# Focus on Math

## Helping children become mathematicians!

### Use What You Know to Figure Out What You Don’t KnowMarch 7, 2014

I was working with some students this week who were learning their “basic facts” in multiplication. These are generally considered to be those one-digit times one-digit problems that we use when we figure out the products of multi-digit problems. I was going over some different strategies and ways of thinking that can be used to help students learn those facts.

There are a number of strategies that can help in the learning of basic facts, but one phrase sums up many of those individual strategies: “Use what you know to figure out what you don’t know.”

This phrase actually applies to FAR more than just the learning of basic facts. The truth, however, is that often we condition students to NOT think for themselves in mathematics. We have a long tradition of teaching by telling: the “here’s how to do it now go practice 50” method. In reality, that kind of math lesson programs students to think that unless someone has told them “the way” to do something (and, of course, they must remember exactly how to follow the directions of “the way”). If they forget, they are stymied and cannot know how to proceed. They remain in their “stuck” position until someone comes to rescue them with “the way”.

It is far better to regularly encourage students with the idea that when they are stuck, they need to stop and think about the things they DO know that can be applied. We might ask questions (and teach them to ask themselves) such as these:

• What might be something similar that you do know?
• If the problem had smaller or simpler numbers, how would you try to solve it?
• Why did you choose to do it that way?
• What is important in the question?
• Is there a pattern?
• Is there a way to record what you have done so far so you a pattern might be noticed?
• Can you think of another way to do that?
• Does this remind you of another problem you have done?

In the case of basic facts, “Use what you know to figure out what you don’t know,” might look like this: a student cannot remember 6 x 8. But 5 x 8 is known. So, knowing that 5 groups of 8 is 40, he need only add one more group of 8 to have the needed 6 groups of 8; thus 40 + 8 = 48 is the solution to the unknown fact.

Students may need practice in doing such strategies, but the important thing is that there ARE strategies to help. It removes the case of having to rely solely on memory and sitting there stuck if memory fails.

What are you doing in your classroom today that encourages students to help themselves when they are stuck? Maybe post the title phrase for them (and model for them how it looks): “Use what you know to figure out what you don’t know.”

Strategies make a difference in student learning!

Mathematically yours,

Carollee

### Step Out of Your Comfort ZoneJanuary 21, 2013

Most of us tend to teach mathematics in the same manner as it was taught to us. I think of that as our “default setting”. We are comfortable with it.; it “feels right” to us. Unfortunately, it is often not the best way to teach math (which is why most of the North American population does not understand mathematics!).

To teach otherwise, to use strategies and approaches that we did not experience in or school years, requires real effort to change. It makes us uncomfortable; it does not “feel right”.

I believe that when we teach mathematics meaningfully, we need to have students doing more than just following our instructions. When we show them how to do a particular computation (e.g., 27 x 46), demonstrating each step of the computation that leads to the answer, their subsequent work (i.e., the 50 problems to do on the page) only shows to us whether or not the students could follow all of the necessary sub-steps in order to arrive at the final answer. Such work does not show any understanding of multiplication, nor does it show that the students understand why the sub-steps produced the answer.

I contend (again!) that “understanding” lives in mathematical processes. The National Council of Teachers of Mathematics (NCTM) lists 5 math processes, namely these:
• Communication
• Connections
• Problem solving
• Reasoning and Proof
• Representation

If we regularly incorporate these processes into our mathematics teaching, students cannot help but build mathematical understanding!!

I will add one caveat: you cannot add the processes for a week, examine the results, and say, “this doesn’t work!” The truth is, we must help students build skills in these areas. If they have not been talking and/or writing about their math thinking already, such communication will take time to build. If students have not been problem solving (in the truest sense of the word) then they will need to learn some strategies and approaches to help them solve problems. A similar case can be made for making connections, reasoning and proving, and representing.

But building competency in the processes is worth the time that it takes! When students are doing the hard thinking in math (and not just following rules that are meaningless to them) you will find you and your class enter a new place of teaching and learning!

Will you step out of your comfort zone so the students can go “where the magic happens”?

Mathematically yours,
Carollee

### Chess: A Power-Packed GameNovember 8, 2012

My interest in chess was stirred again recently after reading an article in the on-line magazine Education Week. I learned to play the game as a young girl and have always found it to be a fascinating challenge in logical thinking and problem solving.

It turns out that chess is, indeed, it is an excellent way for children to begin to develop thinking skills that will serve them for life. The game provides opportunity for players to anticipate moves, to think ahead, and to begin to solve multi-step problems. The game can help children improve memory, increase their skill in planning and strategizing, and just generally improve cognition.

Salome Thomas-EL, a teacher and principal in the Philadelphia School District for many years, was also a chess coach to many students from schools in low socio-economic districts with great success. In one of the three schools he was hired as a “turn-around principal, 96% of students were living at or below the poverty level. Yet the students excelled, with much of the credit going to the after-school chess program that had a profound impact on how the children think. Not only did he have students go on to win local, state, and national chess championships, but many of these students from impoverished neighborhoods beat the odds and went on to university and graduate school.

El writes this about the impact chess can have on students: “So many young people are raised to question their intelligence. Chess helps shatter that doubt. Chess teaches our young people about rewards and consequences, both short- and long-term. It challenges young people to be responsible for their actions. It cuts across racial and economic lines and allows poor kids to excel at a game thought to be reserved for the affluent. It boosts self-confidence. It is the great equalizer. Students must learn that they are not born smart, but become smart through hard work and the process of growth.”

Some years back I ran a lunchtime chess club at a Duncan Cran Elementary School. I really did not know much about how to teach the game well (El contends that children as young as grades 1 and 2 can begin using a few pieces, or even all the pieces), but I opened my classroom and invited students to come and play each other and myself. Here in the northern Canada we often have “inside days” in the winter when it is deemed too cold for sending students out for recess and/or lunch, and chess club offered a great activity for those inside days. The group that met regularly to play the game was certainly focused on thinking strategies as they faced each other across the chessboards.

I hope you will consider teaching your students (or other children in your life) the game of chess. And to help you do that, you can download some ideas for teaching chess here. There are, of course, many resources available to help you out, many availale with the click of a mouse.

Don’t miss the opportunity to help students learn that they can become smart through hard work – and have some fun while they are at it.

Mathematically yours,
Carollee

### Place Value not Face ValueApril 20, 2012

One cannot understand the numbers in our number system without understanding place value. Simply put, where a digit is situated in a multi-digit number, or its place in the number, determines the value of the digit. Thus, although 32 and 23 both are made using the same two digits (a 2 and a 3) we know that 32 and 23 are not equal. The 3 in 32 is in “ten’s place” and actually carries a value of 30 (or 3 tens) whereas the 3 in 23 is in “one’s place” and has only a value of 3. Each of the 3’s have a “face value” of 3, but one has a “place value” of 30. The same can be said for the other digit 2 in each of these numbers: in 23 the 2 carries a value of 20 (or 2 tens) whereas the 2 in 32 has only a value of 2 ones or just 2. Since 3 tens has a greater value than 2 tens, 32 is greater than 23.

We have traditional algorithms for doing each of the four main operations (addition, subtraction, multiplication, and division). An algorithm, according to Wikipedia, is a “step-by-step procedure for calculating.” It goes on to say, “More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating.” Thus, for each operation there is a list of rules or instructions which, when followed, effectively produce the solution. That is a good thing, right?

Well, the answer to that depends on whether the goal is to produce answers quickly or to understand mathematics. Following rules requires little or no understanding of the mathematical concepts underlying the procedure. Worse, all of those who do research regarding the brain and learning report the same thing: the brain remembers things that have meaning; isolated bits of knowledge that are not meaningfully connected to other knowledge that we have are discarded. Sadly, the rule-based approach to mathematics is rather devoid of meaning. The user often has no idea why the rule works. It is no wonder, then, that teachers spend weeks in the beginning of a new school year “reteaching” things that have been forgotten over the summer. Without meaning behind the rules, the rules are easily forgotten.

Let’s look at the thinking processes involved in some computational strategies using the problem 26 + 39. Most of us were taught in such a manner as to think the problem through like this (see example a): “6 plus 9 is 15, so put the 5 under the 9 and carry the 1. Now 1 + 2 + 3 equals 6, so put the 6 under the 3. Reading the bottom numbers together, the answer is 65.” Certainly that set of rules works, but do you notice that the “1” being carried does not, in reality, have a value of 1. It has a value of 10 — we just neglect to refer to it that way. In like manner we ignore the fact that the 2 and the 3 are worth 20 and 30 respectively, but we again never refer to this fact. You might argue that this is not really a problem — we are adding two 2-digit numbers and it makes sense to get a 2-digit number as an answer. But I will say, from many years experience as a classroom teacher and as a mathematics specialist, that the more complex the adding (or whatever operation) gets, the further removed from logic students get. It is, in fact, quite easy to make a mistake leaving out a whole place value spot and get an unreasonable answer, but students do not notice such things. They are not looking at the place value of any of the digits in the problem, and thus have no reference for really thinking about the answer.

I like to have students work with non-traditional methods of computing that keep the place value of digits. I offer three other such methods for your consideration (all of which, by the way, have been used to good effect by young students whom I have taught, and as recently this year!). There are many other ways of non-traditional computing — this is not meant to be an exhaustive list, but rather to give some examples of other possibilities.

Looking at example b, the thought process might sound something like this: “20 plus 30 is 50, so I will record that. 6 plus 9 is 15, so I will record that. But I see that 15 includes a ten which I can combine with the other tens, giving me an answer of 65.” For example c, the thinking might go like this: I see that 39 is very close to the friendly number 40. If I add one more I can use that friendly number. Now I can mentally add 26 plus 40 (adding 4 tens and 2 tens) and get 66. Since I added one extra to make the friendly number, so I must remove it**. My answer is 65. Finally, for example d, the thinking might be this: 39 is very close to the friendly number 40 — just one away. I can take one away from 26, leaving 25, and move that one to add it to 39 to get 40**. Now I just have to add 25 plus 40 (adding 4 tens onto 2 tens) to get the answer of 65.

Examples b, c, and d all keep place value of each number, a critical component, especially when first learning about the operations or different levels of the operations. Later, when understanding is firmly in place, students can move to the “traditional algorithms” and not lose sight of what is happening conceptually. But conceptual understanding needs to be developed for the operations — which takes time. It is not something accomplished in flash.

I could give similar examples of meaning-based methods of calculating for each of the other three main operations, and maybe in a later post I will do so. The point, however, is that we want to students to be thinking about the meaning of the numbers that they work with, to be thinking about the place value of each digit in the number and not just the face value of each digit. It is an important component of understanding mathematics.

What will you do tomorrow to ensure that your students working meaningfully with numbers?

Mathematically yours,
Carollee

** Both of these examples involve using the “zero principle” — that for any number when you and and then subtract the same amount, you do not change the number. Thus 39 + 1 – 1 = 39, because + 1 – 1 combine to be zero. This zero principle is actually a very powerful principle in addition and subtraction and can be used effectively to simplify many problems.

### Basic Facts: Mental Math as a Foundation for Multiplication Fact StrategiesMarch 16, 2012

Basic Facts are still very important. Although newer curricula put a greater emphasis on problem solving, communication, reasoning, and representation of numbers, basic facts are still an integral part number sense in students. If a student is a good “memorizer”, then learning the multiplication facts will not be difficult. However, for many students the random bits information we call “facts” don’t stick well in the brain (the brain tends to remember information that is personally meaningful!), and thus it is important that we support those students in their learning by teaching and rehearsing thinking strategies.

Before we look at those particular strategies that are useful for learning the multiplication facts, there are some “prerequisites” to consider. Many of the strategies I will be suggesting use some kind of mental math to help students go from a known fact to an unknown fact.

The mantra for students is this: “Use something you KNOW to get to something you DON’T KNOW!” This is a comforting thing for students, particularly those who have struggled with learning their facts. They tend to feel that there is no hope for them. In some cases they have worked for a very long time, even several years, to memorize these facts, and at this point they feel like it is a hopeless task. We need to offer hope in the notion that they can begin their learning with things they do know, and build from there.

Consider working with your students to build these kinds of skills, remembering to tie them to concrete and/or visuals (such as ten frames or 100 dot arrays):
• Subtracting a single digit number from a multiple of ten (e.g., students use the known fact of 10 – 6 to solve 60 – 6. Tie into ten frames).
• Subtracting a double-digit number from a multiple of ten (e.g., build on the previous skill and have students solve 60 – 16 by subtracting first 10 and then 6.)
• Doubling any 2 digit number, using whole-part-part strategies if necessary. It may be easy to double 12 (think 10 + 10 + 2 + 2), but it will be harder to double 16. Students might consider 16 as 15 + 1, then double each of the two parts, and add back together (think 15 + 15 + 1 + 1 — look for “friendly numbers ending in 5’s or 0’s).
• Adding any single digit number to a double-digit number, particularly when the sum of the one’s place digits is greater than 10. E.g., 35 + 7 can be considered as 35 + 5 + 2; 48 + 6 can be considered as 48 + 2 + 6. (Pull apart the number to be added in a way that makes a group of ten.)
• Subtracting any single digit number from a double-digit number, particularly when “regrouping” would be required. E.g., 54 – 8 can be 54 – 4 – 4. (Again, break apart the number being subtracted into parts that make the work easier.)

It is well worth the time that you invest with students doing mental math. In As well as being a great life-skill, mental math allows students to be flexible with numbers and use powerful thinking strategies.

Mental Math and Basic Facts — don’t skip these important things!
Mathematically yours,
Carollee

### Candy Bars: a grade 2 or 3 questionNovember 15, 2011

I recently gave both my grade two classes at Charlie Lake School a question involving groups of six. We had been working on a variety of problem-solving strategies including these:

• using counters/objects
• drawing pictures
• using ten-frames
• using a 100-dot array
• using a 100 chart
• using a blank number line
• breaking numbers apart
• using operations such as adding and subtracting
• looking for patterns
• using charts or tallies

I posed this question to the students:
Luke is buying candy bars to share with his classes. They come in packages of 6. How many packages will he need to buy if there are 25 students in his class?

The students glued in their question strips and we read the question together to make sure they understood what was being asked.

I will stop here and give my opinion on an issue. I know there are a lot of math text books out there with lots of writing in them, and teachers have told me that some students who are not good readers, but are better with numbers, have little success using books that require a great deal of reading. I like using a problem-based approach to mathematics, and find that, when I am focusing the lesson primarily on a single “rich” question, then I can read the question with the students, make sure students understand what is being asked, and set the students to work. Although I am a great proponent of literacy and want students to be accomplished in that area, I do not want reading to hold a student back in my math classroom. A rich question, again in my opinion, is one with multiple possible solutions OR a single solution with multiple strategies for finding the solution (or both!). I use many of the latter, and encourage students to find as many solutions as they can. In many cases, the more ways they can solve the problem, the greater their understanding of the concept.

So, back to this particular question. The students went to work solving the problem of candy bars bought in packages of 6. The photo shows some of the strategies shared in the one class. The discussion was quite interesting. The students realized that buying four packages of candy bars would get Luke 24 bars, but most did not want him to purchase another full package. They were suggesting it would be better if he then went to a convenience store and bought only one more bar, which, in real life, is a great idea!. I asked the students, what if Luke were in a hurry and had to buy only packages, and they all agreed that he would need to buy 5 packages to have enough. There was then a discussion around the extra bars: he could save them; he could sell them; he could give them to his family. Lots of good ideas!

It is important to have those kinds of discussion around division and any remainders that come up, because in real life things are much more likely not to divide evenly than to do so. The remainder must be considered carefully. If students were doing this in a “standard” way, they might be likely to say that 25 divided by 6 is “4 remainder 1” without ever considering what the remainder of 1 would stand for. In this case, it would be a student without a candy bar! The actual answer to this question is 5, not 4 remainder 1. This is a case of the answer being forced up to the next whole number.

Remember, have your student discuss the remainders!
Mathematically yours,
Carollee

### Missing sheep: a grade one problemNovember 7, 2011

I recently got to do a problem-solving lesson with some wee folks in a grade one class. (Thanks, Alison, for inviting me into your classroom!)
The teacher had been working a lot with 10-frames and various number relationships, but as of yet had not done math in the format of giving a question for the students to solve using strategies of their choice.

Before I gave the word problem, I introduced the “tool” that I had brought along for the students to use: mini blank 10 frame. I showed them the little 10-frames, already cut apart, and explained that they could be glued into their math journals where they would show their thinking.We spent some time discussing that students could either draw circles in the boxes of the 10-frame, or colour in the boxes, to represent a particular number.

I also introduced the word “strategy” (and its plural “strategies”) to the students. I stressed that once a solution was found using a particular strategy, their job was to look for another strategy that would solve the problem.

I gave the students the problem. Each students received a slip of paper with the question on it which they glued into their journals. I had a large copy of the question which I put on the board with a magnet (see photo). We read the question together and I made sure each student understood what it was they were to find. Here is the question:

A farmer had 12 sheep in a pen. Someone left the gate open and some sheep got out. When the farmer counted his sheep, he only had 5 in the pen. How many had escaped?

This is a subtraction question, as something is “lost” or removed, but it is an interesting question because the change number is missing. It sounds like it should be written in this form: 12 – ? = 5.  The missing-change format allows a problem to be solved by both addition (5 + ? = 12) and subtraction (12 – 5 = ?). I should mention here that I deliberately chose 5 as one of the numbers for the question, knowing that 5 is a very easy number to work with on a 10-frame.

I encouraged students to draw a picture to help them figure out how many sheep had gone missing, but I also suggested that the little 10-frames might help them. From there I “turned them lose” to do the hard thinking math sometimes requires. Alison and I circulated during the working time to prompt, question, suggest, and encourage. A number of students had difficulty getting started, and I continued to suggest the two “highlighted” strategies. Most students were able to do one or both of those strategies. A few students used other strategies that used more symbolic notation.

During this work time I was also making mental notes of which students I wanted to invite to share their strategies. I have found, especially with week folks who have the attention span of a gnat, that I need to make the best possible use of the limited sharing time. Pre-selecting students allows me to make sure a wider variety of strategies is shared to the whole group. Students were given about 12 minutes to work, and then I asked several students to share.

As students shared I wrote on the chalkboard. You will see the ten-frame strategy represented. One boy told how he used a 10 and a 2 to represent the 12 sheep. He circled five in a row to represent the sheep who did not leave the pen, then counted the ones not circled as being those that left.

Another student shared about drawing a picture. She drew a rectangle to represent the sheep pen, and then put 12 circles in the pen to represent the 12 sheep. She put X’s on the 5 sheep that stayed, they counted the ones not crossed out as the ones that had left.

One boy I talked to during the work time had nothing on his paper, but when I talked to him he proceeded to tell me how he manipulated numbers mentally to get the answer. I scribed for him as he told me this: “I know that 6 + 6 = 12, and that one of the numbers I have is 5. It is like taking one away from six, so I had to put that one with the other 6.” Thus he turned the double 6 + 6 into 5 + 7. It is important that students who do the work in their head learn to put their thinking down on paper. Mathematicians need to be able to represent their thinking~

One other boy shared that he knew if he did 12 – 5 (which represented the five sheep still there in the end) that the answer, 7, would show how many sheep had left.

Did you notice the word “represent” showing up in the sharing? That is another word I use regularly with students. I actually stress when students are drawing pictures to solve a problem, that we are NOT in ART class and that the pictures do not have to be fancy. We only need to represent the problem so we can think about it. If you use the word “represent” with students, students will use the word, too!

I was delighted with the thinking the class did during this first experience with problem solving! If we teach/model/share tools and strategies with students, they can become powerful problem solvers.

Mathematically yours,
Carollee