10-Frame Flash Cards

Click on the link to download free 10-frame flash cards....read on to find out what they are and why you need them ;)  Set of 10-Frame Flash Cards- Free Download (PDF)

 In John Van de Walle's book, Teaching Student-Centered Mathematics, K-3, he devotes a whole chapter (Chapter Four) to helping children master the basic facts. He writes that Mastery of a basic fact means that a child can give a quick response (in about 3 seconds) without resorting to noneffficient means, such as counting. He doesn't talk about memorizing here, but he does talk about helping students develop efficient strategies based on relationships between quantities (such as anchoring on 5 and 10  to master facts such as 8 + 6 by thinking of 8 as 10 with 2 missing and then getting that 2 from inside the 6 and making a new fact, 10 + 4) and relationships between operations (such as thinking addition to solve subtraction).

There always seems to be some controversy between the "memorize your facts" group and the "develop your number sense" group. But these things aren't mutually exclusive. Memorizing your facts is very helpful. But, developing your number sense is helpful AND important.  In the text series adopted my district, the lessons for subtraction in 2nd and 3rd grade direct students to "think addition" to solve subtraction problems. So for the problem 14 - 8 = ___ students are told, "Since you know that 8 + 6 = 14, then you already know that 14 - 8 = 6."

The only access for students is if they've already memorized 8 + 6 = 14. If they don't have recall of that fact, they have no other strategy for thinking addition to solve 14 - 6. I think this is why people confuse fact memorization as important, instead of just helpful. If it's the only strategy we develop, then it's of the utmost importance!

But think ahead now. Nobody can memorize every fact. I mean, if the child is solving the problem 54 - 28 = ____  is there anybody who thinks it would be reasonable to tell them, "Well, don't you already know that 28 + 26 = 54? " Of course they don't hold this information at a recall level.

But let's go back to the 14 - 8 = ____ problem. If we think addition that 8 + ___ = 14, but haven't memorized that 8 + 6 = 14, I can still derive this fact. I can think about needing 2 more to get to 10; then once I'm at 10, I need 4 more to get to 14. So now I've derived the missing addend as 6, but in a 2 part and a 4 part. Students (people) can do this so efficiently that it can appear memorized.

When students have ample opportunities to develop this strategy of counting up in parts, then they can also use it in the 54 - 28 = _____ problem. If we think addition as 28 + ____ = 54, we need 2 more to get to 30.... 10 and 10 gets us to 40, 50...and 4 more gets us to 54. So the parts we used to count up are 2, 10, 10, and 4, for the answer of 26.

The Common Core Standards are written so that students have both procedural and conceptual understanding. I guess I've heard people say that it's more important to teach procedures first, and worry about the conceptual understandings later. I've also heard people say that you have to develop the conceptual understanding before students can use procedures.  I think everyone must have their reasons for saying so, but in the reality of the classroom, I have found that it happens much more organically than that. You can actually teach the procedures and  concepts in any order or even at the same time. If you are providing a variety of opportunities for students, they tend to take what they need, and cleave to the important ideas, as it makes sense for them.

So here come the flashcards that John Van de Walle suggests.  An old-school flash card has the fact written, such as 9 + 7 and students have to recall 16. But adding the 10-frame quantities to the flashcards help students develop their number sense too. I can SEE now that 9 is 10 with one missing, and I can VISUALIZE moving one dot from the 10-frame with 7 to fill in the ten-frame with 9. Now I'm using the Make a Ten Strategy.

 

Or now, instead of only having access to 8 + 7 if I memorize the answer as 15, I can now use the 10-frame quantities to see that if I just cover up that last dot on the 10-frame with 8, I've created a 7 + 7, and I only need to add back the covered dot to get the answer 15. Now I'm using the Use a Double (Double + 1) Strategy.

Alright there are your procedures right alongside your conceptual understanding! You really don't have choose. Other ways I've used these cards is to have them sort the cards by which strategy they would use...for which ones would you make a 10? Which ones would you use doubles? Which ones make the most sense just to count up? Which ones do you know "by heart"? Can you explain to a classmate why the strategy you are using makes sense for these numbers? Are there any cards you would use more than one strategy for?

Ohhhhh the possibilities are endless :)

More on Math Tools

I get so many questions on how to use math manipulatives when I do professional development on teaching math. I wrote about many of the logistics in this last post, but really, I'm sure I will continue to write about in various forms and contexts. It's that interesting, I promise! ;)

I started out explaining that I present the groups with a tray of a certain tool, and they are given free exploration time. When I was a math coach/specialist for my district, I actually had this process written into the pacing guides. So the first five days of every school year are about introducing tools and procedures, including number talks.

Because I have an abundance of tools (largely scavenged from hallways and supply rooms where they were discarded) I just give everybody the same tools. If I were short of supplies, I would fill each tray with something different and rotate them over the five days. That's still legit for many reasons, actually, but I choose to do it this way because they also learn about the tools by watching each other use them, and talking to each other about what they see happening in different groups.

While they are working with the tools, I am circulating and learning a lot. I make notes on conversations I hear, math I see, and procedures I may (will) need to reteach. You can learn a lot about who your kids are in math class by watching them work in free exploration.

Who are your sorters...

Your artists...

Your pattern seekers...

Your hard core playahs...

Your spatial geniuses...

Your future engineers....

and, of course, Your Mopers

 I'm always surprised at how much math surfaces during these play exploration sessions. And if it looks like play? So be it.

Since they have their tool bags (these are last seasons bags, and the new ones are much roomier for the same number of tools), they are in charge of deciding which tools to use, to solve which problems, and in which way. I'm not going to lie to you...the first year I implemented this, and I saw what they did, I thought I'd made a HUGE mistake.

For example? Do you see the 18?

No? Me either. Until he explained: 1, 2, 3, 4 cubes...2 green tiles...1 yellow cube...2 sticks...1 red counter. (4 and 2 and 1 and 2 and 1 is 10). Now look at the dice....one says 3 and one says 5. 10 + 3 + 5 = 18.  This is certainly not what I envisioned!

But we pushed through this part, and it has become easily one of the most powerful, sense-making policies of our math period. It took some time, but we always highlight strategic use of tools. I'll often ask students doing something efficient and/or interesting to tell us a little bit more about why they did it that way. "Why did you use a 10 stick instead of the red counters?" Or, for a problem involving regrouping for subtraction: "I see you used a stack of 10 unifix cubes...how did that help you do the subtraction?"

It also takes offering an invitation to experiment. "What if you used _____? Would you get the same answer?" The more students used the tools as part of their daily practice, and the more invitations I offered to do the same problems using multiple tools, the more the kids made sense of the tools and they began to actually use them strategically. (Standards for Math Practice #5, CCSS)

Using, Storing, and Managing Math Manipulatives

 I am passionate about teaching math, most especially with Littles. I myself love math. A seven year old is a very imprecise tool with which to do math, and can be particularly baffling, for a lover of math. But the thing is, is that the most profound mathematics are introduced and mastered with 5, 6, 7, and 8 year old kids. The number system, relative magnitude, every operation, geometry, relationship and comparisons,  problem solving and logic...it's the same math that underpins algebra, trig, and calculus...it's the same math that underpins engineering, architecture, and balancing your checkbook. It's deep, important, and complex.

Students have access to these tools at any time. They can come here and get a "scoop" of counters or a stick of unifix cubes, anything they think will be helpful...even when I don't believe it will. True story, I bite my tongue a lot.

The new Common Core Standards are, necessarily, imperfect. But if they do nothing else, they elevate the DOING of math to the importance typically reserved in elementary schools for the LEARNING of math. Math so often happens to kids. They are expected to learn math by watching teachers do math. They are often taught discrete rules and procedures in order to get them through arithmetic, without ever developing their sense of being mathematical. When I was a math coach/specialist for my district, it never ceased to amaze me how little cognitive lifting kids were being asked to do in math class. Teachers, pressured to the point of breaking by NCLB (No Child Left Behind, or as we call it, No Teacher Left Standing) requirements, often shied away from problem solving and cleared a path through story problems wide enough to march their whole class across without ever giving kids room and space to struggle and grow mathematically.

I ask myself: would I try to teach reading by never giving a kid a book to read? To me it is the same thing. I can't fathom trying to teach math without giving kids a problem to solve.
 

I moved into a classroom with very little storage this summer, and I invested in these white buckets ($7 with lids at Home Depot) to provide additional math tool storage and seating at our writing center (shown) and reading table.

The eight Standards for Math Practices all start with the words mathematically proficient students to describe how students (people) behave when doing math. The fifth Math Practice is about choosing tools. It's not just about breaking out the base ten blocks to show kids how to make exchanges for the regrouping procedure in subtraction. In fact, research has shown that when teachers decide which tool to use, and how to use it, the students experience the tool as one more set of rules they have to memorize...instead of contributing to the true goal of conceptual understanding.

Making sure my students have access to a variety of tools, as well as promoting a culture of understanding where students are asked (no, required) to pick their own tools and make sense of those tools in a variety of contexts...this is no small undertaking. The pictures above show how I store the math tools.

In the beginning of the year, we spend time introducing the tools. Partly to make sense of them, and partly to outline expectations. We do a "what does it look like and sound like when we are using our math tools" anchor chart.

During the first five days, I put out a tray of different manipulatives each day on each table group. Go to town! We focus on behaviors and procedures for sharing and putting away the tools.

Finally, when they have "met" the tools, the are each assigned a "math bag". Over the years that I have been doing this, it has evolved from a gallon ziplock back stuffed into an already crammed desk to this arrangement, which I'm calling AWESOME. I used a sheet to make 32 drawstring bags. Nothing fancy. Cut rectangles out of the sheets, folded over the top to make a casing for the drawstring ribbon, then folded over the rectangle in half and sewed along the bottom and open side (don't close the top, you have to feed through the ribbon). PRO TIP: once you've run the ribbon through the casing, bring both ends together and tie them into a single knot. You are welcome! ;) These were considerably smaller in past years, I just this summer redid them. Improvements: they are bigger than the last ones, by twice.

 Also improved: I used little 3M tape mounted wall hooks to create a storage area under the whiteboard. It was fun when the bags spilled out of their desks and pieces flew across the floor two to six times a day but, alas, all good things must end. Each bag space is labeled with a student number (1 to 32, I number my students from the time they walk in the door. With a 35% mobility rate, we remain in alphabetical order usually no later than October. By then there's been movement, but when you come in, you just get the last person who moved's number. It means Montse Rivera is going to end up being number 2, ahead of Luis Chavez, but pay that no mind. Number order please!)

 The labels on the bag space now match the numbers I put on each bag. I used puffy fabric paint that comes in a tube and I freehanded the numbers. Because I'm a gangster like that. haha

Students use these bags during the math time. If they need more or different tools than what are in the bags, they can get it from the bins or buckets. Like Crystal here, who was counting out 60 pencils for her story problem...by counting each 10-stick as one pencil...at the beginning of third grade. Ayyyyy....my aching head.

 When Crystal told me she needed more ten sticks, what I thought was, "No you don't. You need 6 of them." What I said was, "Sure, they are over there." And then I made a mental note that half my class was just like her, and I had some place value work to do. Ahem.

In my second grade class, each math bag has: 40 ten sticks, 25 unit cubes, 25 red and yellow counters, 40 unifix cubes, 40 square tiles, and a die (for centers and games). I'm pretty sure it's exactly the same thing they had in third grade last year...maybe they had 60 square tiles.

Also available in the bins and buckets: more of all those, plus hundreds flats, thousands cubes, cuisenaire rods, fraction pieces (strips and circles), atrribute blocks, centimeter and inch cubes, plastic coins, and pattern blocks.

Available at all times in baskets around the room: rulers, measuring tapes, protractors, ten frames, and balance scales.

It's a process. We go back to review/reteach how to store and take care of our manipulatives regularly. But of all the things I've done in the last years as a teacher to open up the world of mathematics to my students, this is up at the top. So worth the aggravations and inconveniences!

Reading Pointers

My ultra-cheap whisper phones left me with enough money to make some cute "pointers" for reading the room/word wall work.

The painted wooden pieces were around $0.29 each at a craft store. While at the craft store, I picked up two 5 packs of wooden dowels for $1 a bag. So for $5 and in less than 20 minutes, I made all 10 of these fab pointers.

I struggled with how to cover the plain wooden dowels. Well, haha, "struggled", let's not be dramatic. But I did puzzle over it for a day or two. I was busy so I kept passing this little pile on the sewing table and wondering....do I modge-podge tissue paper onto the sticks? Cut and modge-podge scrap book paper (which is not a hobby of mine, and would require a second craft store run)? Eventually common sense (and an aversion to glue mess) prevailed and I wrapped each stick in colorful duct tape I already had on hand. Then I hot glued the pre-painted wooden shapes to the tips (I stopped the duct tape an inch or two from the top of each stick).

Boom! I also made three extra long ones in exactly the same way. The extra long dowels were $1 each, so for $4 I made three extra long reading pointers too. The extra long ones will work for pointing to the word wall (it's tall, baby) and the number line above the white board and the alphabet at the top of the back wall, too.

Go to town, kiddos!

Going For It

I've decided to do the Daily Five method and CAFE menu. I had bought the books last summer but didn't get around to reading them. I read them both this summer (largely because the word "vacation" confuses me) and you know...I am suddenly all in. It seems like there are pieces of various things I've been using (reading workshop, guided reading, etc.) but I'm moving down to second grade next year (from third) and having the full power of their road map under me feels like the right thing to do. Hold my hand?

In preparing for "read to self" I made a class set (for me, that's 32...ahem) of whisper reading phones. These were a breeze to make and exceptionally cheap too.

It looks like the ones you can purchase (don't click that unless you have $5 per phone) are more like 3/4" around for the tube. I used pvc plumbing pipe (a 10 ft length I purchased at a home improvement center for around $1.75) in 1/2" size. It comes in 3/4" size, but the thing is, is that I had a bag of 25 of the elbow joints in 1/2" size ($5 for a bag of 25) and that was reason enough for me. I mocked one up and I had my In House Quality Control Department (my own second grade son) test it out for me. It looked and felt fine for him, so there we go!

I might be the only teacher on the planet who epic fails every time I try to play the "Um...pretty please I am a teacher, can I get this cut/have a small discount/ask if there are any discards" card. So when I asked, "Hi, I'm a teacher and I need this 10 ft pole cut into 3.5" pieces for a class project...is there any way to do that with an electronic saw?" ...I wasn't that surprised that I got a teenager with dreadlocks shrugging his shoulder and saying, "Nah, cuz that doesn't go through the saw." Fine. I'll saw it with a hack saw on the miter bench in the molding aisle. Thankfully my long-suffering husband had come with me and we took turns sawing and nursing our aching shoulders. All in all, for 15 minutes of sawing, it turned out fine.

So, cut the 10 ft. pvc piping into 3.5" lengths. Buy two pvc elbow joints in matching width (for me 1/2" elbows) for each phone. Put an elbow on each end of the pipe and muscle it down as far as it will go. I added a round of colorful duct tape for no other reasons than a) I had some and b) it covered the printing some of the pipe pieces were sporting.

Total cost for 35 phones (gotta make some extras, you know how it is): $1.75 for pipe, $10 for 50 elbows, plus one bag of elbows and duct tape I already had on hand. Don't forget $0.59 for the bag of frozen peas I used to soothe my saw-injured shoulder. That makes 35 phones for 12 bucks.

If you have a lot less kids than me (mazel tov!), there is also an option of buying the elbows at around $0.29 each. So if you need to make like, 27 phones, you could buy two bags and 4 extra elbows so that you don't have a bunch of extra elbows.

Right now I'm deciding between numbering these for my Littles versus having some wipes to just give them a run down when you want to use one. I don't really have the OMG GERMS! bone but...I could use advice if you're feeling that way inclined. Seems to me if they are numbered and privately held, they should just go in their book boxes so they aren't all piled up in one big germ-fest. If you've had any experience with this type of thing, I would love to hear about it, thanks!