I've been watching my six-year-old daughter explore math all fall and now into the winter. At the same time, I have also been reading books about how K-2 kids learn math, and growing my own 'math eyes' (as Maria Drukova of Moebius Noodles and Natural Math says).
It's been fascinating to watch my kid's understanding of numbers develop over the last six months, even with a somewhat hands-off approach. She started this summer with a lot of comparing and measuring. Then she started asking lots of questions about numbers and shapes, mostly on her own time. Sometimes I'd be there during the process, as in this line of questioning about dividing a number in half. Other times I'd run across the results of her thinking later in the day, as with a geometric study she initiated with some tangrams.
In this book, Peggy shares all sorts of fun math games that require no more than some paper, a pencil, some dice to roll, tape, egg cartons, beans...you get the picture. The best thing about this book, besides the very easy-to-implement, high-interest games is how she puts the games into context with stories about working with specific children. I really, really like 'hearing' her have math conversations with children; the activities are pretty simple but the important thing is how she models mathematical reasoning and problem solving strategies while she plays the games with each child.
My favorite games so far are Star Count and Lots of Boxes. Both involve a pen, some paper, and a die to roll and both focus on understanding groupings of numbers as a way of practicing counting and introducing multiplication at the same time. My kid has been asking questions that show she is thinking multiplicitively, but she is not quite ready for or interested in sitting down to discover the patterns in the multiplication chart, so these games are perfect for her right now. Also, How Many Boxes is also a great introduction to perimeter and area since it is played on graph paper.
A couple weeks ago the kid surprised me with some spontaneous backward counting from 100. It was wonderful listening to her find enjoyment in figuring out the pattern, but I didn't really understand the importance of this new skill until last night. That's when I read a section where Peggy Kaye talks about the kinds of number skills kids need to have to learn how to add and subtract. These include:
- Counting forward from one (I keep trying to remember to start counting from zero, but starting at one is just too ingrained)
- Counting forward from a number other than one, like five and going from there
- Counting backward from ten or twenty or one hundred
- Counting backward from a random number
- Skip counting twos, threes, fives, tens...
My kid has been spontaneously skip counting since the early summer, and counting forward for longer than that. In terms of the girl's new penchant for counting backward, it seems this new skill of hers is really important for mastering subtraction, something we haven't done a lot of.
What she and I do do is a lot of adding numbers together with daily UNO games but, just today, the whole 'adding up the UNO score' activity seemed like it had lost it's challenge. When I lose, she whisks the cards over to her space, spots the tens easily and without prompting, adds up 10's and 20's without pause, and takes the extra cards and finds a final total. Sensing a turning point, I suddenly had an idea for another way to challenge her. Here's my new thing:
Instead of adding up the numbers, take all the cards you've won from your opponent and use them as digits. The object? Make the absolutely largest number you can. A five and a one can make 15 or 51, for example. Or, there's this hand I lost to her this afternoon:
Ha! It's probably the largest UNO score EVER!! And, it's a great way to introduce hundreds, thousands and millions as well as 'which number is biggest' on the grandest possible scale. The cards become great manipulatives, so once she gets comfortable reading the numbers I've made (while modeling my thinking process), she'll get a chance to figure out how to move them around to make 'the biggest number' she can.
But back to counting backwards. I don't really understand my child. She is quite independent in her learning, and often resistant to formal instruction. But she really likes it when I give her spelling tests!? And, today, when we were at the library, she asked me to give her a math test. Um, okay.
I eventually obliged and put together a list of equations. As I've said, we haven't done much in the way of subtraction, so this little test was actually a good way to assess where she is right now. Plus, with my new understanding of counting backwards? Wouldn't you know it -- there she was counting backward to figure out the answers!
She also wanted some multiplication problems so I used a strategy from Peggy Kaye's Star Count game to illustrate each problem. "Two groups of two", "three groups of four", "five groups of one" are the way we say it. She got her answers by counting the dots, but I can imagine if she keeps this up we might be ready for coloring in the patterns in the multiplication chart by spring. I really want to do square and triangular numbers with her using visuals but, as always, I just need to be prepared to jump in with it when there's an opening.
In closing, as I was in the process of wrapping up this post, the kid wanted another math test. I learned a lot from seeing how she answered the questions. Next time I'll put the addition, subtraction and multiplication problems in their own sections, not all mixed together. But the biggest thing is that I'm going to get the Cuisenaire rods out again so we can solve the subtraction problems together for a while. 30-2 was just a little too big of a 'problem' for her, but if we can look at it with the rods I think the idea will click into place without much fuss. My final lesson? Math is definitely a morning thing. No more math at 5pm!
[Edit, 1/26/12: Here's what happened the very next day with the subtraction and the counting backward. When it rains, it pours.]
Thank you for this post, Malke. I'm also working on developing number sense and addition and subtraction skills with my son. Actually, just wrote a post for Moebius Noodles with the two games we played recently. Maria suggested playing grouping games as a bridge to multiplication and I will look them up in Peggy Kaye's book. And thank you for the list of the number skills you posted. I tend to skip introductions in books, so I overlooked it, but it's VERY helpful!ReplyDelete
Yelena, your Moebius Noodles post is great and I continue to be so thankful that you and Maria are out there doing this work. My math eyes get sharper every time I read one of your posts!Delete
I have a copy of the book you mention in the post and it makes total sense. (I love the 6-story building activity!) I think I'll take a closer look at it. They are simple enough activities but I can't always be certain when and how I can do them with the kid. My biggest challeng is storing up enough activities in my brain to pull out of my hat when my kid shows me she's ready...and not a second before, lol!
Peggy Kaye's book is deceptively simple and her words of wisdom are hidden inside the activity narratives. It's the first math activity/game book I've really wanted to just take my time with. And, I never thought I'd like and understand numbers as I am starting to do now. I never understood what kinds of patterns you can find in numbers -- I know I'm personally at an elementary level with this (albeit with an adult brain that works with other kinds of patterns professionally), but it's still fascinating and exciting.
It's interesting for me to see what others are doing with their kids. My daughter (5 years old in public K) is a little math whiz and obsessed with equations and big numbers. However when I challenged her with dividing 100 jelly beans between 25 kids, that stumped her completely. We went to the manipulatives (Target counting shapes) and then she got it pretty quickly. One thing that really helped develop her math sense is Dreambox.com. She played it all the way out to the third grade level and finally petered out. I am giving her a break to digest before she ventures there again.ReplyDelete
Here's a couple resources for you and your daughter that I think might be interesting and challenging to think about:Delete
Have you seen this? http://www.mathsisfun.com/geometry/symmetry-artist.html -- it's fun to play around with different variables (color, number, image choice) and see how it changes your design. And, you can print out your designs when you're done!
And, have you been to mathpickle.com yet? You may see some fun stuff there. Have fun!
I love watching how you grow with your daughter. I love how you love math. Thanks for sharing.ReplyDelete
Thanks, Sue! :-)Delete
Malke, you rock! One of the participants in our open "Developing mathematics" course recently came up with a dance-related activity, and I immediately thought of you: http://p2pu.org/en/groups/ed218-developing-mathematics-the-early-years/content/week-2-developing-modern-mathematics-january-23-29/?pagination_page_number=1#10421 What links in your work would be good for her to explore?ReplyDelete
Thanks Maria! I've sent you an e-mail, but I realized I didn't answer your question fully. In terms of links to my work, the ideas I had after reading the activity description and your response were related to what I developed for preschool age kids, which I haven't written much about. At some point I'll need to write about it, but there are only so many hours in the day, lol! ;-)ReplyDelete