What are the three components of relational understanding of place value?
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Data management is becoming an increasingly important topic as our students try to make sense of news, social media posts, advertisements… Especially as more and more of these sources aim to try to convince you to believe something (intentionally or not). Show
Part of our job as math teachers needs to include helping our students THINK as they are collecting / organizing / analyzing data. For example, when looking at data we want our students to:
While each of these points are important, I’d like to offer a way we can help our students explore the last piece from above – central tendencies. Central Tendency Puzzle TemplatesTo complete each puzzle, you will need to make decisions about where to start, which numbers are most likely and then adjust based on what makes sense or not. I’d love to have some feedback on the puzzles. Linked here are the Central Tendencies Puzzles. Questions to Reflect on:
I’d love to continue the conversation about these puzzles. Leave a comment here or on Twitter @MarkChubb3 Advertisement Share this:
Like this:Like Loading... A few weeks ago I was introduced to Jenna Laib‘s game “Number Boxes” and was very interested in using it as a dynamic game to help students learn a variety of new content — Jenna’s blog explaining the game can be found here: “One of My Favorite Games: Number Boxes“. Basically the game involves students rolling dice (or spinning a spinner / drawing a card) to generate a random number and placing that number in one of their empty number boxes one-at-a-time. The game can progress in a variety of ways: As you can see, the game is quite adaptive to the sizes of numbers and concepts your students are comfortable with. As students roll/spin/draw a number, they have to place it on the board. What makes this tricky is not knowing what future numbers will be. In the board above, you can see that there is also a “Throwaway” box that students can use if they do not like one of the numbers rolled/spun/drawn. This game is an excellent example of a “Dynamic Game” or “Dynamic Practice” as students are following the ideals on the right side of the chart below:  Blow is a gallery of some possible adaptations of this game or linked here is a slideshow Metric ConversionsI however, wanted to use Jenna’s game to help students practice a concept they often have difficulty with – Metric Conversions. Once students have had many opportunities to estimate and measure various distances, capacities, and masses, they should be able to start making connections between all of the units. I suggest a good balance between using problems that help students make sense of the relationships between the units, and opportunities to practice conversions on their own. However, instead of randomly generated worksheets or other rote practice, I think Jenna’s game could work perfectly. Take a look at some examples: ReflectionIt is important to offer tasks that allow students to make choices and decisions like the ones offered in this game. Learning needs to be more than handing out assignments, and collecting work… Learning takes time! Students need more time to explore, see what works, have peers challenge each others’ thinking, make important connections… Hopefully you can see these opportunities in this task. Final Thoughts:
I’d love to continue the conversation about “practicing” mathematics. Leave a comment here or on Twitter @MarkChubb3 Share this:
Like this:Like Loading... I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions. Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like. In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:
Knowledge vs. UnderstandingBelow are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding: As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning. Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true! Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:
Application vs. ThinkingBelow are a few examples that can help us delineate the differences between application and thinking: These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:
Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:
Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution. Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems. Final ThoughtsSomehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off! Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples. Questions to Reflect on:
If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics” I’d love to continue the conversation about the questions we ask of our students. Leave a comment here or on Twitter @MarkChubb3 Share this:
Like this:Like Loading... Throughout mathematics, the idea that objects and numbers can be decomposed and recomposed can be found almost everywhere. I plan on writing a few articles in the next while to discuss a few of these areas. In this post, I’d like to help us think about how and why we use visual representations and contexts to help our students make sense of the numbers they are using. Decomposing and RecomposingFoundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and be then recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half… Understanding how numbers are decomposed and recomposed can help us make sense of subtraction when we consider 52-19 as being 52-10-9 or 52-20+1 or (40-10)+(12-9) or 49-19+3 (or many other possibilities)… Let’s take a look at how each of these might be used: The traditional algorithm suggests that we decompose 52-19 based on the value of each column, making sure that each column can be subtracted 1 digit at a time… In this case, the question would be recomposed into (40-10)+(12-9). Take a look: 19 is decomposed into 10+9 The problem is recomposed into (40-10) + (12-9) While this above strategy makes sense when calculating via paper-and-pencil, it might not be helpful for our students to develop number sense, or in this case, maintain magnitude. That is, students might be getting the correct answer, but completely unaware that they have actually decomposed and recomposed the numbers they are using at all. Other strategies for decomposing and recomposing the same question could look like: Decompose 19 into 10+9 Subtract 52-10 (landing on 42), then 42-9 Some students will further decompose 9 as 2+7 and recompose the problem as 42-2-7 Decompose 19 as 20-1 Recompose the problem as 52-20+1 Recompose the problem as 49-19+3 The first problem at the beginning was aimed at helping students see how to “regroup” or decompose/recompose via a standardized method. However, the second and third examples were far more likely used strategies for students/adults to use if using mental math. The last example pictured above, illustrates the notion of “constant difference” which is a key strategy to help students see subtraction as more than just removal (but as the difference). Constant difference could have been thought of as 52-19 = 53-20 or as 52-19 = 50-17, a similar problem that maintains the same difference between the larger and smaller values. Others still, could have shown a counting-on strategy (not shown above) to represent the relationship between addition and subtraction (19+____=53). Why “Decompose” and “Recompose”?The language we use along with the representations we want from our students matters a lot. Using terms like “borrowing” for subtraction does not share what is actually happening (we aren’t lending things expecting to receive something back later), nor does it help students maintain a sense of the numbers being used. Liping Ma’s research, shared in her book Knowing and Teaching Elementary Mathematics, shows a comparison between US and Chinese teachers in how they teach subtraction. Below you can see that the idea of regrouping, or as I am calling decomposing and recomposing, is not the norm in the US. Visualizing the MathThere seems to be conflicting ideas about how visuals might be helpful for our students. To some, worksheets are handed out where students are expected to draw out base 10 blocks or number lines the way their teacher has required. To others, number talks are used to discuss strategies kids have used to answer the same question, with steps written out by their teachers. In both of these situations, visuals might not be used effectively. For teachers who are expecting every student to follow a set of procedures to visually represent each question, I think they might be missing an important reason behind using visuals. Visuals are meant to help our students see others’ ideas to learn new strategies! The visuals help us see What is being discussed, Why it works, and How to use the strategy in the future. Teachers who might be sharing number talks without visuals might also be missing this point. The number talk below is a great example of explaining each of the types of strategies, but it is missing a visual component that would help others see how the numbers are actually being decomposed and recomposed spatially. If we were to think developmentally for a moment (see Dr. Alex Lawson’s continuum below), we should notice that the specific strategies we are aiming for, might actually be promoted with specific visuals. Those in the “Working with the Numbers” phase, should be spending more time with visuals that help us SEE the strategies listed. Aiming for FluencyWhile we all want our students to be fluent when using mathematics, I think it might be helpful to look specifically at what the term “procedural fluency” means here. Below is NCTM’s definition of “procedural fluency” (verbs highlighted by Tracy Zager): Which of the above verbs might relate to our students being able to “decompose” and “recompose”? Some things to think about:
If you are interested in learning more, I would recommend:
I’d love to continue the conversation about assessment in mathematics. Leave a comment here or on Twitter @MarkChubb3 Share this:
Like this:Like Loading... A few days ago I had the privilege of presenting at OAME in Ottawa on the topic of “Making Math Visual”. If interested, here are some of my talking points for you to reflect on: To get us started, we discussed an image created by Christopher Danielson and asked the group what they noticed: We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image. After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture. The picture above was more difficult for us as teachers to see the mathematics. While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table….. the group correctly guessed that many parents do not typically notice the mathematics around them. So, my suggestion for the teachers in the room was to help change this: While I think it is important that we tackle the idea of seeing the world around us as being mathematical, a focus on making math visual needs to by MUCH more than this. To illustrate the kinds of visuals our students need to be experiencing, we completed a simple task independently: After a few minutes of thinking, we discussed research of the different ways we use fractions, along with the various visuals that are necessary for our students to explore in order for them to develop as fractional thinkers: When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about. Data taken from 1 school’s teachers: Above you see the 6 ways of visualizing fractions, but if you zoom in, you will likely notice that much of the “quotient” understanding doesn’t include a visual at all… Really, the vast majority of fractional representations here from this school were “Part – Whole relationships (continuous) models”. If, our goal is to “make math visual” then I believe we really need to spend more time considering WHICH visuals are going to be the most helpful and how those models progress over time! We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem: As you can see, being able to share a story or visual model for certain mathematics concepts seems to be a relative need. My suggestion was to really consider how a focus on visual models might be a place we can ALL learn from. We then followed by a quick story of when a student told me that the following statement is true (click here for the full story) and my learning that came from it! So, why should we focus on making math visual? We then explored a statement that Jo Boaler shared in her Norms document: …and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually? If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually. I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description): Along with a few different responses that students had completed: Most of the students in the class had responded much like the image above. Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2). In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals. If you are wanting to help think about HOW to “make math visual”, below is some of the suggestions we shared: And finally some advice about what we DON’T mean when talking about making mathematics visual: You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers. I want to leave you with a few reflective questions:
I’d love to continue the conversation. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ). If you are interested in all of the slides, you can take a look here Share this:
Like this:Like Loading... A few weeks ago a NYTimes published an article titled, Make Your Daughter Practice Math. She’ll Thank You Later, an opinion piece that, basically, asserts that girls would benefit from “extra required practice”. I took a few minutes to look through the comments (which there are over 600) and noticed a polarizing set of personal comments related to what has worked or hasn’t worked for each person, or their own children. Some sharing how practicing was an essential component for making them/their kids successful at mathematics, and others discussing stories related to frustration, humiliation and the need for children to enjoy and be interested in the subject. Instead of picking apart the article and sharing the various issues I have with it (like the notion of “extra practice” should be given based on gender), or simply stating my own opinions, I think it would be far more productive to consider why practice might be important and specifically consider some key elements of what might make practice beneficial to more students. To many, the term “practice” brings about childhood memories of completing pages of repeated random questions, or drills sheets where the same algorithm is used over and over again. Students who successfully completed the first few questions typically had no issues completing each and every question. For those who were successful, the belief is that the repetition helped. For those who were less successful, the belief is that repeating an algorithm that didn’t make sense in the first place wasn’t helpful… even if they can get an answer, they might still not understand (*Defining 2 opposing definitions of “understanding” here). “Practice” for both of the views above is often thought of as rote tasks that are devoid of thinking, choices or sense making. Before I share with you an alternative view of practice, I’d like to first consider how we have tackled “practice” for students who are developing as readers. If we were to consider reading instruction for a moment, everyone would agree that it would be important to practice reading, however, most of us wouldn’t have thoughts of reading pages of random words on a page, we would likely think about picture books. Books offer many important factors for young readers. Pictures might help give clues to difficult words, the storyline offers interest and motivation to continue, and the messages within the book might bring about rich discussions related to the purpose of the book. This kind of practice is both encourages students to continue reading, and helps them continue to get better at the same time. However, this is very different from what we view as math “practice”. In Dan Finkel’s Ted Talk (Five Principles of Extraordinary Math Teaching) he has attempted to help teachers and parents see the equivalent kind of practice for mathematics: Below is a chart explaining the role of practice as it relates to what Dan Finkel calls play: Take a look at the “Process” row for a moment. Here you can see the difference between a repetitive drill kind of practice and the “playful experiences” kind of practice Dan had called for. Let’s take a quick example of how practice can be playful. Students learning to add 2-digit numbers were asked to “practice” their understanding of addition by playing a game called “How Close to 100?”. The rules:
What choice would you make??? Some students might want to keep all 4 roles and use the 14 to get close to 100, while other students might take the 41 and try to eliminate one of the roles to see if they can get closer. When practice involves active thinking and reasoning, our students get the practice they need and the motivation to sustain learning! When practice allows students to gain a deeper understanding (in this case the visual of the base-10 materials) or make connections between concepts, our students are doing more than passive rule following – they are engaging in thinking mathematically! In the end, we need to take greater care in making sure that the experiences we provide our students are aimed at the 5 strands shown below: You might also be interested in thinking about how we might practice Geometrical terms/properties, or spatial reasoning, or exponents, or Bisectors… So I will leave you with some final thoughts:
I’d love to continue the conversation about “practicing” mathematics. Leave a comment here or on Twitter @MarkChubb3 Share this:
Like this:Like Loading... A few days ago I had the privilege of presenting at MAC2 to a group of teachers in Orillia on the topic of “Making Math Visual”. If interested, here are some of my talking points for you to reflect on: To get us started I shared an image created by Christopher Danielson and asked the group what they noticed: We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image. After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture. I asked the group of teachers what mathematics they noticed here… then how they believed parents might have answered the question. While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table….. the group correctly guessed that many parents do not typically notice the mathematics around them. So, my suggestion for the teachers in the room was to help change this: I then asked the group to do a simple task for us to learn from: After a few minutes of thinking, I shared some research of the different ways we use fractions: When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about. Data taken from 1 school’s teachers: We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem: Followed by a quick story of when a student told me that the following statement is true (click here for the full story). So, why should we focus on making math visual? We then explored a statement that Jo Boaler shared in her Norms document: …and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually? If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually. I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description): Along with a few different responses that students had completed: Most of the students in the class had responded much like the image above. Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2). In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals. We wrapped up with a few suggestions: And finally some advice about what we DON’T mean when talking about making mathematics visual: You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers. I want to leave you with a few reflective questions:
I’d love to continue the conversation. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ). If you are interested in all of the slides, you can take a look here
Share this:
Like this:Like Loading... My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing. Here’s a quick story about one of those times, along with some considerations: A while back Anne-Marie and I were preparing lunch for our three children. It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup. If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water. Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup. Here is how the conversation went: Anne-Marie: How many of these (3/4 cups) do I need to make 4 cups? Me: I don’t know. How many do you think? (attempting to give her time to think) Anne-Marie: Well… I know two would make a cup and a half… so… 4 of these would make 3 cups… Me: OK… Anne-Marie: So, 5 would make 3 and 3/4 cups. Me: Mmhmm…. Anne-Marie: So, I’d need a quarter cup more? Me: So, how much of that should you fill? (pointing to the 3/4 cup in her hand) Anne-Marie: A quarter of it? No, wait… I want a quarter of a cup, not a quarter of this… Me: Ok… Anne-Marie: Should I fill it 1/3 of the way? Me: Why do you think 1/3? Anne-Marie: Because this is 3/4s, and I only need 1 of the quarters. The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making. My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm. To many students, however, division of fractions is learned first through a set of procedures. I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book Knowing and Teaching Mathematics. In her book she shares what happened when she asked American and Chinese teachers these 2 problems: Here were the results: Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:
Visual RepresentationsIn order to understand division of fractions, I believe we need to understand what is actually going on. To do this, visuals are a necessity! A few examples of visual representations could include: A number line: A volume model: An area model: Starting with a ContextStarting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday). Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced. I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn: Design your own problem that others could use to launch a discussion of division of fractions. Share your problem! A few things to reflect on:
As always, I’d love to hear your thoughts. Leave a reply here on Twitter (@MarkChubb3) Share this:
Like this:Like Loading... In the last few weeks I have asked several groups of teachers to indicate where 1 billion would go on this number line: It has been really interesting to me that many have placed the 1 billion mark in a variety of areas and have had a variety of reasons why. Many have attempted to use their understanding of place value digits (there are 12 zeros in 1 trillion and only 9 zeros in 1 billion, so 1 billion should be 3/4 of the way toward a trillion) or their knowledge of prefixes to help (million, billion, trillion… so it must be 2/3 the way along the line). Others thought about how many billions are in a trillion asking themselves, “Is their one-hundred or one-thousand billions in a trillion?” Using this strategy, everyone picked a spot toward the left, but some much closer to zero than others. Others did something interesting though. They started placing other numbers on the number line to help them make sense of the question. Often placing 500 billion in the middle, then 250 billion at the 1/4 mark and so on until they realized just how close to 0 a billion is when we are considering 1 trillion. What’s the point? Really big numbers, and really small numbers (decimal numbers), are difficult to conceptualize! They are hard to imagine their size! Think about this: How long is 1 million seconds? Without doing ANY calculations would you guess the answer is several minutes, hours, days, weeks, months, years, decades, centuries…? Can you even imagine a million seconds without calculating anything? How about 1 billion seconds? Or 1 trillion seconds? I bet you’ve started trying to calculate right! That’s because these numbers are so abstract for us that we can’t imagine them. Because of this little experiment, I am left wondering three things:
What numbers can/can’t the students in our classrooms conceptualize?Before we start working with operations of any given size, I think we need to spend time making sure our students can visualize and estimate the size of the number. Working with numbers we can’t imagine doesn’t seem productive for our young students! In our rush to move our kids into more “complicated” mathematics, we often move too quickly through numbers to include numbers that are too abstract for our students! We think that if a student can accurately carry out a procedure that they understand the numbers they are working with. However, I’m sure we have all seen many students who produce answers that are completely unreasonable without them noticing. Is this carelessness, or is it a lack of understanding of the magnitude of the numbers involved? Or possibly that our students aren’t visualizing the size of and relationship between the numbers??? What practices do we do that gets kids to think about digits more than magnitude?The other day, Jamie Garner shared her frustration on Twitter: Think about the question from the textbook for a second. Students trying to think about 342 pencils (not sure why they would want that many) should be considering a strategy that makes sense. For example, if you had 342 pencils how many boxes of 10 would that be? Thinking this way, student should answer 34 or 34.2, or maybe 35 boxes (if you wanted to purchase enough boxes). However, the teacher’s edition tells us that none of these are the right answer. Take a look: If our students attempt to make sense of the problem, they will be completely wrong! In fact, many students will likely answer 4 because they’ve been trained not to think at all about the mathematics, and instead focus their attention on what they think the text wants them to do. This is one of MANY cases where elementary mathematics focuses on digits over understanding magnitude or relative size. Here are a few others: These, along with pretty much any standard algorithm (see Christopher Danielson’s post: Standard Algorithms Unteach Place Value) tell our kids to stop thinking about what makes sense, and instead focus on steps that help kids get an answer without understanding. What practices could/should we be including that helps our students make these connections?If we want our students to understand numbers, and their relative size… if we want to help our students develop a conceptual understanding of operations… if we want our students make sense of the math they are learning… then we need to:
Below are 2 activities taken from Van de Walle’s Student Centered Mathematics. Think about how you could adapt these to work with numbers your students are starting to explore (really big or really small numbers). A few questions for you to reflect on:
P.S. Here are the answers to the seconds problem I posted earlier: Share this:
Like this:Like Loading... Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding. If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above). Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”. Here are the 2 opposing definitions of the word “understanding”: “Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures. “Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work. Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes. Think of the two types of understanding like this: Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill. Each new skill requires a new set of procedures. However, those who are taught relationally make connections between and within concepts and skills. Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made. While it might seem obvious that relational understanding is best, it requires us to understand the mathematics in ways that we were never taught in order for us to provide the best experiences for our students. It also means that we need to start with our students’ current understandings instead of starting with the rules and procedures. Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn. Take a look: Notice the top right quadrant for a second. If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term. You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves. However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade. On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist. Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics. These students soon disassociate with mathematics and will stop taking math classes as soon as they can. These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning. While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences! I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics. Here are a few I want to share with you: Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools. Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy. While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions. These students “understand” how to get the answer, but in no way understand how the answer relates to addends. On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding. If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:
Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding. For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening. Students can draw a collar and place an egg below, but in no way will this help with future concepts! Teachers focused on relational understanding again use representations that allow their students to visualize what is happening. Connections between representations, strategies and the big ideas behind multiplication are developed over time. Take a look at some wonderful resources that promote a relational understanding of multiplication:
Each of the above are developmental in nature, they focus on representations and connections. So how do we make these shifts? Here are a few of my thoughts:
Goal 1 – Notice instrumental teaching practices.Many of these are easy to spot. Here is a small sample from Pinterest: The rules/procedures shared here ask students to DO without understanding. The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems. Think about something you teach that involves rules or procedures. How can you help your students develop a relational understanding of this concept? Goal 2 – Learn more about how to move from instrumental to relational teaching.I don’t think this is something we can do on our own. We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you). I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally. However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students. Goal 3 – Align assessment practices to expect relational understanding. This is something I hope to continue to blog about. If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand. Looking at developmental landscapes, continuums and trajectories will help here. Below is Cathy Fosnot’s landscape of learning for multiplication and division. While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding. Asking questions or problems that expect relational understanding is key as well. Take a look at one of Marian Small’s slideshows below. Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.
I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students. This will continue to be my priority this year! What does the relational understanding of place value begin with?What does the relational understanding of place value begin with? Counting by ones, making a model and saying and writing the numeral.
What are the three types of computational strategies?Three Types of Computational Strategies. Direct Modeling. Use of manipulatives, drawings,or fingers along with counting to directly represent numbers involved and meaning of operation. ... . Invented. Any strategy other than standard algorithm, that doesn't involve the use of physical materials or counting by ones. ... . Standard.. How are 3 tens and 4 ones represented in a base 10 system?How are 3 tens and 4 ones represented in a Base 10 system? 3 tens = 30; 4 ones =4. Put these values together to get 34.
What estimation strategy focuses on the first number on the left of a computation?The front-end strategy is a method of estimating computations by keeping the first digit in each of the numbers and changing all the other digits to zeros. This strategy can be used to estimate sums, differences, products and quotients.
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