But in the past 16 years as a University of Regina education professor, she has met many teachers who shy away from math.

“I think it’s kind of a cycle,” said Nolan.

“I think when teachers don’t feel comfortable with it and students don’t feel comfortable with it, then it becomes dry and uninteresting, and I don’t think it has to be.”

Nolan hopes the U of R’s new certificate in teaching elementary school mathematics can help fix the problem.

The first two courses of the program were offered in July, and registration is now open for fall courses.

Several years in the works, the program is geared mainly to elementary teachers who enjoy teaching math or want to get more comfortable with the subject.

Kerrilyn Rosengren is one.

“I find that schools in general, they tend to focus on language arts with their professional development,” said Rosengren, a Grade 3 teacher at George Lee School.

“I’m really enjoying sitting and discussing these things with likeminded people who also enjoy teaching math and want to learn more about it.”

In July, Nolan taught about being culturally responsive in teaching math and her colleague Gale Russell taught about making math classes more inclusive — something that didn’t occur to Deb McCloud.

“I definitely got told to teach for your learners, so making sure that your students understand in their own way,” McCloud said of the overall education program, from which she graduated earlier this year.

“I missed the cultural aspect in that and I think a lot of us do.”

“I don’t think anyone thinks that math is influenced by culture and that there is a lot of culture involved in math,” Rosengren agreed.

For example, teachers should think outside of the box when using “real-life” examples in their class, like splitting up a pizza to teach about fractions.

“Instead of dividing up pizzas and chocolate bars and cookies,” said Nolan, “use examples of wealth distribution in our country.”

Drawing on socioeconomic issues or real world statistics to teach math makes the subject more tangible, Nolan added.

“I think if we remove the context all of the time from the mathematics, then we risk having students think that it’s just a detached subject and it has nothing to do with real life,” said Nolan.

The lessons debunk the common perception that “math is just math,” said Nolan, challenging teachers to think differently.

The seven students in Nolan’s July class ranged from recent graduates to someone with 30 years’ teaching experience.

Schedule-wise, the program is designed for teachers, with courses generally happening in the evenings.

Registration is open and fall courses begin in early September.

]]>

1. To increase your chances of being contacted on online dating websites, you should include interesting aspects about yourself which may not be attractive to the opposite sex as well. People who appear too attractive don’t get as many messages because people think they are most likely taken anyway.

2. To optimise your chance of finding a perfect mate, assuming you cannot backtrack and seek out someone you rejected before, you should reject the first 37% of all potential mates, then pick the next mate you find who rates to be better than all the mates you seen so far.

3. What determines your partner’s response to your conversation is your own mood, your partner’s mood, and the influence your partner has on your mood. The same formula is used for countries involved in an arms race, which is why negativity feelings in couples tend to spiral uncontrollably downwards. Couples who have low negativity tolerance for each other interestingly tend to have a higher chance to stay together.

My take is even if these are true, these are just averages, and most people are not average. Even Hannah Fry alludes to this during her talk regarding finding your soulmate. I would suggest taking these advice with a pinch of salt.

]]>**08/18/17**

*“How can we have a self-correcting control system?”*

Open loop control systems may be affordable, but the lack of control over them (pun intended) makes them useful for only select applications. So how can we fix this problem? Well, what if every time our system was to produce an output, we compare it to our setpoint, and then modify the process to achieve our desired result accordingly? This is the fundamental idea behind a **closed-loop control system **and is used in a vast array of controls applications from electric vehicle battery life monitoring to drones and even laundry machine monitoring.

*Let* *be an outer measure on the measurable space* . *Then the λ-sets in* *form a σ-algebra* *on which* *is countably additive, so that* *is a measure space*.

Three aspects of the proof of Carathéodory’s lemma provided in Williams’ book are clarified in this blog post.

**Definition of λ-system**

The concept of λ-system, which is used implicitly but it is not defined in Williams’ book, is introduced in this post.

A collection of subsets of a set is called a λ-system on if

- ,
- (it is closed under complements),
- with for it holds that (it is closed under countable disjoint unions).

Not that the only difference between a λ-system and a σ-algebra is that the former is closed under countable disjoint unions while the latter is closed under countable unions. Moreover, the first condition on the definition of a λ-system could be alternatively set to instead of due to closure under complementarity, i.e. due to the second condition of the definition.

**Lemma**

If a collection of subsets of a set is a λ-system and a π-system on , it is also a σ-algebra on .

This lemma is used without being proved in Williams’ book for proving Carathéodory’s lemma. In what follows, the lemma will be proved before proceeding with the proof of Carathéodory’s lemma.

Although not relevant to subsequent developments, it is mentioned that a σ-algebra on a set is also a λ-system on as it can be trivially seen from the involved definitions.

**Proof of the lemma**

Let be a collection of subsets that is both a λ-system and a π-system on . To show that is a σ-algebra on , it suffices that it is closed under countable unions.

Let . The main idea is to express the collection as a collection of pairwise disjoint sets ( for ) so that . Along these lines, define .

Obviously, . To prove the converse set inequality, let and assume that . In this case, for each , either or . There is at least one such that , otherwise leads to the contradiction . Let be the minimum natural for which . In turn, . Due to being the smallest natural for which , it is deduced that , hence . Thus, , which is a contradiction. Thereby, , and this establishes the equality .

Assume that there are with and . Let . Without loss of generality assume that . Then with , while , which means that with it holds that , so a contradiction has been reached. Thereby, the sets , are pairwise disjoint.

It has thus been shown that the collection consists of pairwise disjoints sets that satisfy .

Notice that . Since is a λ-system, for the various . Moreover, is a π-system, hence the finite intersection is also in . Since the collection is a disjoint union of elements and is a λ-system, it follows that the union is also in .

Since the countable (but not necessarily disjoint) union of any collection of sets is also in , it follows that is a σ-algebra.

**First clarification**

The above lemma explains why the proof of Carathéodory’s lemma in Williams’ book states that it suffices to show that for a countable collection of disjoint sets it holds that . The conclusion then extends to any such countable union of sets, disjoint or not.

**Second clarification**

It is mentioned in p. 197 of Williams’ book that from

follows

.

To see why this is the case, recall that the outer measure takes in values in for any .

Distinguish two cases. If (i.e. if is finite), then the sequence , is a bounded increasing sequence, therefore it converges, which means that the limit exists, so taking limits leads from the former to the latter inequality in the book.

If , then holds trivially.

**Third clarification**

To show that for , notice first that

follows from the countable subadditivity of the outer measure .

Moreover, setting in equation (d) of p. 197 gives

,

which concludes the argument.

]]>Suppose ∇ is a derivative operator on the manifold M. Then there is a (unique) smooth tensor field R^{a}_{bcd} on M such that for all smooth fields ξ^{b},

R^{a}_{bcd} ξ^{b} = −2∇[_{c}∇_{d}] ξ^{a} —– (1)

Uniqueness is immediate since any two fields that satisfied this condition would agree in their action on all vectors ξ^{b} at all points. For existence, we introduce a field R^{a}_{bcd} and do so in such a way that it is clear that it satisfies the required condition. Let p be any point in M and let ξ’^{b} be any vector at p. We define R^{a}_{bcd} ξ’^{b} by considering any smooth field ξ^{b} on M that assumes the value ξ’^{b} at p and setting R^{a}_{bcd}ξ’^{b} = −2∇[_{c}∇_{d}]ξ^{a}. It suffices to verify that the choice of the field ξ^{b} plays no role. For this it suffices to show that if η^{b} is a smooth field on M that vanishes at p, then necessarily ∇[_{c}∇_{d}] η^{b} vanishes at p as well. (For then we can apply this result, taking η^{b} to be the difference between any two candidates for ξ^{b}.)

The usual argument works. Let λ_{a} be any smooth field on M. Then we have,

0 = ∇[_{c}∇_{d}] (η^{a}λ_{a}) = ∇[_{c} η^{a} ∇_{d}] λ_{a} + η^{a}[_{c}∇_{d}] λ_{a} + (∇[_{c} λ_{|a|}) (∇_{d}] η^{a}) + λ_{a} ∇[_{c}∇_{d}] η^{a} —– (2)

It is to be noted that in the third term of the final sum the vertical lines around the index indicate that it is not to be included in the anti-symmetrization. Now the first and third terms in that sum cancel each other. And the second vanishes at p. So we have 0= λ_{a}∇[_{c} ∇_{d}]η^{a} at p. But the field λ_{a} can be chosen so that it assumes any particular value at p. So ∇[_{c} ∇_{d}] η^{a} = 0 at p.

^{a}_{bcd} is called the Riemann curvature tensor field (associated with ∇). It codes information about the degree to which the operators ∇_{c} and ∇_{d} fail to commute.

** Nothing goes to the dustbin in science.**

There is a big debate about whether mathematics can be considered as a science. The modern day notion is that it cannot, for science is about things that can be proved or disproved experimentally. But I have always considered math to be a part of science. For god’s sake, consider it as a **honorary member. **Without it, almost no branch of science would exist.

As some of you might have gleaned from the heading – today’s post is about math. It all started over dinner today, when me and my friends somehow started talking about erroneous proofs in mathematics.

Erroneous proofs in math, also called **Mathematical Fallacies**, have great pedagogic value. Consider this proof for “2 is equal to 1”,

**But is that the only mistake???
**

Also be warned – I do pure math, which is sometimes more dangerous than pure meth. Read on only if you are into it.

It is not often that you know something and Wikipedia doesn’t :D

I have had this discussion with several people, asking them to figure out the second mistake. It’s great fun to see them break their heads with it.

Very few get it, and that too, with hints from me long after I’m bored of playing with their minds.

**However, the mistake itself is nothing complicated. In that sense, it’s like a magic trick. It loses its value once its revealed. **

Now that I have had my fun with you, let me reveal the second mistake.

**To understand why, we need to look at the beginning of math.**

The **fathers of mathematics** started by defining something called numbers. They called them “One”, “Two”, “Three”… – and gave them symbols 1, 2, 3…. They even gave a name to this collection – **Natural Numbers**.

This is the first bit of mathematics we all learn. Much later, we learn that these are called **Natural Numbers**. We learn the symbols and the words – but most importantly – we learn this without asking questions.

The fathers of mathematics were not content with just having numbers. They wanted to do things with numbers. They did this by defining **Addition** and **Subtraction**.

We learn **Addition **and **Subtraction** too. We learn this using our fingers to add and subtract.

Addition was a good kid – never bothered the fathers of mathematics. But subtraction – he had all sorts of difficult questions.

These questions troubled the fathers, until, one day, someone defined the **Whole Numbers** and **Integers**.

But, the fathers didn’t stop there. They defined two more operations – **Multiplication** and **Division**.

We learn these too – Whole Numbers, Integers, Multiplication and Division. We learn Multiplication and Division as shortcuts to **repeated addition** and **repeated subtraction**.

Multiplication – like Addition – created no trouble. But Division, oh Division – he was naughty. He too, asked a lot of questions.

The Pandora’s box was open – the fathers tried to close it by defining another set of numbers called **Rational Numbers**.

But Division was not done. He had gone rogue – he went around the neighbourhood doing things like

The fathers could not look the other when Division was committing such atrocities. They banned such operations and forbade division to ever engage in such things.

After all this, the fathers had learnt not to mess with operations. They were careful this time – and defined only one operation. They created the **Exponentiation**.

But Exponentiation was not happy with n being forced to be * “Natural”*.

He wanted n to be free. He wanted n to be

Cursing themselves for not being more careful, the fathers were desperately trying to answer the questions without having to define new things.

They were able to answer the first two,

But the third question could not be eluded. In particular – the following question could not be answered.

This led to the definition of **Irrational Numbers**. By this point, the fathers were tired and old, and lazily defined irrational numbers as,

But exponent was not done – he found a loophole in our definition. He asked,

And yet again, we had to define another number. We were so fed up at this point that, we decided to use a **letter** as the symbol for this number. We defined i as,

The questions haven’t stopped – but we have to – for this was not the original point of this post.

However, if you want to know who started this **“House of Cards”** – I have a name. **Giuseppe Peano** defined the famous **Peano’s Axioms** – which defined natural numbers.

Before going back the original topic – which most of you have forgotten by now – we have to understand who is **Propositional Logic**. Propositional Logic is another creation of the fathers. They created him to take care of everyone else.

**Propositional Logic to Axioms of Mathematics is like Law Enforcement to Constitution.**

Logic makes sure that nobody messes with the axioms. Axioms are absolute laws of mathematics and cannot be broken by anyone – and if someone tries to – Logic makes it clear to them that they are mistaken.

At the heart of Logic, we have **Statements**. Statements can either be **True** or **False**. All axioms of mathematics are **True** – simply because they are axioms.

We will not dive into Logic more – because its a Labyrinth and we do not have a ball of thread. Instead, we will get back to topic.

All mathematics can be broken down into,

- Start by taking up some
**axioms**of mathematics and some**hypotheses**. - Perform valid operations on these.
- End up with valid, and sometimes astounding results – called
**theorems**.

Since we start with axioms, which are absolute truths, and hypotheses, which are valid assumptions, and perform valid operations, we end up at truthful statements – Logic enforces this.

However, if Logic is forced to make our **hypotheses** false or the **axioms** of mathematics false – he will make our **hypotheses** false. Again, Axioms are Absolute with a capital A.

In our proof, until step 5, we were doing valid operations.

- Assuming a and b to be real numbers, such that a=b
- Multiplying both sides by a
- Subtract b
^{2}from both sides - Factorising

In step 5 we are doing an invalid operation – dividing both sides by (a-b), which is division by zero.

This gives us

**a+b = b**

and from there, using valid operations, we can get,

**2b = b**

Now, we assumed that a,b are general real numbers – we didn’t force any condition upon them. But we made a mistake in step 5.

Logic – the enforcer, has to make a choice.

- Either make 2=1 and allow b to be any real number – which keeps our hypothesis true but destroys all mathematics
- Or force b to be 0 – killing our assumption but upholding the axioms.

]]>

**08/18/17**

*“What is the amplification of a controls system?”*

When a signal passes through a transfer function, its output will be modified in an according way depending on its frequency. So what do we call the **change of magnitude **of the signal? Well, after much debate, controls researchers have settled on the idea of **gain **to describe this. Gain is defined as the ratio of the new magnitude to the old magnitude, such that a system that is twice as large will have a magnitude of 2 and half the size would be 1/2

Download the lecture notes form following link.

]]>

This past week was no different; here’s some of what I caught.

Here’s the toolbar in my browser so I’m ready to go….

**Response to: Five Ways to Damage a Good School**

Only five?

Paul McGuire focuses in on a post from another blogger and manages to use furniture and limited resources in the same thought. Oh, and technology in another thought.

Here’s the thing. Too often, educators get caught up in the latest fad – flexible seating and the expense that comes with this is one of the newest things. In schools with limited resources (I would say most schools in Canada), the purchase of new furniture means that something else will not be bought.

He makes a good point which leads to a good discussion about priorities within a school. I find the interesting point about all this about flexible seating and changing learning spaces to be interesting. If it’s just about some new chair or table, then it’s just an advertisement. If it’s about changing a philosophy with a stated purpose about why you’re doing it and the results that you’re expecting, then I can get excited. I can’t help but throw in a golf quote here…

You drive for show; you putt for dough.

Maybe the question in these times is “Are you driving or are you putting?”

I’d like to see Paul do something like posting an online form asking everyone to add their thoughts about how to “damage a school”. I’ll bet there would be lots of things to learn from and it would give Paul an endless resource for blogging.

This post, by Deborah McCallum is guaranteed to get you thinking.

She leads us to this marriage by focusing on identity. Perhaps this is another way for all to reflect on the message that is present in the mathematics classroom.

What is identity? It is connected to the groups that we affiliate with, the language we use, and who we learned the language from. I believe that we all have different identities depending upon the different groups that we belong to, and that this has implications in terms of the languages and discourses we use.

I’ve seen a number of suggestions about improving mathematics instruction (including some from Deborah). This is a new and interesting take.

This is a post that put me in someone else’s shoes. Ann Marie Luce is taking on the role as a Principal of the Canadian International School of Beijing. This is part of a series of posts talking about her nervousness in the decision and then landing in a different land with different language and different customs and GO!

I absolutely can put myself in her place as she goes about what we would consider a regular routine — shopping, going to a restaurant, going shopping, …

But she’s doing it in a land where she doesn’t speak the language!

So, many of the things that we would expect to do with our regular language have to be done with gestures just to get the message across.

Then, she turns to that new ELL student in our present classrooms. It’s an interesting transition that will give you renewed sympathy for that new student, trying to get along in a new world, and learning how to speak the language in order to get the job done.

I hope that she continues to blog about her experience. This could be very interesting.

We move from a discussion about the reality of China to the reality of Japan. Deborah Weston was inspired by an article in an English language Japanese newspaper about the reality of being a temporary teacher.

I’m so fortunate that I didn’t ever have to go through the hoops of the current reality for Occasional Teachers. I graduated from a Faculty of Education and there was a school here in Essex County that needed a Computer Science teacher. Other than waiting annually for the seniority list and the horror of being declared redundant (which I fortunately never was), my teaching life unfolded as I wanted it to.

That’s not true for all.

It’s an interesting comparison and a similarity of realities of how long it takes before getting that permanent position. She quotes:

- Ontario – 6 or 7 years
- Japan – 5.9 years

It’s an interesting look at another’s reality.

Like everyone, Sharon Drummond is getting ready for September. Her activity is looking and pondering classroom setup. The floors are clean and polished and the room is empty. It’s time to think about setup.

She shares a picture and a diagram of the room with some preliminary thoughts. I was so impressed that green screen is built in before the furniture. That sends a powerful message and she shouldn’t need to rearrange things later in order to take advantage of this tool for video making.

And, she’s not starting with thinking about how to control the flow or maintain classroom discipline. She’s talking about things that she wants her students to do.

- I WANT MY STUDENTS TO TALK MORE THAN I DO
- I WANT MY STUDENTS TO COLLABORATE WITH EACH OTHER
- I WANT MY STUDENTS TO BE COMFORTABLE
- and more – you’ll have to click through to read her post to see them all

She’s asking for input and ideas. If that’s your game, go over to her blog and share.

I hope that there’s a subsequent post to show us all how this activity ends.

If you’re a regular reader here, you know that Kristi Bishop’s blog is high on my list of favourites but had gone missing recently. But, she’s “Back in the Saddle” and ready to blog.

I really like her rationale for blogging and sharing her thinking online.

I don’t think any blogger should apologize for being a bit selfish and using the blog primarily to get their own thinking down in one spot. In fact, I can’t think of a better way of reflecting and geting other people to chip in with their own thoughts.

So, Kristi, it’s great to see you back and I look forward to reading many inspiring posts in the future.

How about you, reader? Do you have a blog that’s playing possum? How about kick starting it?

If you’re looking for a good description about what being connected and how it works, then you’ve got to look at this post from Terry Greene.

In fact, I had looked and commented here on a post that he wrote last week. It was also one of the posts that Stephen Hurley and I talked about on our Wednesday radio show.

But, that was only a small part of the connected educator story.

In this post, Terry gives us the complete story of all the connections that surrounded his one blog post, complete with links, and it’s a testament to why we do this and how you can share the learning love around.

Great summary!

And it all started with one simple request.

Please take the time to click through and read the complete, original blog posts from these wonderful **Ontario Edubloggers.** There should be a little there for everyone. And, if you’re blogging yourself and not already in the collection, just fill out the form and you will be.

The complete collection of these Friday posts illustrating the thinking of Ontario Edubloggers can be found **here**. I’d love to have you become part of it.

Complete Q.5 and 6 of Ex-6.4 as we’d discussed in a class.

Regards

MT

]]>Complete Q2 and 3 of Ex. 5.7 in Nb1. Bring NB1 and NB2 tomorrow without fail.

]]>The moment you open it, if you use the GUI mode, you will see 2 big text boxes – assumptions and goals. Naturally, when playing one starts with simple stuff.

We leave the assumptions blank and set as our goal. If we click Start, we see that it proves it immediately.

% -------- Comments from original proof -------- % Proof 1 at 0.00 (+ 0.00) seconds. % Length of proof is 2. % Level of proof is 1. % Maximum clause weight is 0. % Given clauses 0. 1 P -> (Q -> P) # label(non_clause) # label(goal). [goal]. 2 $F. [deny(1)]. ============================== end of proof ==========================

That was fun, but what stood out for me is that in mathematics we often take axioms for “granted”, i.e. it’s rarely that we ever think about the logic behind it. But with Prover9, we need to express all of it using logic.

For Prover9 there is a general rule, that variables start with u through z lowercase. Everything else is a term (atom).

For start, let’s define set membership. We say denotes set membership, and that’s it. Because that’s all there’s to it, it’s like an atom. So let’s say that member(x, y) denotes that.

Now let’s define our sets A = {1}, B = {1, 2}, and C = {1, 2, 3}.

all x ((x = 1) <-> member(x, A)). all x ((x = 1 | x = 2) <-> member(x, B)). all x ((x = 1) <-> member(x, C)).

Now if we try to prove member(1, A), it will succeed. But it will not for member(2, A).

What do we know about the equivalence of 2 sets? They are equal if there doesn’t exist an element that is in the first and not in the second, or that is in the second and not in the first.

In other words:

set_equal(x, y) <-> - (exists z ((member(z, x) & - member(z, y)) | (- member(z, x) & member(z, y)))).

So, Prover9 can prove set_equal(A, C), but not set_equal(A, B).

Another example is that we can define the set of the naturals with just:

member(Zero, Nat). member(x, Nat) -> member(Suc(x), Nat).

So, it can easily prove:

0: member(zero, Nat)

1: member(Suc(Zero), Nat)

2: member(Suc(Suc(Zero)), Nat) (2), etc.

**08/17/17**

*“How can we convert from the s to time domain without using a Laplace transform?”*

Taking an inverse Laplace transform of a function from the s to time domain is very useful but quite difficult, so is there a way that we can get around it for specific case? Well, after years of hard work, mathematicians have discovered that if you multiply the frequency function by s and take its limit as it approaches zero (lim s–>0 s*f(s)) then it would actually be equal to the value of the time domain function at infinity! The proof is shown in the picture above

]]>And as an example we can take one of the most famous unsolved problems in mathematics,

**Goldbach conjecture: Every even integer is the sum of two odd primes.**

Other terms related to the conjecture are,

**Goldbach number**: All those positive integers that can be expressed as the sum of two odd primes. And the smallest number that can be written in such a way is . So, every even are Goldbach numbers.

**Goldbach Partitions**: The expression is called the Goldbach partition of that number.

1. The partition function I wrote about last month is just one of such problems in additive number theory, which tells us the number of ways an integer can be written as the sum of different positive integers.

This particular problem has a specific name, it’s called **unrestricted partitions**.

Because the number of integers that can be used to write the sum is unrestricted. Repetition is allowed and the order of numbers does not matter.

So, is called unrestricted partition function.

]]>Also recall that the Fibonacci sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

Each number is found by adding up the two numbers before it. For example 5 is found by adding 3 and 2.

German Astronomer Johannes Kepler once wrote that “as 5 is to 8, so 8 is to 13, approximately, and as 8 is to 13, so 13 is to 21, approximately”. What Kepler is saying is that the ratios of consecutive numbers in the Fibonacci sequence are similar.

About a century after Kepler’s statement Scottish mathematician Robert Simson discovered that if you take the ratios of consecutive numbers in the Fibonacci sequence, and put them in the sequence

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34 …

or to three decimal places

1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618 ….

the numbers of these terms get closer and closer to the golden ratio.

What Simson discovered is that the golden ratio is approximated by the ratio of consecutive numbers in the Fibonacci sequence, with the accuracy of the approximation increasing as we move down the sequence.

Now here’s something else, lets consider a Fibonacci like sequence starting with two random numbers and them adding consecutive terms to continue the sequence. So, lets start with 4 and 10, the next term would be 14 and the next 24. Our example would give us:

4, 10, 14, 24, 38, 62, 100, 162, 262, 424 …

lets check the ratios of consecutive terms …

10/4, 14/10, 24/14, 38/24, 62/38, 100/62, 162/100, 262/162, 424/262 …

or to three decimal places

2.5, 1.4, 1.714, 1.583, 1.632, 1.612, 1.620, 1.617, 1.618 …

Try this with any other two terms and you will see the Fibonacci recurrence algorithm of adding two consecutive terms in a sequence to get the next term is so powerful that whatever two numbers you start with, the ratio of consecutive terms will always converge to the golden ratio!

]]>The 2017 frameworks were approved by the Board of Elementary and Secondary Education in March 2017. The revisions to the frameworks were reviewed by a team of Massachusetts educators in partnership content advisors from Massachusetts institutions of higher education and DESE staff members.

Highlights of the revised 2017 Massachusetts Curriculum Framework for Mathematics include:

- Stronger learning progressions for pattern recognition in the early grades, the measurement of circles in the middle grades, and ratio, rate, and proportions in the middle grades;
- Revised language to clarify the standards, including the incorporation of footnotes into the standards, definitions of key terms, and examples to specify expectations for students;
- A revised high school section with an explanation of the alignment between the Conceptual Category Standards and the Model High School course standards in the two model pathways (Traditional and Integrated);
- Added guidance for making decisions about course sequences that includes pathways to calculus and other advanced mathematics courses;
- New and revised Guiding Principles;
- Revised and updated glossary and bibliography.

Highlights: 2017 Revisions to the Mathematics Standards

Summary of Grade-by-Grade Detailed Revisions in Mathematics

Massachusetts is transitioning from MCAS/PARCC to MCAS 2.0 in both format and test design in 2018. Also, the MCAS questions themselves will be updated to reflect the new 2017 curriculum frameworks.

The 2018 grade 3-8 Mathematics MCAS will be based on the newly revised 2017 curriculum frameworks

For the 2018, the grade 10 Mathematics MCAS will align to the 2001/2004 and 2010 Curriculum Frameworks. The Mathematics MCAS will align with the revised 2017 frameworks when MCAS 2.0 grade 10 tests are given for the first time in spring 2019.

Details on the 2017-2018 MCAS Schedule are available at www.doe.mass.edu/mcas/

The DESE is also hosting a series of events in 2017-2018 to support the implementation of the English Language Arts and Literacy Frameworks, the Mathematics Frameworks and the Science, Technology and Engineering Frameworks. These Instructional Support Networks will include professional learning opportunities for educators of English language learners, literacy specialists, elementary educators, middle school educators, math specialists, curriculum specialists and science leaders.

Curriculum frameworks in Massachusetts are used to guide local districts’ curriculum, units and lesson plans. MCAS assessments are based on the curriculum frameworks while instructional decisions are made at the local level to determine the best methods and materials to prepare students based on local needs. Check with your local school or district administration to learn more about what is taught in local schools. All the Massachusetts Curriculum Frameworks are available at www.doe.mass.edu/frameworks.

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