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The Language of Multiplication

Multiplication (2 groups of 3)Recently, my kindergartener, Lilly,  came home and declared, “I’m bad at times.”  Let me clarify a couple of things.  First, when she said that, she didn’t mean that she was occasionally evil.  She meant she was bad at multiplication.  Second, we don’t do, “I’m bad at ___.” in our house.  We work pretty hard to send the message that we’re all capable of most things if we only apply ourselves.  Lilly knows this, but as the youngest kid in her class, and likely her entire grade district-wide, she’s constantly comparing herself to classmates who are much older and wiser.  She’ll tell you all day long, “You just have to practice, practice, practice.”  And at times, she actually believes this.  Over spring break she was determined to skip a bar on the monkey bars and by jove, she did it.  Other times, she needs some convincing.

So your first thought at a kindergartener declaring her deficiency in multiplication might be, “Why are they doing multiplication in kindergarten?”  I assure you, they’re not (at least not formally).  This statement came from a classmate, whose older brother just went through third grade.  Third grade is the year students are supposed to know from memory (and by the end of the year) the product of all single digit numbers.  Needless to say, her classmate spent a year listening to his brother complete math homework and picked up a little – perhaps a lot – here and there.

My first thought was, “You’re not bad at times.  You’re using the wrong language.”  Most of us learned our multiplication facts from drill and practice.  We learned tricks to remember the nines and were always stumbling over the sevens and eights.  No one bothered to explain the meaning behind multiplication until after we’d memorized our facts.  Turns out, if we do the reverse, understand before we practice, memorization comes a whole lot faster.  And better.  Part of understanding is knowing what “times” means.  I sat my daughter down and explained that “times” just meant “groups of”.  We practiced this way:

Me: “2 times 3 just means 2 groups of 3.”Multiplication (4 groups of 2)

Lilly: “Oh, I know that!  That’s just 3+3.  It’s 6!”

Me: “Let’s try another one.  4 times 2.”

Lilly: “4 times 2 . . . so that’s four groups of 2.  So 2, 4, 6, 8!”

When multiplication language is understood, especially the language of operations, competency and fluency increases.  Learning multiplication requires an understanding of the language (times, factor, product, etc.) and an ability to think flexibly about numbers.  Most kids are comfortable multiplying by 1, 2, 5, and 10.  That’s because they all have had practice skip counting in these number patterns.  If they’ve had good instruction, they understand that counting by 2s is just adding 2 each time.  This connects to student understanding of multiplication as repeated addition.  The other facts anchor around knowing these facts and are based on students being able to decompose (break) numbers into parts.  There are classroom appropriate posters that are useful for parents to use as references for helping their children at home too.

Practice with multiplication strategies and work with word problems allows third graders to develop an understanding of multiplication, which in turn supports recall of “basic facts”.  Because kids have had so much practice developing a foundation of multiplication in K-2 in looking for patterns, working with repeated addition and arrays, they are able to more quickly develop fluency for multiplication.

If your little one isn’t yet learning multiplication, don’t hesitate to pose questions that support multiplicative thinking.  As I was writing this post, I took a break to put my daughter to bed.  We finished reading The Lion the Witch and the Wardrobe earlier in the day.  At bedtime, Lilly asked how many pages we’d read.  I wasn’t sure, and since I was in the middle of writing this post, I couldn’t help but respond, “I’m not sure how many pages but, if we read 4 chapters and each chapter was about 9 pages, how many pages do you think?”  She said, “So 9+9+9+9 . . .”  She understood the situation and was thinking multiplicatively.  I’m assuming, with more conversations like this, she’ll be ready to go when she gets some formal instruction in multiplication in three years!

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Count the World

Counting the WorldI know church is supposed to be a reflective place.  Can I help it if I happen to reflect on math?  I hope not!  As we were leaving one Sunday morning, there was a significant back up on the stairs.  This is frequently caused by the very dainty, and somewhat fragile church goers who live in the retirement home next door.  On this occasion however, it was a different kind of dainty parishioner – a toddler.  As she carefully graced each step with both feet, her dad peered behind his shoulder and gave us all the, “I’m so sorry but I know you understand.” look.  He then returned his attention to his daughter and began counting.

The act of counting steps, or anything tangible, is a vital component to supporting young children while they mathematize their world.  Mathematizing is really just about bringing out the math that is inherent in the world and space around us.  That things (anything really) can be counted is mathematizing.  Kids mathematize when, given the option of a portion of cake, choose what they perceive as the larger portion.  They learn about volume and dimensions when trying to build towers with varying sized sets of blocks.  They mathematize when they’ve calculated that there is only one “fun” swing on the playground and the likelihood of loosing it is good so giving it up to play on something else is not an option!

Early counting and grouping is of particular importance.  The action of that father counting steps with his toddler supports her development of cardinality.  Cardinality is the idea that number and quantity are related.  Each number represents a set of that many things.  While this is obvious to you and me, it is not clear to the youngest mathematicians of our world.  Watch a very little child, 2 or 3 years old, try to count a set of objects.  He may understand the idea that he is supposed to say the count sequence while pointing to objects but he may not yet know that each number he says has to correspond to one of the items.  And not only that, each number has to correspond to a different item.  He doesn’t know you can’t count it twice!  This one-to-one correspondence develops over time and through repeated opportunities to practice with guidance.  As the toddler on the steps felt each count underfoot, she was developing one-to-one correspondence and cardinality.

These ideas of cardinality and one-to-one correspondence are components of the kindergarten Common Core State Standards for Mathematics.  They are really quite basic but so vital for mathematical proficiency.  While they “live” in the world of kindergarten, they are skills that can and should be developed much earlier.  At home and at daycare, adults can help children mathematize their world by subtly applying the count sequence to objects.  Imagine all the times you could say, “Let’s count them!”  Rocks, legos, beads, toys, shoes, birds, swings, diapers, grapes, forks, blocks, friends, etc.  Adults can easily teach children to touch and count each object and to help them distinguish between the counted and uncounted by demonstrating pushing the counted collection aside, one by one.  While one-to-one correspondence takes time and fine motor skills to develop (so don’t fret if it takes awhile), modeling of this behavior is invaluable.

One more thought on cardinality.  When children finish counting a set of objects and are done saying the count sequence, we assume that they understand that the last number said represents the total in the group.  This is not necessarily the case.  If you follow up a counting sequence by asking, “So how many?” you may notice that your child repeats the count sequence.  This is a good indication that he does not yet understand this component of cardinality.  You can’t force him understanding, that will take time and experience,  but you can model it yourself.  When you model you can say, “1, 2, 3, 4, 5.  Oh, there are 5 markers!”  This indicates to your child which of the numbers represents the total of the group.

Happy counting!

Common Core Math Resources for Parents & Families

It’s amazing how controversial learning mathematics can be.  It seems like everyone either loves, hates or isn’t interested in the Common Core State Standards.  As someone who spends an excessive amount of time exploring the math standards, I happen to fall on the side of love but am perplexed about how best to educate parents and families about what these standards are all about.  Because of all the politicizing, it’s become difficult to address what is most important – what do these standards say kids should be able to do and how do we help them achieve this?

Regardless of your opinion of the standards, it makes sense to spend some time learning about what they say.  Council of the Great City Schools has created a fabulous resource to help parents and families navigate the terrain of the Common Core math standards.  Fittingly, they’ve named them Parent Roadmaps.

I strongly encourage you to spend a few minutes learning about the math your child is learning in school.  It turns out these standards aren’t nearly as juicy as the news would have you believe!

Checking Your Work

Homework is a tricky subject in our house.  One family member is an elementary teacher, one family member is in kindergarten and one family member has chaperoned a field trip, helped on the weekends and attended elementary school many, many years ago.  You can imagine that a certain someone – ok, it’s the dad in our house – tends to get told what to do all to often when trying to help out with homework.

This week, as I was fixing dinner, Adam sat down to help out with the weekly math homework.  Things are pretty simple at this stage of the game.  We’ve seen lots of tracing numbers 1 through 5 and some matching numerals with dots or cubes.  This week’s homework asked the kids to extend a bug pattern.  Lots of cutting and pasting – right up kindergarten alley.  Since patterning didn’t seem to require too much nuanced discussion, I happily let Adam take over.

Math Practice 6

After assembling all of the materials and lots of cutting, they set about figuring out what bug came next in the pattern.  Then, the magic happened.  I heard Adam say, “Check it to make sure it sounds right.”  Ah, be-still my heart.  Without even knowing it, he was instilling in our barely 5 year old daughter, an overarching habit of mind – oh so important to mathematical proficiency.

Outlined in the Common Core State Standards of Mathematics are content standards (what the kids should know) and the practice standards (how kids should “behave” with math).  The content standards are what many think about and, unfortunately, the practice standards are often brushed aside.  This might be be due, in part, to their location in the standards document but it is also because the wording tends to be a bit complex.  Many teachers have been working on trying to understand the eight math practice standards and apply them at the level they teach.  While the ideas are big, and extremely important, at the earliest grades, these practice standards can look quite simple.

Math Practice 6, Attend to precision, emphasizes precise use of math language and vocabulary as well as accuracy.  This obviously looks different at different levels but with kindergarten, “Checking your work” can elicit this standard.  Every time you ask your child to check his/her work or praise him/her for doing it independently, you are reinforcing the idea that review supports precision.  If you think about it, you can apply this thinking to a variety of subject areas.  How many times did your teacher ask you to reread your writing looking for errors and opportunities to improve?

Just like everything you do with your very young children, establishing routines early can lead to habit.  Maybe I’ll let the novice take over more of the responsibilities with math homework!

You can learn more about the standards for mathematical practice from Dreambox.  If you want more in depth information you can get it from Think Math and Illustrative Mathematics.

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