Latest / Rounding Up / Exploring the Seeds of Algebraic Thinking with Janet Walkoe & Margaret Walton

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Exploring the Seeds of Algebraic Thinking with Janet Walkoe & Margaret Walton · Rounding Up

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  Algebraic thinking is defined as the ability to use symbols, variables,
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  and mathematical operations to represent and solve problems.
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  This type of reasoning is crucial for a range of disciplines. In this episode,
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  we're talking with Janet Walko and Margaret Walton about the seeds of algebraic
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  thinking found in our students' lived experiences and the ways we can draw on
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  them to support student learning.
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  Hello, Janet and Margaret. Thank you so much for joining us.
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  I'm really excited to talk with you both about the seeds of algebraic thinking.
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  Thanks for having us. We're excited to be here. Yeah, thanks so much.
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  So for listeners without prayer knowledge,
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  I'm wondering how you would describe the seeds of algebraic thinking.
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  Okay. For a little context, more than a decade ago,
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  my good friend and colleague, Mari Levine, she's at Western Michigan University.
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  She and I used to talk about all of the algebraic thinking we saw our children
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  doing when they were toddlers. This is maybe 10 or more years ago,
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  in their play and just watching them act in the world.
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  And we started keeping a list of these things we saw. And it grew and grew.
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  And finally we decided to write about this in our 2020 FLM article that
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  introduced the seeds of algebraic thinking idea. Since they were still toddlers,
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  they weren't actually expressing full algebraic conceptions,
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  but they were displaying bits of algebraic thinking that we called seeds.
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  And so this idea, these small conceptual resources,
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  grows out of the knowledge and pieces perspective on learning that came out of
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  Berkeley in the 90s led by Andy Desesa.
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  And generally that's the perspective that knowledge is made up of small
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  cognitive bits rather than larger concepts.
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  So if we're thinking of addition, rather than thinking of it as leveled,
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  maybe at the first level there's knowing how to count and add two groups of
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  numbers.
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  And then maybe at another level we add two negative numbers and then at another
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  level we could add positives and negatives.
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  So that might be a stage based way of thinking about it. And instead,
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  if we think about this in terms of little bits of resources that students bring,
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  like the idea of combining bunches of things,
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  the idea of like entities or non-like entities, opposites,
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  positives negatives, the idea of opposites canceling,
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  all those kinds of things and other such resources to think about addition.
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  It's that perspective that we're going with.
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  And it's not like we master one level and move on to the next.
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  It's more that these pieces are here available to us.
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  We come to a situation with these resources and call upon them and connect them
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  as it comes up in the context.
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  I think that feels really intuitive,
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  particularly for anyone who's taught young children.
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  That really brings me back to the days when I was teaching kindergartners and
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  first graders. I want to ask you about something else.
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  You all mentioned several things like this notion of do undo
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  or closing in or the idea of in betweenness while we were
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  preparing for this interview.
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  And I'm wondering if you could describe what these things mean in some detail
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  for our audience and then maybe connect them back with this notion of the seeds
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  of algebraic thinking.
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  Yeah, sure.
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  So we would say that these are different seeds of algebraic thinking
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  that kids might activate as they learn math and then also learn more formal
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  algebra. So that first seed,
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  the doing and undoing that you mentioned is really completing some sort of
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  action or process and then reversing it.
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  So an example might be when a toddler stacks blocks or
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  cups.
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  I have lots of nieces and nephews or friends' kids who I've seen do this often,
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  all the time really, when they'll maybe make towers of blocks,
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  stack them up one by one, and then sort of unstack them, right?
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  So later this experience might apply to learning about functions,
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  for example, as students plug in values as inputs,
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  that's kind of the doing part,
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  but also solve functions at certain outputs to find the input.
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  So that's kind of one example there.
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  And then you also talked about closing in and in betweenness,
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  which might both be related to intervals.
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  So closing in is a seed where it's sort of related to getting
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  closer and closer to desired value.
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  And then in former algebra and maybe math leading up to former algebra,
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  the seed might be activated when students work with inequalities maybe or
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  maybe ordering fractions.
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  And then the last seed that you mentioned there in between this is the idea
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  of being between two things. For example,
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  kids might have experiences with the story of Goldilocks and the three bears and
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  the porridge being too hot, too cold or just right.
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  So that just right is in between.
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  So these seats might relate to inequalities and kind of the idea that solutions
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  of math problems might be a range of values and not just one.
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  So part of what's so exciting about this conversation is that the seeds of
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  algebraic thinking really can emerge from children's lived experience,
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  meaning kids are coming with informal prior knowledge that we can access.
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  And I'm wondering if you can describe some examples of children's play or
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  even everyday tasks that cultivate these seeds of algebraic thinking.
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  Yeah, that's great.
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  So when I think back to the early days when we were thinking about these ideas,
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  one example stands out in my head,
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  I was going to the grocery store with my daughter who was about three at the
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  time, and she just did not like the grocery store at all.
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  And when we were in the car, I told her, "Oh, don't worry.
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  We're just going in for a short bit of time,
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  just a second." And she sat in the back and said, "Oh,
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  like the capital letter A.
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  " And I remember being blown away thinking about all that came together for her
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  to think about that image, just the relationship between time and distance,
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  the amount of time,
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  highlighting the instantaneous nature of the time we'd actually be in the
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  store, all kinds of things.
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  And I think in terms of play examples, there are so many.
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  When she was little, she was gifted a play doctor kit.
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  So it was a plastic kit that had a stethoscope and a blood pressure monitor,
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  all these old school tools, and she would play doctor with her stuffed animals.
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  And she knew that any one of her stuffed animals could be the patient,
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  but it probably wouldn't be a cup.
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  So she had this idea that these could be candidates for patients
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  and it was this, but only certain things.
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  We refer to this concept as replacement,
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  and it's this idea that you can replace whatever this blank
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  box is with any number of things, but maybe those things are limited.
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  And maybe that idea comes into play when thinking about variables in formal
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  algebra.
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  A couple other examples just from the seeds that you
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  asked about in the previous question.
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  One might be if you're talking about closing in games like when
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  kids play things like you're getting warmer or you're getting colder when
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  they're trying to find a hidden object,
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  or you're closing in when tuning an instrument,
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  maybe like a guitar or violin. And then for in between this,
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  we talked about Goldilocks,
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  but it could be something as simple as I'm sitting in between my two parents or
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  measuring different heights and there's someone who's very tall and someone
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  who's very short, but then there are a bunch of people who also fall in between.
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  So those are some other examples.
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  You're making me wonder about some of these ideas, these concepts,
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  these habits of mind that these seeds grow into during children's elementary
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  learning experiences. Can we talk about that a bit?
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  Sure. Thank you for that question.
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  So we think of seeds as a little more general.
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  So rather than a particular seed growing into something or being destined for
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  something,
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  it's more that a seed becomes activated more in a particular context and
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  connections with other seeds get strengthened. So for example,
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  the idea of like or non-like terms with the
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  positive and negative numbers,
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  like and non-like or opposites can come up in so many different contexts and
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  that's one seed that gets evoked when thinking potentially when thinking about
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  addition. So rather than a seeds being planted and growing into things,
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  it's more like there are these seeds,
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  these resources that children collect as they act on the world and experience
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  things. And in particular contexts,
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  certain seeds are evoked and then connected. And then in other contexts,
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  as the context becomes more familiar,
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  maybe they're evoked more often and connected more strongly.
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  And then that becomes something that's kind of connected with that
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  context and that's how we see children learning as they become more expert in a
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  particular context or situation.
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  So in some ways it feels almost more like a neural network of sorts.
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  The more that these connections are activated,
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  the stronger the connection becomes.
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  Is that a better analogy than this notion of like seeds growing?
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  It's more so that there are connections that are made and deepened,
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  for lack of a better way of saying it.
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  And pruned in certain circumstances.
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  And we actually struggled a bit with the name because we thought seeds might
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  evoke this, "Here's a seed, it's this particular seed,
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  it grows into this particular concept." But then we really struggled with other
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  neurons of algebraic thinking news.
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  So we tossed around some other potential ideas
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  to kind of evoke that image a little better. But yes,
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  that's exactly how I would think about it.
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  I mean, just to digress a little bit,
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  I think it's an interesting question for you all as you're trying to describe
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  this relationship, because in some respects it does resemble seeds,
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  meaning that the beginnings of this set of ideas are coming out of
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  lived experiences that children have early in their lives,
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  and then those things are connected and deepened, or as you said, pruned.
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  So it kind of has features of like this notion of a seed,
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  but it also has features of a network that is interconnected,
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  which I suspect is probably why it's fairly hard to name that.
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  And it does have ... So if you look at, for example, the replacement seed,
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  my daughter playing doctor with her stuffed animals, the replacement seed there,
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  but you can imagine that that seed, it's domain agnostic.
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  So it can come out in grammar, for instance, the ad libs, a noun goes here,
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  and so it can be any different now. It's the same idea, different context,
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  and you can see the thread among contexts,
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  even though it's not meaning the same thing or not used in the same way
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  necessarily.
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  It strikes me that understanding the seeds of algebraic thinking is really a
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  powerful tool for educators. They could, for example,
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  use it as a lens when they're planning instruction or interpreting student
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  reasoning. Can you talk about this, Margaret and Janet?
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  Yeah, sure, definitely.
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  So we've seen that teachers who take a
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  seeds lens can be really curious about where student ideas come from.
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  So for example, when a student talks about a math solution,
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  maybe instead of judging whether the answer is right or wrong,
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  a teacher might actually be more curious about how the student came to that
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  idea. So in some of our work,
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  we've sort of seen teachers who have a SEEDS perspective can look for pieces of
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  a student answer that are productive instead of taking an entire answer as right
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  or wrong.
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  So we kind of think that SEEDS can really help educators intentionally look for
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  student assets and build off of them. And for us,
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  that's students informal and lived experiences.
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  And kind of going along with that,
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  one of the things we really emphasize in our methods courses and is emphasize in
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  teacher education in general is this idea of excavating for student ideas and
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  looking at what's good about what the student says and reframing what a student
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  says, not as a misconception,
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  but reframing it as what's positive about this idea.
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  And we think that having this mindset will help teachers do that.
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  Just knowing that these are things students bring to the situation,
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  these potentially productive resources they have, is it productive in this case?
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  Maybe. If it's not, what could make it more productive?
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  So having teachers look for these kinds of things we found is helpful in
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  classrooms.
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  I'm going to ask a question right now that I think is perhaps a little bit
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  challenging,
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  but I suspect it might be what people who are listening are wondering, which is,
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  are there any generalizable instructional moves that might support formal
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  or informal algebraic thinking that you'd like to see elementary teachers
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  integrate into their classroom practice?
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  Yeah. I mean,
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  I think honestly it's listen carefully to kids' ideas
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  with an open mind. So as you listen to what kids are saying,
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  really thinking about why they're saying what they're saying,
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  maybe where that thinking comes from and how you can leverage it in productive
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  ways.
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  So I want to go back to the analogy of seeds.
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  And I also want to think about this knowing what you said earlier about the fact
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  that some of the analogy about seeds coming early in a child's life
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  or emerging from their lived experiences,
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  that's an important part of thinking about it,
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  but there's also this notion that time and experiences allow some connections to
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  be made and to grow or to be pruned. What I'm thinking about is the gardener.
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  The challenge in education is that the gardener who is working with students in
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  the form of the teacher and they do some cultivation,
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  they might not necessarily be able to kind of see the horizon,
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  see where some of this is going, see what's happening.
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  So if we have a gardener who's cultivating or drawing on some of the seeds of
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  algebraic thinking in their early childhood students and their elementary
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  students,
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  what do you think the impact of trying to draw on the seeds or make those
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  connections can be for children and students in the long run?
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  I think a couple of important points there. And first,
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  one is early on in a child's life,
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  because experiences breed seeds or because seeds come out of
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  experiences, the more experiences children can have, the better.
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  So for example,
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  if you're in early grades and you can read a book to a child,
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  they can listen to it, but what else can they do?
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  They could maybe play with toys and act it out.
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  If there's an activity in the book,
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  maybe they could pretend or really do the activity.
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  Maybe it's baking something or maybe it's playing a game.
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  And I think this is advocated in literature on play and early childhood
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  experiences, including Montessori experiences,
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  but the more and varied experiences children can have,
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  the more seeds they'll gain in different experiences.
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  And one thing a teacher can do early on and throughout is look at connections,
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  look at, "Oh, we did this thing here.
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  Where might it come out here?" If a teacher can identify an important seed,
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  for instance, they can work to strengthen it in different contexts as well.
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  So giving children experiences and then looking for ways to strengthen key
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  ideas through experiences.
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  One of the challenges of hosting a podcast is that we've got about 20 to 25
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  minutes to discuss some really big ideas and some powerful practices.
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  And this is one of those times where I really feel that.
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  And I'm wondering if we have listeners who wanted to continue learning about the
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  ways that they can cultivate the seeds of algebraic thinking,
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  are there particular resources or bodies of research that you would recommend?
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  So from our particular lab, we have a website and it's
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  notice-lab.com,
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  and that's continuing to be built out. The project is funded by NSF,
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  and we're continuing to add resources. We have links to articles,
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  we have links to ways teachers and parents can use seeds,
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  we have links to professional development for teachers,
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  and those will keep getting built out over time. Margaret,
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  do you want to talk about the article?
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  Sure. Yeah.
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  Janet and I actually just had an article recently come out in mathematics
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  teacher learning and teaching from NCTM,
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  and it's issue five,
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  and it's called Leveraging Early Algebraic Experiences.
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  So that's definitely another place to check out. And Janet,
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  anything else you want to mention?
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  I think the website has a lot of resources as well.
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  So I've read the article and I would encourage anyone to take a look at it.
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  We'll add a link to the article and also a link to the website in the show notes
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  for people who are listening who want to check those things out.
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  I think this is probably a great place to stop,
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  but I want to thank you both so much for joining us. Janet and Margaret,
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  it's really been a pleasure talking with both of you.
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  Thank you so much, Mike. It's been a pleasure. You too.
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  Thanks so much for having us.
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  This podcast is brought to you by the Math Learning Center and the Meyer Math
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  Foundation,
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  dedicated to inspiring and enabling all individuals to discover and develop
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  their mathematical confidence and ability.