kitchen table math, the sequel: Can you spot the error?

Sunday, June 29, 2014

Can you spot the error?

Can you spot the error?

This is the EngageNY 8th grade, module 7 Teacher's Guide for part of the lesson on irrational numbers.  You can find the rest here:
Hint: extremely egregious error. 


Hainish said...

Well, they're dividing the segment between 1 and 2 into equal parts, whereas the square root function is not linear. Whoever wrote this should have really known better.

Anonymous said...

What Hainish said is the egregious error, but there is a second issue which is either an error or a poor choice of words, depending on your perspective.

"We said that the length of the diagonal was s^2=2."

Should read: We said that *the square* of the length of the diagonal was equal to 2.

ChemProf said...

So this is why I have to teach college students to recognize that the square root of 2 is about 1.41, huh?

Why on earth do they think dividing the number line section into three is a useful explanation?

Obi-Wandreas, The Funky Viking said...

The very first lesson of Module 1 for 7th grade states: "A ratio is an ordered pair of non-negative numbers, which are not both zero." This statement, besides being wrong, has absolutely nothing to do with the definition of a ratio. From there, it only gets worse.

I could see arguing that it's between 1 & 2, and closer to 1 than 2, but this is just idiotic.

Allison said...

Chemprof, I think the person who put this together for EngageNY thought it was true. Worse, I think they misunderstood another argument being made about how to locate sqrt 2 as this argument.

It is what they wrote: since the endpoints of the unit segment were square root of 4 and square root of 1, then square root of 2 and square root of 3 must fall equally in between.

Not only should the person have known better, so should everyone above and below. How many of the 8th grade teachers in NY followed this lesson?

Anonymous said...

If they were going to use a linear approximation, I'd go with y = 1/2(x-1) + 1.

Allison said...

For those reading and not confident in seeing the error:

An irrational number is a real number that cannot be written as a fraction. (A fraction is defined as follows: the fraction k/n, for k, n whole numbers, n not equal to 0, is the location on the number line when each unit segments is divided into n equal length parts, and then take k, concatenated from 0.)

So the idea that you find the location of sqrt(2) by breaking the unit segments into thirds is absurd: that is how you find a fraction, in this case, the fraction 4/3 (1/3 past 1). But the irrational number can't be a fraction. So the lesson writer was wrong, deeply wrong, when s/he showed sqrt of 2 was 4/3.

The person who put this lesson together was supposed to understand a proof by contradiction for why there was no fraction s for which s^2 =2. Perhaps that alone tripped them up--they read about s as a fraction and then defined a fraction, unaware of the contradiction pending.

That person was also supposed to understand that sqrt(3) could be found as the length of the hypotenuse of a right triangle with lengths sqrt(2) and 1.

I surmise that that was far too difficult, but the picture of circles with radii sqrt(2) and sqrt(3) respectivsly looked enough like thirds for them not to notice.

Barry Garelick said...

The very last line states that Given that reasoning, students shall be able to estimate that SQRT(2) is 1 1/3."

That was the first mention of estimation in the whole discussion. Would it have made a difference if they had instructed the student to "estimate" where sqrt(2) and sqrt(3) fall on the number line, and cast the whole discussion as a way to estimate these values?

The way it's presented makes it sound as if these values ARE the square roots of 2 and 3.

Auntie Ann said...

"Estimate" is in the instructions in the first paragraph:

"Let's look at the number line more generally to see if we can estimate the value of sqrt(2)."

Barry Garelick said...

Yes, I see "estimate" does appear in the beginning. While this makes the intention clear, the phrasing "sqrt(2) would be at" is extremely bad articulation of the intention. It gives the impression that the point 1 1/3 IS the square root of 2. Poorly articulated texts lead to poor mathematical articulation skills.

Michael Weiss said...

This is a nightmare.

And speaking of nightmares, this reminds me: Did I ever post that page from the book that explained the difference between rational and irrational numbers with the example of 18/23, which (according to the authors) is an irrational number because its decimal expansion just goes on and on with no pattern?

No, I'm not kidding. I can post the pages if you want.

Barry Garelick said...

Allison, Do you have a listing of errors in the EngageNY modules? I may be doing an assignment in a school that is going to rely on EngageNY modules for their CC aligned math courses.

Allison said...

No I don't have a list. I wasn't even looking for errors. I was actually looking for how they dealt with irrationals, so I read this module and found this.

But this is my consistent anecdotal experience. I look at a lesson, find an error.

I did once look specifically to see how they taught the area model for fractions, and that was more egregious my wrong. It had unit squares drawn with lengths of 1/5 and widths of 1/8.

I doubt anyone has the time. But a website to catalog the errors would be nice. I may so that....if we can crowd source the effort.

Allison said...

Yes please. Even if you have posted it before.

Allison said...

Auntie Ann,
Yes it does say estimate occasionally. But two huge problems: first, it tells the students to "Place the numbers on the number line". Not approximate them, not place their approximations.

But worse, the so called reasoning for why they should be placed where they do is atrocious. It goes: since sqrt(2) and sqrt(3) fall between sqrt(1) and sqrt(4), (something not justified here), we will just assume we can equally space them (something COMPLETELY unjustified here.)

Why is *approximate* linearity even a reasonable assumption? yeah, it works with these numbers, within 6% and within 4%. But why? No reason is given. 3/2 is just as good an approximation for sqrt(2) by that argument. It could have been true, as far as the kids know, that sqrt(2) is close to 1 and sqrt(3) is close to 2.

The assumptions are teaching a laziness that makes doing real mathematics impossible.

How well does it work between sqrt(16) and sqrt(25)?

ChemProf said...

Right, it is the lack of reasoning -- saying you can estimate that sqrt(2) is more than one and less than sqrt(3) is reasonable. Saying that you can divide the number line into three is not.

And even if you say the estimation is not bad, in an introduction to irrational numbers you should IMMEDIATELY follow up with "Irrational numbers are not really exact fractions; that is why we call them irrational numbers!" and then give the more precise value. Otherwise some kids will have the incorrect value stuck in their heads.

Auntie Ann said...

Allison: Auntie Ann,
Yes it does say estimate occasionally. But two huge problems: first, it tells the students to "Place the numbers on the number line". Not approximate them, not place their approximations.

Sorry, but, no it doesn't. Look I'm no apologist for crappy curricula, but what's missing from the discussion above is the fact that this is from the *teachers' materials, not the students'. At the top of the page it states that this is part of a "Discussion" which should take approximately 15 minutes, and backing up 2 pages shows that this is part of the "Classwork" for the unit.

This problem does not exist in the student materials.

The students will only see this crap through the filter of their teachers. Hopefully, a large number of them will teach this well, despite the poor wording of the EngageNY materials.

SteveH said...

Auntie Ann is right. This is not for students. EngageNY is for teachers, and its use is only "optional and supplemental".

Actually, a linear approximation works quite well for slightly higher numbers. I tested sqrt(32)and found an error of only about a third of a percent. However, this is not about an estimation skill because nobody can do that in their heads. What about sqrt(5692)? Is this just a concept idea - that the sqrt(3) has to exist somewhere between sqrt(1) and the sqrt(4)? It's not presented that way.

That doesn't mean I like the explanation or am justifying EngageNY. I think it's a big waste of money. Does NY require subject certification for teachers in 7th and 8th grades? Don't teachers have textbooks with teacher supplements? What are the exact problems being fixed? Are teachers struggling to find just the right explanation or approach to teaching? Are their textbooks and teacher supplements really that bad? Is this a failure of the publishing industry? I doubt it. I liked the Glencoe math series my son had in 7th and 8th grades. Lack of good textbooks and teacher manuals is not the problem.

Before our state required subject certification in 7th and 8th grades, teachers with higher seniority could bump others out of their grade/class if their class was eliminated. The certification law stopped that, but it still exists in K-6. One year, we had a chain reaction bumping of 4 teachers in K-6. Parents were very upset at the disruption of teacher/grade expertise. One bad year in math can cause long-term problems.

In K-6, is EngageNY just a cover up for the need for more teacher content training in college or the need for math certification? I would rather argue the assumptions of EngageNY rather than their published details.

froggiemama said...

EngageNY is supplemental material for teachers who are using materials that are not up to Common Core standards

froggiemama said...

We're in NY. My two middle school kids use something called "Coach" as Common Core supplements in math, and I think in ELA. They have books that say "Coach" and a website with tons of practice problems. This may be a dumb question, but is that something put out by EngageNY, or something separate?

Allison said...

No, this argument *exactly* is in the student lessons. lesson 2.

You can find all of the student lessons in a zip file.

what I showed here is there, word for word.

I am flummoxed that you think the fact that these are teacher materials makes this not an issue.

This is the teacher's guide. This is the new Common Core textbook, free to everyone, NY has decided to create. This is what a teacher is supposed to do in class on this day.

The teacher was supposed to lead the students to place irrational numbers on the number line using this argument. It isn't "in" the student module because there is no content in the student module. Just problems. You need to learn how to do the problems by following the teacher.

EngageNY was/is the biggest effort nationally to write new curriculum straight for CC standards, rather than just cutting up bindings and re-sorting.

From their web page:
" is developed and maintained by the New York State Education Department (NYSED) to support the implementation of key aspects of the New York State Board of Regents Reform Agenda. This is the official web site for current materials and resources related to the Regents Reform Agenda. The agenda includes the implementation of the New York State P-12 Common Core Learning Standards (CCLS), Teacher and Leader Effectiveness (TLE), and Data-Driven Instruction (DDI). is dedicated to providing educators across New York State with real-time, professional learning tools and resources to support educators in reaching the State’s vision for a college and career ready education for all students."

Yes, officially, it was "supplemental"--because EngageNY was not finished when CC was enacted. But the goal was that EngageNY would become the future curriculum of New York state. No more publishers, no more hard copy books.

This is the exemplar for the nation.

Allison said...

shows modules for student and teacher and lessons. modules are chapter overviews, and workbook pages. lessons are daily lessons. you'll find this in the teacher'a module and the student lessons.

Cassandra Turner said...

Barry Garelick said...
Allison, Do you have a listing of errors in the EngageNY modules?

Heavens to Betsy, Barry. Have you looked at the modules? Who has the time to go through all of that!

Barry Garelick said...

Just asking, Cassy. The school where I was teaching will be using it starting in the fall.

Cassandra Turner said...

I'm not a fan of the order of content in EngageNY. You'd be better off using the new Primary Mathematics Common Core Edition than either EngageNY or Math in Focus, which slapped a sticker on their cover "Common Core Aligned! and added the standards to each lesson. The EngageNY module order is more aligned to Common Core than Singapore. For example, Grade 3 module 1 is Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10. This content is covered in the Grade 2 materials in both Primary Mathematics and Math in Focus.

Then module 2 is Place Value and Problem Solving with Units of Measure and Module 3 is Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10. Part of the idea of Singapore's spiral, mastery-based curriculum is that it allows incubation time for concepts. Time to practice, manipulate and solidify skills. The EngageNY modules give you 25 days (on Measurement and Addition & Subtraction algorithms) between Module 1 & Module 3. That's not enough time to truly master multiplication by 2-5, before piling on facts for 6-9, but it's what we've done in the U.S. for years.

Long division is introduced in EngageNY in Grade 4 Module 3, yet in Primary Mathematics it is introduced in the first half of grade 3. The algorithm is a challenging topic that takes time to master. The lessons in Grade 3 Module 3 are hard for teachers as they introduce this idea: 54 ÷ 6 = (30 ÷ 6) + (24 ÷ 6) - using this language:

Lesson 6 builds on Lesson 2 with a formal re-introduction of the distributive property using the 5 + n pattern to multiply and divide. Students understand that multiples of 6 can be thought of as (5 + 1) × n to make 5 and 1 more groups, or 6 groups of n. Similarly, multiples of 7 can be thought of as (5 + 2) × n to make 5 and 2 more groups, or 7 groups of n. In division students decompose the dividend using a multiple of 5, and then add the quotients of the smaller division facts to find the quotient of the larger unknown division fact.

How many 3rd grade teachers were either taught this way, or had more than one math class in their pre-service coursework?

One of the biggest benefits to the Singaporean materials is the thoughtfully written Scope & Sequence of topics and skills. Concepts lay the foundation and build understanding for future concepts. The curriculum has been vetted for over 30 years. The same cannot be said for either the Common Core or the Scope & Sequence Engage NY has authored.

Catherine Johnson said...


Where are the negative numbers?

(I've only just looked at the beginning...haven't read the thread.)

Catherine Johnson said...

I've just skimmed a bit...and I don't think it's right to say that engageny is supplemental material (though this may be).

My district is using it as the entire curriculum, and from what I've seen of engageny math it is intended to serve as the curriculum.

I know a couple of years ago, when I went to see David Steiner speak, he told the audience that the state intended to write an entire math curriculum, start to finish, that schools could use for free in place of the curricula they would have had to purchase.

I'm pretty sure this is the result. (Not this particular page, but the 'modules' for students.)

Catherine Johnson said...

"In order to assist schools and districts with the implementation of the Common Core, NYSED has provided curricular modules and units in P-12 ELA and math that can be adopted or adapted for local purposes. Full years of curricular materials are currently available on EngageNY, for grades Kindergarten through 9th grade in Mathematics and Kindergarten through 8th grade in English Language Arts (ELA). NYSED is working with our partners to deliver high quality curricular materials for all remaining grades in both Mathematics and ELA. In Mathematics, full years of instruction will be available for all remaining grades this summer. In ELA, full years of instruction will be available in 9th and 10th grade this summer and 11th and 12th grade this fall."

froggiemama said...

Our district is not using EngageNY as its curriculum. Unfortunately, I am blanking on the name of the curriculum, which they just adopted this year. My kids use something called "Coach" though, as a supplement, and I have no idea if that has anything to do with EngageNY or not

Cassandra Turner said...

Catherine - I have heard from some schools in NY where admin says follow EngageNY to the letter and I have other schools in NY where admin says "adapt, not adopt". Have also heard from schools in LA and other states using the EngageNY as their main curriculum.