Understanding vs. Memorizing

When I was in elementary school my teachers regularly asked us to complete Math Mad Minutes.  These were sheets of math problems, usually just 1-digit numbers, and we had to complete as many as we could in just one minute.  Some years we did addition and subtraction, some years multiplication and division.  Sometimes we even had to do a Mad Minute that had a variety of operations on it.  When I first started learning how to become a teacher, my mentor teacher used these.  Children started with a sheet that had 20 problems, and if they could do all of those in a minute they upgraded to a sheet that had 30 problems!  The super fast kids got a sheet with 50 problems.

I hated doing these.

I remember having only one strategy:  I went through the Mad Minute, week after week, and did all the problems that had 0, 1, or 2 for an addend, subtrahend, or factor.  If I saw a number along the way that was a “double” I would do it (3+3, 6×6).  Basically, I memorized the location of the problems for which I knew an answer.   I have a clear picture of myself sitting in Mr. Goodrow’s 6th grade class and reciting to myself the answers to the top two rows of problems.  I was certainly memorizing a bunch of stuff but I wasn’t actually memorizing anything useful beyond the Mad Minute.

True confession:  In my first classroom as a teacher, I finally “memorized” the times tables for good. Nobody gave me a sticker when I could recite them all, but I did it anyway.  I was teaching math on a rotary to 90 fifth grade students every day.  I have a clear picture of myself standing at the whiteboard writing answers to multiplication problems and realizing there was a pattern to the answers.  I was 27. I was university-educated.  I feel quite confident nobody had every told me about these patterns.  It opened a door for me.

What if I had understood this sooner?  Sticking with the multiplication example (though I could also talk about how understanding addition and subtraction is equally important!) if I had understood these connections and patterns I’m sure division, fractions, decimals, algebra and statistics would have all come much easier for me.

I’m listening right now to a Ministry of Education “Town Hall” call.  People are advocating for spending the Primary grades memorizing facts. The thing is, nobody ever says, “In the Primary grades kids should just memorize words.  We’ll teach them to understand words, read sentences, and write sentences once they get to the junior grades.”  Sounds ridiculous, right?

So if you are at home at night and want to work on helping children memorize math facts, then go for it.  But in class, I have some really important foundations of understanding to build. I have concepts to connect, I have patterns to point out, and I have number sense to build. You will not find any Mad Minutes.  Do I want them to have facts memorized?  Absolutely!  Are we actively working toward that?  FOR SURE! But I’m not going to focus on this at the expense of spending time on building understanding.

Real Geoboards vs. Virtual Geoboards

This past week we’ve been doing some geometry work in class.  The grade 2 curriculum expectations for geometry are fairly simple:  name, sort and make 2D and 3D shapes.  In general, children arrive in grade 2 already knowing most of these.  The more common the shapes are in the natural environment, the more likely this is true.  Octagon and hexagon usually give everyone a hard time with their tricky names, but by the end of grade 2 few children can’t recall these names.  In grade 3, we have to do a few more things. The vocabulary is increased (quadrilateral, angles) but again other than folding and unfolding nets of 3D solids, it’s nothing too complex.  Of course, I say that from this point of view – some kids do find it a bit tricky.  In all, however, it’s about 1 week’s worth of expectations.  I like to teach them early on because there are a lot of problem solving opportunities that can involve geometry and once we have the vocabulary learned the problem solving comes more easily.

This week I had rubber bands on hand.  That’s not actually something that happens all the time.  Since we had them, I pulled out the old geoboards.  Lack of rubber bands is actually one of the main reasons I don’t always pull them out.  The virtual geoboards, available here, here and here, are so much more reliable.  And nobody can shoot a virtual geoband across the room at somebody.

In the activity shown below, our Friday lesson, students were asked to make some shapes according to a rule.  Then their classmates had to figure out the rule.  Was the rule: shapes that have 3 sides?  Shapes with 4 corners?  Shapes I enjoy making because they create cool patterns? Here are some of our results:

A few years ago I remember reading an article about how important it is for students to have a real experience with a manipulative before they move to the virtual version.  I tried to find that this morning and couldn’t.  My brain doesn’t remember the source!  So I put it out to my virtual PLN (Professional Learning Network) on Twitter, and found a lot of teachers agreeing with my thinking. Reflecting on our week today, I was so glad that I had used the real geoboards.  There was some really interesting stuff that happened.

First, students were making shapes of different sizes over and over in different ways.  On the apps, they tend to get busy playing with other tools – like changing the colour of their geobands, or colouring the shapes in with a variety of colours.  They get focused on the wrong things and come up with rules like:  Shapes that are orange.  And I’m sorry, but orange is not a geometric attribute.

Second, I noticed that some children were struggling to stretch the bands across the pegs.  Some of the rubber bands are smaller than others, so this became a problem solving challenge.  I feel like they were motivated by the task to work through the challenge and find a new rubber band or change the perimeter of their shape.  This simply doesn’t present as a challenge to be worked through in the virtual environment. I had forgotten about this part!  Developing learning skills needs to be embedded in every part of our day, and I’m glad they got this chance to work on some problem solving skills.

Finally, there were some social things that we could work on.  Sharing is always an issue for 6 and 7-year-olds. They had to cooperate and collaborate to share the rubber bands on their table, and to decide who would get which geoboard.  I tried to make sure that every board at least matched the other boards in a group, but I didn’t always make that happen.  Again, students had to talk through this because everyone wants the one that is different, and therefore special.

Now that we have spent some time with the geoboards,  they can become one of the activities students can do during a Math Workshop session.   I can put them on a table with some task cards, or the students can request them to help solve a problem.  When we move on to perimeter and area (after we spend some time working with number lines for the next 3 weeks!) I can incorporate questions about shapes and feel confident that everyone knows the shape, and can work with the shape.  And I can add some

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Games

Last week I wrote about using the 100 board in class. There are many games that can be played with a 100 board, and all of them help strengthen a child’s number sense.

I recently bought some 100 board games from Arnold Tutoring, and we’ve been having a great time with them at home. We like to play all sorts of games, so it was easy to convince my children to play these!

Both games were definitely worth the money! I can download a 100 board for free, but by the time I copy, laminate, buy all the game supplies and a nice container to keep it all in, it will have cost me a bit. Plus: I have no time! So the games were definitely worth the \$— I spent. And I didn’t have to pay for shipping. And I love internet shopping. Really, this was more like an investment in my quality of life. (Seriously!)

“What’s my number?” is like one of our favourite games, which is called “Guess Who?” Except this is played with numbers instead of people. “Add to 10” is a bit tricky for us right now, but we easily changed it to suit us (my son is good with numbers, but he’s only in grade 1, so not quite ready for double digit addition.). We rolled the very nice 0-9 die, and then talked about our strategy for moving forward on the board. If nothing else the game would have paid for itself just using it this way. But I know we’ll get to more involved math before the end of the school year. This game set will grow with us.

These are games meant to replace math worksheets. They are for families that want to strengthen their child’s math skills, and have a good time doing it.

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The 100 Chart

It used to really bother me when my students filled out a hundred chart in random order. I wanted them to start at and finish at 100, demonstrating that they can count in order, and that they could print all the numbers. But I’ve changed my thinking on this.

Here is some work we did this week:

In each of these examples, the child is following a pattern. It’s not always the standard counting pattern usually it’s the patterns in the one’s place.

This tells me more about my growing mathematicians than whether or not they can count to 100.

Next week we are going to fill in the numbers from 101-200. I can’t wait to see what patterns they continue to use to fill out the chart! And I really can’t wait for the conversations we’ll have around this!

Games

This past week I had a chance to think about, and talk to colleagues about, how my family did math at home when I was a kid. My answer: we played games. I don’t recall having much homework until high school. I always got off the bus (5th grade and up) by myself and had about an hour before my mom came home. I typically did my homework while watching Days of Our Lives.

After dinner, however, it wasn’t uncommon for us to play Rummy, Gin Rummy, or Uno. We also enjoyed Yahtzee and Monopoly. If nobody was into it, I’d play Solitaire by myself (yes, I was a kid before it was possible to start a game online with someone.) When I was a teenager my dad taught me to play Black Jack. (It’s much less stressful with my dad than it is in Atlantic City for sure!)

Now I like to play games, in class and at home with my own children. Some of our favourites are War, Addition (or multiplication) War, and Tens Go Fish. You can play any of these with a standard deck of cards. Remove the Face cards if your children aren’t ready for adding, or multiplying, 11-13!

Today in class we did 100 chart puzzles. I copied 100 charts on card stock and cut them apart. Since I made these I have been using a 120 chart in class, but I couldn’t let the game go to waste. I suppose I should pass it off to the grade 1 class and make a new set. It was easy, and cheap. I’ve got nothing to lose. You can see here that some of my puzzles are more challenging than others.

Besides practicing math, games are a great way to practice taking turns, and losing with grace. These are important skills for kids to learn too!