Association of Teachers of Mathematics in Maine
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Chapter 2 Response Choice 2

10 Dec 2018 6:31 PM | Anonymous member (Administrator)

Learning intentions can help students make connections between current learning and previously learned content. Identify the learning intention for a lesson you have recently taught. What previously learned content is connected to this learning intention? Did your students see the connection? If so, how did this impact thier engagement in the learning? If not, how might you modify the learning intention and experience to bring more attention to this connection?

Comments

  • 10 Dec 2018 9:39 PM | Anonymous member
    I am currently teaching a unit on ratio and proportions. Here are some of the learning intentions (they are written as I can's and can definitely be improved!)

    I can use ratio language to compares two different quantities
    I can differentiate between part-to-part and part-to-whole relationships
    I can calculate a unit rate in simple context
    I can calculate a unit rate in complex context
    I can solve a proportion using a variety of methods
    I can use a proportion to find an unknown quantity in complex content

    As the book talks about, I think I do have a bit of an "expert blind spot". It seems obvious to me that this unit builds upon last year's unit on fractions and percents. But interestingly enough, students don't realize it at first. I'll suggest that they think of equivalent fractions and they look at me and ask how since it's a ratio! As the book suggests, I have been trying to use the warmup time to build on the prior fraction knowledge and that does seem to be helping a bit.

    I also went back and looked at some of the learning progressions for the fraction unit and noticed that I should probably add them as the "foundational" level of my ratio learning progression. Such as:
    I can use fraction strips to estimate fraction size.
    I can use pictures or other visuals to show or recognize equivalent fractions
    I can use common factors to simplify fractions to lowest terms
    I understand a simplest form fraction as a ratio of all of its equivalent fractions.
    I can write a percent from a given fraction

    Adding those previous learning intentions from the start and sharing them with students will help them realize how the learned content is connected.
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    • 21 Jan 2019 8:15 AM | Anonymous member (Administrator)
      I agree that repeated practice, fluency, is a great place to review prior year's expectations. I notice that in my book, we start prepping for the next unit like a week in advance, by review last year's content. I think being sure to front load helps student confidence tremendously. Our attitude about math is huge, too.
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  • 11 Dec 2018 8:54 AM | Anonymous member
    My Advanced Algebra 2 class is working through a Polynomials unit. Last week, I had these learning targets on the board:

    "I can use the factor theorem to determine the factors of a polynomial."

    "I can factor the sum and difference of two cubes."

    Looking back on these learning targets, and thinking about chapter 2, I realized a few things. First of all, these learning targets are really only clear to those students who are currently taking the course. That's not necessarily a bad thing, but it is important to be mindful of making the learning targets inviting to students. These targets, while directly addressing the concept of the lessons, are not exactly user-friendly. They imply that students will already know multiple pieces of vocabulary, like what a theorem is, what factoring is, and even what sum and difference is.

    I think it's important for students to speak and understand in a mathematical language. But I also think it's important, based on the reading for this chapter, to have learning intentions that seems attainable for students. When a student reads that first learning intention, perhaps a student who was absent and didn't see the material, that's a pretty scary task to undertake. Looking at the language, perhaps it would be better to approach with:

    "Now that we know synthetic division, and how it can create output values for a function, let's extend that to find the places where a given polynomial will cross the x-axis. If the points where a graph crosses the x-axis are all points with output values of zero, we can use synthetic division to find those points."

    Learning targets or intentions were introduced a few years ago in our district, and thus far our students have struggled to find meaning in them. I think by making them clearer and more inviting, students might be able to look at the intention for a given day and feel like they have direction and clarity in their approach to the lesson.
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    • 21 Jan 2019 8:25 AM | Anonymous member (Administrator)
      One thing I do with learning targets is try to use as few words as possible, and state them very directly. Less words makes it more approachable to me.
      Also, I repeat the learning targets at least twice in a lesson - at the beginning (and I ask the students how they are the same, how they are different from yesterday, and if there are any new words that they need to know.) Sometimes I review them right after teaching one part of the target, and say, "Okay- so now we'll work on this part." Then I review them just before I send them off to practice something or work on something collaboratively.
      Even just asking, "What do you notice? What do you wonder?" while looking at the targets can help me formatively assess where to start. Some days I end on a reflection, and then choose tomorrow's groups based on how students rate themselves on our "learning ladder."
      All of these strategies have taught my students to pay more attention to the target at the beginning of a lesson to help them focus their minds.
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  • 11 Dec 2018 10:38 PM | Rachel A. Johns
    (grade 4) Logically, multi-digit addition and subtraction are 'taught' (reviewed) before multiplication. Multiplication can be taught as repeat addition or regrouping. This concept keeps addition in mind (where the success is). Division is then taught as reverse multiplication (fact families, commutative property, multiples). The concept of repeat subtraction ties in nicely with repeat addition. In the bigger picture, you want students to have success from addition to long division--but it's a tough journey for many. One key is being specific in lessons, as the book suggests--'the intention of today's lesson is repeat subtraction'. Another key is remembering for whom and how success was seen in the previous lessons. Using phrases like, "...remember how Bob showed us repeat addition when he added by 3s all the way to 63?!" can help remind students of their own successes and what tools they used to achieve success. It also engages their brains to reach back to remember. Currently I am returning to "repeat subtraction" to hopefully re-engage in previous successes to approach long division differently (as a reteach moment).
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    • 21 Jan 2019 8:31 AM | Anonymous member (Administrator)
      What a great idea! I try to do that too- reminding students of which classmate did what. They take such pride in being recognized especially a day or two later! This builds such confidence and willingness to take more risks.
      I also love how your learning intention is simple and direct. So key!
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  • 17 Dec 2018 8:42 AM | Ann Luginbuhl
    I have a pretty solid example how NOT making this connection was very ineffective. One of my math groups tackled a chapter in their books on Equations with two variables. It introduced Constant Rate of Change, Slope, Slope intercept, Point slope form and solving systems of equations. To me, the progression of concepts was very clear and each built on the previous concept logically. Perfect example of expert blindspot and my perception was not the same as my students and when it came to the end of chapter test- it was a disaster. So I went back to the beginning- carefully explaining (in as inviting a tone as possible) the learning objective of each section.(I actually cribbed from the teacher Mr. Turnbull in the book) I placed each goal on a step, one above the other- culminating in solving systems to find solutions for both equations. Once the students asked questions and discussed the connection between the concepts, practiced some skills with some success criteria, they took another test. They did MUCH better demonstrating their mastery of the concepts.
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    • 21 Jan 2019 8:34 AM | Anonymous member (Administrator)
      Amazing how just highlighting the connections made such a difference.
      Thank you for going back and reteaching. Sometimes I'm worried about getting behind others in my grade in terms of pacing, but isn't the most important thing that they truly learn? I think we need to keep a balance of perpetual review of concepts as we move forward, and reteaching the whole class when necessary. The latter probably has a lot to do with teacher clarity- which is what you demonstrated.
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  • 19 Dec 2018 10:15 AM | Anonymous member
    We are finishing up our Trig unit in Algebra II this week and one of our two content standards is on the Unit Circle. I've created a learning map to identify the small chunks that will help students to be successful on the full standard, one of those is: I understand the relationship between sine, cosine, tangent and the unit circle. This learning intention builds on the prior knowledge of right-triangle trigonometry, knowing what Sine, Cosine, and Tangent are in a right triangle. My students did remember learning trigonometry, but unfortunately their prior experience was apparently very negative... this has carried over to the current course, where they have been extremely resistant to building on this learning. I'm not sure how to modify the learning intention as the use of the trig ratios is integral in understanding the unit circle, so I can't really ignore the connection nor do I really want to bring more negative attention.

    I'm thinking that in the future if I see this negative energy building I may need to back up and try to create a new more positive experience with the basic trig ratios before continuing on... of course, this can be very difficult given the very short time I have to cover a lot of material.
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    • 19 Dec 2018 4:02 PM | Carolyn Dupee
      This week I was reviewing a Math Meet question with seventh graders related to exponents. Students were expected to understand (without a calculator) the periodic nature of exponents. In class, I took a different examples 2^2018 and asked what number would be in the units digit of this number. I asked students to see the pattern created from the exponents. They were able to see and notice that the numbers were doubling, but didn't make the connection that the numbers were were all even in the units digit, or that the numbers made a four number pattern. I asked them to struggle through finding the first 8 exponents so that they could see the pattern better. After modeling this process, I asked students to find the units and the tens digit of 7^2019. More students were able to see the connection. As I was consulting in another teacher's classroom, the teacher had the AHA moment that the students didn't really understand the conceptual importance of exponents despite being able to use order of operations.
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      • 21 Jan 2019 9:55 AM | Anonymous member (Administrator)
        It's awesome that you didn't point out that the answers were even, but had the students discover that for themselves, as well as the rotating digits. I do the same with grade 4 students to notice divisibility rules. And then they were able to transfer to the next situation!
        I think that using contexts for solving exponents could help them bridge to understanding what they are, and why to use them, not just in practice.
        These are such great AHAs.
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    • 21 Jan 2019 9:48 AM | Anonymous member (Administrator)
      I think the learning intention is fine- it's more about how to change their perceptions. Can you do some kind of a project, collaborative work, or some other delivery that helps the students build connections to one another as they struggle through the material together? Those social learning intentions can be huge.
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  • 26 Dec 2018 4:39 PM | Jenny Jorgensen
    I am currently teaching a "no credit" math class at a community college. The students have struggled in math during their public education and now find themselves needing to take this course in order to be able to take a college math class for necessary credit. The course focused mostly on typical 8th-9th grade math content. The learning intentions for one of my recent classes were: Identify approximately linear data and write an equation to model approximate linear data. Students had been working with linear data in the previous lessons and were now expected to understand that data can have a linear trend and an equation could be used to approximate this trend. Most students did see the connection, some struggled to recognized that their line of best fit, needed to represent all of the data. Students were engaged in that they were interested to see that the could still make predictions even though the data wasn't linear. In chapter two, pg. 51 discusses the focus on social learning intentions and how they can be used to "foster effective collaboration and communication" which was definitely part of the math class that I'm teaching. Students often commented on the positive value of being able to collaborate with at least one other person during class.
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    • 21 Jan 2019 8:29 AM | Anonymous member (Administrator)
      It's so good that these students get to collaborate, because expressing ourselves verbally and listening to ourselves helps us process and revise our own thinking. I'm guessing that's crucial for these students. I also wonder about how much to share social learning intentions. Sometimes I do, but not as part of our math class intentions.
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  • 28 Dec 2018 9:00 AM | Anonymous member
    The Learning Intentions that I teach are specifically outlined for us in the program that we use to teach math (Everyday Math). AS some may know, this is a "spiraling" curriculum, which I haven't completely developed an understanding for, so I do not always make the connections myself. It often feels like I am on a roll, then I come to a screeching halt and teach something different, that will relate down the road. So, in order for students to find the connections, I have to be able to make them as well.

    Prior to the break, I had just finished a unit on decimals, percentages, and ratios. After the assessment, I found that students were struggling because they didn't see how they would use these in their own lives. Yes, they had been given problems to solve that were deemed as "relatable", but they were just going through the motions and solving the problems without seeing the true application. So, I jumped ship (because I didn't want to start a new unit right before the break), and decided to do some projects to help students make the connections. We decided to do some "cooking" for our holiday party, with extra credit is they made the item and brought it in to share for the class.

    Our learning target was:
    I am learning to multiply and divide fractions.
    I am learning to add decimals.
    I am learning to find the percentage of a number.

    Success criteria:
    I can divide a recipe in half.
    I can triple a recipe.
    I can add decimals to find the total cost of the ingredients.
    I can use percentages to find the tax on items and include it in my budget.

    Based on evidence from assessments, students were still having difficulties with understanding the difference between multiplying and dividing fractions. I felt that this was an area that they needed more practice, and what better way than to make it applicable to their current lives. Students had been talking about making cookies with their families, and the cooking that had already started to take place for the holidays. So we decided to jump back a bit and revisit this, and really do a deeper dive to make connections that were personal for the students. Students really became involved in this project. students that have difficulty engaging because the connections were not "reality" for them, suddenly took on this task and completed it within the time constraints given. They were thoroughly engaged in their learning, and it showed through their efforts. And having the clear guidelines outlined in the rubric, most students went for the exceeds category, and were not satisfied just getting a "partially meets" or "meets". Having the clear learning targets, guidelines, and success criteria really made students "step up" and take ownership of their learning.

    I think what it comes down to is that my students are most engaged when they can make a direct connection to their learning and see how they can apply it in their own lives. I think that the learning intentions need to be worded in such a way that it is clear what is expected for them to learn and be able to demonstrate by the end of the lesson, but students also have to have buy in by "accepting these goals as legitimate" (Hattie, Fisher and Frey, 2017, 61). Looking to the future, even though I have a curriculum that is required, I truly believe that making the time for these mini projects during the units would be a successful way to truly engage my students and help them make more personal connections to engage them in their learning.
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    • 21 Jan 2019 8:48 AM | Anonymous member (Administrator)
      I think you should feel completely empowered to change a learning intention to make it more accessible for your students. I would take those I can statements and break them down even further into success criteria from the book the way you have done for your own project.
      I also agree that project based learning gives students a chance to sink their teeth into deep learning (next chapter in our study).
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  • 28 Dec 2018 9:38 AM | Deleted user
    In my third grade classroom we have just begun a chapter on understanding division. For the second lesson in this chapter my learning intention was "Understand that division means equal sharing." Students used cubes to model different "sharing" situations. Most were successful when the situation was concrete--for example, sharing cookies. As we discussed how we solved some of these problems I pointed out that we were doing the opposite of what we had been doing in multiplication where we were combining equal groups. Instead, we were making equal groups. Because this is just the beginning of my instruction on division, I am not sure that students have yet realized the inverse relationships of multiplication and division.
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    • 02 Jan 2019 8:40 PM | Anonymous
      I am going to piggy back off of your post. I teach fourth and fifth grade. We just finished our division unit. For some of my students it was like they had never been introduced to the topic, while others could do it fluently before day one and make connections and references to prior learning.

      I can introduce ideas or or challenge my advanced fourth graders with content from the fifth grade text, and I can remind fifth grade students about how topics were introduced or practiced the previous year as I know them well because I am teaching them to other classes. What I am noticing is that I really need to spend some time looking at the third grade textbook and sixth grade. Generally I know what standards they will be working on, but I am not sure which order and how things will be introduced which is a disadvantage to my students because I can't make those references or lead them to those connections as I have not made them myself. Something to look into...
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  • 29 Dec 2018 2:51 PM | Anonymous member
    I'm going to slightly deviate from this question to get at what I think my chief area of improvement is: making students understand the learning intentions. I have them up on the wall, I walk over to them and point to them and discuss them, and that all guarantees that (dramatic pause) *I* know the learning intentions :)

    If I want the kids to understand them, I need to make sure my spoken words have taken root inside their heads. Probably a simple question or two would do it. "Ok, class, turn-and-talk: what is the learning target for today? Next kid, answer this: how does it connect to yesterday's learning target? Ok, raise a hand if your pair had trouble answering." That little one-minute activity would probably go a long way to making sure kids knew what the point of today's class is.

    While I'm riffing on non-question topics, another thing I wanted to talk/ask about was success criteria. Anybody else have a hard time distinguishing success criteria from learning intentions? It seems like a learning intention could be, "I can identify and interpret the rate of change in a linear relationship." But then the success criterion for that learning intention would be, well, "I can identify and interpret the rate of change in a linear relationship." Is that right? Any help?

    One last thing, did anybody else notice on p57 the walloping effect size of "self-reported grades/student expectations"? It's 1.44 - holy kamoly! Maybe I missed it, but I don't think the book talked to much about HOW to implement such a giganto-effective teaching strategy. I'm guessing it could take the form of a Google Form self-assessment, maybe where the success criteria are listed and the students rate themselves on a red/yellow/green scale or something?
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  • 30 Dec 2018 11:36 AM | Anonymous member
    I teach a group of fifth graders who I also had as fourth graders. These students are identified as gifted and talented in math. We are covering the fourth, fifth, and sixth-grade curriculums in two years. We started fraction multiplication and division and will return this week to dividing mixed fractions. We covered all of these topics in the fourth and fifth-grade curriculums and I considered these lessons to be a review.

    The learning intentions for the previous two lessons and the upcoming lesson are:
    I can model fraction multiplication and fraction division.
    I can compute the multiplication of two fractions.
    I can compute the division of two fractions.
    I can multiply and divide mixed numbers.

    All of the students recalled how to multiply fractions. One or two needed to review fraction division. We had spent a great deal of time creating models for both operations in the fifth-grade curriculum but we reviewed this also. The area where I anticipate the most emphasis will be required is on simplifying the improper fractions before the final multiplication operation. The students just learned about least common multiple and greatest common factor so they have the tools needed. They will need to apply these here.

    I will be rewriting the learning intention to incorporate reducing the fractions after turning the mixed numbers into improper fractions and setting up the multiplication. Our math textbook offers many relevant word problems that provide context for the division of mixed numbers. Additionally, I will be asking the students to create their own word problems to share with their classmates. I have found this kind of assignment really engages the students.
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    • 21 Jan 2019 9:05 AM | Anonymous member (Administrator)
      I love having students create their own problems and giving them to each other to solve. It's very engaging.
      I also love how they are constantly modeling these!
      To me it seems like a lot of learning intentions at once, but I suppose that these students are up to such a complex task.
      I also like to take each learning intention and write two success criteria- one performance based and one analytical (explain, model, interpret, etc.) It forces me to deepen the learning from the skill to the understanding.
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  • 01 Jan 2019 3:50 PM | Anonymous member
    Similar Polygons
    I can determine if the polygons are similar.
    I can write a similarity statement and state the scale factor of similar polygons.

    Our work with congruent triangles is connected to this learning intention. I have an activity that gives the students two similar polygons and it asks them to compare corresponding angles and corresponding sides. The comparison reveals that the angles are congruent and the sides are not. Students are asked to find the ratios of the corresponding sides, which reveals that they have the same ratio. I feel that the activity has led the students to see the connection, but I feel that connection has only been made by my leading them there. I think I want to take it to a deeper level by giving them several congruent, similar, and neither polygons and ask them to sort them. If they don't sort the way I expect, I can ask them to find other ways to sort them and have them write the parameters used for each sort.
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    • 21 Jan 2019 9:01 AM | Anonymous member (Administrator)
      Great idea to formatively assess and adjust your instruction!
      You could even take the second target and break that down into smaller success criteria like I can compare two side lengths, and two angle measures of two polygons to describe how they are similar. The more transparent we are, the more we make the learning visible.
      Also, on page 92 there are a list of focusing questions that can help us dial back helping too much, and putting the learning back on the students. I have seen a big difference in my students this year. I also catch myself when I start funneling questions to them. I try to stop myself and take the time to really make them learn.
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  • 03 Jan 2019 7:53 PM | Anonymous member
    I hope it isn't too late to jump into this post! I will be more timely with my comments from here on in :)
    I have been working to improve this each year, it is a major goal for me. We always write our learning targets on the board, but "learning targets on the board are not learning targets in the head" resonates with me! I talk about the learning targets a lot and try to show students how they connect. Here is an example:
    Unit 1 focuses on introduction to linear equations. Students are not required to write an equation from two points yet, but are expected to be able to write equations from graphs, tables, and given a scenario. While we are accomplishing these small goals, I am constantly reminding students of what they are trying to do. "Here we have a table, so remember one of our goals is to be able to write an equation from this table." Once these goals are clear (even if they can't do them yet!) I will start eluding to Unit 2 (Bivariate Data) where they will have to write an equation given two points. I will say things like, "Right now we get the whole table to write our equation. We can find the slope, we can expand the table to find the y-intercept. Next unit, we will be able to write the equation just from two points. Isn't that crazy?!" Once in unit two they are ready for this target, they know its coming and it is so much more meaningful to them. Many of them resort back to the table by putting their two points in a table and expanding it to find b. This shows me that they have made the connection between the two skills.
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    • 21 Jan 2019 8:53 AM | Anonymous member (Administrator)
      I love the familiar tone you use when talking about the learning targets. It makes it much more understandable for students. Alluding to upcoming targets helps them see important connections. Great ideas!
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  • 05 Jan 2019 5:22 PM | Anonymous member
    I recently finished our first of 3 fourth grade multiplication units. This unit focused on the relationship between addition and multiplication, multiplication and division, and division and subtraction. A few of the learning intentions were:

    I can explain how addition and multiplication are related. I can explain the relationship between multiplication and division, and use multiplication and fact families to solve division math facts. I can explain how addition can be used to solve a multiplication math fact. I can explain how repeated subtraction can be used to solve a division problem.

    Using a pre-assessment, I was able to determine that most of my students were able to solve, but not able to explain, using addition to solve multiplication, using multiplication to solve division. They could complete tasks for these learning intentions, but they could not explain. In addition, the pre-assessment showed my students had never been exposed to the idea that division is repeated subtraction. I reviewed the first couple lessons more quickly to allow more time to work on explaining our math thinking in multiplication and exploring the idea of division as repeated subtraction.

    Students verbalized that they had discussed multiplication as repeated addition the previous year and that addition and subtraction are inverse operations, and so are multiplication and division. We connected to that previous lesson. But they had never made the connection or explored the idea of division as repeated subtraction. Without previously being exposed to or discussing this concept, which is totally developmentally appropriate that they had not covered this, we needed to spend more time making these connections to move forward and have a better understanding of division, as well as how the operations to relate to one another.
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    • 21 Jan 2019 8:44 AM | Anonymous member (Administrator)
      That is the very best way I can think of to use a pre-assessment. Wow!
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  • 12 Jan 2019 8:26 AM | Kim L.
    I'm catching up! Bit late posting this one ...

    We have been working with ratios and proportions for a few weeks now. I thought students would pick right up on the similarities between proportions and finding equivalent fractions / simplifying fractions. Not so much! Many of my students view math as distinct topics and steps to be memorized, so new thinking is simply new steps to memorize to several. One student asked me to “just give him the formula.” They are not yet proficient problem-solvers.

    They can be quite overwhelmed by the language of math, and I think this plays into their thinking that math isn’t connected, that every new problem is a whole new set of steps. Our school uses restorative practices, so we have community circles in class once a week or so - one popular topic we discuss often is “math panic,” that feeling you get when you look at a problem and don’t immediately know how to solve it. Just giving that feeling a name helped several students. We also talk about struggle, and that struggling is part of learning.

    In the end, I directly taught the connection between simplifying fractions and finding equivalent fractions. I should have just started out with that. I think I had an “expert blind spot,” figuring kids would make the connection on their own and that finding proportions would be easy for them.
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    • 21 Jan 2019 8:42 AM | Anonymous member (Administrator)
      I'm so glad that you responded! You can respond to the prompts at any time that works for you.
      What a great social learning intention you address with the restorative circles.
      As we look at the difference between surface learning and deep learning (over the next month) I hope you find some additional strategies to help your students struggle more productively - especially in problem solving. I call them challenges instead of problems (trying to change their perception of attacking the situation.)
      This week, for one of our lessons, we will work in small groups and all work on a single challenge. They will have ten minutes to work together to solve, revise and make a final copy before presenting the solution to the class. They are in ranked heterogenous groups (on page 156 in our book.) I have found that this really increases group and individual success with the content.
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