This next week, I will be giving two presentations at the Ontario Association of Mathematics Educators conference in Toronto. One will be on using fluid hundred charts, which is essentially using 100 numbered tiles to teach about number sense concepts.
I’ve made a video that describes how to make a set of 100 tiles.
My plan is to post about the different ways to use the tiles, and to link those posts with other videos.
As part of a Math AQ course, I put together this Dos and Don’ts list for things to consider when working with students with autism. Many of the suggestions can be applied to more than math instruction.
However, there is a saying amongst those who work with these children. “If you’ve worked with one child with autism, you’ve worked with one child with autism.” Generalizations about this disorder, thanks to the spectrum nature of it, are impossible to make. Therefore, when referring to this resource, keep in mind that, while these are pretty sound suggestions, the degree of application of any particular suggestion must be tailored to each child individually.
In this list, the first Do corresponds with the first Don’t, the second Do with the second Don’t, etc. The downloadable PDF pairs them in columns.
I hope you find this useful.
- refer to general resources that support your math planning for students with ASD.
- get to know your student’s interests, because those will be the entry points to the math.
- make the math stimulating, fun, and interesting.
- use the same kind of differentiated methods of instruction as with other students.
- use visual aids and concrete materials to make the abstract real and accessible.
- break learning up into manageable parts.
- help the student to generalize math concepts and procedures.
- incorporate rote learning into your program: Many children with ASD are strong in this kind of learning.
- assessments that allow the student to be most successful in sharing thinking, playing to strengths.
- use precise praise for effort and success.
- assume the resource is definitive, e.g. most recommend visual aids, but some students with ASD are stronger auditory and/or kinaesthetic learners.
- be afraid to broaden the student’s horizons by introducing new things.
- over-stimulate, distract, or agitate the student: know the triggers and warning signs that tell you if things are about to take a turn.
- forget the communication skills such as vocabulary, and being able to explain the math, be it with pictures, numbers, or words.
- forget the hierarchy of visual aids: from most effective to least are real objects or situations, facsimiles or models, colour photographs, colour pictures, black and white pictures, line drawings, graphic symbols and written language.
- do the lesson without relating it to the other small parts and the whole.
- let manipulatives take on one fixed meaning or purpose: They are variable models that can represent many ideas.
- limit your lessons to rote learning: It’s important that this knowledge be connected to, and is in service of understanding the big ideas and concepts of math.
- forget to take the student’s weaknesses into account: be sure to consider stamina, mood, and external factors to decide when, how, and how much you assess at any one time.
- be stingy with the non math-based positive reinforcement that will foster a positive attitude and ideal brain state.
So, I was driving to work, listening to the CBC (Canadian Broadcasting Corporation – for U.S. followers), and they were talking about the Heartbleed bug.
Well, they were actually talking about changing passwords, and how important it is to make them fairly complex, differentiated, and not in any way predictable or guessable.
I am SHER locked.
Of course, this level of complication makes passwords extremely forgettable.
Wait. I want to digress a bit. Let’s play a game. This is how you figure out your Movie Star name. Your first name is the the name of the first pet you ever had. Your last name is your mother’s maiden name. My Movie Star name is Snoopy Masters. What’s yours?
And while you’re at it, why don’t you just save me some time and tell me your father’s middle name, the street you grew up on, and your favourite fictional character, because clearly you want to be the victim of identity theft.
Most webpages let you reset your password, or at least get a hint, if you can accurately answer a security question that you set up when you registered. You know what I mean. “What is the name of the first street you grew up on?” “What was the name of your first pet?” Imagine my shock when I heard the CBC hosts talking about how con artists set up these posts and threads, collecting people’s information with innocent little internet games.
Not as advanced as the scene in Now You See Me, but just as effective.
So, NO MORE “What’s your Rocker Name” games for me. Or you, I hope.
Winnie Finn, Worm Farmer
Winnie is not your stereotypical picture book heroine. She is industrious, creative, and she has a penchant for slimy things that eat chicken droppings. She is an expert on everything worm, thanks to the hours she spends studying them, rescuing them, and racing them.
Now, Winnie has a dilemma. She needs a new wagon, her efforts at repair being stop-gap at best. She’d love to win some prize money at the local fair, but there are no prizes for worms. Only for corn crops, chicken eggs, and puppy litters.
No matter. Winnie also knows a thing or two about business. Look for a need and fill it. And, as the saying goes, if you can make money doing what you love, you will never have to work a day in your life.
Winnie Finn is a great example of childhood ingenuity and resilience, as well as a role model for delaying gratification and setting goals. With her worm farm and her diligence, she helps her neighbours achieve their dreams, and then she shares in their success.
New wagon, here she comes!
Adding to the story and character building, the illustrations by Ard Hoyt are cartoony and quirky. He fills in the gaps that must be left out from a picture book manuscript, such as the fact that Winnie’s parents own a flower shop, and that Winnie isn’t all tom-boy. She likes a bit of bling too. Plus, Ard likes to put a few sight gags into his work, letting Winnie’s cat provide some background comic relief.
Winnie Finn, Worm Farmer was selected for the 2014 Illinois Reads literacy initiative. Carol Brendler is also the author of Not Very Scary, and the novel Radio Girl. Check out her web page.
I recently posted instructions for a hands-on activity wherein the students make their own meter measuring tapes. I promised I’d explain how I’ve used them so far. Below is a description of ways to use the tapes to teach patterning. This will be followed in another post with ideas for using them with number sense and then, of course, measurement.
Patterning is the act of analyzing the relationships between elements in a string of repeated occurrances of those elements. 😛 In this case, we are looking at the relationship between one number and the next after something has been done to that number. We can focus on the individual numbers, or on the change happening to those numbers.
Have you ever noticed that your students can skip count, no problem, rambling on and on until they reach the biggest numbers they know? Have you noticed that they don’t actually know what 10+2 is, despite being able to count by 2s, or what 15 + 5 is, despite knowing how to count by 5s; they don’t actually know how these patterns are constructed or how they work?
This is because they have memorized these patterns the way they memorize the lyrics to the latest Katy Perry song; they know the words, but have no idea what they are actually saying!
To avoid this meaningless regurgitation, it is important to teach the students how the pattern works at the same time as teaching them the pattern. Better still, is teaching them about how changes happen in patterns and letting them figure out the different patterns that can happen because of those changes. Specifically, we can continually add on, or take off a fixed quantity, and say the new number we get, and eventually, we notice the same numbers happening over and over until we see a pattern.
To accomplish this learning, the skip counting must be accompanied by a concrete understanding of the quantity we start with, the quantity being added, and the quantity that we end up with, and the students must understand that it is a recursive process wherein the quantity we end with becomes the quantity we start with.
These measuring tapes (and any other number line example, such as thermometers, hundred charts, etc.) provide both a concrete example of quantity, and a way to figure out, check, and keep track of the pattern until it is memorized with meaning.
With the students sitting in a circle, with their personal measuring tapes in hand, start at zero, and go around the circle, adding the number you’re skip counting by. As you go around, the children mark the place on the number line with their fingers, and they move to each new number in sequence as they hear it called out.
If you followed my advice in the first post, and numbered only the 5s and 10s, then you should start the students practicing this activity with 5s. This will help those who struggle with counting, as the numbers are clearly marked.
Then you can do 10s, noting that you skip the 5s.
Then you can do 1s, easy because they should know the sequence orally, difficult because the numbers aren’t marked on the strip.
The students will lose their places on the tapes. That’s fine. Once they hear the number that came before, they will be able to catch up. Either, they will be forced to go right back to the beginning and count out (because they haven’t yet realized that the number on the tape always represents the same fixed quantity), which is valuable practice, or they will, hopefully, eventually, realize it’s easier to go to the closest anchor of 5 or 10, and work out from there where the next number is.
For example, “The previous number was 34, I can find 34 because it is one down from 35, and now add 2 more, that makes 36.”
Over and above developing a true understanding of the skip counting process, this activity builds estimating skills, builds fluency with number and the distributive property, and sets the ground work for operations with ‘friendly numbers.’
Follow the same activity to count by 2s.
In Ontario, grades 1 and 2 have to count by 1s, 2s, 5s, and 10s, starting from a multiple of those numbers. Grades 3 and higher have to be able to count by those numbers from any starting point, and be able to count by other factors as well. It’s easy to adapt the circle counting activity by changing the factor being added, and/or changing the starting point.
Using the tapes forces the students to figure out what comes next, by adding/subtracting the right amount to/from the previous number. This is better than just hearing the pattern and parroting it back. Also, as patterning becomes algebraic study in the later grades, the students become familiar with the recursive process, or the algorithm of skip counting (n=n+1), so the formula will have meaning when they eventually learn it.
What concrete thing are we counting exactly? Make sure that the students understand we are counting centimeters, “the spaces between the lines.” This might require some groundwork on what centimeters are, in measurement terms.
If centimeters are too abstract, use counters of some kind, small enough to fit between the lines on the tape. Place the correct number of counters, directly on the tape, with each new iteration of the pattern.
In and Out
There is a game played in the Junior grades where the players give a number to put in to the ‘machine’ and the leader gives the number that comes out (input and output). For example, a student says 5, the leader writes 10. A player says 10, the leader writes 15. You can use these tapes to figure out the algorithm for what happens to the input to get the output. By marking on the number line where the input is, and the output is, commonalities will reveal themselves, and a rule will be developed.
The patterning in this case lays in the repeated operation. What is always happening to the number given to get the new number. Incidentally, this is an A pattern. +1, +1, +1, etc. is A, A, A, etc..
Essentially, any number pattern game you can think of can be done with these measuring tapes. If you have games you use to explore the numbers on 100 charts, you might be able to adapt them to these number lines, taking advantage of the linear nature as compared to the array layout of the 100 chart.
If you have any other ideas for ways to use the measuring tapes to build number pattern knowledge, please share them in the comments below.
How do you feel about number lines? Are they only for the weak kids? Do you use them in your math instruction?
I never used to, until last year when I was discussing them with a colleague who’d been doing some extensive reading on early number routines to use in her grade 1 class. She told me about what she’d read, and how she’d been using these counting models with her students. By the end of the talk, I was sold.
I had a grade 2 class, which is expected to count in various ways up to 200, and I always need to make sure I’m getting as much bang for my buck as possible. So, I took the concept of number lines, tied in the research I’d been doing on patterning instruction, and decided to bring in measurement, for good, ahem, measure.
The project I came up with was to have the students make their own meter measuring tapes. This is a very rich, hands-on task that provides 100 repetitions to reinforce the length of a centimeter. It gives practice printing numbers and skip counting. And it allows for many opportunities for you and the students to assess and problem solve.
Here are the instructions for how we made the measuring tapes, complete with cautions.
In my next post I’ll share ideas for how to use them, and I’ll invite you to add your own suggestions too.
Narrow rolls of masking tape
Bristol board or Cash Register Rolls
Fine tip permanent marker/Ball point pen
1. Prepare strips of paper that are over a meter long. You can do this with lengths of cash register paper, or by cutting strips of bristol board, about 5 cm wide. You’ll need to cut the bristol board sheets width-wise, and tape or glue two short strips end to end to make one strip that’s long enough. The bristol board will be sturdier and less likely to tear when the measuring tapes are being used. And besides, we have about 500 sheets of pink that no one wants to use.
2. Run a line of masking tape along the edge of a meter stick, leaving a portion of each of the centimeter marks showing. You should do this if expedience and materials are factors. Straight taping is not the goal of the lesson. Be thoughtful about how you’re placing the tape. If a particular student is a bit “rough” with her things, don’t let the tape hang over the edge, or it will be twisted, bent and torn before long.
And, a word to the wise, if the meter stick is also broken down into half centimeters and millimeters, cover these marks entirely with the tape so that only the centimeter lines are showing. Otherwise, some of your more industrious and less attentive students will mark every single dash on the ruler.
3. Have the students use the marker or ball point pen to mark only the centimeter lines on the tape. Demonstrate that they should go only about to the middle of t
he tape with each line. Some guiding questions are “What number do we start the dashes at? (Zero) How many dashes will you have when you’re done? (101) What is the word we use to describe the distance between each dash? (centimeter)”
I promise that some students will skip dashes. Others will do all the millimeters anyway, because they can still sort of be seen through the tape. Whatever the error, each is an opportunity to reinforce the concepts of centimeters and standardized measures. I.e., Dashes that are closer together are smaller than a centimeter. Missed dashes mean that there are some spaces that are more than 1cm long. Not following the dashes exactly means that some “centimeters” will be bigger than others.
If an error happens, it will be better in the long run to pull the tape off and start again. Otherwise you’ll be dealing with those stray, scribbled out lines every time the student tries to use the ruler. But keep note of the mistake for your records.
4. Once you’ve checked all the dashes, peel the tape off the ruler and stick it down the center of a prepared paper strip. Don’t pull the tape too taught, or the whole thing will curl. If the tape tears, just carefully place the two ends together on the paper strip. You’ll never notice.
5. Have the student mark the desired numbers under the appropriate dashes, along the paper.
NOTE, I said along the paper. This is because there is every likelihood that your students
will miss a number, repeat a number, print the numbers backward or in reverse order, or partition the numbers too close or too far apart. IF they write the numbers on the tape first, you will have to redo the whole dash process as well as fix any number situations.
This process is an excellent opportunity for you to assess and remediate your students’ number knowledge. You can consider having them use charts or exemplars in the room, or on the actual meter sticks, to find and fix their errors. For example, rather than saying “your fives are all backwards,” you could say “you made one kind of number backwards every time – see if you can figure out what to fix.” If you’ve pre-questioned the child about how many dashes she had to draw, and she’s written “100” five dashes too early, question her about why she still has five dashes left.
6. Transfer the numbers to the tape. After you have confirmed that the numbers were all written properly, in the right order, with the right spacing, it is now just a matter of the student copying the correct numbers onto the corresponding places on the tape, using the correct numbers on the paper as the guide. Note, that one or two students might start from scratch, missing the whole point of writing the numbers on the paper first. Does anyone come to mind?
7. Put each child’s name on the tape itself as they complete the work. Make any notes about their knowledge or learning in your mark book. Then cut the tape away from the paper strip.
You might want to do the cutting yourself, unless you are very confident in your students’ scissor skills.
Here are some points to consider when making these.
Decide what increments you want the students to use. Last year I had them do it by 1s. This was very time consuming, it was not easy for the students with poor fine motor control, and there were a lot more errors. This year I had them work by 5s. This gave me the chance to see who could count by fives (three could not), and I have the added advantage of forcing the students to think in terms of anchors of five and ten when they use the rulers in the future.
Some students will need a lot of hand holding, either due to physical or intellectual issues. Anticipate who these students are, ahead of time, and allot time accordingly.
Make a couple of extras in case a tape gets damaged when you start using them.
Check back for the next installment where I tell how I’ve been using these tools.
by Michael Ian Black
Illustrated by Debbie Ridpath Ohi
Not plain, yet simple, this book is FUN!
A little girl, let’s say a precocious 4, beautifully rendered with simple wood-cut lines, is bored, bored, bored, blahhhhh, bored. And so is the disagreeable potato she has befriended.
“What do you like to do?” asks the little girl. The potato makes a suggestion. But, seeing how flamingos are scarce, she must do her best to entice the spud with other distractions.
She demonstrates games and tumbling, to no avail.
So she must resort to her imagination.
Finding the ballerina and lion tamer banal, the starchy little antagonist goads the child on, each new page turn revealing more and more elaborate scenes that fill with detail, colour, and emotion, until she’s HAD IT with the tuber’s bad attitude, and storms off.
Dynamic use of text layout and vibrating line work make this picture book a funny and entertaining read, but the funniest bits come from the incorrigible potato.
I’m big on cross-curricular integration. The Ontario curriculum is huge, with more than I feel is possible to cover in any meaningful way. And I hate gym. So, if I can
kill cover as many birds as possible, yay for me.
Here is a game I call Don’t Throw Your Junk in My Backyard (after the song), and some ways to integrate math and physical activity.
First, here’s the game.
Have the class divide itself into 2 or 4 groups, each team getting a half or a quarter of the gym. Or you can divide them, unevenly, and see how fast they figure out about equal division and same sized fractions.
Put out “Junk,” which is any piece of equipment that can be lobbed from one side of the gym to the other, safely, such as bean bags, soft balls, foam frisbees, etc. Have a fixed amount that can be shared equally among the groups.
Blow the whistle and have the students pick up and lob or slide (whatever you feel is safest – unless you enjoy filling out insurance forms) the “Junk” from their side to the other team’s side.
Give a time limit and blow the whistle again when the time is up.
Have each team count the “Junk” in its yard. Get ideas for the fastest way to count these, so they can GET ON WITH THE GAME ALREADY!
Determine which team has the least “Junk” And how much less (many pieces fewer). Working with a friendly number, like a ten, or even 100 can build number sense.
Discuss the winning team’s strategies for clearing their junk quickly, and challenge the other team to incorporate those strategies next round.
In The Garden
by Peggy Collins
Published by Applesauce Press
This is a joyous concept book that takes us through three seasons and the life cycle of the backyard garden. It stars a preschool-aged boy who is discovering the wondrous changes that happen between putting seeds in the ground and enjoying the fruits of his labours.
This book is full of big, bright illustrations. Each mixed media drawing teems with life, every page crawling with the little and large creatures that we share our small corners of paradise with. The art is cartoony and exaggerated, fun and energetic.
The writing is full of gentle humour, in a voice that is easily grasped by young children, but still holds appeal for the adults doing the read aloud. And it has some lovely observations that reflect a real understanding of the way a little person sees the world.
Teachers, use it in units that explore life cycles, plants, the interaction between man and the environment, and our food sources. It is also great for studying descriptive language, and text features like the creative use of font colour and size.
Collins has another book in this series called In The Snow, and several others that she has authored and/or illustrated. Her blog is listed in the side menu.