like, you might understand the relationship between a table and a graph, and how they connect. That would be a nice connection there.

And seeing math is connected, not isolated rules. I know why we factor and how we use factoring, instead of just being like, we just factor because the teachers told us to do so. So that’s conceptual understanding.

Now we’re going to get into the three that people really don’t emphasize, because it’s not as common in teacher prep programs and things like that.

So, one is strategic competence.

This is the ability to formulate, represent and solve math problems. This is students being able to pose problems. And actually, being able to ask good problems is a big part of mathematics, and as we saw in the mathematician’s own words, being able to ask good questions is incredibly important, and it’s probably more important than solving problems, honestly.

So, what this looks like is you can translate real situations into math, and vice versa. You can choose useful representations, and you know how to plan different solution pathways. So that’s strategic competence. It’s actually really important, being able to ask good questions and mathematicize the world around you.

And again, if you’re just joining us, if you have questions, put them in the Q&A, and we’ll get to them at the end. And also, if something resonates, you can share in the chat.

Productive disposition is what we all hope our students develop, as many of us love mathematics, because we teach mathematics.

It is the habitual inclination to see math as sensible, useful and worthwhile.

So, we did a listening tour at the Lassinger Center two years ago now in mathematics, and we asked a lot of people about math.

The only uses of math that they could really name are balancing a checkbook or reading a clock.

Which is unfortunate, because if you think about math more comprehensively, it’s really about logical thinking, reasoning, sense-making, those kinds of things, which are really, really, really important. And so, if we want to build a productive disposition among our students, we can’t just say math is only relevant when it comes to money. It’s more than that. It’s more than personal finance. It’s a way of understanding our world, quantifying it, understanding data, representing data, making arguments etc.

So, it looks like confidence, that effort leads to understanding.

Belief that math makes sense, which we hope our students believe. A willingness to persist. A number of you said in the chat earlier that persistence and grit is important, I agree.

And also viewing yourself as capable. Like, I can do this. Imagine if we started every problem, and students were like, I can’t do this yet, but I think I can do this. It would be a game changer for how we work with our students.

And then finally, adaptive reasoning. So, adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification. This is probably my favorite. You shouldn’t have a favorite, like children, but I do have a favorite strand of math proficiency, and I think it might be adaptive reasoning in a lot of ways.

So, it means explaining why something is true, justifying your solutions, recognizing flawed reasoning so that you can address it, like error analysis, and making logical arguments. And I think this is a really powerful use of mathematics that I think is underutilized in our classrooms.

So there’s all five strands together.

So my next question is, which strands do you feel like you do the most of in your classroom? So if you want to put it in the chat, out of these five strands that we just talked about, which would you say you’re doing really good at, like, fostering with your students?

And you can just put it right into the chat. Which strands do you feel like you’re doing a good job emphasizing?

Thank you. Procedural fluency? Yep.

A lot of curriculum materials really emphasize procedural fluency, so that one’s usually one of the easier ones to kind of get at.

Oh, nice, Elizabeth. Conceptual understanding and productive disposition, that’s great. Yeah, productive disposition is an important one.

It’s the one that you would never say, “I’m not a math person” publicly, right? Because you don’t want them to think that they don’t have a positive identity around mathematics.

Conceptual understanding, I see that, yep.

Oh, strategic confidence, awesome! That’s great. That’s probably the one that, among this, strategic competence and adaptive reasoning are probably the ones I see least emphasized, I would say, in classrooms.

So, here are the 5 traits again for brilliant mathematicians. Here’s the five strands.

And just look across and see what you notice in terms of relationships and connections. So the mathematician said a deep curiosity and passion for understanding.

That feels a lot like a productive disposition and also strategic competence.

If you look at the others, they didn’t really emphasize procedures. I think they took it as a given, probably, but that’s not what distinguishes a brilliant mathematician. It’s not the procedures. You do need to know it, you need to have some intuition and some conceptual understanding. 

But these other strands that are less emphasized are actually the ones that a lot of mathematicians are like, hey, these are super valuable to becoming great at mathematics. And so I think that says something about how we might approach mathematics in our classrooms with our students.

Now, the other question I have for you, and you can put this in the chat, is in terms of the Florida best standards (if you’re not in Florida, you can think about it, whatever state you are,) but in terms of the Florida best standards, where are these? Are they in there? Which ones are in there?

Put in the chat any thoughts you might have about where in the best standards might you find these?

Claire is on it! The MTRs, okay? And she got 100% on that, and a shout-out.

Other thoughts? Okay? You’re getting, like, a lot of, yes, affirmation.

Yeah, also, I agree. Procedural fluency and conceptual understanding in the content standards, we see a lot of this, right? We see some justification in things, I would say, as well.

But if you ignore the math, thinking and reasoning standards, I think you’re gonna lose out on all strands of math proficiency. They’re really important for math proficiency. 

I would say, if you’re teaching, print the MTRs out. Keep them close to your heart or on your wall so you remember it, because a lot of the useful mathematics comes from us emphasizing and teaching in a way that really honors the MTRs and brings them into our classrooms. We want students to participate effortfully in collaborative learning. We want them to understand representations that are presented in multiple ways.

Task fluency, of course. We also want them to be able to reflect on their mathematics and others’ mathematics, use patterns and structure, assess the reasonableness of solutions etc.

So you can kind of see there’s, like, a nice combination between these as well. So I showed the five traits of a brilliant mathematician. Now I’m showing you the MTRs, and you can kind of see they help complete the picture of math proficiency. If we just do procedures or common problems, we’re probably not going to get at all the strands very well. They’re going to end up like me, and I was very successful throughout K-12, and then I really struggled after that, which for some people, you might be like, that’s a good aim, but I would argue that we can do better, and we can support more students in really loving math and understanding it deeply so they can go even farther and keep their opportunities open.

In this next portion, I’m going to show you some problems and kind of think about, like, how these problems highlight one particular strand. Ideally, you have problems that hit at multiple strands, but a lot of these webinars are going to be looking at, like, okay, whatever curriculum you have, what are you kind of looking for if you really want to foster these practices?

So, procedural fluency. I think this is the one that we all felt pretty confident about, but if we want to help our students to think about, or to develop procedural fluency, when we’re looking at curriculum materials, these look-fors are for your curriculum materials. 

So say you’re, like, in an adoption cycle, which we’re fixing to go into soon, or you have lots of different resources to choose from as a teacher. What might you look for that would help them build procedural fluency beyond just speed drills? Because, like, we really know that speed drills are not going to help you be flexible or efficient with your strategies necessarily, or be discerning of procedures.

So, you would look for strategy-based practice, like, can you practice some of the different strategies so you gain proficiency in them? So you’re not just stacking all the time when you’re adding numbers. Are you also thinking about decomposing, making tens? Maybe if you’re multiplying, you’re not just doing the algorithm, you’re also doing partial products, which is more efficient sometimes, things like that.

Do your curriculum materials give you the opportunity to choose efficient strategies? Do they encourage you to think about more than one approach and choose the one that’s more efficient?

And do they help practice connected to concepts? Like, do they give you practice that is connected to the underlying concepts, not just procedures for no clear purpose?

So I’m going to give you a minute to look at this problem.

Okay, I picked the same topic for all these examples, so they’re all multiplication, and then you’ll just see how they bring out different strands based on what the strand is. So this one’s a multiplication problem, so the math goal is multiplication.

And it says, solve each problem using an efficient strategy. Again, what’s efficient for Jose might not be what’s efficient for Zandra. And so you have to think through and have students think about what is most efficient for them, and then introduce more strategies so they can understand what’s efficient when.

So then, explain which strategy you used.

So, not only are they allowed to choose, which I’m a big proponent of student choice in mathematics, but they’re also talking about which one is efficient. So that’s a nice way to think about procedural fluency.

Conceptual understanding, the other one that we feel like comes out a lot in the classroom, what would you look for in materials?

So, if you know about cognitive demand, higher cognitive demand tasks typically get at more conceptual understanding, but not always.

So you would look for multiple representations of a concept. You would look for prompts that say things like, why does this work? Explain your thinking, which students don’t really like, but are very important for their learning.

And you would look for things like visual models, because you want to see them show this in different ways to make those connections.

You would kind of be dubious about just problems that are naked all the time, because they’re, like, you know, naked computation problems. They’re not going to give you a lot of conceptual understanding, or help students develop conceptual understanding.

So, here is a more conceptual understanding-based approach to this multiplication, so look at that.

So it says, draw an array to represent six times four, and then it asks some follow-up questions, like, what does each factor represent in your array? How does it show multiplication? What addition sentence matches your array? 

So these are nice questions to connect to in terms of, like, how does what they’re doing connect to different representations? Do they understand what they’re doing, and can they explain it?

Strategic competence. Again.

Does this encourage problem-solving? It’s the problem-solving kind of strand. So you would look for rich word problems, modeling tasks and the opportunity to choose a strategy that makes sense.

And you would be aware of curriculum materials that just say, like, Friday Challenge, and that’s the only problem solving, or problem solving is only for the above-grade level students. We want all students to engage in problem solving all the time. 

So, if you see things that are just, like, problem solving Fridays only, that’s not great, right? Because we want students to develop a complete idea of math proficiency, or math proficiency in full, and if we just limit problem-solving to special days, it’s not really thinking about math more comprehensively as a problem-solving activity.

So here’s an example, back to math multiplication. A baker packs nine boxes with six muffins in each box. Write an equation, solve it, draw or model the situation, and then create a new story problem for a multiplication problem. 

So it’s really nice that they get to pose a problem, create a story problem for this, and a lot of kids really enjoy making their own problems, and you probably know that if you’ve ever done it with them.

Adaptive reasoning, again, my favorite.

Do students have opportunities to explain and justify their reasoning? So you’re going to look for explain why prompts, error analysis tasks, opportunities for math discussions. 

If your curriculum materials don’t tell you how to lead math discussions, or never talk about kids talking, it’s probably not great, right, for developing adaptive reasoning, because we want guidance on how to do this productively.

Here’s an example. So, this is still multiplication, it’s kind of getting at the commutative property here.

So, explain why seven times eight is the same as eight times seven. Is this always true? I love that question. Use words, drawings or equations.

And then, productive disposition. This is the one where we want kids to have positive math identities. So you’re looking for opportunities that encourage a growth mindset, which somebody put in the chat earlier, multiple entry points, real-world connections and you’re gonna beware of things that just suggest that math is for some students, or you have to be fast to be good at math.

So here’s a question that could help foster a productive disposition toward mathematics. It’s still multiplication.

It says, think of something in your life where multiplication is useful, and it gives some examples.

Write what the situation is, Write a multiplication equation. And tell me about what helps you find out about that situation. I think that’s a nice prompt that we can adapt.

So you’ve seen multiplication tasks five different ways, kind of like emphasizing different strands at each time. Ideally, you’re connecting these strands across a problem or a set of problems, and thinking about it comprehensively.

And a good question to ask yourself is, over the course of this week, what opportunities have I given students to develop these various different strands? Has it been balanced? Has it been not balanced? How might I balance it out?

And I’m not saying everyone is 20%, right? We know that state testing often emphasizes procedures and concepts, but you can’t have 0% on some of those strands. I think that’s where the issue really lies.

Okay, so I’m gonna do one more example of a task, and then we’re gonna open up for any questions that you have.

A lot of you already have textbook materials. I’m not saying you have to go out and look for new things. There’s little adaptations you can do to foster more of the strands.

So, here’s a pretty typical middle grades level, linear algebra task. So you’re just solving these for X.

If you teach middle grades or Algebra 1, you’re probably like, oh, I know which ones they’re gonna struggle with. Those negative signs, probably, I would say.

Okay. you can think about what strands this is emphasizing. So it says, solve each equation, make sure you show your work.

It’s probably emphasizing mostly, encouraging or fostering procedural fluency.

There’s not a lot of opportunity to really engage with the rest. I mean, you could do some persistence to get through all of these and things like that, but mainly it’s kind of getting at procedures.

So, some simple tweaks that you can kind of do, and you can do this to any number of tests that you’d like.

Create a story problem that could be represented by one of the equations above.

So if you were able to include just one, you could even just take out one of the problems, and then just say, can you create a story problem for one of them?

That would go a long way. It’s allowing students to pose problems in some kind of way, be a little more creative with their thinking in terms of creating a story problem and a little more enjoyable. It’s like a break in the monotony of class.

Second one, and if you’ve seen me talk about nudges before, this is one of our instructional nudges.

You could ask them to create some wrong answers. You’re not telling them they’re wrong. You’re saying, hey, what’s something that somebody could arrive at that would be wrong?

So, create two wrong solutions. You know they’re wrong. You’re not asking for a right answer, you’re just saying, give me two wrong answers.

And could you think of one that might be super wrong, and another one that could be kind of right, but it’s still a little wrong? You know, so you’re kind of thinking about, like, can they discern mistakes and errors, and think about really severe errors, or lots of errors versus, like, you know, minimal computation errors, potentially things like that. Can they reason and justify it? 

So that’s really getting at that adaptive reasoning, because they’re justifying, like, why they know they’re wrong, and also how wrong they think they are. So it’s a nice way to kind of, like, change up the task also, is, like, just give me a wrong answer and tell me, do you think it’s, like, a super wrong answer? Why? And the kids kind of like it because they also feel comfortable sharing wrong answers because the pressure to get it right is off. Which is nice.

And then, finally, reversals. One of my favorites, I wish I had come up with it, I have not. It’s old work, from Russia, I believe, but it’s just reversing a problem.

All problems kind of have a flow. In algebra, we usually solve for X, but that’s the flow. Very seldom do we reverse it and say, create an equation that has a solution of three. That doing and undoing is very important to mathematics.

And so, if you just reverse a problem from time to time, it’s really helpful for students to develop more flexibility and adaptability in their thinking.

In elementary grades, they always add fractions. You give them the fractions and they add. We seldom say, give me two fractions that have different denominators that sum to a half.

That’s a great task!

There’s lots of different opportunities for answers, and it’s a different type of reasoning than what we typically do. And so I would encourage you to reverse problems sometimes, like, just here and there, now and then.

So that’s three ways you could enhance a task that’s, like, a typical kind of task, but you’re getting more mathematics out of it.

Alright, so that brings me back to the top. We started by talking about five characteristics of brilliant mathematicians in their own words.

Then, we walk through the five strands of math proficiency.

I shared some examples of what that looks like in problems and curriculum, what to look for. So I’m just gonna give you a chance to reflect.

I talk fast, so, hopefully you got a lot of it. This will be recorded.

You can put in the chat, or you can just jot down on your own, what do you want to take away from this session? So we talked about math proficiency, we talked about different problem types, things to remember, so what’s something you want to take away from this lunchtime webinar today?

To bring you on the rest of your day, whether it’s in the classroom or elsewhere.

And you can share in the chat, or you can just jot it down.

Thank you for sharing. Go deeper with MTR and implement it into my coaching practices. Love that.

Yeah, the MTRs are really great.

I love the number one, super wrong, kind of right. Yeah, actually, it’s from an NSF grant that I’ve been working on, but we have a lot of those, like, simple little things at practicedrivenpd.com.

If you go there, assuming I typed it in right, there’s lots of little nudges for your instruction that you can share with folks. And we’ll be using some of those in the subsequent webinars as well.

Ways to utilize given problems in the textbooks and altering them, yep, enhancing them, for sure.

I’m a big proponent of not making teachers’ work weeks longer, just, being more efficient or thoughtful about what we already have instead of doing a lot.

Yeah, the wrong answer. Cool.

Well, with that, if there’s any questions, you can put them in the chat, and I will answer them in the chat.

Or you can either put them in the chat, or you can put them in the Q&A box. We’re monitoring both.

Just in terms of some announcements, the Math Matrix is closed for this school year, but next school year, we will run it again. If you ever have questions about the Math Matrix, it is 60 hours of professional learning online, asynchronous. You get $1,000 if you complete all 60 hours. If you’re interested, you can reach out to Amy Strickland. She will put her email in the chat if you’re interested. It’s also on the screen.

And our next webinar will be February 17th at the same time. I’m gonna try to vary the time, because I know not everybody said that this time works well.

We have recorded this. I wish I could fix my typo in the slides about part of the division, but I can’t, but I will put a note on it. But yes, we will share the recording if you registered for this, so you can share that with your colleagues if they were unable to join us.

Any other questions? I will give it three more minutes, and then if not, we will adjourn for today. I very much appreciate you spending some time with us today.

What would be a good diagnostic test that aligns to these strands and also mass skill deficits?

I don’t think there’s any good standardized diagnostics that I’m aware of, and it’s all going to be concept-by-concept kind of situations overall.

I would say for productive disposition, you could ask students, like, how they feel about mathematics. Like, even small students can do emojis for you, like, their emoji about their feelings about math. That’ll let you see, kind of, how the feelings are about math, and you can go from there. 

You could ask them, like, do you think everybody can be good at math? If they say no, then they don’t have a productive disposition, right? So, like, we need to work on that. 

Otherwise, I think it’s not really diagnosing, it’s more of, like, how are you fostering them? So I would say, like, we don’t need to assess these, we need to ensure that our instruction is like, giving opportunities to enhance and foster these strands of proficiency with students. So maybe that’s, like, my overall kind of recommendation is, at the end of the week, you think to yourself, “did my students have opportunities to show me that they have a productive disposition, or did they show that they had some opportunities to work toward that?”

If not, then, like, next week you should probably think about, like, what is something I want to try with my students that would help them foster a productive disposition, etc. So I think that might be a way to approach that. Thanks for the question.

Yes, we will share the recording, and you can share it with your math team. Yes, yes, yes. I believe it’ll be on our website also, so I appreciate that.

Yeah, it’s a math matrix refresher, so yeah, good. And some people are sharing some ideas in relation to that first question, so please look through the chat. You guys are sharing great stuff, so thank you so much.