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Lesson 5-6: Question
We’d like you to reflect on two (2) planned lessons you delivered back to back in the past where you proceeded as planned instead of pivoting your plan.
We want you to reflect on this experience by first outlining the two lessons and then think about how you might pivot the next time you teach this concept.
Clearly, when you planned these lessons, you thought there would be a smooth transition from one lesson to the next, but something just didn’t work. Maybe you anticipated students would approach the concepts with greater ease or maybe the concept was much more complex than you had realized the first time through your grade standards or curriculum. Be sure to use the 5.6 PROCEED OR PIVOT REFLECTION template to help you.
Share your experience and any questions, comments, or thinking here.
Let’s start a great discussion.
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78 Replies
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I thought that moving from adding simple fractions using grids and counters, then arrays (lesson 1) to adding simple fractions as well as mixed numbers using the algorithm (lesson 2) would not be a stretch but found that I needed to go back and do activities where they struggled with arrays with mixed numbers to cement the idea of the more concrete before the idea of the adding mixed number.
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So much thinking to unpack in that area! Thanks for sharing your work!
Have you checked out the wooly worm race unit? Could be helpful for this work:
learn.makemathmoments.com/task/wooly-worm-race
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This whole idea reminds me of 3 years ago where my administrator thought it would be a good idea to turn all learning goals into SMART goals. Which made a learning goal (which we had to post every day) into long phrases that students didn’t care about or understand. But the idea was if we did not achieve the measurable goal, we had to reteach. So really, it is an excellent practice, but I think putting the practice of pivoting into the learning goal went a little far. Also, the teacher complained about how this practice expected them to add to a timeframe that was already short. As you stated in the presentation, when pivoting to ensure student abilities to create concrete/visual representation will save time because it will allow students to understand that will fit into future learning goals. But I feel the fear is that every lesson will need to pivot, but that fear I think is never justified. But I digress and will go into the task at hand.
With my early experience with CPM, I felt I proceeded a lot when I should have pivoted because CPM trainers told me that the textbook would spiral back to concepts that students do not master. (I do think CPM is a great textbook, but that does put a lot of trust in a book that has no clue who my students are) I love the problem because it always gives me apparent struggles with the students when solving the Linear System of equations. The problem states,
“The Alpine Music Club is going on its annual music trip. The members of the club are yodelers, and they like to play the xylophone. This year they are taking their xylophones on a gondola to give a performance at the top of Mount Monch. The gondola conductor charges $2 for each yodeler and $1 for each xylophone. It costs $40 for the entire club, including the xylophones, to ride the gondola. We know two yodelers can share a xylophone, so the number of yodelers on the gondola is twice the number of xylophones. How many yodelers and how many xylophones are on the gondola?”
Great problem to break apart and slowly release information to the students (Especially since it is wordy, like my response). But the issue becomes that the following lesson has students solve the system using the equal values method. This is an excellent progression, but students often struggle writing the linear equations from situations like the yodelers and xylophone players.
So I feel a great pivot lesson would be a simple lesson where you are given starting values and rates of change to write two variable equations in slope-intercept form and then go back to uncovering different methods for solving linear systems. Sorry for the lengthy response.
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Lots of great points and reflecting here.
In regards to the SMART goals for learning goals – love that idea if we had more planning time to commit to an endeavour like that. Maybe looking at big ideas from the SMART goal lens is a better middle ground?
One thing to consider is balance at all times. When we share whether we fan proceed or pivot, keep in mind we aren’t suggesting that students master or have a high level of expertise (yet) as that simply isn’t possible in just a few short days. So the trouble becomes trying to determine when is it appropriate to pivot or when can I proceed? Spiralling back is always key and if we are building out ideas conceptually starting with low floor (as you suggest BEFORE the word problem you shared), then it gives students an opportunity to revisit ideas.
The problem based unit that pops in my mind as I read your systems of equations scenario is the Shot Put unit:
learn.makemathmoments.com/task/shot-put
It starts with a scenario as a single equation and builds towards a system. Again, giving students opportunities to build their understanding as they go.
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This year teaching proportions to 7th graders I started with the chocolate milk task on Desmos and thougth it would automaitlly be engaging and informative. Students did not understand the double number line and I got discouraged and moved on. I tried the balloon task as well – hoew many balloons are needed to lift you. I did a notice and wonder with David Blaine and again thought this would fix the problem. I kept missing the sense making piece and then moving on to the next thing rather then strategically pivoting using data from student answers and instead followed my path of “intersting” activities.
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Hey @david.diehl We were definitely guilty of this for many years. When you experience this again what will you do differently?
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I am nearly one term into our new school year and I have had to pivot a number of times already. I have many students that do not have a conceptual understanding and I have started to play with the spiralling idea. Using visual, concrete strategies has helped my students and I am sure I will do a lot of pivoting with my students.
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So key to go slow so you can go fast!
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ORIGINAL PLANNED LESSON. STEP 1
Linear and affine functions
Solving proportional problems
representing in a chart
making a table of values.
ORIGINAL PLANNED LESSON. STEP 2
Linear and affine functions
Solving proportional problems
representing in a chart
making a table of values
finding the algebraic formula
POSSIBLE PIVOT LESSON
POSSIBILITY 1: it has happened this year. Some of them had problems putting the coordinates in a plot.
Plotting coordinates
We play how to sink the float. With coordinates, first the numbers should be integers and then decimals.
POSSIBILITY 2: They have problems to understand how they have to proceed to make the table of values and they have very few strategies to solve proportional problems.
The pivot lesson will be a lesson with a friendlier numbers and that the deduction of a unit reduction strategy is more obvious than in the normal lesson.
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Recently while teaching Volume and Surface area to a group of ELL students who were hybrid (In person and remote), I noticed that they were having a hard time differentiating between shapes, and also as to why volume was three dimensional and SA was two dimensional. Even though I had created several graphic organizers I realized that the students needed more real world visual representations. This was especially true of the remote students, which created even more challenge. However, just by going back and re teaching the lesson with the inclusion of real world examples in videos, and also physical representations that myself and the in person students held on camera made a difference. Students needed to be able to make the connection between the shape, it’s dimensions, and their prior knowledge. In the future when teaching these lessons, I will be sure to provide more representations that students can handle themselves, and also connect to their own life experiences.
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I did a review of area/surface area and I thought the kids “got it” (meaning I thought they already “had it” from Gr. 7). I gave them a short task and 1/3 of the class didn’t get it at all. Some mixed up perimeter and area, some just didn’t do the task at all, some figured out one side and thought they were done. My plan had been to have a “boxing match smackdown” where they compared surface areas of various sized boxes to determine the “winner” of each match. I was really flustered that I could not move on, leaving 1/3 completely baffled or defeated. In the “old days” I probably would have moved on anyways, hoping the others would ask questions or miraculously “get it” but with some nods to Damian Cooper’s writings and your own voices in the back of my head I decided to do two things: give the 2/3 who “got it” the boxing match independently (round 1) and have a small group lesson with the 1/3, using the boxing match materials to gap fill and answer questions. They did not ask many, but working collaboratively they caught on to the differences that had meant disaster on the pre-activity. Pivoting to work with this small group allowed them to join in the subsequent “rounds” and we all finished together. Some of the kids in the 1/3 group actually out-performed some of the 2/3 group by the end of the unit, which was pretty satisfying.
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That is so awesome to hear! What a great pivot by allowing the 2/3 to continue on and working in a small group. You should be proud of the learning your students have gleaned from this experience and note that your timely shift in practice surely had a great influence on the outcome! Bravo!
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In third grade we introduce area of rectangles and composite shapes. The first few lessons go through area using square unit tiles and comparing square centimeters to square inches. After a few days of using the concrete manipulatives we move to solving the problems with just the measurements. Then we are presented with composite figures with measurements. Every year my students struggle going from the rectangles to the composite figures but because we are on a time table for completing lessons I follow the lesson sequence.
I think that I need to pivot and prove concrete examples of area with composite figures so that students can see the square units and how the composite figures are made up of a variety of rectangles. Students really struggle when they are not given all the side lengths of the rectangles in the composite figures because they don’t see the rectangles within the composite shape. Before teaching the lessons on area of composite shapes, I think I need a task where students are looking at the shapes within a composite figure.
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The two lessons I taught back to back were completing the square and the quadratic formula. Students can derive the quadratic formula from completing the square of a quadratic function. I taught completing the square with Algebra tiles and I modeled the function x^2+8x and I walked through this example with the students. Then I had students practice with x^2+4x and asked them to complete the square with it. Students could imitate what I did in class but the understanding of an array would have really helped. I didn’t slow down but went straight to the abstract with the lesson.
The following day I derived the Quadratic Formula using completing the square and the students were stymied. They couldn’t participate in the lesson because they didn’t understand first how to complete the square. Usually, a student completes the square with a numerical problem and I model how to derive the Quadratic Formula algebraically.
Had I pivoted at completing the square by spending more time with the students and connected the Algebra tiles with an array, students would have felt more comfortable with how completing the square derives the Quadratic Formula.
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Great realizations here. Both completing the square and deriving the quadratic formula are two pretty abstract concepts (at least how we typically taught it). Getting concrete and ensuring students are seeing the “why” and “how” behind both is so key.
Check out this task if you haven’t yet… it’s really helpful on getting “to” completing the square:
tapintoteenminds.com/3act-math/magic-rectangle
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The transition from teaching multiplication followed by division has led to frustration and lack of understanding for many of the students. With pivoting I can picture the using the spiral method to move between multiplication and division.
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Moving too quickly from solving one-step equations to solving two- and multi-step equations is something I have done more than once (not proud of it!). Every time, I find that after another not-great outcome, I promise myself that I will change things up and learn from my mistakes before teaching it again. Invariably, I do take the time to rethink and make plans to do things better/differently, but somehow end up right back where I started. Honestly, it makes me so mad that I continually let myself be pressured (both by myself and outside forces) into doing things I know are not in the best interest of my kids.
This time, I am hopeful in a different way because I feel the planning aspect that you have laid out encompasses so many more possibilities than I have ever thought through in the past. Additionally, now I can see that if I plan thoroughly enough to specifically see how following through will really save time in the end (and more importantly have a greater impact on student learning and retention) I am much more likely to not make the same choices of just moving on in the future. Feeling optimistic!
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The argument can be made in support of either to proceed or to pivot.
Many a time I proceed and tend to go back and reintroduce the concept, when the chance presents. To a large extent, unit rate is present in solving equations and and also in proportional relations.
Pivoting on the other hand, has an advantage, because it provides opportunity to support the struggling students with multiple representation of the concept, while exposing the students to more concepts.
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Sounds like you’ve come to the conclusion that there is never a one size fits all answer or path to take. There are pros and cons to both and we need to use our professional judgement to try and determine which path is most helpful at the time.
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At the beginning of this past year, I started with a lesson on proportions/ratios and wanted it to lead into a lesson about writing an equation of a line. I had students plot 2 points using desmos graphing calculator. Then their partner tried to write an equation that connected those two points. Reflecting on that lesson, I should have pivoted to many more lessons that require proportional reasoning, but I would do a much better job of connecting to different representations. I also realized that students could easily extend to given whole numbers but did not have equal success with decimals as in your defrosting problem. Much more time should have been spent helping students develop their number sense.
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When I was teaching my first year in a new grade level in a new state, my partner and I had planned for the series of lessons with the standard that says: Solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, and stem and leaf plot. Talk about a loaded standard!! It expects kids to not only read and interpret the three types of data representations, but also requires them to solve two-step problems with them! This would include adding, subtracting, and comparing fractions, decimals and whole numbers.
I think we were really crunched for time or something, because we had planned to simply use the consecutive pages in the student math book, which goes something like:
Day one – Read and interpret frequency tables, with all three number forms, and do the multi-step problems at the bottom of the page.
Day two – Read and interpret dot plots, with all three number forms, and make SURE you do all the multi-step problems at the bottom of the page.
Day three – Read and interpret stem-and-leaf plots, with all three number forms, and of course, do the multi-step problems.
Day four – Read and interpret all three types of data representations mixed up on the page, and solve ALL problems on the both pages. Ta-da! We’re done with this and can move on next week!
I knew going in that this was WAY too fast, but since we were crunched for time, I planned to proceed in this ridiculous way. To make it even more embarrassing, I used the pre-printed problems in the consumable math book, which had NO connection to the kids at all. Sometime in the middle of the first day, I thought, “What am I doing???!!”
I wasn’t thinking the actual words, “Should I proceed, or should I pivot?” but I did realize that this was ridiculous. So, in hindsight, I now know that what I did was to pivot.
The next day, we collected data on the simplest thing I could think of so everyone could access it. I think I had them count how many jumping jacks they could do in one minute. We collected the data, put it in a frequency table, dot plot, and stem-and-leaf plot. Then we used their actual data to write and solve problems. For example: How many more students did greater than 15 jumping jacks than did fewer than 15 jumping jacks? We repeated with different student-collected data the next three days, and they learned so much more. By the end, they could transfer data from one form to another, and write pretty sophisticated questions.
It actually took only one day longer than the original four-day plan, but it was so much more “real!”
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Last year, quite a few of my students struggled with identifying zero pairs on an Expression Mat, which lead to difficulties when moving to an Equation Mat. As we moved into simplifying expressions algebraically, many also struggled with identifying zero pairs to make simplifying easier.
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When I taught 5th grade math, I spent one day teaching students how to convert mixed numbers and improper fraction. Due to time constraints, I went strait for abstract thinking. The next day we moved onto applying this when adding fractions (LCM and GCF were also in the mix). In hindsight, I should have pivoted or even backtracked by helping students gain conceptual understanding of how/why converting mixed numbers and improper fractions help can help us solve more complex problems.
****Pivot Lesson: How would you represent a number greater than one using colored tiles?**** By using manipulatives, the progression to abstraction would have been organic.
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I tried NCTM’s Unusual Baker task with my grade 5 this year. They were all successful with identifying the fractional amount for each piece of cake, but they really struggled with the proportionality of what to charge for each piece of cake. The total cost of each cake had to be exactly $10.00. So they would tell me that a half a cake could be $8.00 and a fourth of a cake could be $1.00. I realized that we needed more work in fractions as parts of a set, like the apples/oranges task in the book Student-Centered Mathematics. We really needed concrete experiences finding fractions of whole numbers and then fractions of whole numbers that would lead to decimal fraction amounts.
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In many ways, the need to pivot came when I realized the way math was taught, when I returned to the classroom, was so different from what I had taught +20 years ago or even what I had experienced as a child. I had to learn so many new representations of numbers, how to use manipulatives more effectively (though I had used them before), and how to put an understanding of numeracy first and foremost. Many of my first lessons were more “old school” in that I “taught” techniques and tricks instead of real understanding.
Subtraction of multi-digit numbers is a good example. “More on the floor, go next door and get 10 more.” “More on the top, no need to stop.” Great sayings, right? Only for the kids who already got it. Those who didn’t just switched the digits in the numbers to subtract the smaller digit from the larger digit. I needed to step backwards and make sure they had a clear understanding of place value and quantity (conservation?) before I could teach subtraction using more concrete representations and pictorial models first.
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Rates and Proportional Reasoning: 2 tasks, Which is juicier? Comparing juice boxes’ ratios of juice concentrate to water and then Kayak, comparing kayakers speed and time given in fractional form. Both lessons were rushed and in hindsight, failures.
My students incorrectly set up juice fractions using part to part, not part to whole ratios. I should have pivoted back to a fractions refresher, looking at equivalent fractions and deepening my students’ understanding. Then moved into the kayak task.
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At the beginning of the school year we learn about fractions and decimals, and how to convert back and forth between them. Then towards the middle of the year we learn about equivalent fractions, decimals, and percentages. I always feel like the second lesson should be very easy since we are only learning one more component, but it seems to be the biggest struggle and it scaffolds into Data when they work with relative frequency. I’m usually very good about pivoting when needed, but this one usually gets the best of me.
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I spent a day where students explored exponents to figure out for themselves the rules for multiplying exponents. This was a YouCubed activity where students identify, for example, that 2^3*2^5=(2*2*2)*(2*2*2*2*2)=2^(3+5)=2^8. They also do it for raising exponents to exponents, dividing exponents, and so forth. The next day, I had them trying to combine several of the operations at the same time. It went too quickly for many of them and they couldn’t see how what they had done the previous day applied to this assignment. I should have pivoted to help them understand that things can be combined as well, though I don’t know what kind of task would have helped with that.
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Recognizing a pivot would be helpful is very important. The hard part is what to do in between. This might not happen right away, but do take note of it and think on it so you can provide a better progression next time around. Something to ask yourself is “what specifically were they getting hung up on? How can we smoothen that path a bit without doing the work for them?”
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I can think of a few examples of this, the most prominent in my mind is when my students start to solve equations with 1 or 2 steps.to solve for X. I know I have moved on to 3 step equations knowing full well students were still floundering. I know I need to PIVOT, get visual, have math talks and figure out the stumbling blocks so we can work through the difficulties…
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Right now, specific lessons evade me, but I suppose lessons that needed to be more fully explored is substitute and calculate.
Many of the students that are not good at math do not feel confident with calculation.
It must be BEDMAS that seems daunting or, should I say, has not stayed in front of all other lessons when needed to be recalled.
The need is for them to be able and do this by default, but this isn’t the case for many but we move on.
A pivot lesson when we see evidence that they are not strong in this area before we need this skill for formulas is definitely necessary.
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This reply was modified 1 year, 7 months ago by Velia Kearns.
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This reply was modified 1 year, 7 months ago by
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While teaching the unit on fractions, we go from finding equivalent fractions to comparing fractions. They student often struggle with idea of what a fraction looks like based of the numerical representation. We could pivot to look at models of fractions and talk about fractions that are less than one-half and fractions that are greater than one-half.
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In the past I’ve had one lesson where we use 2-sided counters to model adding integers and then in the next lesson we add integers without them. That first lesson there is always a group that wants to jump and so I get excited, however the 2nd lesson can bomb. Instead I can pivot and do more visual representations, going into a more abstract way, but with still some visual. Students could use number lines or drawings to show their understanding.
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Fractions! When teaching fractions with grade sevens they are always all over the place for understanding. I started the lessons while online, due to the way in which the year was, so I started with equivalent, then common denominators then multiplying, (this was suggested by another teacher), then adding and subtracting. I found the kids were very confused and I have to back up and reteach a lot.
Reflecting now, I would have pivoted at the start and found some students did not have a clear understanding of whole and part, then started with adding to make whole.
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I pivot constantly! However, there were a couple lessons I did on volume of spheres, cylinders, and cones where I couple it with a lesson on the area of a circle so that students could. And I think I still pivoted! I do bring these two up though because I think the reason I didn’t have to pivot as much as normal is because I read a blog on teaching volume that mentioned standard connections and common misconceptions so I put in the work beforehand to anticipate student needs and sequence the concepts accordingly to facilitate connection.
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Sounds like you’re learning and growing if you are constantly planning ahead and learning / shifting your thinking before heading into the lesson. Love it.
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I was co-teaching a lesson in a Grade 5/6 classroom and we were exploring equivalent fractions. The task was to draw an outline around the area that represents one half. I guess I did not give clear instructions and they started drawing in the white space to make halves. I was maybe too open to not intervening. I was exploring how much to say in the beginning. It was also tough since the co-teacher and I did not have an established dynamic. Anyway the task did not go very well, and the discussion was weak.
So we pivoted to a task of making 2 and 1/2 as many ways possible with pattern blocks. The goal is to show equivalent fractions, but that learning goal adds on the concept “it matters what you define as your whole” Which came out naturally. That lesson was successful and in discussion you could see the student really understanding and lighting up and even articulating their understandings of equivalence and also defining the whole. So that worked much better. With very little teaching, only directing conversation between the students for the whole class.
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In my third unit lesson 4 is “Comparing Proportional Relationships” and has students using graphs and tables of things like lemonade mix and water or money and envelopes. Lesson 5 is “Introduction to Linear Relationships” and one of the earlier tasks is predicting how much a stack of cups is going up. I remember that this lesson 5 tends to be quite difficult and the majority of students really had a hard time moving on to lesson 5.
I’m thinking of going back to a problem they have seen before but applying it in ways that work more for these topics. They have already looked at how much a park charges for people entering and figured out that they had a charge for each vehicle regardless of the size and a charge per person in the vehicle.
I would need to do more but I also think I need to see what the current group of students is doing. The second half of the pivot might need to be crafted for areas where this group needs more work. It could be something more concrete like blocks or a copied shape and recording how many squares it goes up when you add another brick or whatever. They could be encouraged to represent this with pictures or tables rather than just algorithms.
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@denny.nelson sounds like you’re doing some great on the spot reflecting and learning!
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In the grade 8 program, students tend to run into difficulty problem-solving with percentages. In particular, this occurs when (a) they need to find the whole (unknown number problem) when the percentage for the other number is provided, (b) the change with percent increase/decrease, and (c) calculating the unknown percent increase/decrease. The activity you carried out with the tree drawings in Module 5 Lesson 1 could really help to solidify how proportions work using percentages which in turn would be beneficial to helping students understand how percentages work in situations a-b-c above. I will definitely pivot after the first planned lesson before moving on to the second planned lesson. This should save a lot of time and grief for students. I can feel the confidence building in them already.
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I struggle to have the courage to pivot especially when working to stay at pace with teachers in the same subject since we give common quarterly and final exams. There are many times when some students need additional time to gain a deeper understanding while several incorrectly placed students are already there. I struggled to find enough extensions and additional tasks for the students who grasp the concept quickly the first time. My goal this year is to focus on developing a conceptual understanding and I think the concrete and visual models will be very helpful in achieving this goal.
Last year I was able to develop students visual understanding of multiplying binomials and factoring trinomials in the form x^2+bx+c but I struggled to find a visual model for factoring trinomials in the form ax^2+bx+c when a>1. I tried having my advanced students work on it as a challenge but we had a difficult time explaining the why between the trick to find the two terms that add up to b and multiply to ac. The visual works so well with the simple version. Any ideas, thoughts, or suggestions would be greatly appreciated.
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I can’t think of specific examples but I do know I had to pivot a lot in 2021 because my class in general was not strong mathematically. I had a lot of “Maths refusers” (eg take 20 mins to find a pencil and another 20 to rule up the page and , oh look at that, it’s the end of the lesson) and a lot of them just weren’t ready to tackle the Year 7 concepts so I found myself pivoting back to Year 6 concepts frequently. I tried to do this in engaging ways (explorations, competitions, situations which gave them the chance to wrk together and talk) but some of them were really hard to engage because they only came to school for the social side. What made it really hard was having 2 or 3 students who were years AHEAD of the curriculum so trying to keep them progressing in this environment was hard! I stocked up the classroom with hands-on manipulatives (which most students refused to use because they were ‘babyish’), whiteboards and markers to try to make learning more concrete and visual and didn’t worry too much about formulae if it didn’t seem like they would understand or remember them. I’m sure I’ll cop it from their Year 8 teachers this year but hey, you do what you need to to survive!
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I pivoted many times. I slowed down and made time for the students to share their thinking. I love it!
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Be sure to keep using that tool of pivoting when necessary as well!
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This reflection fit a situation that recently occurred in our classroom. As we were wrapping up our ADDITION and SUBTRACTION unit, we realized that our students were not able to apply their learning to real-world problem solving. Since we have to “move on,” I am using problem solving in my warm-up, spiraling, and RTI (intervention) time. I also tend to build in a week each 6 weeks that’s strictly for review.
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Glad to hear you have some strategies in place to address this challenge.
I wonder: what if we started with contextual (ie: problem solving) instead of pre-teaching to lead up to problem solving?
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I think this year has been a lot of pivoting for me. My students and teachers tried really hard but learning during the pandemic was so challenging. They did not retain concepts like they usually do and they are left with unfinished learning. Since I am aware of that, I try to be flexible and pivot as needed to reteach those concepts maybe in the same way but maybe in a more student directed way to help them. The lessons I chose were from the beginning of the year and I really don’t feel that I pivoted when i should have. The students did not really remember the Pythagorean Theorem which I was using as a quick warm up for the Distance and Midpoint formulas. Since I was caught surprised, I just did a quick review and moved on. However, I think it would have been better if I had pivoted to a problem that Jon described at the beginning of this course that I think could help my students to explore more and hopefully build on the prior knowledge they do have around distances and Pythagorean Theorem.
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This is a great reflection and so very common when we have planned out our long range plan and feel the need to “keep moving”! That ability to stop, zoom out and think about where students are is so key to helping pace your course at the speed of learning.
Have you considered using a math talk structure like a string of related problems to help you build more fluency and flexibility with concepts like Pythagorean theorem?
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After working through Triangle similarity and postulates, my original plan was to get moving with problem-solving and equations involving similar triangle setups. However, given the confusion or difficulty I noticed students were having, I decided to pause and do a recap on proportions and solving with fractions (something they consistently have trouble with). This could bolster their confidence with the solving piece. Additionally, I decided to use physical demonstrations and cutouts to convince students that these are indeed similar by rotating/transforming.
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Definitely common for those struggles to arise. Sometimes stopping to pivot – even if just for a bit – is the right move.
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This is an area where I struggle. I have had groups that struggle with a concept, say Dividing fractions for example. My lesson planning has me scheduled to move on to adding fractions, but it is apparent that a group of students does not have a grasp of the topic. I don’t often have time to replan for the next day, so what do I do? I have spiralled in coures before but I get bogged down in the overwhelming task of tracking learning and reporting learning to parents and students. I inevitably fall back on the traditional style of units. So 2 questions: Does the pivot have to happen immediately after the initial lesson? How do you deal with tracking evidence of learning while spiralling? I am booged down with multiple pages to enter assessment of learning for multiple curriculum expectations, I get so overwhelmed.
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Pivoting could also mean spending another day working on the same concept instead of proceeding to the next concept in your original plan.
When it comes to spiralling and tracking, we find assessing by learning goal can be really helpful to keep you and students/parents informed of where they are in the journey with less stress.
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This is a really boring example, but in a recent coordinate geometry unit, I had a plan to do a
First lesson: problem solving/ reasoning lesson with a variety of problems involving lines, circles, and quadrilaterals on the coordinate plane
Next lesson: partitioning line segments in ratios, which is one of the trickier skills in the unit, and the last new skill the students learn.
From some individual formative work during the first lesson, it was clear that the majority still needed some practice on basic skills like distance, slope, etc, so going to the second lesson would probably have been a disaster.
Pivot lesson: students who needed it worked on practice problems from DeltaMath or Khan Academy, with my support or in groups. Students who didn’t worked on a Desmos activity to apply some of the skills they new to unusual situations. The entire class then moved on to partitioning together. It was a needed break from the rushed pace, and got us all back on track together.
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I recently had a “planned” pivot to my lesson plan, not exactly what was taught in this lesson.:) It helps me see the value of pivoting to ensure students learn core concepts before moving on. I taught this curriculum for a couple years and I felt that the curriculum was weak in explaining the connections between linear models. Because of the things I’ve learned in this class I remember an activity from a different curriculum that was very good at making connections between growth and beginning points. I inserted the lesson this week and I felt like it did such a good job at helping students make those connections.
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Nicely thought out! Planned pivoting is just another example of helping our students with what we KNOW they need.
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I just taught a lesson on area of a circle where I definitely had to pivot. I hadn’t taught this standard before and it was way harder for the kids than anticipated. There were misconceptions from previous years that I had to spend time on that I wasn’t expecting. Pivoting made all the difference!
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I was teaching credit recovery first quarter and was to teach Rational Exponents then Exponential growth and Decay. To some degree this works, but it can also be very confusing with the square roots when converting expressions with rational exponents to square roots. Perhaps I am missing something here, but this was my experience.
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This year I have classes with big gaps in basic skills and understanding of relationship between multiplication and division. I hadn’t encountered this before and it really threw off my ability to anticipate students responses and where they would have difficulty. So I’ve had to pivot. Rather than doing big tasks to fill in the gaps, we’ve done some number talks whose discussion have helped student comprehension.
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Number talks are great for this! As a heads up, most day 2 and day 4 lessons in our units have number talks so maybe consider viewing some of our earlier units to pull those specific number talks to help. They are very visual which is also super helpful!
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This school year, students were learning integer operations (to be applied to rational number later). While subtracting negative numbers, I could tell many students were not able to make sense of the visual math we were doing with positive/negative counters. I pivoted (perhaps too soon!) encouraging students to use number lines to make sense of subtracting negative values. It was clear then that many students weren’t able to make sense of the concept still. I wish now I had pivoted once more and/or gave student more time with that concept before plowing (proceeding) forward. I also think it would have been more valuable to avoid encouraging students to approach the problem in a certain way, and rather give them a chance to make sense of it more on their own. I think this would have proven far more valuable for my students.
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The key is reflecting and realizing what might be better in the future, so good on you for that! There is always an option to pivot to a different concept or skill and come back after some time to see where students are at then… sometimes you feel like you’re just spinning wheels. Sometimes time can help!
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My two areas were exponents and then moving to polynomials. After seeing the dismal lack of success working with polynomials, it became clear that while the students had given correct answers on the exponents unit (even enough to pass the end of unit test), they really had no conceptual understanding of exponents. But not only were they lacking in understanding of exponents, there was a huge disconnect when it came to working with polynomials. Looking back, it would have been good to pivot to like terms and helping them have a deeper understanding of that concept (which I did not realize at the time that they did not have). I definitely have to get better at knowing what the students actually understand and what they manage to act like they understand when they do not.
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A few questions:
I do want my students to be confident interactive problem solvers. I am wondering do I set aside the text/worksheets/etc? Do I look at concepts and spend 5 days working through a 3 action task or a problem solving activity (such as tasks you have on the website)? Or do use these tasks to introduce concepts and then give work from text? Unfortunately state assessments are not fun tasks and the problems look more like a textbook question.
How would I structure working through the units of the textbook using resources that have been shown in this course?
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I talked about the pivot of starting with the abstract…which I won’t do anymore…instead starting with context problems to ground the students in what the pieces of a linear equation mean.
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I planned to move from angle relationships formed when parallel lines were cut by a transversal to (interior) angle sum theorem. I was following our curriculum, and thought that the guided notes would help me let all students keep up/catch up. Time was tight. I think using the Sliding shelves lesson first or to pivot, would have been really helpful.
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I reflected on two lessons involving fractions. I teach special education math, so all of my students are behind grade level. I do try, however, to cover as many grade-level concepts as I can. Before teaching anything involving fractions, I gave a pretest to see what my kids knew. Over half of them were not able to show 1/4 on a picture of a circle cut into fourths. That means we spent a lot of time reviewing fraction basics before doing any operations with fractions. I should have known they would continue to need this level of support before jumping into algorithms!
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Always good to get a sense of where they are ahead of time. Do you have any units of study you plan on leveraging to help them deepen this understanding ?
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I definitely plan on using more manipulatives and other hands-on materials in my classroom this year. (That’s what I focused on in the organizer I attached above.) Could you explain what you mean by units of study? Are you suggesting to add in additional units to my course outline to help deepen understanding? (Just want to make sure I understand.)
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I am using the unit we did last year for trigonometry.We felt very rushed, as it was the end of the school year, but there were lots of opportunity for pivoting to ensure that students had better understanding.
For specifics, I chose our original path of going from the unit circle (students using triangles to come up with the 5 points on quadrant one) to the introduction of the parent function graphs for sine and cosine. Later on, students really struggled with evaluating simple sine and cosine expressions. I think it as because of the lack of true understanding between the connection of the unit circle and the sine and cosine waves.
I think a great pivot would’ve been to spend time using the unit circle and mapping it out on an x and y axis over time to see how it naturally makes the sine and cosine waves. There are great activities that are already set up to do this. Like one I really loved that Laura (my colleague and mentor) did with pasta.
Memorizing a parent graph worked great for tests, to connect it to all the other parent graphs, but I do not think this was effective to make connections within the unit.
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I taught factoring with Algebra tiles. I showed my daughter (on video) trying to make a rectangle with specific area but she was struggling. Stopped video for students to notice and wonder. They picked up that the square was 100, the strip was 10 and the small square was 1. Then we switched it to x^2, x and 1. My one lesson was too much for this group of students. I should have pivoted to just work with numbers the first lesson making base times height = to area. Then the next day go into algebra representation. So many kids do not have the number sense to then cram down their throats Algebra.
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This exercise was interesting as I now know about the curiosity path and have to dramatically improve upon my lessons in general to increase student engagement. I definitely would add more hands on puzzles with fractions to both spark curiosity, but also to deepen understanding of unit fractions instead of just continuing on the path and not addressing how students had difficulties decomposing fractions into units.
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This past school year I found myself really struggling to have student conceptualize multiplying fractions. I will fully admit already spending so much time on Fractions I moved on, and quickly used one slide that showed of = x. I see now though that taking a Pivot stance would have actually been more beneficial in the long run, where if I would have went back to even a simple task of Reeses Peanut Butter Cups, the idea of multiplication meaning an amount of another amount would have stuck. This would have further helped out as I proceeded to division of fractions where I unfortunately still had so many students lost.
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Given the craziness of the past few years in the classroom, I had presumed my students had worked with rates, unit rates, and proportions in a basic manner in the previous grade as they generally have done. However, I found out very quickly in my first lesson on proportional relationships that students did not know what a proportion even was. We did a pivot to review what ratios and rates were then discussed the concept of the unit rate which led into a conversation about equivalent ratios. This did take some time but in the end it made the subsequent conversations about proportional relationships that much richer. I wish that I had been in more of a “spark curiosity” type of mode in these past few years as I think it would have assisted greatly with the disengagement of students who had spent a year at home doing “remote” learning. So many of those students told me that they did nothing while at home.
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I thought that identifying terms in an algebraic expression would go smoothly after students had had lessons on that in the previous grade. I then thought that combining like terms after it would be even smoother. BUT, I found that students needed more practice with the kinds of terms and what they even mean (what does a constant mean? or why does the coefficient and the variable change value?). They also needed more work with properties, like the associative and commutative property. They were comfortable with 3 + 5 = 5 + 3, but not x + 5 = 5 + x and especially not comfortable with moving things around when there were terms being aded in parenthesis. This year, I am thinking I will need to take some time to delve into these properties a little more. I have been bringing them up as we add and subtract integers and rationals, and look at student strategies, but I think this year it will be worth giving those strategies a reason (the property) for why they work.
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This reply was modified 4 months, 2 weeks ago by Victoria Murphy.
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This reply was modified 4 months, 2 weeks ago by
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When I teach solving complex equations, we go on to teach Special Solutions (no solution and All Real Numbers). One solution students have a hard time with is when the answer is zero. Many students look at this as no solution. I would like students to better understand the WHY an equation is No Solution, All Real Numbers and Zero. We look at the structure of the equation once it is simplified to see what is the same and different about each equation once it is simplified on each side but this does not seem like enough for them to understand. Students still get tripped up. One way I could pivot on these lessons is to have some word problems that would work out to be no solution, all real and zero so students can understand the logical reason why a problem would have one of these answers. For example if there are two gyms and they both have the same monthly fee but different joining prices, there will never be a time that they will cost the same, therefore being a no solution.
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I recently progressed through an intro trig unit where we spent time unpacking pythagorus and then working with the theorem. I noticed that students were ok with the visual representatios but really struggled with the algebraic notation and sequencing the steps for solving the equations.
I then made the mistake of moving into the tan ratio with what I thought was a great hook of building some physical right angle triangles with string all over the classroom (inspired by Jon’s task… I think it’s called connecting corners or something). Once again the issue was not the visual, but the algebraic representation and the steps to solving to a solution.
Anyway… I’ve now decided to pivot and circle back to reviewing two step algebraic equations. I’m hoping that this fundamental concept is the piece that this holding them back. We’ll do one of the tasks from MMM or perhaps some of the visual options from Kyle’s mathisvisual.com.
As usual… thank you for your inspiration… and the reminder that what we’re really trying to do is to build creative, flexible and resilient problem solvers… not check boxes on a curriculum.
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Sounds like an interleaving / spiralling strategy which is great.
Your question about how much to do before moving on is key… I think it varies by topic and by group of students. You want enough there to make the experience a moment they can go back to through context, but not too drawn out where it just seems like “one moment” when you’ve squished 250 moments together! Hard thinking!
