An “Equalizing” Question: Moving Sheep May 26, 2011

Earlier this week I wrote a question for my grade two classes to work on. I had decided to try an “equalizing” question with them, where they would have to use addition and subtraction to equalize two groups. The final draft of the question was this:
A farmer has 12 sheep in one pen and 28 sheep in another. He wants to move some sheep so both pens have the same number of sheep in them. How many does he need to move? How many sheep will be in each pen?

I had originally put different numbers into the problem. I thought about using 14 and 36, or 18 and 44. But in the end, I went with 12 and 28 knowing that a) the end number of sheep in each pen would be the “friendly” number 20, and b) that when students represented the number using ten frames or on a 100 dot array, they would see clearly the eight from one pen could be moved over to the eight “open spots” in the other pen.

Even with presenting this easier set of numbers, most of the students in both grade two classes that did the problem found it challenging. Some just wanted to add or subtract the two numbers and use the resulting sum or difference as the answer. Some split the 28 into two groups of 14, but thought they were done. I strongly encouraged the children to draw a representation of the sheep in the original pens and work from there. Though not all students were able to arrive at a solution in the time frame (in my Wednesday primary classes I have only 30 minutes with each class to set up the problem, let them work, and share solutions!) many did find at least one strategy that worked for them. The students also figured out during our discussion that the method of splitting the 28 sheep into the two pens would have worked had they gone on to split the 12 other sheep into the two pens as well. We had 5 good strategies shared/figured out even in our limited time frame.

If you do this problem with grade one students, I would recommend you present fewer sheep (say 8 and 14, or whatever combination you think your students can handle). For grade three students you might also try larger numbers, but make sure the students have some strategies in manipulating and/or representing numbers so they can be successful with the problem. Grade four and five students could be given a problem that has them equalizing with decimal numbers (so obviously sheep are out!).

It was clear to me as my students worked on the problem that more equalizing problems are needed to help them make sense of what is happening in such situations. And next year I will be starting earlier doing “equilizing” questions with my classes.

Mathematically yours,
Carollee

Flowers in Pots: A Multiplication Problem May 20, 2011

This week I gave my grade three classes a problem solving question that was a double-digit by single-digit multiplication problem. This level of problem is clearly above the mathematics for grade three here in BC, but I knew it was a good problem because it got the students thinking deeply about multiplication. A number of misconceptions surfaced during the lesson and we were able to talk about those both during the working time (when I visited the small groups) and in the sharing solutions time. The problem was this:

Clara planted 13 pots of flowers. She put 6 flowers in each pot. How many flowers did she plant?

The students used a variety of tools in working to solve the problem. They used tally marks, drawings, hundred charts, 100-dot arrays, and blank number lines (student drawn, and only important numbers marked). Some used only symbolic form (digits only) but there were a number of different strategies used to add up all of the flowers.

This problem is “rich” in that, although there is only a single correct solution to it, there are multiple ways to arrive at that solution.

I like to tell students that being able to come up with different ways to solve a problem is like flexing math muscles, and that it makes them “strong like bull” (which I say in a heavily accented voice while I strike a “he man” pose! The kids love it, but more importantly, they get what I mean!

Help your students become “strong like bull” by having them do some problem solving.
Mathematically yours,
Carollee

Math Toolkits for Students — More Stuff to Add (part 3) May 19, 2011

There are more items that can be added to the toolkits for students, but these I will separate by primary (gr 1-3) and intermediate (gr 4-7) levels. Again, it is hard to just mention the contents without going into activities that use the tools to help students build mathematical understanding. Hopefully the tool itself will prompt you to think about some ways to use it.

Primary Tools:

• 25 chart, laminated (usually created in 5 rows of 5)
• blank 5-frame (with spaces big enough to put counters on)
• blank 10-frame
• blank double-10-frame (two blank 10-frames on one card)
• set of filled in 10-frames (1-9, multiple 10’s)
• bead bracelet (10 beads in two colours, 5 of each) to be worn draped over the fingers so the beads can be manipulated. Two bracelet may be worn to use for numbers in the teens.
• large flattened paper plate or cut out paper circle for making dot plate configurations with bingo chips
• mini bags of small coloured wooden sticks or other small materials for patterning
• teeny-tiny Hundreds Tens and Ones (HTO’s) — miniature place value pieces cut out of large plastic canvas (found in crafting stores)
• place value cards — overlapping cards that show, for example, 425 can be pulled apart to reveal 400, 20 and 5 (click on image above to print)

Intermediate Tools:

• booklet of mini 100 charts to be coloured in to show multiples (x2, x3, x4, etc.)
• metre tape (purchased or created by taping photocopied paper lengths together)
• fraction-bar card (a card with a fraction bar in the middle — students use numeral cards to place as the numerator and denominator)
• fraction percent circles (two different coloured circles partitioned off in hundredths each cut along one radius and then placed together so they “spin” over each other to show different percent values)

As you can see, there are many things that can be used as “tools” in the teaching of mathematics. Creating a toolkit with students is a wonderful way to make lessons engaging.

Mathematically yours,
Carollee

Math Toolkits for Students: The Basics (part 2) May 17, 2011

In the first toolkit post I set the background for toolkits, so now let’s look the the really important part — what goes into the tookits. I will first list items that I have used and recommended for all students in grades 1 to 7. (Please note that the toolkit idea may even be useful in grades 8 and 9, but I have not personally used them in those grades or carefully looked at the curriculum to see which tools might be useful in a toolkit.)

Toolkit Contents for grades 1-7:

• A response board and appropriate tool for writing
• A large, laminated 100 chart and/or 0-99 chart
• A large, laminated 100-dot array (see illustration — I LOVE this tool!!!!)
• A mini-deck of cards (playing cards ace to 10, one of each)
• A set on numerals 0-9 (two of each is best) along with symbols for “greater than”, “less than”, and “equals” — also a decimal point for older students
• Bingo chips or punched-paper circles (in a snack-sized zip baggie)
• A piece of string (random length for each child), wrapped around a piece of box-board to keep it “tidy”
• A ruler marked in cm (also mm for older students)
• Pattern blocks (either real, or die cut out of construction paper, in a snack-sized zip baggie)
• Blank spinners, with pre-marked sections — paper part only
• Paper clips to use as the spinning part of “fast spinners”
• small mirror(s) — hinged ones are fabulous!!

In reality, it is hard for me to put this list out there without stopping at every item and going through a set of activities that uses the particular tool — thus I do a workshop about creating and using toolkits! But for now I will just post the list and elaborate if someone has a question about a particular item.

Mathematically yours,
Carollee

Math Toolkits for Students (part 1)

When I worked with a group of teachers last week, one of them asked about math toolkits for students. I have talked a lot about this idea in my district, and even did a workshop on the topic a couple of years ago. It has come up again as a topic of interest in a number of teacher groups, and I thought it would be a good idea to post some ideas about toolkits here.

I first heard about the idea of creating a toolkit about 15 years ago when I went to a workshop offered by Kim Sutton. I loved the idea then, and still do. In fact, as I work with children all day at Charlie Lake School each Wednesday, I have a toolkit in each desk for students to use during the day. When I had my own classroom, each student individually had his or her own toolkit. The way I have my one-day-a-week room now is that a toolkit must be shared with all the children who happen to sit at that desk during the day (I teach 8 groups of students during the day ranging from grade 1 to grade 5). I confess I do not like the shared tookit quite as well as each student being responsible for his/her own toolkit, but it is still working out pretty well.

I have used some of Kim Sutton’s ideas for the toolkits, and added other ideas of my own, all with the purpose of helping children “do mathematics”.

So this post does not become unwieldy, I will talk about some general components of a toolkit, and separately post some specific ideas for primary and intermediate toolkits.

The Toolkit Container:
I have always used a large, Ziploc-brand freezer bag. I prefer the “squeeze to close” kind, without the little white slider, as I found that those sliders come off too easily with consistent use. We call them “large plastic bags” only on the very first day when they are passed out. After that they are always referred to as “toolkits”. I liked that the bags were basically flat, and I always have had students store them right in their desks. For me, this is part of the “power” of the toolkits: they are easily accessible for students to use, whether in an activity directed by me, or by their own choice as they work to solve problems.

Although plastic bags are my personal preference, some of the teachers in my district have used other things. One teacher would go out at the beginning of the school year when school supplies are on sale and purchase large, plastic cases for her students. These became the math toolkits and they were stored at the side in the students’ cubby-holes. A variety of containers would work — they just need to fit the kinds of things you are going to put into the toolkits.

Building the Toolkit: My personal philosophy about toolkits is that I do not give them to students packed with tools. I remember hearing about a mechanics program in which all of the would-be mechanics were given empty toolkits, and as they went through the program and learned how to use a particular tool, then the tool became part of each student’s toolbox. At the end of the program the toolboxes were full, but the students knew how to use all of the tools inside. That is my thinking for the math toolkits. On the day I give the bags and introduce the toolkits, I also give out the first tool and we use it together in class. Thereafter each time a new tool is introduced, we work with it as a class and then it goes into the toolkits.

I’ll begin to talk about the contents in my next post.

Mathematically yours,

Carollee

In the Dog House May 16, 2011

I found this question in Everybody Counts (Mathematical Sciences Education Board, 1989, p. 32) and thought it would be a great one to do if your class is learning about area.

Design a dog house that can be made from a single 4 ft. by 8 ft. sheet of plywood. Make the dog house as large as possible and show how the pieces can be laid out on the plywood before cutting.

This is a great place to say that even though here in Canada we are officially a metric country, we are bombarded in real life with the Imperial system of measurement. I, for one, think students should be “bilingual” in both metric and Imperial! If Imperial measurement are not in your curriculum, it is still fine to work with those units. It just means you don’t assess for a grade regarding Imperial measures.

Mathematically yours,

Carollee

What Does it Mean to “Do Math”?

We all sat through many, many hours of math classes, and we “did math” daily through many years of school. But have you ever stopped to ask yourself what it really means to “do math”?

In 1989 the Mathematical Sciences Education Board (MSEB) published a landmark book titled Everybody Counts, and on p. 31 there appears this wonderful quote: “Mathematics is the science of patten and order”.

Most of us understand that “doing science” is more that just getting quick answers. Rather, we associate with science many verbs such as these:

• investigate
• solve
• justify
• predict
• verify
• explain
• represent
• develop

Of course, there are many more verbs that could be added to the list, but what is important in all of them is that they indicate a process of “figuring out” or “making sense” of the science. All of the verbs require active participation and deep thinking about the ideas. It is virtually impossible to be a passive observer and “do science”. Science is really about figuring things out, and it always begins with some problematic situation.

Mathematics should be this way also. Students should regularly be required to figure out the mathematics of a situation, to work to solve mathematical problems. And what is amazing about this process is that, in doing that hard work of problem solving, students come to build an understanding of the mathematical concepts. My personal term for this is “noodling” — I guess I derive that from the idea of “using your noodle.” 🙂

Traditionally in North America we show students how to do something (e.g., multiply decimals with tenths such as 3.5 x 2.3) and then have them practice a page of practice problems. But rarely do we them give questions that require them to show an understanding of the mathematics behind the procedure, to demonstrate with models or with diagrams what is going on in the multiplication. Just why does multiplying tenths times tenths give us an answer in hundredths? Sadly, many students (or adults!) can’t answer that — they just know they are supposed to “count the decimal places in the numbers being multiplied and count that many in the answer”.

Later, as students go on to use calculators more, few often stop to look at the answer displayed and ask themselves if the answer is reasonable. Clearly the answer to 3.5 x 2.3 should be between 6 (3 x 2 — both numbers rounded down) and 12 (4 x 3 — both numbers rounded up). But if a wrong number was accidentally entered or a decimal left out in putting in the data, few students ever notice.

A single mathematical problem for which students have to “noodle out” the answer is almost always a better plan than having students to a page of practice problems that has no conceptual meaning for them. We need to provide ways for students to build an understanding of mathematics.

Let’s give students the opportunity to really “do math” in a meaningful way!
Mathematically yours,
Carollee

A Small Change but a Big Result May 13, 2011

Yesterday a teacher in our district, Kevin, shared with me a wonderful story about math in his classroom. Kevin and I go back a ways — I was the Faculty Associate when he completed his Professional Development Program (student teaching) through SFU, and he had also taken the “how to teach math” course from me. He was hired in my district, and he has spent most of his career teaching at a rural K-12 school where for the past number of years he has taught the gr 8-12 math courses.

Kevin told me of his frustration, as well as his students’ frustration, in the math classes over the last years. He would present a lesson, have the students begin working on the problem set in the text book, and then have the students do the remainder of the assigned problems for homework. However, invariably the students had difficulty with the homework problems and would become increasingly frustrated with trying to solve those problems. The next day Kevin often felt he needed to get on with the new lesson, but clearly time was needed with these homework questions which, being later in the practice group, were often the more difficult problems in the set.

Kevin remembered the emphasis I had put in the “how-to-teach-math class” on the power of students solving problems, and he decided to change up the class time with his students to see if he could incorporate more problem solving in his class.  So, instead of this:

• teach a concept
• do the first, easier problems in the practice set in class, students working together
• send home the later, harder problems in the practice set, students working alone,

Kevin changed his time with students to look like this:

• do the later, harder problems from yesterday’s lesson in class, students working together
• teach a new concept
• send home the first, easier problems as homework, students working alone.

Kevin found this to make a profound difference for his students. Where many of them had felt unsuccessful in math, never being able to complete the homework on their own, they were now able to do so. This began to build their confidence. When they worked on  the harder problems together in school, most students found the knew most of what to do. Sometimes they were forgetting only a small step. Kevin realized when the students had tried the problems at home, if they could not arrive at the answer at the back of the text, most would erase all of their work, believing they were completely off track. Working together in class students had the opportunity to make connections to previous lessons, to communicate their thinking, to reason about the logic of what they were doing, to justify their answers. By engaging in these processes day after day, the students began to build a set of problems solving skills and strategies than empowered their mathematics thinking.

The three “chunks” of the lesson really did not change, but in changing the order in which they happened (which in turn changed which problems were addressed through the mathematical processes**) Kevin facilitated a change in student understanding and success. Way to go, Kevin!

I hope you will look at how math is going in your classroom and see if you need to turn it “upside down”!
Mathematically yours,
Carollee

P.S. The NCTM lists the process standards as these: connections, communication, problem solving, reasoning and proof, and representation. More information about these can be found at this link:
http://www.nctm.org/standards/content.aspx?id=322
In BC, and for the members of the WNCP in Canada, the mathematical processes are defined as these: communication, connections, mental math and estimation, problem solving, reasoning, technology, and visualization. More information about these can be found in the “front matter” of the BC IRP curricular documents found at this link:
http://www.bced.gov.bc.ca/irp/subject.php?lang=en&subject=Mathematics

Get “Illuminated”!

I was talking to some math teachers yesterday, and one of the things I mentioned to them was a great site run by the NCTM (National Council of Teachers of Mathematics) called Illuminations. The site is a treasure trove of lesson plans and activities for grades K-12. Many of the lessons have interactive applets embedded in them and, as such, are suitable for using with a white board.

Searching for particular lessons is easy. You can create a specific search by choosing a grade band, math strand, and then adding specific terms (for example, gr 3-4, geometry, with the words “right angles” typed in) or choose to cast a wider net and use broader criteria in your search. There are some really wonderful ideas there, and I encourage you to bookmark the page and try some of the ideas presented.

Remember as you look at lessons and/or activities that they are grouped in four grade bands: K-2, 3-5, 6-8, and 9-12. You will need to be aware of the curricular requirements of your province or state and check the lessons and/or activities against your curriculum. It is, of course, never a problem to “stretch” students beyond the given curriculum, but for assessment purposes, it is important to assess against the standard set by the grade level curriculum.

So, here is the link:
http://illuminations.nctm.org/

Go have a look! I know you will find some great ideas there!
Mathematically yours,
Carollee

Fraction Question for “Thirds” May 12, 2011

First, let me say what a wonderful year of collaboration it has been with the teachers from Alwin Holland Elementary School. Thank you, ladies, for your hard work in mathematics this year. We all came away having more informed than we were about mathematics education.

Part of our last day together was spent in two different classroom where I did demonstration lessons. In the grade one classroom I demonstrated the teaching of the +9 addition strategy (as learned in the basic facts addition blog) using 10-frames.

In the grade 2/3 split classroom, the teacher was interested in a fraction lesson that centered around problem solving.

I started the lesson by talking about a class which was made up of 1/2 boys. I asked the children if they thought the class had only two students in it, one of which was a boy. They all agreed that, no, the class was larger than two, and they offered suggestions of how many might be in the class, and then how many of them would be boys. At one point I asked if they thought 23 might work for the class number, and they knew that being an odd number, it did not work to divide the class in half. I then presented this question that I had written for the lesson:
Marcie grabbed some bingo chips from a bag on the table. After sorting her chips, she told her friend that 1/3 (one-third) of them were red. What might her chips have looked like? Find as many different solutions as you can. Each solution should use a different number of chips altogether.

The students had bags of bingo chips to share in their table groups, and they used the chips to find a variety of solutions. All of the children were able to come up with some solutions, and a few of them realized they could keep adding one more chip to each of their sets to find more answers.

When we shared our solutions I put the information on a T-chart with one side labeled “number of red chips” and the other “total number of chips”. Students gave their answers in random order, and after we had quite a list, I suggested we look for a pattern. I made a new T-chart with the same headings, but we listed our answers in order of number of red chips: 1, 2, 3.. No one had given an solution with 7 red chips, but it was easily figured out by looking at the pattern that presented itself. The students recognized the pattern in the second column was “skip counting by threes”. Then I proposed some big numbers of red chips such as 20, 100, 1000 and they could answer the total number of chips to be 60, 300, and 3000 respectively. Finally we looked for the “n-rule”: for n number of red chips, how many chips altogether. At first students guessed other large numbers (500, a million, etc.), but finally one boy said “It would be n + n + n.”
It was the first look at the n-rule for the students, but the teachers were motivated to go back to some other patterns and play with that idea with children. Anytime we help kids move to that point, we are laying the foundation for algebraic thinking.

All in all it was a fabulous day! Thanks again to the collaboration group from Alwin Holland School!

Mathematically yours,
Carollee