# Focus on Math

## Helping children become mathematicians!

### Math Camp 2013 Reflections…August 28, 2013

Wow! Math Camp 2013 was a resounding success! The focus each day was on how we can structure routine activities for our students that will allow them to build number sense. We also talked about Carol Dweck’s research about mindsets and looked at how we could help our students build a ‘growth mindse’t in mathematics and not be stuck in a ‘fixed mindset’. (If you have not read Dweck’s book Mindset, I encourage you to get a copy asap!)

We looked at visual routines, counting routines, and routines involving number quantity, and discussed how each of these can be utilized for learning.

Our visual routines involved using 10 frames, dot cards, dot plates, 100 dot arrays, fraction pocket charts, percent circles, base-10 grid paper, and number lines (I always have students draw these rather than use ones that are pre-drawn and pre-marked). See end of post to download the various tools.

Our counting routines involved choral counting, counting around the circle, and stop and start counting, and counting up and back.

Our routines for number quantity involved mental math, number strings, “hanging balances”, and decomposing numbers.

It would take too long to write here in one post about how best to use/do each of these ideas, but over time I will get to them. Are you interested in something in particular? Email me and let me know and I’ll get to that one right away!

All of the “math campers” went away with lots of ideas that can be implemented in the classroom right away. I’ll be excited to hear from them how it goes it their classrooms.

I’ll leave you with my favourite definition of number sense: “Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Hilde Howden, Arithmetic Teacher, Feb., 1989, p.11).

There is much food for thought in that quote alone!

Mathematically yours,

Carollee

### Math Camp (gr 2-5) 2011September 2, 2011

Once again I am delighted to say that math camp was a success! We had a productive day exploring learning and the brain, mental math, strategies for all 4 operations (addition, subtraction, multiplication, an division), and equality.

We discussed that both cortisol and adrenalin, when present in the blood stream, tend to “shut down” thinking and memory in the brain. Adrenalin induces the “fight or flight” reaction, while cortisol induces a state of stress when other things are more important than learning. If we want students to learn well in our classrooms, we must take the time to build a safe learning community and do all we can to reduce students’ stress.

The mental math we talked about was similar to what was presented in the Grade 6-8 math camp, so I will direct the readers there for some notes about that topic.

As for the basic operations, it is critical that students do these in ways that are meaningful for them. We shared many strategies for addition and subtraction, some based on numbers only and some based on tools (e.g., 100 dot arrays, blank number lines, base to blocks, etc.). One of the most important things that students should know about the operations of multiplication and division is that they are always, always, always about groups. In representing multiplication, the area model is very effective and can help students understand multiplication beyond basic facts. (Base 10 grid paper is useful for this).

Equality is an important for students to develop in the elementary years. Studies show that when students see the equal sign in an equation, they do not think of equality. Rather, they think it means, ‘put the answer here’ or ‘now do what the sign says to do’. I have had students tell me, when I wrote an equation such as 8 = 2 + 6 on the board, that I wrote it “backwards”. Students expect the single number to be on the left because that is how they always see it! Not only should they see equations written “backwards”, but they should explore equalities with “multiple parts” on both sides, such as 5 + 3 = 2 + 6. Here is a “balance scale” which can be useful in exploring equalities.

I wish you all a wonderful school year!
Mathematically yours,
Carollee

PS: I just remembered that I told all of you at the workshop to write on your 100-dot arrays because there would be clean copies available! So here are the links to the 100-dot arrays:  large 4 small, 6 small, 12 small.

### Math Camp (gr 6-8) 2011August 31, 2011

What a great group of participants we had at yesterday’s gr 6-8 math camp! I was delighted with all of the group sharing that we were able to facilitate — there is always so much that teachers can learn from each other.

The workshop yesterday focused on several things: mental math, integers, fractions, and algebra. These areas, I believe, are important for students to master if they are to be successful in subsequent levels of mathematics.

Mental math is a skill which is only developed if practiced, and we discussed some particularly useful strategies that might be incorporated into regular practice sessions. First, mental math begins with basic facts! From there we can have students practice thins such as these:

• adding multiples of tens (MOT’s) (20 + 50)
• subtracting single digits from MOT’s (50 – 8 )
• adding single digits to non-MOT’s (39 + 6)
• adding to get 100 or – from 100
• adding any two-digit #’s (47 + 39)
• doubling numbers
• halving numbers
• multiplying by 10, 100
• dividing by 10, 100
• multiplying by 20 (by doubling and then multiplying by 10)
• multiplying by breaking up numbers (using “nice” numbers)

Response boards are a great way to do immediate full-class assessment during mental math practice.

Our focus on integers was in using “chips” to have students learn about positive and negative numbers in a very visual way. It is helpful for students if they understand the power of zero in adding and subtracting integers, and zeros can be visualized by an equal quantity of positive and negative chips. Using the chips to solve addition, subtraction, multiplication, and division problems involving integers allows students to build a conceptual understanding of integers which goes far beyond memorized rules.

For fractions, we discussed the need for students to develop number sense regarding them. Using benchmarks to estimate fractions is one way to help facilitate this. We talked about making pocket charts for both the teacher and the students to use in practicing this. (Pocket chart directions.)

As for algebra, once again, visualization was the key to helping students make sense of this generally abstract area. By having students display and manipulate equations in a concrete way on a “balance scale”, they have the opportunity to learn what are acceptable or “legal” moves in solving algebraic equations. (Balance scale.) A hands-on, visual approach to algebra allows every student to be successful in this area!

One of the teachers at the workshop is going to email me a rubric that she used in her math classes last year, and I will up-date this post with that rubric once I have received it. (later) Here is the link for the rubric. When we discussed this, the “traffic light” part was really important. Remember that students can self-assess their understanding and record it as red (“I am totally stuck.), yellow (“I am able to work some on the problem but not I am not really sure about it.), or green (“I understand this well enough that I could teach someone else.)

In the meantime, I hope you will think about how you might better teach these areas of mathematics that are critical for students in these grades.

Mathematically yours,
Carollee

### Math Camp: K-1 (2011)August 27, 2011

Thank you to all the wonderful participants in yesterday’s Math Camp session! Judging by the sense of excitement that was in the room by the end of the workshop, I know you were taking away with you some great ideas for the new school year.

Remember that much of what you do in Kindergarten and Grade 1 needs to centre around number relationships (primarily these: whole-part-part*; anchoring numbers of 5 and 10; one and two more/one and two less; and visual-spatial relationships.) It is as children have numerous opportunities to explore these relationships that they begin to develop number sense.

* Do note that I prefer the term “whole-part-part” to the more common “part-part-whole”. The emphasis in this number relationship is the pulling apart of a number, not the pushing together of two parts to make a larger whole. Primary teachers may make a connection to a similar distinction in reading, namely the difference between decoding and encoding words. They denote two very different processes. Traditionally in math the emphasis has been placed on “encoding” numbers, or adding them together, with little or no emphasis given to to “decoding” numbers or pulling them apart. Children need repeated practice in pulling numbers apart in different ways. We want them to notice that in different circumstances, different parts are more beneficial.

As promised in the workshop, I am posting links for the blackline masters that we referred to during the workshop. I hope up you make good use of them! (Click on any item below to download the file.)

dominoes (large) template
small dot cards template
large dot cards – 1       NOTE: these large, demonstration-sized dot cards appear to go off the page.
large dot cards – 2       That is normal. There are only two large cards fully on each page.
large dot cards – 3       Ignore the stuff on the sides! The two that matter are there!
student ten frames
teacher ten frames
blank 5 frames
blank 10 frames
folding whole-part-part cards

As always, let me know if I can be of more specific help.

Mathematically yours,
Carollee