Focus on Math

Helping children become mathematicians!

Suspended Sense Making… February 23, 2012

Filed under: General Math — Focus on Math @ 11:15 am

“Suspended sense making” – that was the phrase that stood out to me recently at the BCAMT New Teachers’ Conference that was held in Richmond, BC. Ray Appel, the keynote speaker, spoke how students plow through math questions focused on trying to do the right operation, to get the right answer, but in their quest they often take leave their sense making.

Research from the last decades points to this particular problem (e.g., Radatz, Freudenthal, Schoenfeld, Verschaffel, and DeCorte). Radatz, in a study with 300 children in grades K -5, gave them some unsolvable problems such as this one: “Katja invites 8 children to come to her birthday party, which takes place in 4 days. How old will Katja be on her birthday?”
While only about 10% of the K and grade 1 students tried to solve the problems, the percentages of grade 2 students (about 30%) and grade 3 and 4 students (about 60%) were much higher. The percentage for grade 5 students went down some (45%) but still reflected a poor showing.

Similar results have shown up in other research. In another study other unsolvable problems were given individually to 7-9 year olds and 9-11 year olds. They were asked things like this: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?” Each problem was followed by the question, “What do you think of the problem?” Only 12% children aged 7-9 and 62% of the children aged 9-11 reacted by saying they could not respond properly to the questions. All of the others performed some kind of arithmetic operations on the numbers in the questions without expressing any doubts. One pupil’s response went something like this:
“125 + 5 = 130 …this is too big, and 125 – 5 = 120 is still too big… but 125 ÷ 5 = 25 … that works… I think the shepherd is 25 years old.” While the student did use some sense-making in thinking about how old the shepherd was likely to be, he did not make sense of the big picture: namely, that knowing how many sheep and dogs were in the field could in any way indicate how old the shepherd was.

Lest we dismiss this as being absurd, I experienced a bit of this “suspended sense making” just last week. I had given a class of grade 3 students this question:
The principal at Charlie Lake School was dividing up all the students into 4 house teams. If there were 157 students in the school, how many students would be on each house team?

My students went to work splitting up the students in the question into the 4 required groups using a variety of strategies. (I was impressed with the overall thinking about splitting up 157 and shall have to write a separate blog about the strategies they used!) Partway through the working time one particular boy had arrived at a solution, and before beginning to work on another strategy he asked me, “Is the answer 39 remainder 1?”

His class had been working on basic facts in division and other division questions “close” to those (e.g., 25 ÷ 5 = 5 and 27 ÷ 5 = 5 r 2) so his answer of 39 remainder 1 would have been reasonable had he been given just the numerical problem 157 ÷ 4. But the numbers were in context, and the ‘157’ in the problem was not a vague, meaningless number, but the number of children in the school. I asked the boy what the remainder of 1 represented, and he responded, “That’s the leftovers”. “What was actually left over?” I asked. It took several moments for him to realize a child was left over. I asked what the principal would do with a leftover child, and the boy suggested that one house team would have to have an extra person because the principal would not just leave one student without a team.

Allowing children to do pages of division problems where the remainders are popped into the answer unquestioned as to their meaning seems, in my opinion, to reinforce the practice of “suspended sense making”. Similarly, when students do a page of any algorithmically-based set of problems with no understanding as to the mathematics behind the rules, they are suspending their sense-making. Chants such as, “Ours is not to reason why, just invert and multiply,” come to mind here — a chant that was just said to me this very week!

What do you think? What other kinds of things might we traditionally do that invite children to set sense making aside?

I believe we have to continually provide opportunities for students to make meaning in mathematics, and we have to expect them to do some explaining, some representing of the ideas in question. We want them to be critical thinkers who realize when things just don’t make sense, and not be lulled into only looking at ‘available’ numbers and randomly trying a variety of operations on them. Students should be asking themselves all the time, “Does this make sense?”

The bottom line: math should make sense for every student, every day!!

Mathematically yours,


We assess what we believe is important. February 21, 2012

Why are some things perceived as a “waste of time”?

I visited a high school math class today where the ‘Foundations of Mathematics and Pre-calculus’ students were beginning a unit on measurement, including conversions between imperial and metric systems. The teacher spent quite a bit of time facilitating a discussion with the students about the concept of measuring: the various aspects of things that we measure, how we might measure things in “difficult” situations (e.g., finding the surface area of an irregularly shaped puddle), and when in life we use particular imperial units of measurement.

It was a great discussion, with the students putting forth many ideas, some wonderfully “out of the box”. As it got to the point that the teacher was setting the students to work doing some of the questions from the text book, he came over to me and made the comment, “I think the kids think this discussion was a waste of time. I know that when I was in school, I would have felt a discussion like today’s was a waste of time.”

I have no doubt that both statements are true – that the teacher remembered feeling discussions about the “big ideas” were a waste of time, and that his students, although they found it interesting, also felt it was a waste of time. The question, however, is why are such discussions perceived as such?

Personally, I think it comes down to the fact discussions about conceptual understanding are not given any honour, any value. Students look at how marks are derived and figure out pretty quickly that those things which are important are what show up on tests, quizzes, and such. If nothing about the big ideas, about the conceptual understanding, about the meaning behind the mathematics is asked when it “counts”, then it seems clear that those things are not valued. They are not important.

We, as teachers, have to decide what is really important in mathematics, and if that includes conceptual understanding, then we must include questions and/or tasks that get to the heart of that conceptual understanding in our assessments. It is not always easy to do – such items will likely not fit into a multiple-choice kind of test to be marked quickly on a Scantron. But again, we must ask ourselves what we value, what we believe is important.

Our assessments will reflect our true beliefs.

Mathematically yours,


100 Day! February 16, 2012

It is that time of year again when primary classes are celebrating the 100th day of the school year. Many primary teachers count the days of school each day in their “calendar” time with children, creating groups of tens with straws, Popsicle sticks, 10 frames or such. So it is a “bid deal” when there are finally 10 groups of 10 to make 100.

I have been part of a math collaboration group with teachers at Alwin Holland Elementary School here in Fort St John, and today they were celebrating 100 day. I joined in one of the activities that students rotated through in groups. Mrs. Hollman and I had arranged for students to make “Fruit Loop necklaces” by stringing the cereal “o’s”. In order for the students to keep track of how many they had put on their strings, we had them string 10 of a single colour, then switch to a different colour for the next 10. The floor was crunchy with crushed cereal by the time all four groups had cycled through, but it was a delightful afternoon! I am sure all of the other stations were equally successful.

And remember, if you have missed counting school days, you can still catch up by counting days in the calendar year. The 100th day this year will be April 9th since it is a leap year! For older students it would be a great day to focus on hundredths and percents.

For more ideas about what kinds of things to do for 100 day, check my blog from last April 5 and 7.

Mathematically yours,


Books! Books! Books! The Literature Connection (part 2) February 7, 2012

It seemed appropriate to follow up the last post with some more of my favourites. The books in these two posts are the ones I use over and over again when I go into different classrooms. I have many more books in my collection, but these are the “go-to” books that I keep reaching for.

Every Minute on Earth by Steve Murrie and Matthew Murrie is a great book to use with students in the grade 4-9 range. This book is chock full of interesting facts about the earth, space, the human body, technology, animals. food, pop culture, and sports. Many pages use many big numbers to tell about the event. For instance, on the page telling that more than 34,000 plant species are threatened with extinction each minute, the authors also tell the readers that seeds for more than 6 million different plant species are stored in 1,300 sites around the world. Of those being stored, about 15% are seeds for wild plant species. There are so many questions that can be generated from the numbers from this one-page story. Sometimes I ask the questions, but often I ask the students to generate (and solve) questions from the information.

Bat Jamboree by Kathi Applet can be used when you want students to explore adding sums of consecutive numbers. The story tells about the number of bats in different groups that are performing at the jamboree, starting with 1, then 2, … up to 10 bats. At the end all of the bats make a pyramid, and I always stop reading so students can figure out ways to calculate the number of bats in the pyramid. Elementary students do not generally come up with the sophisticated algebraic formula that some of you may have encountered [f(n) = n(n+1)/2], but it is rather amazing what patterns they can find.

How Many Feet in the Bed by Diane Johnston Hamm is a wonderful book for using with primary children. In the story a family of five (mother, father, young daughter, young son, and baby) get in and out of the parents’ bed in the course of a morning. It is a counting by two book on that level, and primary children tend to enjoy this. I have followed it by asking children to draw their family gathered in one bed and and telling how many feet. I have also given grade 2-3 children a scenario of a family with dogs and cats as well as children. Then, if I tell them there are 12 feet in the bed, I have them find different combinations of people and animals that make the requisite number of feet. It becomes a great patterning question if the people, animals, and feet are recorded on a chart.

I bought 365 Penguins by Jean-Luc Fromental and Joelle Jolivet sight unseen. I was intrigued by the name, especially having grown up with a sister who was a penguin fanatic (actually, she still is!). I figured any book about penguins and numbers would have to be fun, and I was not disappointed. The premise of the story is that a family is being anonymously sent 1 penguin each day for a year, an oh! the numbers that are generated to play with. There are numbers about pounds of fish, cost to buy the fish, and storing the penguins, just to name a few. Questions can easily be created for addition, subtraction, multiplication, and division. Students love the book — and so do I!! This one is a must-have. (Incidentally, the story was originally written in French, so it would be great to track it down in that language for any French Immersion classes.)

The 512 Ants on Sullivan Street by Carol A. Losi is a rhyme about ants carrying off the food set out at a picnic. It is basically a book about doubling. At the back of the book are some suggestions for activities written by Marilyn Burns. One Grain of Rice by Demi is an Indian folktale dealing with this same principle of doubling numbers. I have used both in the classroom many times.

Although most elementary and middle school curricula do not specifically include a logic component, I am fascinated by how well Akihiro Nozaki and Mitsumasa Anno handle the topic in their book Anno’s Hat Tricks. The reader is part of the book, called “Shadowchild” (since the shadow always can be seen on the page). Two children are in the story, always wearing hats. We are told how many hats there are (always in either red or white) and we can see the hats on the two children. From what we see, we, as Shadowchild, must deduce what colour of hat is upon our own head. (We can, of course, see in the shadow that we are indeed wearing one, but we cannot tell its colour in the shadow.) Each time the hats are changed the challenge level goes up. It is a great book to use with older elementary and middle school students.

I hope you have the opportunity to use one or more of these books in your classroom. I know your students will enjoy the literature connection, and I’m betting you will, too!

Mathematically yours,


Books! Books! Books! The Literature Connection February 3, 2012

Recently the Literacy Support Teacher in our district and I (the Numeracy Support Teacher) joined forces and did a workshop for teachers. Much of what we did that day connected literacy and mathematics through the use of picture books. I am sure that many of you are aware of some great books to use in math lessons, but I shall share some of my favourites here (in no particular order).

Marilyn Burns’ book Spaghetti and Meatballs for All is a classic and can be used in a number of ways in the classroom. It is particularly useful for playing with the concept of perimeter as tables are rearranged to accommodate guests coming for dinner.

Amanda Bean’s Amazing Dream by Cindy Neuschwander is a charming story about a girl who loves to count, but learns that multiplying is a much faster way to count things in rows or groups. I have read this book to many different classes and followed the reading by a multiplication word problem, usually at a “challenge” level for the students. They multiply by using what they know about numbers (breaking them apart, using repeated addition, etc.) to build an understanding of multiplication.

One Hundred Hungry Ants by Elinor J. Pinczes is another classic. It is, in fact, one of the first literature books I ever used in my classroom years ago. It makes use of factors of 100 to divide the ants into even rows, and is a great introduction to factoring.

The Right Number of Elephants by Jeff Sheppard should not be missed. It uses rather fanciful situations and describes the right number of elephants needed to fit the bill. For instance, “When you go to the beach with all of your friends on a very warm day and you simply must have shade, the right number of elephants is 8.” I love reading this book to young children (K and/or grade 1) and then following the story with an activity in which each student thinks of a situation and the right number of “something” for that. For instance, children have written and illustrated these: “the right number of pets is 3” (since she had 3 pets); “the right number of pencils in a pencil box is 5”; “the right number of shoes is 2”. It is a great way to talk about numbers all around us.

Measuring Penny by Loreen Leedy is the perfect book to use to introduce a measurement unit. In the book, Lisa’s homework assignment is to measure something, and she chooses to measure her dog, Penny. The wonderful thing about this task is that there are so many attributes to measure and thus many units of measurement are explored (standard and non-standard, metric and imperial). Students will want to go home and measure their own pets!

If your students are learning about angles, circles, and circumference, you will want to read them The Librarian who Measured the Earth by Kathryn Lasky. This book tells the story of how Eratosthenes calculated the circumference of the earth more than 2000 years ago. His calculation was correct within 200 miles, or 99.2% accurate — pretty good for being done in a very “low-tech” manner, don’t you think?

These are only a few of the books I love using with students. I’ll post more of my favourites later. In the meantime, at the side of the blog you will find an “interesting link” which will take you to a site filled with scores of possibilities for using literature in your math class. It is definitely worth your time to explore the list there.

Happy Reading.
Mathematically yours,