At the end of June, I had the pleasure of spending a week learning from Kathy Richardson at the Math Perspectives Leadership Institute in Hutto, Texas. I’ve been a fan of Kathy Richardson ever since my first week on the job as elementary math curriculum coordinator in Round Rock ISD. That week I sat in on a summer PD session on early numeracy led by Mary Beth Cordon, one of our district instructional coaches. She had us read a little out of Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases. I was hooked from the little I read, so I asked if I could borrow the book.
I devoured it in a couple of days.
Since then I’ve purchased multiple copies for all 34 elementary campuses, led campus and district PD sessions on the critical learning phases, and led a book study with over a hundred math interventionists. The book is so eye opening because it makes tangible and explicit just how rigorous it is for young children to grapple with and learn counting concepts that are second nature to us as adults.
I was so excited for the opportunity to learn from Kathy Richardson in person this summer, and she didn’t disappoint. If you’d like to see what I learned from the institute, check out this collection of tweets I put together. It’s a gold mine, full of nuggets of wisdom. I’ll probably be referring back to it regularly going forward.
As happy as I am for the opportunity I had to learn with her, I also left the institute in a bit of a crisis. There is a HUGE disconnect between what her experience says students are ready to learn in grades K-2 and what our state standards expect students to learn in those grades. I’ve been trying to reconcile this disconnect ever since, and I can tell it’s not going to be easy. I wanted to share about it in this blog post, and I’ll also be thinking about it and talking to folks a lot about it throughout our next school year.
So what’s the disconnect?
Here’s a (very) basic K-2 trajectory laid out by Kathy Richardson:
- Throughout the year, students learn to count increasingly larger collections of objects. Students might start the year counting collections less than 10 and end the year counting collections of 30 or more.
- Students work on learning that there are numbers within numbers. Depending on their readiness and the experiences they’re provided, they may get this insight in Kindergarten or they might not. If students don’t have this idea by the end of Kindergarten, it needs to be developed immediately in 1st grade because this is a necessary idea before students can start working on number relationships, addition, and subtraction.
- 1st Grade
- Students begin to develop an understanding of number relationships. After a year of work, Kathy Richardson says that typical 1st graders end the year internalizing numbers combinations for numbers up to around 6 or 7. For example, the number combinations for 6 are 1 & 5, 2 & 4, 3 & 3, 4 & 2, and 5 & 1. Students can solve addition and subtraction problems beyond this, but they will most likely be counting all or counting on to find these sums or differences rather than having internalized them.
- Students can just begin building the idea of unitizing as they work with teen numbers. Students can begin to see teen numbers as composed of 1 group of ten and some ones, extending the idea that teen numbers are composed of 10 and some more.
- 2nd Grade
- Students are finally ready to learn about place value, specifically unitizing groups of ten to make 2-digit numbers. According to Kathy Richardson, she says teachers should spend as much time as possible on 2-digit place value throughout 2nd grade.
- Students apply what they learn about place value to add and subtract 2-digit numbers. By the end of the year, students typically are at a point where they need to practice this skill – which needs to happen in 3rd grade. It is typically not mastered by the end of 2nd grade.
And here’s what’s expected by the Texas math standards:
- Lots of number concepts within 20. Most of these aren’t too bad. The biggest offender that Kathy Richardson doesn’t think typical Kindergarten students are ready for is K.2I compose and decompose numbers up to 10 with objects and pictures. If students don’t yet grasp that there are numbers within numbers, then they are not ready for this standard.
- One way to tell if a student is ready is to ask them to change one number into another and see how they react. For example, put 5 cubes in front of a student and say, “Change this to 8 cubes.” If the student is able to add on more cubes to make it 8, then they demonstrate an understanding that there are numbers within numbers. If, on the other hand, the student removes all 5 cubes and counts out 8 more, or if the student just adds 8 more cubes to the pile of 5, then they do not yet see that there are numbers within numbers.
- My biggest revelation with the Kindergarten standards is that students are going to be all over the map regarding what they’re ready to learn and what they actually learn during the year. Age is a huge factor at the primary grades. A Kindergarten student with a birthday in September is going to be in a much different place than a Kindergarten student with a birthday in May. It’s only a difference of 8 months, but when you’ve only been alive 60 months and you’re going through a period of life involving lots of growth and development, that difference matters. It makes me want to gather some data on what our Kindergarten students truly understand at the end of Kindergarten compared to what our standards expect them to learn.
- 1st Grade
- Our standards want students to do a lot of adding and subtracting within 20. Kathy Richardson believes this is possible. Students can get answers to addition and subtraction problems within 20, but this doesn’t tell us what they understand about number relationships. If we have students adding and subtracting before they understand that there are numbers within numbers, then it’s likely to be just a counting exercise to them. These students are not going to be anywhere near ready to develop strategies related to addition and subtraction. And then there’s that typical threshold where most 1st graders don’t internalize number combinations past 6 or 7. So despite working on combinations to 20 all year, many students aren’t even internalizing combinations for half the numbers required by the standards.
- The bigger issue is place value. The 1st grade standards require students to learn 2-digit place value, something Kathy Richardson says students aren’t really ready for until 2nd grade. And yet our standards want students to:
- compose and decompose numbers to 120 in more than one way as so many hundreds, so many tens, and so many ones;
- use objects, pictures, and expanded and standard forms to represent numbers up to 120;
- generate a number that is greater than or less than a given whole number up to 120;
- use place value to compare whole numbers up to 120 using comparing language; and
- order whole numbers up to 120 using place value and open number lines.
- I’m at a loss for how to reconcile her experience that students in 1st grade are ready to start putting their toes into the water of unitizing as they work with teen numbers and our Texas standards that expect not only facility with 2-digit place value but also numbers up to 120.
- 2nd Grade
- And then there’s second grade where students have to do all of the same things they did in 1st grade, but now with numbers up to 1,200! Thankfully 2-digit addition and subtraction isn’t introduced until 2nd grade, which is where Kathy Richardson said students should work on it, but they also have to add and subtract 3-digit numbers according to our standards. Kathy Richardson brought up numerous times how 2nd grade is the year students are ready to begin learning about place value with 2-digit numbers, and she kept emphasizing that she felt like as much of the year as possible should be spent on 2-digit place value. If the disconnect in 1st grade was difficult to reconcile, the disconnect in 2nd grade feels downright impossible to bridge.
I’m very conflicted right now. I’ve got two very different trajectories in front of me. One is based on years upon years of experience of a woman working with actual young children and the other is based on a set of standards created by committee to create a direct path from Kindergarten to College and Career Ready. Why are they so different, especially the pacing of what students are expected to learn each year? It’s one thing to demand high expectations and it’s another to provide reasonable expectations.
And what do these different trajectories imply about what it means to learn mathematics? Kathy Richardson is all about insight and understanding. Students are not ready to see…until they are. “We’re not in control of student learning. All we can do is stimulate learning.”
Our standards on the other hand are all about getting answers and going at a pace that is likely too fast for many of our students. We end up with classrooms where many students are just imitating procedures or saying words they do not really understand. How long before these students find themselves in intervention? We blame the students (and they likely blame themselves) and put the burden on teachers down the road to try to build the foundation because we never gave it the time it deserved.
But how to provide that time? That’s the question I need to explore going forward. If you were hoping for any answers in this post, I don’t have them. Rather, if you have any advice or insights, I’d love to hear them, and if I learn anything interesting along the way, I’ll be sure to share on my blog.