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  • Katrien's Progress Log

    Posted by Katrien Vance on May 31, 2019 at 9:25 am

    For a bunch of reasons, my colleague and I have decided to shake things up next year and combine students who would normally be in Pre-Algebra and Algebra into one group.  We each will teach a mixed group.  So it’s time to create the curriculum for this particular group having this particular experience!  Hello, summer goals!  We’ll be figuring out which are the core skills we need, in what order to introduce them, and how.  No textbook.  Hands-on discovery as much as possible.  Wheeee!

    Katrien Vance replied 2 years, 8 months ago 5 Members · 22 Replies
  • 22 Replies
  • Katrien Vance

    May 31, 2019 at 9:45 am

    Specific goals:

    First we have to finish this year and write student evals.  June 1 – 15

    Week of June 17 – Meet with Maggie to confirm plan

    Week of June 24 – Begin identifying skills — research various curricula

    Week of July 1 – Meet with Maggie to share lists of skills (units)

    Week of July 15 – Meet with Maggie- finalize road map of year

    Week of July 22 – vacation

    Week of July 29 – Continue finding materials, investigations, projects, hands-on activities for the earliest units


    • Katrien Vance

      June 19, 2019 at 9:19 pm

      Goal one finished:  year finished and evals written.  Goal two finished:  met with Maggie and created plan!  New plan: Maggie and I are mixing 2 groups who are uniquely suited to be combined and mixed.  Inspired by some ideas from Krall’s book, I thought, how fun would it be if we taught math through projects and problems we know we love.  I color-coded the NCTM 5 strands (jumping ahead to week of June 24).  Then I took a week-long math summer camp I ran a couple of years ago, listed everything I do, and then color-coded things according to those strands.  Turns out I hit every strand without even thinking about it. So, now I am looking to craft 3 more rich “units” that are not Algebra units but are collections of projects and activities and tasks connected by a thread but which allow me to hit all kinds of math strands.  The hope is to have math class look more like what mathematicians do–exploring, analyzing, problem-solving.  @jon and @kyle, my goal is to NOT preteach but instead to present situations that the kids work through, noticing patterns and finding shortcuts, which I can then codify for them.  I have questions, which I’ll pose at the water cooler.  But I’m interested in hearing if it sounds like I’m just exchanging one kind of unit for another . . .

      • Kyle Pearce

        July 2, 2019 at 11:02 pm

        I’m super intrigued.
        When you say the NCTM “5 strands” are you referring to the 5 strands of mathematical proficiency (i.e.: conceptual understanding, procedural fluency, adaptive reasoning, strategic competence, productive disposition) or are you referencing content specific strands? Super curious.

        As for your question re: trading in 1 unit for another… don’t worry too much about that. The key is ensuring that we never shut the door on any particular concept or idea. Ensure that you’re coming back somehow/some way and you’ll be golden!!

        • This reply was modified 3 years, 7 months ago by  Kyle Pearce.
  • Katrien Vance

    July 4, 2019 at 1:57 pm

    I can’t figure out how to make a new post, so I’ll keep replying to my original post.  🙂

    The 5 strands I was referring to are the content strands:  Numbers and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability.  These feel very similar to the TIPS4Math overviews I looked at.  I’m trying to call units “cycles,” just to keep that re-cycling image in my mind.  

    My colleague Maggie and I are each creating month-long units filled with all kinds of problems, tasks, etc., and then swapping classes.  So I will do each one of these, except Bootstrap, twice.

    Cycle 1 (Sept & Oct) — done twice, with 2 different classes) – “Roads, Ramps, and Rockets”- builds off of a unit from the MathScape curriculum, adding rockets to it and a question about trailer’s ramp and whether it could/should be moved, since it ends in a huge pond-like puddle when it rains.

    Cycle 2 (Nov and Dec) – “Candy Crisis” – inspired by Halloween, this takes the candy task I did last year with my class (Are there the right amount of each kind of candy in the Hershey’s minis variety 12-ounce bag?) –tons of data analysis, writing to Hershey’s, etc.  Spiraling through pentominoes, algebraic expressions and equations, and more while we wait for the data to collect on Google forms. Would like to pull in integers here.

    January – Bootstrap 2 – stand-alone coding unit that uses algebraic ideas to help kids write code for games or images; kids did Bootstrap 1 last year

    Cycle 3 (Feb & Mar) – Growing Things – all about functions, sequences, word problems — Zombies for exponential growth — Would like to fit absolute value in here — There are many problems and tasks on this topic; I just need to sift and choose.   Connection also to Humanities class – Black Plague spread

    Cycle 4 (April & May) – Breaking Numbers – inspired by John Sangiovanni’s phrase, this is about factoring, polynomials — I don’t have the problems, tasks, or projects for this yet

    Cycle 5 – Game Time – creating a “fun fair” for elementary-ages students – probablity, data collection and organization — I don’t have time for this one!  Perhaps it becomes an Elective.  I really love the idea of a math elective.<br><br>

    Keeping the door open is a great image.  For example, I plan to use the stacking paper task to talk about proportions the first time we see it but then come back to it later in the year to talk linear relationships.  

    I plan to share this quote with parents early in the year:  “We want to present a universe of math that is interconnected and logical, rather than arbitrary. ”  (Geoff Krall)  

    ALL thoughts, ideas, suggestions, task ideas, cautions, etc. welcome!

  • Katrien Vance

    August 2, 2019 at 4:36 pm

    It’s August!  I just finished Module 2 of Jon’s “Picky” mini-course, and I am so excited about Desmos.  I have:

    -mapped out my first unit (September and then repeats in October with a different group of kids) in terms of learning goals.  There are no dates on this map, which makes me a little bit nervous.  I think I have way too much.  I get to do it twice, with two sets of kids, so I’ll have a chance to revise and try again.

    -begun to plug in activities for each goal, using 3-act tasks from various sites, Desmos activities, @jon activities, and more.  Since this is a unit I’ve run as a camp before (approximately 15 hours of instruction), I have a skeleton to work from, but I am being much more intentional about the way I introduce ideas.

    I admit that I am still nervous about “covering” the curriculum.  I know that this approach and these routines will build problem-solving, curiosity, independence, and resilience in my kids.  I also need to make sure they learn 4 ways to factor a quadratic function, etc., so that they don’t have that terrible feeling of never having heard of something everyone else has heard of when they get to high school.  But wait — if the students are resilient problem-solvers who know how to think for themselves and are confident exploring and finding their own way, maybe that doesn’t matter as much as I think it does?  Thoughts?

    I should probably also think about the history, English, and music I am teaching this year.  Huh.


    • Kyle Pearce

      August 2, 2019 at 5:48 pm

      This is fantastic to read!

      Thanks for clarifying re: content strands vs. proficiencies or other strand types. While strands are helpful for us to organize our math courses and wrap our heads around content, they can lead us to silo – clearly you and your teaching partner have thought this through quite well. 

      I’m loving the big ideas you have planned. The best part is that you can “cover” little ideas by simply extending some of the contexts you’ve already planned out. Super excited for you both. 

      So, clearly you’ll both be team teaching, right? I love this. @yvette Lehman has team taught for years and I’m totally jealous. 

  • Katrien Vance

    August 3, 2019 at 9:56 pm

    Thanks for the support!  

    We’re actually teaching separately and flip-flopping groups of kids.  So we won’t be together in the classroom, but we’re thinking through the year together, seeing how our own strengths and favorite units can be combined to create a full year for the kids.

    I have been very fortunate to be able to team-teach in my Humanities classes for the past few years.  It is definitely the way to go.  As a very independent person, I didn’t think I would like it, but I have loved the way we better serve the kids by noticing different things, presenting different things, and giving our kids a well-rounded experience. 

  • Katrien Vance

    September 7, 2019 at 12:58 pm

    The school year has begun!  The rubber has met the road.  The money is where the mouth is . . . etc.

    Math class is completely different this year for me.  First, I begin with a number sense warm-up every day.  I silence all the internal worry about not having enough time (I have 45-50 minute classes), and I just do it, because it’s that important.  Lucky for us, Steve Wyborney has “20 Days of Number Sense” completely mapped out, so ready to go and easy to use, that this is painless to accomplish.  I’ve done 2 days, and it’s already so interesting.  The best part about this warm-up, which has so far focused on estimating, is that the kids get practice being WRONG over and over.  In 2 days, we have taken more “sting” out of being wrong than I’ve ever managed to before.

    After the warm-up, the kids have some kind of task.  I give them a question; they work in pairs at a whiteboard at our tables or a VNPS (Wipecharts are the bomb!); we share, discuss, and consolidate.  I “teach” only for a very few specific moments, as needed.  For example, we were measuring angles, and the kid didn’t know how to use a protractor.  I let them productively struggle for a minute or two, and then I showed them.  The gasp of excitement when they saw that the tool confirmed the answer they had gotten by other means was amazing–I said, “Never has a protractor garnered that much excitement in any math class ever!”  

    And then we’re finished.  And I assign no homework.  This is completely new for me, and I’m very interested to see how it goes.  Will they retain what we’re doing?  Can they really do it independently?  I’ll find out soon!

    Fridays are “Fraction Fridays,” where I do one of Nat Bunting’s “Fraction Talks” with them.  My colleague and I identified fractions as being one of the weakest areas for our students, year after year, so we’re making sure to give them practice thinking, writing, and talking about fractions. 

    We’ve had only 5 days of school, so it’s all very new, but it’s really exciting.  I am committed to having each and every student build confidence in math this year and become a more resilient problem-solver, knowing what to do when they don’t know what to do.

    Thanks, Jon and Kyle, for lighting the path on this journey!

  • Katrien Vance

    October 12, 2019 at 11:33 pm

    It’s October.  I’ve taught one unit and am now teaching that same unit for the second time to a new group of kids.  Lessons from the first round:

    1.  Planning the structure is not enough.  I need to plan each activity more specifically, so that I not only know what I plan to do, but so that I know more precisely how I will do it, anticipating stumbling blocks because I have done the activity myself.  I cannot emphasize that enough:  do the whole activity yourself.  Don’t just look it over and think, “I got this.”  Do it, thoroughly, so that I have everything I need and have an idea of where kids might get stopped.

    2.  I need a little homework.  I did the first 4 weeks with zero homework, and I felt as if I didn’t quite pull concepts together well enough or have enough repetition to help kids pull concepts together.  At the end, I wasn’t sure kids really got the connection between slope as an angle, ratio, and percentage.  This time, I am giving just 3-4 problems on 2 or 3 nights per week.  This weekend, kids are measuring their stairs in anticipation of slope as a ratio.

    3.  I love making the kids WANT the learning. They were doing the Desmos activity “Exploring Length with Geoboards” and finally said, “How can we measure these lengths?!”  I said, “Would you like know a way I know?”  “YES!”  Enter the Pythagorean Theorem!  They were so happy to learn it because they NEEDED it.  Robert Kaplinsky calls this making math the aspirin that relieves the kids’ headache.

    4.  The kids love Desmos activities, and I get better at facilitating these tasks with each one I do.  The “pause” button is genius.  I am still not good at collecting snapshots and getting them up on the screen, but I am getting better at getting a sample screen up there and having the kids explain their thinking. (I don’t understand how teachers of large classes manage it!)

    5.  I really like having my learning goals listed and using that format to keep track of students’ progress. I’ve never been this specific in my recording, and it is making me be very honest with myself about what we’re accomplishing.

    6.  Variety is wonderful.  We’ve been working at wipecharts, outside measuring, on paper, and on the computer.  Working in many formats gives them different experiences interpreting directions and doing different kinds of problems.

    7.  Productive struggle takes some getting used to — for all of us.

    8.  It works well to have extensions planned for every activity.There are always students who work through problems quickly, and I can sometimes use them to coach classmates, but sometimes it’s better to let them move on.  Having another activity ready, another question to ponder, keeps them learning and stretching.

    I’m excited to plan unit #2 for November/December with these lessons in mind.

    • Kyle Pearce

      October 19, 2019 at 1:56 pm

      Great reflections here, @KatrienVance!

      While there are a lot of pieces to discuss, I’m wondering about #1 and your idea to be more throrough. I would agree that this is a really important part.

      Something that myself and @Yvette (a great friend and colleague) tend to say to educators in our district is that the anticipating stage is really hard to do alone. Finding a group (or even a single colleague) to solve problems in a variety of ways can really help with broadening your perspective.

  • Katrien Vance

    November 2, 2019 at 10:42 pm

    Now I have taught the first unit twice, and I felt I did a much better job the second time.  Humbling, but good to know. You can’t always teach things twice, but you can prep so thoroughly that it feels as if you’ve taught it already.

    I prepared better and anticipated better this time.  I re-ordered things and did a better job building concepts piece by piece.  This let me make better use of the resources I had flagged, such as Desmos activities and others.  

    My colleague and I then did a 3-day review activity with both groups.  First, we gave them a list of all of the vocab, skills, procedures, and concepts we felt we had covered.  Then we did a “Math Cafe,” an activity I saw modeled at the PEN conference in Minneapolis.  Students made groups and made a “reservation” as a “party of 3” or “party of 4.”  We set up tables with tablecloths and “Party of 4” signs on them, with rulers, calculators, protractors, and graph paper where the condiments would be.  

    Students came in and sat down at their tables and looked at a printed menu of math problems, broken into Appetizers, Main Courses, and Desserts.  The problems were listed by learning goal, so, for example, Appetizer 1 was “I can use a protractor to measure an angle.”  Students chose a problem to work on as a team.  They had to do 3 or 4 appetizers before moving on to main courses and then 3 or 4 main courses before moving on to dessert (it took two days for everyone to get to dessert).  My colleague Maggie and I went from table to table taking “orders” and checking answers when parties said they were finished with a problem. We asked every person at the table to explain some portion of each answer.  More than once, the team at the table taught one of their classmates how to do something right before our eyes (always something that we thought we had taught, but, you know…).  We sent a few problems home as “leftovers” with each student, to check in on some independent skills. 

    When I asked the kids how they liked it, they said, “That was fun!”  The class period went by really quickly, and none of them realized they had done a test’s worth of math problems. The teacher who showed me this activity uses it as a scored assessment; we don’t give grades, but it was a great way to see what kids knew and didn’t know.  

  • Krystyna Sibilska

    November 3, 2019 at 8:23 am


    Your “Math Cafe” activity sounds like a lot of fun. My Skill Day is coming up, so I think I am going to try this format. Thank you for a thorough explanation of the activity.


    • Katrien Vance

      November 7, 2019 at 7:04 pm

      I’m so glad!  Let me know how it goes.

      • Katrien Vance

        November 7, 2019 at 7:09 pm

        Here are some pics to show what it looked like. 

      • Kyle Pearce

        November 7, 2019 at 10:02 pm

        This looks fantastic! Love the table cloths!

  • Katrien Vance

    December 29, 2019 at 1:46 pm

    End of 2019, and I am taking stock.  I have so far taught 2 units, one on slope (as angle, ratio, percentage), and one on linear functions.  (I am teaming with my colleague @maggie who has half the class, so the kids have had 4 units–she has taught a unit on ratios/rates and one on 2-dimensional geometry.)

    I am 100% happy with our approach in terms of bringing in more activities, allowing for more questions and more productive struggle.  I still struggle with not telling kids too much and with allowing kids to take the time they need to work their way to their own understanding of a concept.  I got away from using VNPS, but I do use whiteboards for teams at their tables a LOT.  The kids love Desmos.  Using learning goals and knowing each day or activity’s goal has helped me be a lot more specific in knowing what individual students understand.

    For the second unit, we grouped the kids into one group who tends to pick things up quickly and one group of kids who take a little more time.  It was fascinating to see the difference.  If I taught only one of those groups, I might blame or credit the “Curiosity Path” approach for their failure or success.   It took me more than three weeks to get the first group comfortable with y = mx + b in such a way that they could draw a graph from the equation, make the equation from a graph, etc.  The second group was like, ok, cool, we got it, and I had time to do some exponential functions and play with a zombie apocalypse activity.  The approach did not make the math easier for the group that finds math challenging, but I do believe that it allowed everyone to feel some success and feel like a part of the process, and that, over time, this would have a cumulative effect on students’ confidence and willingness to engage in math thinking. 

    I am very fortunate and do not have to give kids a grade or prepare them for any standardized test.  The pressure, if there is any, is simply to say what course they should take when they enter high school.  Some parents really want to make sure their 8th graders complete Algebra 1 and take Geometry in 9th grade.  Some of our kids complete Geometry and go into Alg 2 in 9th grade.  Some of our kids need Alg 1 in 9th grade–there is a wide range.  My approach this year has been inspired by this particular group, who I do not think are ready for all of Alg 1 yet, so @maggie and I are trying to lay a strong foundation of concepts and problem-solving and curiosity and ENJOYMENT of math.  Luckily for me, the parents this year are not concerned when I say that their child will probably take Alg 1 in 9th grade–that what we’re doing this year is an intro.  And I know that trying to cram all of Algebra into this year for them would only set them up for failure next year.  I know everything we’re doing is worthwhile.  But.  I also know that by not completing Alg 1 in 8th grade, they will be viewed by some as being “behind.”  I feel responsible, even as I know that they simply CAN’T complete Algebra 1 this year.  This conversation saps the joy out of teaching math for me (when I let it).

    Onward.  In January, my colleague @maggie is teaching Bootstrap, a curriculum that uses Algebraic principles to teach kids how to write the computer code for a very simple videogame.  It’s a GREAT program, and I encourage people to find out about it.  The materials are 100% free online; they have workshops to teach you how to teach it. The founder Emmanuel Schanzer came up with the idea as a way of making Algebra more concrete for kids; they LOVE creating the videogame.  While she does that with the 7th graders, I am going to use the Algebra textbook for 3-4 weeks to reinforce concepts my 8th graders need, including integers.  I plan to use a lot of ideas from Kyle and Jon’s workshop about making the textbook work for inspiring curiosity, as well as Kyle’s “Math Is Visual” resources.  We’ll see where we get, but I hope to give them a real understanding of these tools so that when they are in Algebra 1 next year, it will feel EASY and NATURAL.

    February/March will be a unit I plan to call this “Breaking Numbers.”  It will include factoring, GCF, prime factorization–all things that should be review–and go to polynomials and factoring quadratic expressions if we can get there. I have about 3 weeks for the unit (doing it with one half of the class in Feb and the other half in March).

    April/May could be a probability unit.  I feel as if this will give the kids some real hands-on practice with skills they should have mastered but have not, including fractions, percentages, equivalents.  I can also bring in dependent and independent events, compound events, etc., and get into more sophisticated aspects of probability.  Again, very hands-on.  The kids will create a set of games for younger students in the school to play (we are a PreK-8th school) and then use that data.  This can also allow for practice organizing data into graphs and charts, mean/median/mode, etc.  Great cross-age activity, too, and a chance to have younger kids see “big-kid math” as fun.

    I don’t know if this is interesting for anyone else to read, but I appreciate the opportunity to clarify my own thinking about what we’re doing and where my challenges are. 

  • Katrien Vance

    February 18, 2020 at 11:27 pm

    Update. I spent the last 6 weeks teaching “Algebra Basics”–basically, Ch. 1 in an Algebra 1 textbook. I spent about half of it using a hands-on approach, using Algebra LabGear to make integers more concrete, create and write algebraic expressions, combine like terms, and understanding parentheses. For the second half, I used the note-taking guide that complements a traditional textbook, filling in blanks, writing down definitions, and trying examples. We then did a traditional Chapter Review and then the Chapter Test. Afterward, I asked my students to write reflections on the unit and how they learn. Most of them prefer the activities to the worksheets, but they see the value of the worksheets in cementing the concepts. I do have a couple of students who prefer worksheets and who can get confused when there are multiple ways to solve a problem.

    It is clear that our new approach–math as a conversation–has made a difference in the confidence of our students. Every one of them reports liking math and feeling confident this year. They can’t always point to what is different this year, but they FEEL that something is different. They are thinking about what they understand, and they know the difference between following steps and understanding. Why is a negative times a negative a positive? They know that it is, but, because it’s one thing we didn’t model concretely, they aren’t satisfied with that. This is really exciting.


    1) What do we do when the informal conversation and hands-on approach does not lead to efficient problem-solving? (An obvious answer is that we’re not consolidating the learning clearly enough, but we’d like to think beyond that–that even when we show the group several methods, moving from less efficient to most efficient, they go back to doing the method they started with. Do we let them stay there–drawing models, for example, rather than using common denominators?)

    2) What about the students who get confused in this informal approach? I know that by using the worksheets they are memorizing, rather than understanding. But in some cases that might lead to more success for them. What is my responsibility to the student?

    3) What are other people discovering about the time it takes to allow students to discover and understand why something works the way it does? I know we all have a worry about having enough time. I have the luxury this year of not needing to “cover” Algebra 1–I’m just trying to introduce the concepts and lay the foundation.

    4) How do you give enough practice with fractions, integers, and negative fractions while using real-world contexts? I love using tasks to bring out the concepts, but then the Algebra text immediately throws fractions and integers into the mix. The kids will need to know those to succeed in high school and on standardized tests–just knowing the concepts and being able to problem solve isn’t enough.

    I get to talk to you guys, Jon and Kyle, about these very questions in a few weeks, on your podcast.

    Right now, I’m building a unit on Solving Equations, using all hands-on and task-oriented problems. I’m also going to have the students who just completed the “Algebra Basics” unit teach their younger counterparts, who did not do this unit so that they can explain some of the properties we’ll use.

    You can tell I’ve re-written the plan for this year several times, but my final unit, the Probability unit, is still how I plan to end the year.


  • Mary Manske

    February 24, 2020 at 9:58 pm

    This really resonates with me–I have a lot of the same questions. I definitely feel like my students feel more comfortable, but I’m not sure they are always getting the point. (I know that’s mostly on me). I teach students with IEPs and 2 is always a concern for me, as is 3. Looking forward to hearing you on the podcast again!

    • Katrien Vance

      February 29, 2020 at 12:52 pm

      Thank you!

      • Katrien Vance

        April 8, 2020 at 2:37 pm

        Well, now that we’re all teaching remotely, the conversation with Jon and Kyle went a different way. It also went a different way because when I switched away from teaching with a “curiosity path” lens, I realized just how dull it was! It was deadening my students’ enjoyment of math, something I have worked to build all year. It felt efficient to work through examples in the textbook, but, as you’ll hear us talk about, that efficiency seems misleading. The students can mimic the steps, but they have no ownership of the information. It would be really interesting if I had some way of objectively comparing the students I taught pre-remote learning, where I taught solving equations in a completely hands-on way, asking them to use manipulatives and tell me what they noticed, create the rules, etc., and the students I taught for 3 weeks of remote learning, which was all textbook. Same material, vastly different approaches. Who can do it more accurately more consistently? Who better understands why they are doing what they are doing? I’ll be watching them in April and May and see if I can tell. I think often it will come down to the individual student–some could fly through the textbook individually and be great. Others need the hands on. But one thing I do know is that when asked, my students will say, to a person, that they feel more confident in math this year and like math better this year. That doesn’t mean they are always doing everything right, but they are moving away from the idea that if they get something wrong it’s because they are “bad at math” and will always be “bad at math.” Math has become a conversation that they feel a part of, and that is HUGE. So I really hope the podcast doesn’t disappoint because it doesn’t answer those original questions. Issues of the moment kind of took over.

        I’m still doing Probability as my last unit–gathering all kinds of hands-on things that can work with remote learning as we speak.

      • Jon

        April 8, 2020 at 8:36 pm

        It was a great discussion @katrien

        and I’ve also wondered about doing controlled studies with students to see what effect my teaching can I have but I can’t ring myself to let a group of students not get the teaching I know is valuable for the sake of “evidence”. I’m sure this podcast episode will be well received.

  • Katrien Vance

    June 3, 2020 at 10:08 pm

    And now it’s the end of the school year. Time to assess.

    First, this was not a normal spring. We all know that. When we switched to remote learning, I was about to begin a 3-week unit on solving equations. I had already taught the unit to half of the class using Algebra Lab Gear exclusively. By playing with the blocks on workmats, students learned about expressions, evaluating and simplifying them, equations, and all the properties and rules one needs to do so. There was no day when I said, “Today we’re learning about equations with variables on both sides.” We worked with the blocks and discovered what happened when we added to both sides, etc. This group had no sense that having variables on both sides of the equation was “harder” than not because we started there–we just put blocks down and figured out what to do.

    When we moved to remote learning, I chickened out. I didn’t have enough manipulatives to give to each student–we always worked in groups of two. I didn’t have the camera set-up to demonstrate the manipulatives. I left my homemade workbooks at school and sent kids home with the textbook instead. We started with one-step equations, then two-step, then equations that included parentheses, and so on. I tried to explain the way the textbook wants them to “clear the fractions” from an equation, and that left them completely confused–one of those things that is meant to simplify but just confuses. All it convinces them of is that fractions are really hard. Argh! This felt like a very clear, logical way to teach, ticking off boxes of “concepts covered,” but it had NO LIFE. It had no connection to anything real. It was a perfect example of stuff they memorized without understanding.

    (It would be so wonderful if I had some clear comparison of these two groups and how well each of them understood and remembered the material when we reviewed at the end of the year. I don’t. We did a quick review, but students got to choose which units they took quizzes on, and many did NOT choose solving equations. Those who chose it, from either group, were kids who felt comfortable and did very well!)

    For the final unit of the year, I had a little more time to prepare and create my own Probability unit. I found that over videochat, it was really hard for information to get through and sink in, so I created a page for note-taking for each day of class. I used that page as a script as I used video, activities like rolling dice or picking cards, and real-world examples to talk about each new concept. Students who followed along and wrote down definitions as we talked did much better than those who zoned in and out. Each day also had its own activity, including the Spin to Win game on this site and, later, Spin to Win 3. Some activities had us all trying something–rolling a die–and contributing our results to a common chart. Some activities involved playing a game with family. Each one was active. I threw in a lesson about the probability of certain game show games after watching an opportune For the final activity, students designed an experiment and charted the theoretical probability, created a hypothesis, ran the experiment, recorded the results, and analyzed those results. In this way, we brought the scientific method into our math unit and had the kids use the concepts in an experiment that they created, so it meant something to them–will my cat go to the blue paper or the red paper? Do twins think more alike than regular siblings do? Can I choose my favorite color candy from the bag of candy?

    The limitations of remote learning and videochats meant that Maggie and I did not get in a full year of hands-on, “curiosity path” teaching. But in a way, being forced to change our plans and go back to “regular” teaching was the best way to show us how much we loved teaching in a more responsive, organic, curiosity-driven way, and the benefits that held for the kids in terms of their confidence, their engagement, and their understanding of the material.

    Next year, I will be teaching both Algebra and Pre-Algebra, and while I have textbooks to help structure the units that I want to teach, I know that I want to teach through tasks and discovery. I am hoping that we will be back in school and in person, and I’m already thinking about how I want to jump in to the year.