Focus on Math

Helping children become mathematicians!

Calgary City Teachers’ Convention: PS February 10, 2016

It is my pleasure to present this session “Power Up Your Problem Solving” to the participants of this session.

Regular problem solving is a powerful way to help students develop conceptual understanding in the various strands of mathematics. Since there is a tradition in North America of “teaching by telling” (the “here’s-how-to-do-it-go-practice-50-of-these” method), it may take many weeks to develop a culture of deeper thinking in a classroom. Students need a variety of thinking tools and strategies to work with, as well as skills and practice in talking about math problems, but the time it takes to help students gain these needed things is time well spent. The payoff is huge!

I hope many of you will be encouraged to begin building a regular problem solving program with your students. It works at every grade level!

All the here are the downloads for the problem solving session:

I would love to hear from you how it goes in your classrooms!

Mathematically yours,

Carollee

 

Simple Definitions Too Simple? August 10, 2015

Even

Math definitions matter! There are many words we use in mathematics that have one meaning in that discipline and another in ordinary life. Take for instance the word “difference”. In regular conversation, if I ask you to find the difference between two things you are looking for some way in which the items are not the same. However, in mathematics, finding the “difference” specifically refers to finding the answer to a subtraction problem.

But we as teachers might be sending some confusing messages to students, sometimes even when we think we are right on track with our definitions. One example of this is the seemingly easy-to-define term “even”. How would you explain to a young child what an even number is?

There are two popular ways this property of even is explained to primary students: First, many teachers suggest that we can do an “even check by examining whether a particular number of items can be split into two equal groups. Armed with this definition, children should see that six is even since there can be two groups of three, but five is not even because there are two groups of two and one left over.

Alternately, teachers often suggest that students look at the value of the one’s place digit of the number in question. If there is a 2, 4, 6, 8, or 0 in one’s place, then a number is even. Using this method children should conclude that 74 is even since there is a 4 in one’s place, but 73 is not since there is a 3 in one’s place.

The problem is that both of these simple definitions are not fully correct. There are exceptions to them that, in fact, that are incorrect.

Concerning the “two equal groups” definition, young children figure out quickly that when sharing 5 cookies between two friends, each can have 2 ½ cookies. There are two equal groups, but the number 5 is still an odd number. What important detail have we failed to communicate here?

Concerning the “one’s place digit of 2, 4, 6, 8, or 0” definition, a student can declare that 74.3 is even since it fulfills the definition stated. Again, what important detail have we failed to communicate here?

Some might argue that these exceptions above the student; that we need not muddy the waters, so to speak, by giving extraneous information that we think is above our students’ heads. I disagree. I feel that if we just mention the restrictions as we talk about the definition, that it becomes part of the language the students are used to. We often underestimate how much students can understand, and we “dumb down” the language as a result. I would love to challenge your thinking along that line. In the case of primary children in particular, they love to learn what I refer to as “27-syllable” dinosaur names, yet we are afraid of using good math language with them!

I hope you will stop and think about the simple definitions you are using with your students and reflect on whether or not there are some hidden exceptions that need to be teased out and exposed.

Mathematically yours,

Carollee

 

10 New Year’s Resolutions for the Math Classroom December 31, 2014

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1. praise effort, not correct answers
2. make sure my students know their intelligence is not fixed: hard work pays off
3. make my classroom a safe place for students to take risks
4. encourage students to take risks
5. give my students rich problems that require they engage in problem solving
6. build a class repertoire of strategies
7. have “thinking tools” handy
8. give regular attention to basic facts (for students who do not know them)
9. give students lots of opportunity to talk to each other when solving problems
10. support math vocabulary learning with a word wall chart

Mathematically yours,

Carollee

 

Welcome to SUCCESS! July 31, 2014

Screen Shot 2014-07-31 at 12.36.30 PMAs I write this the summer is half over — at least for students and teachers in BC, Canada. Schools are generally out for July and August, and then begin after Labour Day in September. I know in other places school will resume in mid- or late-August.

Whenever it begins for you, my question is this: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

For me it was simply this: WELCOME TO SUCCESS!  I had cut the letters for that saying out of construction paper 12 inches high (one letter for each page) and I stapled the message above the chalk board at the front of the classroom.

I talked about student success many times each day for the first few weeks. I basically inundated the students with the message that they would succeed in my classroom because I would not let them fail. I would do whatever it takes to work with them to be successful throughout the year. Failure was NOT an option — this was a classroom of successful students! I even went so far as to tell them that it was their lucky year getting  me for a teacher! Oh, there would be work involved along with lots of learning, talking, thinking, wondering, solving, thinking, testing, proving, thinking, recording, demonstrating, thinking… But we would be working together as a class and each and every student would be successful.

I was especially vocal about success in mathematics. I was teaching grade 6/7 in the early years of my “WELCOME TO SUCCESS” campaign, and it was clearly evident that a large proportion of the class came to me telling me they did not like math and that they were not good at it. I knew the real story was that they did not UNDERSTAND the math and they were not good at remembering all the rules. My plan was to work continually with the concepts in the mathematics knowing that once they understood they would get better and be more confident. My promise was to help them be successful even in an area of study they thought they could not be successful in.

The wonderful things was, of course, that my statement of declaration proved true year after year. All of the students WERE successful! They believed me when I declared it (I guess I said it and they read it so many times that they could not help but believe it!) and ultimately their personal belief regarding their personal success was a turning point for them.

I will ask my question again: what tone do you set in those first days/weeks of school? What is the most important message that you relay to your students?

Mathematically yours,
Carollee

 

Breaking Apart Numbers: Practice Sheets May 27, 2014

Screen shot 2014-05-27 at 10.57.33 AMI have mentioned before about the importance of breaking numbers apart, of having students understand that every number can be broken into smaller numbers. I have included this practice on all of the Number of the Day sheets (level l, level ll, level lll) that I have posted, but it warrants adding these other sheets that focus on this Whole-Part-Part number relationship.

Regularly practicing this skill can change students’ thinking. They will be much more likely, in any given situation involving numbers, to look for alternative ways of thinking if they have spent time pulling numbers apart in many ways.

Remember, the students are not starting with a new number in every circle. Rather they are using one particular number for a line or for the whole page! In different situations it may be more beneficial to break a number apart in one way than it is in another way.

Mathematically yours,

Carollee

download the breaking into two parts sheet here

download the breaking into three parts sheet here

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Host a Parent Night in Math March 9, 2014

Parent night pic

As I have worked with teachers both in this district and in other districts regarding changing their math practice, there is often another element that needs to be addressed. Parents of the students in the class begin to wonder and ask questions about how things are being done in the classroom. Parents notice that instead of a page of problems all done using the exact same “formula” or algorithm, a lesson may be structured quite differently, possibly around a single question! It seems so foreign and strange, and parents cannot help but ask, “What’s going on in math? Why does it look different than when we went to school? The other method worked for me – why, I passed math, so shouldn’t things just stay the same?”

One of things I do to support both teachers AND parents is to hold a “Math Night” for the parents of a given class or school. This is NOT meant to be a fun “Family Math Night” that is set up like a carnival with a variety of stations, all with activities centered on math topics. Those are wonderful events and can be an exciting way to expose parents and children to many interesting components in math, and they certainly have their place. I would encourage any class or school to host such an event!

However, there is a need to actually address mathematical issues with parents, so I am talking about a parent meeting that is meant to be something deeper, something to challenge the “why?” of how we have long taught mathematics. Such a meeting is meant to invite parents to think about what it means to “do math” and why it is “better to do one problem five ways than five problems one way” (Polya). I am asking parents to challenge the notion that just because they were taught a certain way does not make it an effective method of teaching.

Knowing that we are all busy, I keep the time frame to a minimum, but I usually plan for about an hour.

My Math Night plan looks something like this:

  • Welcome and other necessary starting info (e.g., washrooms for young children)
  • Introduction of me – who I am and how I am involved with the class/school/district (done either by the teacher/principal hosting the meeting or by me). If you are hosting for the parents of your own students, this step is, of course, unnecessary!
  • Posing a problem: how many ways can we find to solve a problem
    • Doing the problem (parents actually doing the kind of work I ask students to do!)
    • Sharing our methods for solving the problem
    • Drawing conclusions about the thinking that was taking place
    • Rethinking philosophy about the teaching and learning of mathematics: why it is better to really think in math class and not just do pages of (usually) meaningless problems
    • Questions and Answers

Parents just want what is best for their children, and we want to help parents understand something about mathematics curriculum, and in so doing, grasp a vision of deeper mathematical understanding for their children.

I’d love to hear from you if you host your own event!

Mathematically yours,

Carollee

 

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

Screen shot 2014-03-07 at 10.36.15 AMI 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