Is a swing and a miss better than no swing at all? I used to coach my learners to step up to the plate, just step up. You can do it. But now I realize that just stepping up is not what it takes to learn. It is a great first step, but it is just a first step. Will you ever hit it out of the park if you don’t swing?
Today I got to co-teach Algebra I with D. Dietrich with her learners. It was day two of Stopping Distances. After reviewing what we accomplished during the last class, we stopped to ask them to write a complete sentence about what the graph of reaction distance vs. velocity represented. 
When I asked for volunteers, no one responded. Yikes! Now, they all had written something. Finally, one sweet, brave learner volunteered. It was a sentence that needed some coaching. We talked about the structure of the sentence as well as the need to be more specific about units. Since the distance is in feet, what are the units of velocity?
We collected this sentence with our TI-Nspire Navigator system. The screenshot above allowed the entire class to have multiple representations of the sentence. They could see it as it was being read. The screenshot is evidence of a swing.
I asked for another volunteer… … … … So, I finally asked the one child that was making eye-contact with me why he didn’t want to read. “Because I know it is not perfect.” Isn’t this what learning should be about? Can’t we have a culture where it is okay to not be perfect? How will I ever learn if I am not brave enough to share my thoughts with others? Being in school should be about drafting and making revisions, shouldn’t it? Have we built a culture where we are afraid to try because the only correction I get is criticism? (double-Yikes!!)
We edited and coached and then tried again for the second graph, braking distance traveled vs. velocity.
We got, without question, the best sentences of the day for the second writing. The power of feedback and revision was demonstrated. Unknown to me, the learner that volunteered for the second writing rarely speaks in class and never volunteers. Isn’t that great? She found the confidence to write and share.
But, as the facilitator of the learning, do I know how many of these learners actually took a swing? Not really. I love that one student wrote his sentence on the screen of his calculator. But it is the only one I could monitor.
Which makes me wonder…how many times are my learners telling me they understand when they have not actually taken a swing? Are they so conditioned to tell us that they understand that they are not testing their understanding and seeking feedback? WOW! That is a problem!
We stopped and talked about trust. We talked about risking being wrong to grow and learn. But, again, were they just agreeing with us because they know it is what we want to hear? Would they take action…to swing and possibly miss? Would we celebrate the swings and give quality, safe feedback for a miss? Risky for everyone, right? Trust is the key.
Yesterday we wrote the equations for both of the graphs above. y = 1.1x and y = 0.05x², where x is velocity in miles per hour and y is the distance traveled in feet.
New learning for today…How do we find the model, the equation for the data, of total distance traveled in feet based on velocity in miles per hour. Important questions must be asked.
- How will we calcuate the total distance traveled based on the given data?
- What pattern does this data follow?
- Is the pattern exponential or quadratic?Okay, so let me stop here and say that the question exponential or quadratic is an important question. If you can’t answer it, how will you know what type of model to write? This is Algebra I; we are not using regressions. We are hand fitting data.Aren’t the results interesting? Q for quadratic, E for exponential.
Before you boo, a bunch of learners came without their technology or had login issues.
But notice, even when anonymous, one learner did not take a swing. It was a 50-50 shot at being right and this learner would not take a swing.
New and MUCH more interesting questions are now possible.
- Is the majority always right?
- Can we listen to an opposing view and try to understand their reasoning?
(Shouldn’t Democrats and Republicans try this occasionally?) - Can each side make a reasonable argument for why they made their choice?
- Are you willing to consider that the other side might be right?
The minority view, quadratic functions, explained first. Then a member of the majority party raised her hand and said “I voted exponential, but I can now give another reason why it is, in fact, quadratic. Is that okay?” WOW! We stopped and voted again.
Progress! But, what do we do about the 2 remaining e’s? Remember, the majority is not always right. More discussion ensued. The two e’s identified themselves, AB and SS, and explained why they were now in the quadratic function camp. Also notice, they everyone took a swing this time around.
Let me stop here and say that this is where and why I prefer using my TI-Nspire Navigator over Poll Everywhere and the SMARTBoard clickers. While it appears to be anonymous, I can click on Poll Details and see which learners need intervention if they don’t self-identify during the lesson. I can also see what is on every learner’s graph which I will show if you keep reading.
Now that we established that we are pretty sure the function to write is quadratic, we again returned to the data.
How was the data in the total column computed? “Everyone knows, Ms. Gough, that you just add the reaction distance to the braking distance, duh!” Eyes rolling as only 14 year-olds can do.
So, if that is true, write the equation that fits the total distance traveled in feet based on the velocity in miles per hour. You have all that you need. You can do it. Take a swing.
No one got it right the first time. Some took a swing and missed; some got close; some just stood at the plate. At least two students were still in the dugout – maybe the parking lot; Caswell and St. Cloud were absent the day before. They did not ask for the file. St. Cloud was working to catch up; Caswell was just observing. And we absolutely would not have known if we didn’t see their calculator screens.
Vuckovic looks like he took a swing, but no. He connected his scatter plot; he had not attempted to construct a function. Gibson has tested a function that is close, but does not fit very well; he took a great swing which we celebrated.
So we asked again, how was the data in the total column calculated? Can you use this information to write an equation that will fit the data?
Before class was over, everyone took a swing and got a hit. It was very collaborative. We then took the model and began to interpolate and extrapolate. If I was in a car going 65 miles per hour, how much distance would I need to not crash? Easy, but we needed to talk about how to document our work and thinking so that future physics teacher would know how we arrived at our answer WITH units.
If I am a stunt car driver and know that I have a football field’s length to stop my car, what should my top speed be when I hit the brakes? This question was not so easy. Learners fell in to three camps.
- Camp 1: I substitute 100 in for y. They got feedback quickly from their peers that 100 was in yards and the units were feet!
- Camp 2: I substitute 300 in for x because that is where I substituted 65 before. They also got quick peer feedback that x was in mile per hour and to pay attention to your units!!
- Camp 3: I substitute 300 in for y, but what do I do now? Unfortunately, in this class, they got no peer feedback. We facilitators were informed to say the least! Not one child connected this to using the quadratic formula to solve. (triple YIKES!) Good formative assessment that they are not connecting the skill to the application.
Once they realized that they could use the quadratic formula to solve, they were successful. We could monitor the swings, and lack of swings, using our Navigator. By now, everyone was swinging because they knew we were supporting their effort and learning.
The big take-away for me: Formative assessment that offers feedback and support in the moments of learning are critical for success and confidence.
The big take-away for my learners: You can do this; we will help!
You will not hit it out of the ballpark if you do not swing. A swing and a miss is so much better than no swing. Step up to the plate; dare to swing. Miss; swing again. You can do this; we will help!
Experiments in Learning by Doing 
Great lesson. Graph interpretation is not easy for middle schoolers. I thought you did a fine job trying to get everyone to swing, and to use that as feedback for both you and your students. Discussion and revoting is awesome. And your openess and honesty (the “Yikes!” moments) are to be commended, too. Breaking down the fear of “not being right” is so tough to do, especially at that age. Your students are so lucky to have you!
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Thanks, Frank. I think I was successful with the “take a swing” lesson. I wonder if I have helped them learn more about graph interpretation and function composition. I love the lesson because it is simple data that they can interpret that has a context. I’ll be interested in what @fencersz thinks if/when she comments since she is a JH learner in the class. I appreciate your feedback. I think almost all that I have been doing is pseudoteaching. I’m inspired by the work that you and @occam98 are doing to have teachers reflect on their practices. Thanks for prompting my thinking.
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Jill,
I am absolutely blown away by what I have been reading in your blog. You have found a way to embrace all I have been reading about here in Kentucky and take it from theory into practice. What are you doing in May? I’d really like to come be a fly on your wall.
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Hi Gloria…You are welcome at Westminster at any time. Come spend the week. You can co-teach with me if you’d like. My learners are fun, challenging, and active. Thanks for the taking it from theory to practice. I have been working on it; I need the ground-level view of all of that great research.
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“A swing and a miss is so much better than no swing.” Great post. Taking that swing is vital, that’s for sure, and it’s imperative that students see taking that swing (and missing on occasion) is part of learning. It’s a hard process for teachers to get students outside of their comfort zone, because it’s only then that learning can take place.
This is also a fabulous example of how formative assessments should be used in the classroom; too many teachers take in the data from the formative assessments, but don’t do anything to correct student learning or to change their instruction.
I am inspired. Thanks for this!
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Hi Terie…Thanks for the feedback. I want to build a learning community with my students that encourages risk taking. If they can’t or won’t tell me what causes them to struggle, I can’t intervene. If I don’t know they need enrichment, they could become bored. I hope that I am providing feedback rather than judgement. it is a fine line to walk, particularly when the feedback is delivered in public.
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Jill,
I too will chime in and say that I think this is a great lesson. I also see the confusion between exponential and quadratic all the time, and I wonder why it is necessary to introduce students to exponential functions at such an early age. Exponential growth is an exciting topic, and there are great lessons that I’ve seen math teachers use to teach it, but I wonder what makes it an essential learning for 8th grade students first starting in algebra, and really struggling with the idea of exponents in general. Might students see more success if they simply didn’t have the work exponential to confuse themselves, and were simply trying to distinguish whether this graph is linear, quadratic or cubic?
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Good questions, John. I believe we teach exponential functions because of the patterning and the strong connection to arithmetic and geometric sequences. A young learner can pick out the difference between adding a constant and multiplying by a constant.
Linear (1, 3) (2, 6) (3, 9) (4, 12)
Exp (1, 3) (2, 6) (3, 12) (4, 24)
Using simple exponential functions is one of the easiest ways to show an increasing rate of change. Graphing the above two sets on a graph dramatically illustrates the difference between a constant and non-constant slope. So I suppose it is about numeracy too. Using the patterning will also help us find the y-intercept.
I know that you know the above, but I wanted to illustrate how easy it is to find the pattern. Algebra I should be about pattern finding rather than polynomial function expertise. We don’t teach cubic functions in 8th grade. Do you use cubic functions in physics?
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I don’t use cubics in physics (linear and quadratic functions are the bread and butter of most of what we do), and I see your pattern finding point, but I still find so much confusion with my students in distinguishing exponential vs quadratic functions to be so difficult that even though they can recognize patterns, I don’t think they have the skills to distinguish the graphs when they see them. Maybe this is a skill we could directly target somehow—I’m not sure.
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Hi John…Good thinking and questions here. Thanks! I appreciate the gentle push back (in the best sense). I do think that identifying linear, quadratic, and exponential patterns is a new universal essential learning for my team. I also don’t know what the Honors Algebra I team considers to be essential. We have taken a stronger approach to pattern recognition rather than memorization. We hope that you will see improvement in our current learners when the arrive in your care. We need this level of conversation between our teams, disciplines, and people. We are collectively responsible for our learners; we need each other’s thinking and support. Our conversations will continue to refine the work and expectations for our teams.
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Enjoyed this post and comment exchange very much. I don’t understand why Gibson fit is not a good one, so that might keep me up thinking. Or I may just need you to tell me if I do t have access to all the info I need to discern that. Clearly, the students are intensely engaged because they are ALL actively participating and gaining confidence about swinging. Without swinging, we have no experience on which to reflect, gather feedback, and adjust our muscle for better trajectory and future chance. Fantastic! Was the assessment formative? How did you and DD adjust your path? What were you planning to do that you decided to change course on?
Also loved post on the bridge to summer camp. Hope you have lots of takers and we do more of this type of learning.
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Hi Bo…
I think it was continuous formative assessment because the learners directed the path based on the results of the quick polls. We did adjust our approach several times during the lesson. The writing prompt at first where we worked on peer editing to improve their sentences was adjusted because they needed more practice. The discussion about exponential or quadratic was a lengthy period where the learners brainstormed and discussed to reach consensus. We learned that this particular set of learners did not automatically know to use the quadratic formula to find the speed of the car if the distance was 300 feet. We did not introduce the ball bounce data because they were not ready to move forward. We decided to back up and practice where these learners need reinforcement.
Gibson’s model is not bad; it looks like a better fit in the screen shot than it does live on the graph and in the spreadsheet. I’ll let you think about the mathematical model a little while longer. ER from my 7th period discovered it independent of direct instruction. He did need the question: how is the total distanced traveled in feet by the car found?
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[…] We are studying quadratic functions. We started with the Stopping Distances data to look at quadratic data visually. Our hypothesis was that the distance required to stop while braking is proportional to the square of speed, d=k·v². Many of our learners had trouble fitting the curve; they were hesitant to take a swing. […]
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“Do not let the fear of striking out ever get in the way of hitting a home-run.”–Babe Ruth
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THANKS Chad! I LOVE this quote. I am a huge Babe Ruth fan.
I need a poster of this quote for my classroom.
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[…] Using the TI-Nspire Navigator system, we polled our participants to assess their disposition on formative assessment which caused us to change our plan on the fly. […]
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[…] safe for our learners to step up to the plate and ask their questions? We want them think “swing even if you miss.” The human brain is far too complex an organ to determine that x can’t be taught. ~ Frank […]
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Sorry I missed this one when you originally posted. I’d love to know what arguments the quadratic camp used to convince the exponential camp. What did the exponential folks say & why weren’t they able to convince any quadratics over to the “dark side”? Given the very little that most Algebra I learners understand about physics, I expected the roughly 50-50 split in the initial voting, but I’m really interested in knowing how the conversation went. Was it really as one-sided as the post seems to suggest? (Thanks so much for all of your sharing, Jill!)
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Gosh, Chris. I’m not sure that I remember much. It was DD’s class. We co-facilitated the lesson. I’m sure that we asked leading questions to guide the discussion, I do remember being shocked that even with anonymous, the kids seemed hesitant to answer if they did not know they were right. We need more culture-hacking in our classroom. It is not about being right or perfect, it is about persistence and perseverance. (But you already know that.)
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[…] Swing even if you miss; stepping up to the plate is not enough […]
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