What is "New Math"?

Last week my father emailed me to ask what I thought of Mr. Anthony Quinn who is running for one of our School Board's Trustee positions. I told my dad I would look up Mr. Quinn and then get back to him with my opinion. I found the following from one of Mr. Quinn's campaign newsletters that I would like to discuss here.

This "news" piece really bothers me because it is rife with misconceptions and incorrect information. The current Math curriculum that we use in Ontario was written in 2005 and is an updated version of the older curriculum written in 1999, so why it is being called "new math" is something I am having trouble understanding. An analysis of curricula of high-achieving regions around the world indicates that our Math curriculum is aligned with those that are the most successful in the world. But in addition, it is my understanding that the gap between our highest achieving students and our lowest achieving students is smaller than anywhere else in the world. 

I am also struggling to determine where the term "discovery math" comes from. I have been an Elementary school educator for 13 years, and until now, I have not heard that term before. I did a search of our Math curriculum document and I found the word "discover" in two places:

Students who are willing to make the effort required and who are able to apply themselves will soon discover that there is a direct relationship between this effort and their achievement in mathematics. Pg. 4 of the Ontario Math Curriculum, 2005

and 

Graphs and statistics bombard the public in advertising, opinion polls, population trends, reliability estimates, descriptions of discoveries by scientists, and estimates of health risks, to name just a few.

Pg. 9 of the Ontario Math Curriculum, 2005

Certainly, as you can see, the term "discover" does not figure largely in our current Math curriculum in Ontario. However, Mr. Quinn seems to be under the misguided notion that students are expected to "discover" math concepts on their own, which is very much NOT the case, and he would know this if he were to have a look at our Math Curriculum document. In fact, if he were to scan the verbs used in our overall and specific expectations for Math he would find terms such as: identify, describe, construct, create, analyse, compare, connect, extend, and determine, just to name a few. 

Mr. Quinn also has an italicized quote that suggests that students are not expected to know their multiplication facts. Quite the opposite is true. Here is an example of an expectation from our Grade 4 Number Sense strand:

– multiply to 9 x 9 and divide to 81 ÷ 9,

using a variety of mental strategies (e.g.,

doubles, doubles plus another set, skip

counting);

If someone were to ask me what is the difference between the expectations in the Math Curriculum currently being taught from what was taught in the 1970's I would have to say that when I was growing up, I was expected to have rote memorization of my multiplication facts whereas students today are expected to have conceptual understanding of mathematical operations and can represent them in a variety of ways, as well as use them to solve problems. 

Students today are not only expected to solve questions like ¾ ÷ ½ = ?, they are also expected to be able to represent a real-life situation where that expression would be needed to solve the problem. I wonder how many people educated in the 1970's are able to do that? I was taught "Yours is not to reason why, just invert and multiply". Students today ARE expected to reason and they are expected to be able to explain why multiplying by the inverse fraction provides the solution for the division of fractions. 

I have spoken at several schools' Family Numeracy nights as well as at the Halton Catholic Parents Conference and each time I begin by asking how many parents in the audience think of themselves as "Math People". Invariably, less than half of the people in the room raise their hands. I want ALL of our students to see themselves as "Math People". Learning Math the way I did simply did NOT achieve that result. So I disagree strongly with Mr. Quinn's statement about "fixing something that wasn't broken"; there was something very "broken" in the way that Math used to be taught thirty and forty years ago. 

As Dr. Christine Suurtamm said at a recent symposium I was fortunate enough to attend, the mathematical thinking we are teaching is so complex, we definitely do not support "discovery" learning. But we do support the generation of student algorithm. We are supporting students actively participating and thinking, not just being passive consumers regurgitating and performing rote procedures. Kids need to do the math to learn the math. They need opportunities to makes sense of the mathematical skills they are learning. If you look at the curriculum, you will see it includes traditional algorithms and mental math, and in addition, the thinking involved in doing the math. 

Our current EQAO data indicates that students are actually doing quite well in using procedural knowledge. The area they are having difficulty with is in solving multi-step multi-strand problems. No amount of rote procedural knowledge is going to help them think their way through these types of problems. Students need to have deep conceptual understanding of number sense, including operational sense, place value, and proportional reasoning to be able to successfully solve the types of problems they are currently struggling with. They need to develop a facility in composing and decomposing number.

Our students need procedural fluency, which implies much more than merely knowing their Math facts. Yes, they need to know those facts, but that is not enough. Procedural fluency is the ability to perform math operations flexibly and see the connections between those operations. Rote memorization of a procedure does not mean that you have conceptual understanding of that procedure. 

Do we need to learn more about teaching and learning Math in Ontario? Absolutely! We need to support our teachers in developing their own conceptual understandings and pedagogical knowledge in Math because they are victims of what I will call the "old Math". We need EVERY student to achieve at high levels in Math in Ontario and we are not there yet. I'm sure if Mr. Quinn took the time to read the Board's Improvement Plan, he would see that a philosophy of continuous improvement exists and that no one is suggesting "there is nothing we can do".

I am sure that if Mr. Quinn is elected as a trustee he will dedicate himself fully and devote himself to representing the voice of our parent population. I hope that he, and all of our trustees, will take the time to look at our Math curriculum so that they can provide the informed support that is very much needed if we are to improve student learning in Math.

I will leave you with a typical question from the Junior EQAO Math Assessment. I ask you to consider - could you have solved this question when you were in Grade Six? Can you solve it now? 

Exploring Fractions with a Growth Mindset

I love summer. I love it for so many reasons. One of the reasons I love summer is because it gives me the opportunity to tutor students at my own pace with no "curriculum" other than what I believe to be good instructional practice to follow.

This summer I'm tutoring a 10 year old who has just finished Grade Five. I have worked with this student before, (I will call her Grace because she's an incredibly graceful young lady), and I know that she has struggled with Math for the last few years. Grace is pretty typical of many students that I have taught. Traditional math instruction is not that effective for her. She is quiet, and if something doesn't make sense to her she won't ask questions, preferring not to single herself out. 

Grace also has some issues with her short term memory. Many of her teachers have expected her to learn her addition and multiplication facts by rote memory. That is just not a realistic expectation for Grace. She has difficulty memorizing facts. Last summer, Grace and I explored repeated addition, skip counting, and making groups. Although she doesn't know her facts with lightening speed, she can figure out any multiplication question using strategies that makes sense to her. She understands what multiplication means. But some of her teachers don't value this, and as a result, at the tender age of 10, she does not really see herself as a "math person". 

I also love the summer because it gives me time to work on my own professional learning. In past summers I have taken Marilyn Burns' Math Solutions course and attended our Ministry's Math Camppp. This summer is no different and I am currently taking Jo Boaler's MOOC How to Learn Math. (You can watch the Youtube videos here). There is a common thread through all of the learning - Math is not a textbook subject! You don't develop deep understandings of mathematical concepts by completing worksheets or workbooks. In order to develop conceptual understanding in math, in order to see connections between concepts, in order to love math and think creatively in math, students need multiple opportunities to explore math. I love the summer because I can work one on one with students, give them those opportunities to explore and construct their own understandings, and watch and listen to learn how they learn. 

I knew that Grace had had trouble with fractions this year, so that is where we started this summer. 

This was an SOS I got from Grace in the Spring

More than ever, I've been trying to use Growth Mindset language. I've been using "traffic light comprehension" with Grace, asking her frequently if she is red, yellow or green light in her understanding, and asking her to really pay attention to her own learning. I'm being careful about the language I'm using, drawing attention to how successful she has been with her persistence and hard work. I'm also making sure that if she doesn't get something, we add the word "yet". I give her lots of time to explore her understandings and allow her misconceptions to "float" out there, merely asking questions that allow her to re-evaluate her own beliefs and re-adjust her understandings. 

We've spent three 90 minute lessons just on representing proper fractions using fraction circles, fraction strips, sets, area models and number lines. We compared these representations looking at the connections between them. I've been encouraging Grace to name her learning so that she can see her own growth and take ownership for her understandings. Grace has concluded that:

- the numerator counts how many parts you have (or are discussing)

- the denominator tells how parts make up the whole

- a proper fraction is always less than one

- in a proper fraction the numerator is always less than the denominator

- different fractions can represent equivalent amounts e.g. 1/2 is the same as 5/10 which is the same as 0.5

- the equal sign (=) means "the same as"or "is equivalent to" and not "the answer is..."

Grace doesn't have to memorize these things because she came to these understandings on her own. 

During one lesson while exploring proper fractions Grace said "My teacher kept giving me questions like this: 2/3 = ?/6.  I didn't know how to answer those". Those questions had absolutely no meaning for Grace at all.  She did not know what the teacher was asking or looking for. I pointed to the number line we had created and how we had divided it up many different ways. I pointed to the half and asked "How many ways could we name this fraction?" She said we could call it "1/2 or 5/10". Then she went to the fraction circles we had on the fridge and said "It's like 1/2 is the same as 2/4 and 3/6". I explained that is what the teacher was asking, that she was asking what fraction with a denominator of 6 was the same size as a fraction of 2/3. All of a sudden the question made complete sense to Grace but she needed to connect the question to a visual representation. 

 Today we began working on improper fractions. I could not believe how quickly she picked it up.  I guess it was easy after all the work we had done on proper fractions. Grace had such a deep understanding about the role of the numerator and the denominator she quickly deduced that if the numerator was greater than the denominator we were talking about a fraction greater than one. She had no trouble representing them in any format. Next week I will show her how to write an improper fraction as a mixed number. This should be easy for her to understand since she's already been naming them out loud as "two and a quarter" for example because she can see them pictorially as a combination of wholes and fractions.

Along the way we've been comparing fractions. As we put the fractions on the number line, as we use the fraction circles on the fridge, and as we draw our area models, I'm always asking "What do you notice about these two fractions". Eventually, I will teach Grace about common denominators, but not for a while, not until she has a really solid understanding of fractions, and has an idea of benchmark fractions on a number line so that she can estimate the relative size of a fraction. I want Grace to have many ways to compare fractions. She has already noticed that the larger the denominator, the smaller the fractional piece. I want her to realize that 7/8 is less than 9/10 because each fraction is missing only one piece but the tenths are smaller pieces, so 9/10 represents more. I want her to know that 9/20 is closer to half than 4/10 is, and I want her to know this without having to use a common denominator because she understands fractional parts.

I'm really enjoying the Jo Boaler MOOC. One thing that Jo said was that intuition is an extremely important part of math competency. I've often thought that having mathematical intuition was a genetic gift - I guess you could say I had a Fixed Mindset about math ability. But Jo has conducted research to show that it is mathematical understanding that helps a person to develop mathematical intuition. And having mathematical intuition, in turn, helps a person to develop their mathematical understandings. Jo Boaler explains it as an iterative process. This makes total sense to me. I'm helping Grace to understand fractions. I'm hoping this will help her develop an intuitive sense about proportional reasoning. Once she has honed this intuition, it will help her solve problems and make sense of problems involving fractions, decimals, percentages, rates, and ratios. It is my goal that Grace sees the connections between all of these beautiful math concepts.

Graces always texts me before she comes over. I told her I had bought her some fraction circles she could take home. She wrote "Do the fractions stick on the fridge?"She wants some like mine so she can play school at home and use them for math homework. Then she wrote "I can't wait". She clearly loves math, she just doesn't love it at school. Isn't that a shame?