
BLOW UP Math Education with Cognitive Science: S2 E15
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Joni,
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hello everyone. Welcome to tier one interventions podcast. I am your host today, Jonily Zupancic, Jay Z in the house, and we are going to continue our conversation on the science of math in a previous podcast. We extended the podcast to about 90 minutes, which is not typical for a podcast, but we had captured a session recently with our followers and some of our certified coaches, really getting ahead of the wave that is coming called The Science of math. In our words, we like to call it cognitive science of math, and basically it's the neuroscience of learning and specifically geared toward mathematics, mathematics teaching and learning in general, typical traditional mathematics teaching and learning works exactly against the way that The brain actually learns information, and at this time, in certain parts of the United States, Canada and in some parts of the world, there's a controversy right now in reading called Science of reading, which is the research evidence based focus. There's been much initiative going back and forth on the pendulum in reading for a number of years, but there are a number of programs in reading that have been proven to help students read, learn to read and then read to learn. There really isn't a lot about mathematics and at minds on math. We have created research based, evidence based models, frameworks and processes for the delivery of instructional facilitation that matches the neuroscience the cognitive science, ultimately what we call the science of math learning. Through this journey, we've created so many resources that are the math recovery that has completely transformed math understanding, joy, focus, engagement, motivation and commitment of not only our teachers, but also our students. So in the previous podcast episode, it was a lengthy podcast episode, we laid the groundwork for what is the cognitive science of math. And upcoming in June of 2025
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we are having our full day cognitive science of math event, training and seminar. We look forward if any of you are listening to the podcast before then contact minds on math. Jonily j o n, i, l, y Joni at minds on math.com
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to email for any questions and to grab the flyer for the event. The event is the event is going to be an in person event. Just outside of Central Ohio in the States, there will not be a virtual option. So mark your calendars and plan now.
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As a follow up to the previous podcast, we'll call this the science of math part two, and I'm going to give you two more examples of how to implement the science of math in your classroom. In the previous podcast, we talked about the outcomes that we want. We also talk about the barriers and the frustrations that teachers have working against these barriers, which are the deficits that students have mathematically,
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one of the biggest deficits that we say, is number sense. Number Sense in the making math and ears Book, Number Sense in the making math veneers, book, is the conceptual development and understanding.
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The conceptual development and understanding
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of mathematics of the magnitude of numbers, magnitude, meaning how large and small numbers are. The making math and ears book is the first step to unveiling the science of mathematics. It's available anywhere that you purchase books. It's also available on audio and also available on Amazon, making math and ears
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by yours truly, Jay Z Jonily Zupancic
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in the book, we separate procedural and conceptual learning for students in terms of mathematics if we want kids to get better at procedures, long division, stacking and subtracting, solving equations, vertically balancing equations, some of those rote procedures or just understanding simply automaticity with math facts
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that.
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Procedural camp, that procedural
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pocket, that procedural focus, has a different instructional facilitation for educators than the conceptual focus. If we want to improve number sense, Number Sense is conceptual, not procedure. Number Sense cannot be explicitly taught. It can only be experienced, but through explicit, deliberate intention and intentional processes, frameworks, formulas and models, the minds on math model encompasses what we do in our tier one, regular, general core classroom, which is what we focus on on this podcast, tier one interventions, but our minds on math model also has a program and framework for tier two math interventions, as well as a specified assessment and intervention process For Tier three mathematics we differentiate those because the clientele and the needs of students are different at each of those levels, and the instructional facilitation is different as it should be different. However, some of the same essentials and power standards work with
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any and all of the above, tier one, tier two, tier three, the best tier two and tier three intervention programs are based on a strong core tier one, regular general classroom environment and experiences with a high quality, highly effective math instructor, educator and teacher, we all want the same outcomes. We want kids to have number sense. We want them to have computational fluency, math Fact Fluency. We want them to understand equations, equations not just with whole numbers, but understanding fraction, decimal percent as well as integers. And finally, we want kids to understand shape and how shape relates to numbers.
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If we do this in a neuroscience way, we have four strategies to focus on in our instructional deliver, delivery, interleaving, spaced repetition, retrieval, practice and metacognitive feedback during our science of math sessions, we're going to focus on these four strategies of teaching and learning and how they directly connect with how and what We teach in mathematics. Before I give the two examples that I'm going to leave us with today on how to utilize those strategies in our day to day lessons, I want to talk a little bit about how we have, over the last 20 years, unpacked the essential and power standards of mathematics.
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One thing that we have done is we've surveyed algebra two teachers. So Algebra Two is a named course. Algebra Two that is typically a third year high school course. And Algebra Two Content assumes that students have content from algebra one, which is linear, exponential and quadratic functions, as well as factoring polynomials, maybe absolute value functions and square root functions.
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We surveyed a number of algebra two teachers a few years ago, and our question was, what do students need entering algebra two to be successful in this math class. And interestingly enough, when the results came back surveying many algebra two teachers in lots of different school districts and different types of school districts, there were five main essentials that algebra T, algebra two teachers said, Look, here is what kids need to be successful in Algebra Two mathematics, it would be ideal if they were able to come in with these things to have more success with our course and our content. Again, the big arch is number sense, which I've just talked about. Underneath that is math facts and fraction. Typically, when we say math facts, we mean single digit multiplication, automaticity. So having that here, which is general among all teachers, not just algebra two teachers, the algebra two teachers funneled the essentials into five areas, radicals, and this one is what we talked about on the previous podcast. Radicals being square root, understanding square root and really understanding squares, what numbers make squares, and how square root operates and functions and the behaviors of square root functions, as well as simplifying just basic square roots, you've got to have that math Fact Fluency to have fluency with understanding radicals. The second essential was being a.
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To solve equations, and solving and working with systems of equations, meaning two different relationships that we compare and find where they're the same or where they overlap.
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The next is graphing, graphing on a coordinate plane. But this is grounded in understanding number line. Number line is the most abstract concept. If we have students that don't understand numbers on a number line, that actually is is fairly typical, fairly normal. We can't really make an assessment from that. But if we have students that highly understand numbers on number line, that's not a false positive. You can't accidentally understand number line. It is probably the ultimate understanding in terms of number and having number sense. So if a student understands number line, they have a very high level of understanding for numbers and mathematics in general. For most students, they don't understand number line, but that doesn't necessarily mean they have significant deficits or math gaps. So we have radicals, solving equations, solving systems, graphing, which really is grounded in number line.
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And I said there's five big ones, five big ones, radicals. So then try to think, I want you to try to guess radicals, solving equations, solving systems, graphing. What's the big one? I've actually already mentioned it factoring. And what algebra two teachers are talking about are factoring trinomials and polynomials. But factors are absolutely essential. Meaning, does the number 18 have more or less factors than the number 91
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that's how much we want students to understand. Number We want students to understand just because the number itself is larger doesn't mean that it has more factors. Number Sense is really understanding the magnitude of number and magnitude meaning the size of number, how large or small numbers are and where they are on a number line. But magnitude and number sense is also about understanding the characteristics of a number, being able to look at the number 51 and identify that 51 is not a prime number when our gut tries to tell us that the number 51 is prime. Now, as we uncovered these essentials, and we've done surveys and some research and lots of conversation, and there's more to the story than this, but I wanted to give you the shortest version of the story possible. My question in all of this about 12 years ago, when we did this, was, if those things are so essential for Algebra Two success, which is typically like year 11 in school, or year 10 or year 12, what do those things look like as early as second grade? What do they look like in kindergarten? How can we be preparing our kids in preschool for these concepts?
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The answers to those questions are in the minds on math model of instructional delivery. One of our specific models is called the achievement formula, a course that we have that encompasses the tier one model is called Mastery math, or in other words, tier one interventions. The full course goes into detail on what this looks like at every single grade level, applied, connected and paralleled to the standards that we have to teach. Based on these essentials, we also have developed a 15 day math formula. Yes, you heard it here at any grade level. In any course, there is a 15 day formula that at the end of 15 days, students have been exposed to more than 90% of the standards at that grade level or that age level, again, based on this cognitive science, approach of mathematics, how we deliver it, and how we've chosen the essentials and the power standards. I just want to share two other examples today on this topic. And the two other examples comes from this first 15 days. I'm going to show you some slides if you're listening to this podcast audio only, you're going to want to catch the podcast on YouTube. When you go to YouTube, you're going to want to search tier one interventions, podcast on YouTube. On YouTube. You can see my screen now and see some of the slides that I'm going to share. The two examples from today,
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the 15 day formula. There are two slide decks, and we presented in the previous podcast the first two days of the 15 day formula for grades three through 10, and it started with the square root of 50. So to listen to that previous podcast, you can see how we handle the square root of 50 in as early as second or third.
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Grade. What we didn't mention in the last podcast is the 15 day formula, specifically for grades K through two. So instead of the square root of 50, we're going to ask kids, tell me about 50. Tell me about the number 50. They might say things like, it's a five and a zero. It's two digits, it's 510s
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it's 410s and 10 ones. Another question we're Another prompt we're going to give is tell me about 50 cents or 50 cents as a decimal. So this the slide decks and the stimulus and the prompts are almost identical. But when we get to the grade two, three through 10 slide deck, and we're looking at the square root of 50. It's the same stimulus and same prompt. Every single year, we just increase the complexity and understanding for the student of that same number and that same symbol. What
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I'm going to share with you today are a couple new
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examples.
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And one is I'm going to go to as I scroll here, day three. Day three. The symbol that is here is a big two with an exponent of negative three. Now in typical, traditional standards, third and fourth graders aren't going to see a negative exponent, but as third graders are learning about repeated addition, We notate that with symbols of multiplication. Repeated addition is multiplication
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at an exposure level, not a mastery level. Thinking about the cognitive science of math, the science of math and the neuroscience of learning, what we know is exposures are more important than mastery. Exposures and multiple exposures over time, even without meaning,
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give kids a better chance at success and at the higher levels of mathematics.
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So when we show kids to with an exponent of a negative three, my third graders latch on to that because they're like, I know it's an exponent. It might mean that I have a negative number. And then we actually have a task called paper folding that shows visual and in a concrete way, how to represent, how to act out and how to experience two to the negative three, not going to go through that process today. I just want you to know that we have the process for that
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memory increases and retention of content increases when we experience multiple things at once, meaning interleaving.
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Interleaving is out of context, random mathematics with weird symbols that we tell students you're not supposed to understand this, and you're not going to understand it by the time we're done. But I want to expose this to you, because you're going to see it later in your upper level math courses, and we don't want you to be afraid of it, but what we're going to do today is use a strip of paper, and we're going to fold paper so that you can experience it today, but you don't need to fully understand it. And what happens is our second, third, fourth and fifth graders start to experience what some of these notational symbols mean in mathematics, even at grades 789,
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or whether, if you're taking if you're a student taking algebra,
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the most typical wrong answer for students when they see two to the negative three is negative six.
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I don't like the wrong answer of negative six. I would actually much rather, students answer negative eight. Negative eight is a better wrong answer than negative six. However, two to the negative three doesn't have a negative answer at all. Now I'm coming at you from this very high level to tell you that it is possible to do so much more in mathematics in so much less time teaching and exposing all of our standards and more. And yeah, you guessed it, in 15 days or less, through this science of math model,
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we also, if you look at day one through five and look at day three four grade K through two, I am not going to show my kindergarten and first graders an exponent of negative three. That would that would just be a little bit of malpractice. There's no reason they need to see that. But kids as young as eight, 910, years old need to start seeing what math truly is. So on day three, the stimulus is a little bit different. For students in kindergarten and first grade, I have them look at these numbers, 2468, 10, and I simply say, tell me about these numbers. But what I do say and show to my kindergarten, first graders and my third, fourth, fifth graders, when I do this stimulus with them,
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is that this number pattern comes from a high school math program.
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Problem, and the high school math problem looks like this. A sequence is shown, 02468,
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the prompt is write a function, f of n that and the finish of that sentence is that describes this function, this sequence, this pattern such that we have a rule or a formula, an explicit rule, an explicit formula, to find any number in this pattern.
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The reason we do that is because we want students to understand that some of these most simple, basic things that we're learning now, 02468,
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10, even numbers just get more complex over time with the notation and the symbols we use,
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we want students to have exposures to as much as possible, and what ends up happening in the end is they begin To master more and have more fluency and automaticity through this framework process model,
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the minds on math. Model encompasses tier one interventions as well as tier two and tier three. This podcast focuses on tier one, general core classroom, because ultimately, the general core classroom is the foundation for tier two and tier three interventions.
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If you want to keep learning more, like and subscribe on YouTube to tier one interventions podcast. But what we would really like you to do is in the comment section, give us any of your thoughts or comments, and, most importantly, put your questions in the comment section so that we can continue to have dialog about the science of math. This is Jonily with Tier One interventions podcast. We'll see you soon. You.
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