Focus on Math

Helping children become mathematicians!

“Why am I doing this?” August 25, 2016

Filed under: General Math — Focus on Math @ 6:51 pm
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Screen Shot 2016-08-25 at 6.36.45 PMIt is nearly the beginning of the school year here in BC, Canada. New beginnings are always special with untold opportunities and challenges before us. As a teacher, educational assistant, or parent who is home schooling a child, it is important to keep one question at the forefront and ask it of yourself daily: “Why am I doing this?”

I am particularly interested in this question in reference to math lessons because I think the intent behind a lesson matters greatly. For most of us, in our past experiences in school, the main point of doing anything in math was to get the answer, and usually with the added hope of getting it quickly and efficiently. Full stop. The answer was what was important. Indeed, it was the only thing that was important.

But is producing an answer quickly really enough? Consider some other intentions you might have for a lesson:

  • I want my students to be able to explain how they arrived at the answer.
  • I want my students to be able to explain why the answer they got makes sense.
  • I want my students to see how this particular aspect of mathematics fits into the bigger mathematical picture.
  • I want my students to understand this math concept and be able to explain the concept with concrete materials and/or visual representations.
  • I want my students to understand how this mathematical concept relates to the real world.
  • I want to make sure that all of my students have an opportunity to link this new knowledge with previous math knowledge.

I hope your answer to the question is not, “This is the next lesson in the book,” or “The curriculum mandates that I teach this.” Both of those things may be true, but I urge you to rethink your lesson and focus on an intention that will make a difference for the learning of your students.

Mathematically yours,

Carollee

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Seeing Dots: NCTM 2014 New Orleans Presentation April 11, 2014

Screen shot 100 dot arrayI am excited to be here in New Orleans at the 2014 NCTM conference. Yesterday was a great day of sessions for me, and I am delighted to be presenting a session in just a couple of hours! “Seeing Dots: Using Arrays to Add, Subtract, Multiply and Divide” will focus on all the different ways the 100 dot array can be used to help students visualize and represent numbers — something which leads to a deeper understanding of numbers.

I am posting the handout from the workshop as well as links to 100 dot arrays is the different sizes.

I hope you try using the 100 dot array in your elementary classroom!

Download the conference handout here.

Download a 100 dot large array here.

Download 4 arrays on a page here.

Download 6 arrays on a page here.

Download 12 arrays on a page here.

Mathematically yours,

Carollee

 

A Thought for Today April 2, 2014

Screen shot 2014-04-02 at 8.43.24 AM“A typical classroom narrows our thinking strategies and answer options. The teacher insisting on a ‘right answer’ is NOT healthy for growing a smart, adaptive brain. Good quality education education encourages the exploration of alternative thinking, multiple answers, and creative insights.”Eric Jensen

What kind of classroom do you have?

Mathematically yours,

Carollee

 

Number of the Day – Level III March 10, 2014

Num of day tally picToday 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.numero du jour III pic

 Download the English version (tallies) here.

Download the English version (equations) here.

Download the French version here.

Mathematically yours,

Carollee

Num of Day III eqn pic

 

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

 

 

Calgary City Teachers’ Conf 2014 February 17, 2014

Screen shot 2014-02-17 at 9.04.21 AMIt 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

Screen shot 2014-02-17 at 11.27.57 AM

 

Wanted Posters Revisited February 4, 2014

wanted poster picDoing wanted posters for numbers is a great way to have kids think about specific properties of particular numbers. The posters make a great display, too —  and I am always looking for ideas for math bulletin boards!

The idea is useful at lots of grade levels. You could have older students choose a proper fraction (e.g., 5/8), a mixed number (e.g., 4 1/3), or a square or cube root (e.g. the square root of 50).I wrote about these posters before — click here for the link to the previous write-up where you can download the template.

Give them a try and send me a picture of your display!

Mathematically yours,

Carollee