Math Relearning/EngageNY/GPK/Module 1: Counting to 5

These are my comments on EngageNY's Prekindergarten Module 1.

I'm expecting to go through pre-K in a hurry, but I'll take some time to think through things more carefully at the beginning to make sure I don't neglect lines of thought I'd regret missing.

Overview
To help me wrap my mind around all the prose, I'll list the central points I gather from the material for each module, mainly from the module overview. I'll also list concepts from recent modules that are being applied in the current one. These lists will help me get a sense for the progression of concepts in the curriculum. In fact, to save time, for some modules the lists might be the extent of my comments, since grasping the progression is one of my few major goals. The standards mainly give skills and sometimes include concepts to understand, but in these lists I want to focus on the concepts. I don't want to take the time to make the statements rigidly formal at this point, but they'll be a bit more formal than descriptions in the curriculum. I'll call these statements propositional concepts to contrast them with nominal concepts, would would be nouns, such as attribute or counting. You'd find nominal concepts in a concept map or ontology. Speaking of ontologies, I've found a few for math I want to examine.

New propositional concepts from this module:


 * Objects have attributes.
 * Objects can be compared according to the values of their attributes.
 * Objects can be matched by their attributes.
 * Objects can match at different degrees of similarity: exactly the same or the same in some attributes but different in others.
 * Objects can be grouped together based on their matching attributes.
 * The quantity of items in a group can be identified by a number.
 * The quantity of items in a group can be discovered by counting.
 * Number conservation: The quantity of items in a group remains the same when their spatial arrangement changes.
 * Two groups can have the same quantity of items even if they contain different types of items.
 * Rote counting: In the standard sequence for counting, each number is one greater than the previous number and one less than the following number.
 * One-to-one-correspondence: In counting a collection, each object is paired with one number using the standard sequence.
 * Cardinality: In counting a collection, the last number paired indicates the quantity in the collection.
 * Decomposition: Numbers have smaller numbers embedded in them.
 * Successively removing one from a collection is equivalent to a series of quantities that move backwards through the counting sequence.

The topics in the number core (rote counting, one-to-one correspondence, etc.) match the kinds of issues raised for this age group in my other sources (Chapin and Johnson, etc.).

It's interesting that even though counting is the first major goal, the learning actually begins with pattern recognition in terms of comparing and grouping objects by their attributes. This does seem like a fundamental skill. Recognizing attributes prepares students for counting, measurement, and geometry. I believe the immediate point for these lessons is that matching allows you to recognize a collection of objects to be counted.

It's also interesting that the curriculum doesn't follow the standards in order. It starts with #2 in Measurement and Data. But that's a feature of CC. It gives a paradigm for teaching math and a set of benchmarks for measuring student progress, but it otherwise leaves the details of implementation up to the teachers.

Learning to count on your fingers using a piano template is the best part of the whole curriculum. Next they need to develop CC standards for music that require everyone to learn the instrument. Okay, joking, half. Of course, the other reason it's important is that it gets students ready for the number line, which just shows how carefully these curriculum developers have thought through everything, which is one reason I love the CC.

Topics E and F are already preparing the kids for arithmetic.

The "1 more" pattern reminds me to include my earlier thoughts on counting somewhere, such as "the pattern embedded in the counting sequence," as they put it.

The CC standards start with Kindergarten, so the PK ones came from somewhere else. It would be nice to have a plaintext list of the PK standards, and all the rest, so I or someone could use them more flexibly, say in CSV format. I imagine that exists somewhere. Okay, after searching, I found this (PDF), this, and this (PDF).

Topic A: Matching Objects
It would be nice to analyze the standards by the concepts they contain and the relationships between those concepts as well as the kinds of tasks. It might help to translate the standards into some kind of formal language, such as a programming language.

It's tempting to copy these coherence links into a spreadsheet for diagramming purposes. I'm curious if their links are different from Jason Zimba's.

What does it mean for an object to have an attribute? What does it mean that multiple different attributes exist and than an object can have more than one?

Fluency Practice
This would take way too much time if I did it for every lesson, but if I were going to analyze the fluency practice, it would go something like this:

This exercise illustrates the abstractness of numbers by associating them with multiple objects (fingers, claps, people). The connections are somewhat implicit. The teacher doesn't say, "We're counting fingers." The students are supposed to intuit that holding up a finger means it's being associated with the number.

Concept Development
I like CC's emphasis on vocabulary. I always try to nail down terms and use them somewhat consistently. It helps keep communication clear, and it contributes to a feeling that I'm doing things right.

If I comment much on the PK lessons, it'll probably be on combinations of lessons, since each one covers so little ground. For example, I'd group the grouping lessons and discuss their connections and distinctions.

Why would I pay any attention to the math concepts and skills preschoolers have to learn? Even though adults don't normally have to think consciously about them, sometimes unusual situations come up that require some conscious consideration about these basic concepts and skills. For example, let's say you're counting drops of water for some reason, and some of them combine while you're counting. You suddenly have to think a bit about conservation: The amount of water is the same, but the number of drops has changed. What are you going to consider a drop? How will you know you've counted every drop only once?

These kinds of basic issues especially come up in programming, because you're having to think about unconscious, intuitive thought processes and spell them out in detailed, logical, repeatable steps so the computer can reliably reproduce them. This is certainly true in the field of AI. For example, if a program is counting moving objects, how will it keep track of each one to make sure it gets counted but only once?

But I'm not creating computer algorithms for all these procedures right now, so for this project I'm only making a note of the issues that occur to me as I'm reading.

Matching is easy to think about. When I was reading the Progressions, I found that I didn't get out of elementary school before starting to strain my brain, at least with mental math involving word problems. It'd be good to pay attention to the point at which math becomes an unnatural way of thinking for me.

General questions I would ask for each lesson if I were analyzing everything:


 * What is the meaning or nature of the task of the lesson or topic? For matching I'd talk about things like the nature of objects having the same attribute and the workings of human perception and categorization.
 * What is involved in the objective possibility of the task? What's involved in the human activity? In this case, what is it about objects that allows them to be grouped? What capabilities and actions do humans need in order to do the grouping?
 * How does this lesson relate to the lessons it prepares for?
 * Why are mathematicians interested in this concept, and why do we select it for teaching? This is a question that came up a few times during the Progressions, and it reminds me I want to reread my Progressions notes and list my other recurring questions.

Analyzing things to death is fun, but for this project the important question is, what am I looking for from the lessons in this curriculum that would help me achieve my goals? My tentative answer is that I want to know all the math concepts and skills I need, and I want mental aids for understanding them deeply enough to apply them to problems flexibly. An added bonus would be to contribute to people's thinking about math as a result of my ruminations while learning. So generally I'm listing concepts, skills, and mental aids. Even if I don't analyze them now, the lists will make it easier to think about them later.

Student Debrief
Math misconceptions help me think about the concepts. I run across discussions of them here and there, some of them more organized and complete than others. I'll probably link to some of the better ones somewhere in this project at some point, probably on the intro page. It's too bad they don't talk about specific mistakes (so far) in the curriculum, since they have the teachers listen for them to correct them.

Lesson 3: Match 2 objects that are the same, but…
Lesson 3 is the same as Lesson 2 but uses different words.

Lesson 4: Match 2 objects that are used together.
Interesting. This Lesson expands the idea of association beyond visual attributes.

Topic D: Matching 1 Numeral with up to 3 Objects
At about this point I settled on a rhythm for reading these early-grade lessons. I'm reading the module and topic overviews, the note paragraphs from the lessons, and the questions in the student debriefs. I'm also skimming the assessments.

End-of-Module Assessment
It's interesting that even at this point in the curriculum the students are solving for unknowns, though the material doesn't call it that. That's one reason I'm reading the student debriefs. The questions there approach the lesson's concepts from different angles to make sure the students can think flexibly about them. For example, you could state the idea of the Topic A, Lesson 1 as "(1) Objects A and B (2) are (3) exactly the same (4) if they are the same shape, size, and color." The student debrief asks the students to solve for each part of that statement as an unknown: (1) specific objects that match ("Do you see any things in our classroom that match?"), (2) whether two objects match ("Are these 2 students exactly the same?"), (3) the vocabulary of matching ("These counters are _________."), and (4) the conditions for matching ("How did you choose things that were exactly the same?").

Since matching and counting are things I already know how to do, there isn't much to learn, so it's hard not to see this material as a big wall of text. It's helpful to have a framework to fit the content into so it means something to me. There's something of a framework in the Module 1 overview (the number core: rote counting, one-to-one correspondence, cardinality, and written numerals), but it's a little clearer in the Counting and Cardinality Progression, and it's even clearer in Hatfield et al, or at least spelled out in more detail. Counting is a surprisingly complicated activity. You have to know the sequence of counting numbers and the pattern that each number is one more than the previous one. You have to match each number with each object you're counting without missing or double-counting any objects, and you have to know that the last number you say is the quantity of objects in the collection. Then if you're going to write the number down, you have to know the numeral that corresponds to that number name. And if it's over 9, you have to know the base 10 system. Kids aren't born knowing any of this, and it takes lots of careful instruction and practice to learn.