BCTF New Teachers’ Conf: Seeing Dots February 27, 2016

I am delighted to be here in Richmond, BC, today presenting at the BCTF’s New Teachers’ Conference. I am doing a similar workshop to what I did at the Calgary City Teachers’ Convention two weeks ago, but it is well worth the repeat in this city!

I cannot say enough how important it is for students to be able to visualize and represent numbers in many forms. This tool, the 100-dot array, offers one tool for students to be able to use regularly and thus internalize the number relationships that can be seen when using it.

As before, I am making the handouts available here for downloading:

• the session handout
• large 100 dot array
• 100 dot arrays – 4 per page
• 100 dot arrays – 6 per page
• 100 dot arrays – 12 per page
• “Anchor to 100” sheet

I will upload the extra large dot sheet (a quarter portion of the regular sized one) which can be made into a poster-sized array once I am home with access to my scanner. Watch for that in the next few days!

Let me know how things go with your students!

Mathematically yours,

Carollee

Calgary City Teachers’ Convention: Seeing Dots February 10, 2016

The 100 Dot Array remains one of my favourite tools for helping students visualize numbers. This session at the CCTC focuses mainly on its use with students in grades 2 and 3, although it can be used at many other grade levels. We will be talking about the best way to introduce the tool to students, showing an early activity to help with general number sense, and using the number in problem solving situations. A variety of problem are included to show its diverse use.

Here are the downloads available from the session:

• the session handout
• large 100 dot array
• 100 dot arrays – 4 per page
• 100 dot arrays – 6 per page
• 100 dot arrays – 12 per page
• “Anchor to 100” sheet

Please let me know how it goes with using the 100 dot arrays with your students! I love to hear about kids using tools and strategies in math.

Mathematically yours,

Carollee

GAD Workshop, Surrey, BC October 23, 2015

Thanks to the teaching staff of GAD Elementary in Surrey, BC, for their warm welcome and heartfelt participation as we delved into problem solving, math tools and strategies, and math processes (especially communication). Changing our teaching practice is not an easy feat, but if we commit to some small changes, practice them regularly, add more changes, practice those regularly, and keep on going in that manner, we can end up making a significant and lasting change that will benefit students greatly.

Remember, “math talk” does not just happen. We have to plan ways to incorporate it into each math lesson. It is a good idea to create math partners so students are responsible to talk to someone about their math thinking. Modeling (letting students hear YOU talk through a demonstration problem) is always a good idea. Responding to students with proper math language/vocabulary (when they have not used such) is helpful. Posting “sentence stems” is a great way to give them an easier start in speaking math. Additionally, try creating a “math words” chart with the students that they can use as an on-going reference in both their speaking and writing (click here to see an example of a “math words” chart.)

As promised, I am adding links from this post to the handouts from today’s session (see bottom of the post) and some that we just talked about.

I would LOVE to hear from any of the GAD staff of how things go in your math lessons in the next weeks. You all listed something that you could begin to do right away in your classrooms, and I hope you will share what you are doing and the effect it is having on the students.

Remember, understanding “lives” in the processes! Reflect on your teaching regularly to see if you are embedding those processes into math classes. It will make a big difference in students’ understanding if they are immersed in the processes!

Mathematically yours,

Carollee

 

Download materials here:

100 dot array (teacher size)

100 dot arrays 4 per page

100 dot arrays 6 per page

100 dot arrays 12 per page

break apart number sheet – 2’s

break apart number sheet – 3’s

problem solving assessment rubric

10 frames (teacher size)

10 frames (student size)

10 frames blank mini’s

“Number of the Day” Sheets: Choosing the Number January 5, 2015

I recently received an email from Stephanie, a grade 2 teacher in Newfoundland, inquiring about choosing the number for the Number of the Day sheets:

“I really love the Number of the Day sheets you have produced and the opportunities for differentiating the instruction. Just wondering how you set this up? Do children do this everyday or on designated days? How do you decide on the number for the day?”

I thought others might be asking this same question, so have decided to post an edited version of my response to Stephanie’s question:

As for setting up the Number of the Day sheets, things are really flexible. There is no one right way — you want what works best for your students and your time constraints. That being said, I have found that if you are able to have the kids do them really regularly (daily if possible) over a good number of weeks, the students are able to really get into the meat of them. By sharing about them after they have worked them, students get to hear what others have tried and will often stretch themselves to try to match what others are doing. They have a chance to really play and explore the number relationships that are brought out in the sheets’ activities.

The number can be picked in a variety of ways — everything from you choosing, a student choosing, drawing a number from a jar or dropping a bean on a 100 chart! Sometime I have chosen specific “repeats” (e.g., every number that week has a 9 in one’s place) sot the kids to see and compare what happens in such cases. What is the same as before? What is different? Or I might pick several numbers within a “decade” (e.g., 33, 37, 31, 35, 38) and again have students compare/contrast over those days.

No matter what number is chosen, one question that is really great to ask is “What do/did you notice?” When that is asked often in the math classroom, students get in the habit of paying attention to details, looking for patterns, making comparisons, and such.

I am happy with random numbers, too, but sometimes choosing numbers with a particular relationship is good so you can really draw out the depth of the relationship.

I hope you will try the Number of the Day sheet(s) with your students. More information about the sheets as well as the downloadable versions, can be found in the links below.

Mathematically yours,

Carollee

Number of the Day Sheets to Download:

Level 1 (English and French)

Level 2  (English and French)

Level 3 (English and French)  (pictured above)

 

 

 

What’s Important to Have in a Grade 1 Classroom? October 2, 2014

I was recently contacted by a former colleague, Dawn, regarding what manipulatives a grade one classroom might need to have on hand to support effective learning math. It seems a friend of Dawn’s is in a classroom which really has nothing for the children to use for hands-on math learning and they were wondering what was needed.

First off the classroom needs counters — counters in different shapes, sizes, etc. They can be purchased ones (such as mini plastic teddy bears) or ones gathered from home (such as bread tags, but†ons, etc.). But the need to be abundant and available.

Students need a way to count efficiently, especially in tens and ones. Egg cartons cut down to 10 holes, blank 10-frames printed on paper or card stock, or commercially produced 10-frames can all be used. I even like using cookie sheets (non-aluminum) and marking them with coloured tape as a giant 10-frame for use with magnets.

Base 10 blocks are also great for young students. These a generally in the form of small 1 cm cubes for “ones”, sticks for the “tens”, and flats for the “100’s”. I do want to make a critical point here: students may be engaged in a game of trading 10 cubes for a stick, or 10 sticks for a flat with every appearance of understanding the “ten-ness” of our base-10 number system. But be careful here. Student can be following your rule of trading 10 for 1 without that understanding. They might be just as happy to trade 8 for 1 or 12 for 1. The manipulatives give a opportunity for students to develop that important base-10 understanding, but moving blocks around correctly does not necessarily indicate that the understanding has been built in the student’s mind.

I think a grade one classroom needs “pop cubes” (multi-link cubes) — those blocks about 1inch in each dimension that can be attached together. I like to store them sticks of 5. If students need a particular amount for an activity, say 18, we discuss how many sticks each student will need, and then go get them. I also use these in many quick number-sense building activities. If I have students hold up a certain number of blocks, I want them to do so to model a ten frame. If I ask for 9 blocks and a student were to hold up a single stick of 9, I, as the teacher, cannot tell from a distance if the student is holding 8, 9, 10, or 11 blocks. But if he holds up a five stick beside a four stick, I can tell at a glance that he has the correct number. Pop cubes can be used in a multitude of math activities and should be on-hand for regular use.

Another must-have in my book are pattern blocks. They are particularly great for patterning activities for exploring symmetry, not to mention the creativity factor! I love them!

There are a number of things that I think should be in the classroom that are “make-able” such as dot cards, dot plates, printed ten-frames, even printed dominoes (click for more info on these)— all useful in exploring numbers, in building number sense, and  in helping students develop the skill of subtilizing. Students need to SEE the numbers in math, and these materials can help develop that “seeing” in the children.

Of course there are many other things that are fun to have in the math classroom, such as dice, dominoes, blocks, playing cards, geoboards, plastic coins, bingo chips, square tiles, Cuisenaire rods, and two-colour counters, to name a few. But lots of math learning can take place with some thoughtfully crafted lessons and activities and just the basics.

I hope you will focus on the math understanding with whatever materials you have at your disposal!

Mathematically yours,
Carollee

Math in Nature July 15, 2014

When you are spending time in the outdoors, one of the mathematical things you can look for is the connection in nature to the Fibonacci number sequence. Of course, one must know the sequence in order to recognize when it shows up in nature, and it is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34…
Do you see the pattern? It begins, with 0 and 1, and after that each successive number is the sum of the two previous numbers. Thus the number after 34 is 55, derived by 21 + 34 = 55.

The sequence shows up in nature in a variety of ways. For many flowers, the number of petals that they have is a Fibonacci number. Looking at the bottom of pinecones, the number of spirals formed, whether left spirals or right spirals, is a Fibonacci number. This is true for pineapples and sunflowers as well. The number of seeds found in fruit is often a Fibonacci number. The next time you are eating an apple or orange, or squeezing lemons for lemonade, stop and count the seeds! Even when looking at how plants branch off a stalk or grow their leaves we can see the Fibonacci sequence.

Click here for one great site for delving into this further— there are many others out there, too!
Happy number hunting!
Mathematically yours,
Carollee

 

Mathematics is Much More than Numbers on a Page May 22, 2014

One of my favourite quotes about mathematics: “Mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music, but the notation and the music are not the same; the music itself happens when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive; it exists not on the page but in our minds. The same is true for mathematics.” — Keith Delvin

Are your students really DOING mathematics, or just working with notations on a page?

Mathematically yours,

Carollee

Early Counting: the Foundation of Math May 12, 2014

The meaning attached to counting is the most important idea on which all other number concepts are developed.

Counting Involves at Least Two Separate Skills:

• A child must be able to produce the standard list of counting words in order: “one, two, three, etc.” This must be learned by rote memory.
• The child must be able to connect this sequence in a one-to-one manner with the items in the set being counted. In other words, each item must get one and only one count. This important understanding is called one-to-one correspondence.

Meaning Attached to Counting:

There is a difference between being able to count as explained above and knowing what the counting means. When we count a set, the last number word used represents the magnitude or the cardinality of the set. When children understand that the last count word names the quantity of the set, they are said to have the cardinality principle.

Give a child a set of objects and ask, “How many”? After counting, if the child does not name how many are there (as, “There are 7 of them,”), then ask again, “How many?” If a child can answer without recounting, it is clear he or she is using the cardinal meaning of the counting word. Recounting the entire set again usually means that the child interprets the question “How many?” as a command to count.

Almost any counting activity will help children develop cardinality.

• Have the child count several sets where the number of objects is the same but the objects are very different in size. Ask the child to talk about this.
• Have the child count a set of objects, and them rearrange the objects. Ask, “How many now?” (If the child sees no reason to count again, likely the child has a good sense of number and has developed cardinality.)

Happy counting!

Mathematically yours,

Carollee

Number of the Day – Level III March 10, 2014

Today I am posting the third Number of the Day sheet. I cannot overstate that I believe that elementary school students should be involved with numbers everyday they are in school!

Level III is one to primarily use with numbers to 100. The section “100 chart tic-tac-toe” will not be familiar to most. I had devised that math game based on the positioning of a number on the 100 chart. For instance, if 26 is written in the centre of the chart, then the middle line is to show one more and one less than 26. (25, 26, 27 across). Above the middle number is 10 less, in this case 16. Below 26 is 10 more, 36 in this case. The corners can then be filled in using the horizontal or vertical relationships already established. (For more on the use of 100 chart tic-tac-toe, see my previous blog post.)

When using 100 dot arrays, I have students use highlighters to colour the numbers. I also stress marking efficiently – we do NOT colour each individual dot; rather a line or partial line is coloured with a swipe of the marker.

At every level breaking apart the number of the day is an important component of the sheet. Quoting John Van de Walle once again, “To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers.” [Van de Walle, J. and Folk, S. (2005). Elementary and Middle School Mathematics: Teaching Developmentally (Canadian Edition). Pearson: Toronto.]

I did have one teacher ask a question about the breaking apart section. She was used to having students only break apart numbers according to tens and ones. Thus 26 could be broken apart as 20 and 6 or 10 and 16. But sometimes it is easier to work with numbers when we break them in ways other than ten and ones. Consider the thinking that might happen when adding 26 + 27. If a student knows that 26 comes apart as 25 an 1 and that 27 comes apart as 25 and 2, it is easy to put the 25′s together to get 50, then add the 1 and the 2 —total 53. Students who use the 100 dot array often get especially comfortable with 25′s. Also consider adding 97 and 36. If a student notices that 97 is just 3 away from 100, it makes sense to split 36 as 3 and 33. Breakng apart in tens and ones are definitely useful, but so are other “break-aparts”. If students do not practice this kind of thinking they are not likely to ever do it!

I had one teacher here in my district that was using this sheet and her students were getting tired of making tallies for large numbers. So I am including a second English version of the sheet asking for equations for the number instead.

Again, a French version is offered as well with thanks to my friend and colleague Lynn St. Louis for her translation.

 Download the English version (tallies) here.

Download the English version (equations) here.

Download the French version here.

Mathematically yours,

Carollee

Calgary City Teachers’ Conf 2014 February 17, 2014

It was wonderful to share the Friday morning session at the Calgary City Teachers’ Conference with so many new friends! I hope you walked away with some ideas for helping your students understand mathematics is a deeper way. Congratulations to Shannon Muir who won the math coaching session in the draw!

If you remember one idea from the morning, I hope it is one about building understanding in math. Students need to make sense of the concepts using first concrete materials, then with pictorial representations, and then with symbolic (or numeric) representation. Rules for manipulating numbers are not remembered well if they are not based in meaning. Caine and Caine report from their brain research, “The brain resists meaninglessness.”

As promised, I am posting here the tools we used and referred to for your easy access.

100 dot arrays (1 large)

100 dot arrays (6 per page)

100 dot arrays (12 per page)

ten frames (teacher size)

ten frames (mini blank ones, 40 per page)

base 10 grid paper (enlarge as needed)

fraction & percent circles

fraction pocket chart    (link here for more discussion about these)

I think that is everything. If I have missed something let me know. And I would love to hear how this all makes a difference for your students!

Mathematically yours,

Carollee