May 292015

Wow, it has been a while since I posted anything, and I need to share a ton of things I have done. I predict that I will post a lot in the next several weeks. The school year is winding down, but my learners are ramping up. Grad school is down for the summer (with the exception of an independent study on activity theory) so I have much more time to write.

My learners are working on their final exam / projects, and they are hating me right now. They realize that the stats has a purpose, and that it is far harder than they thought. The handout for my assignment is here if you want to use it, or see what I required.

The only reason I veto projects are because it is too easy, too hard (and it is my opinion for that, although we discuss the reason why so they have an opportunity to revise and make it appropriate) or if the subject matter is just too sensitive / personal and it is in the realm of professionals, not high school learners.

Below is the list of surveys / observational studies / experiments that my learners have decided to undertake this year, broken up by period. It is a rather impressive list!

———Period 2———–

  • Social media use / grades
  • How do adults / teenagers differ in choosing restaurants
  • Does quizlet or flashcards help more in learning vocab (using ancient Sumerian words?!)
  • Does education really affect income (using census data from several zipcodes in the city)
  • Is there an association between a school’s weightlifting records and win/loss at sports?
  • Is sex ed successful?
  • Does involvement in club cheer affect grades (4 different age groups)
  • Do taller people run faster, looking at high school, college and Olympic atheletes?
  • Does appearance have an impact on grades?
  • General questions about tobacco use and quitting
  • Which costs more, male or female beauty products?
  • Quality of life of the parents / learners in school
  • Which area of the city has more trash on the sides of the roads?
  • Does music affect memory (experiment)?
  • How do you use social media?
  • What is your perception of LBGT issues?

———Period 3———–

  • How does sports affect grades?
  • Are oreos really double stuffed? (I never showed my class the story on this, this team came up with it on their own! Love it)
  • Is bullying an issue, how large?
  • Are drivers more likely to stop at a stop sign when they are being watched?
  • An experiment on what type of information changes learners opinions on drinking age.
  • Does work hours affect GPA?
  • Does being exempt from an enrichment class at school affect GPA?
  • Are cheetos packaging regarding number of pieces correct?
  • Does social media use hurt GPA?
  • Is the dress code at school appropriate?

———Period 5———–

  • Does the sugar content of cereals affect the placement of the cereals in the grocery store?
  • Does music affect memory?
  • An experiment on whether or not gender effects whether or not people help with dropped books in the hallways.
  • How does our school compare to other schools in the community service of the learners?
  • Does sexism exist in the high school population?
  • Does the perception of animal rights change from learners to adults in the building?
  • What kinds of social media is most prevalent & how should the school use social media?
  • Who is bullied most over social media, males or females?
  • How does M-M vs. F-F & hair length affect the attitudes towards GLBT youth in stores (a very daring observational study)
  • What drugs are prevalently used in the high school per grade level?
  • Is marijuana use a problem in the high school?

As you can see, there is a huge variety (and some major overlap) between the different classes and projects. Each group is working their way to answering their questions, with the final exam being a presentation of their results.

Always exciting.

Poverty, Education & a local angle

 APStats, Personal  Comments Off on Poverty, Education & a local angle
Jun 222013

In my AP Stats class a learner wanted to see if there was an association between poverty or income and graduation rates in the local Washoe County high schools. This project was done in May and it is an initial project that does have some shortcomings. First, I will explain the methodology the learner used to construct the data, explain the results, and then the shortcomings that can be addressed in a future study.

First, the learner gathered a list of high schools in Washoe County and collected data from the Census Bureau’s “American  Community Survey” to gather information about the corresponding zip code. The learner did ignore the schools that select their learners such as Truckee Meadows Community College High School, the Academy of Arts, Computers & Technology and charter schools. Including those schools would skew the results, especially since they are not associated with a specific zone.


The learner then used JMP to create graphs and calculate statistics on the data, focusing on only 2 associations; median income on grad rate and %below poverty on grade rate.

image image

R is –.836 for the poverty, graph and r = .727 for the median income graph.

Grad Rates = 89.03 – 1.055*(Percent Below Poverty Line)

Grad Rates = 52.69 + 0.0003837*(Median Income in dollars)

Both of these regressions had statistically significant slopes, but the interpretation of the slopes makes the case even more apparent. For the poverty graph, as the percent below poverty increases by 1 percent, the graduation rate drops by 1.06 percent. Clearly, poverty is having an outsized impact on graduation rate.

For the median income graph, for every $1000 change in median household income, the graduation rate increased by only .38%. Clearly, the larger impact on graduation rate is the percent below poverty, not the rising median income.

The problem with this initial analysis is that not every zip code in Washoe County is represented. I know my school zone covers 3 different zip codes, while some of the other schools split a zip code between them. As an initial analysis the results are very interesting, and shows that a more detailed study needs to be performed to get a better handle on impact poverty has on Washoe County’s graduation rates.

Why bring this up with such a long explanation? Why post this project in such detail? Because it needs to be seen, especially in light of the recent national and international news / discussion on the impact poverty has on education.

Michael Pershan did a very good analysis of the PISA data and it shows that in the US, poverty matters, a lot.

Secretary Paul Reville also did a take down of the idea that poverty doesn’t matter. The association between poverty and low educational outcomes is well established, but the deniers and the ideologues won’t allow for reality to impact their thinking. Mr. Reville’s statement that,

“Some want to make the absurd argument that the reason low-income youngsters do poorly is that, mysteriously, all the incompetency in our education systems has coincidentally aggregated around low income students.”

is as close as it gets to a perfect description of the state of the political attacks on education.

The problem as I see it is an AP Statistics learner in Reno, NV can figure out the reality of the situation with publically available data. When will we stop denying the obvious and start acting on the good data?

Functions, GeoGebra and a question for ourselves

 Alg 2, Lesson idea, Technology  Comments Off on Functions, GeoGebra and a question for ourselves
Apr 112013

It is spring break, so what am I doing? I am attending AP workshops and volunteering at my local university. All in all, a great spring break.

So, Let me start with the question first. Why do we make it so hard to learn functions? I mean really. We treat each topic; linears, quadratics, cubics, transendentals, etc, as if they are a new and unique idea. And they definitely are not. I have discussed this before when I was thinking about the Exeter materials, and I have to keep coming back to it for good reason.

What brought it to me today is the fact I am presenting at UNR for the professor of Math Methods to pre-service teachers. I was asked to present on calculator technology, and I will also branch out into GeoGebra, Desmos, and the MathTwitterBlogosphere.

As I was running through what I was going to say and planning my lesson I made a short video on what I wanted to show with GeoGebra. This only scratches (heck I probably doesn’t even leave a mark) on the surface of what GeoGebra can do but it is worth discussing to present it to teachers who will be immersed in Geometer’s Sketchpad in college.

GeoGebra & Functions


And then I turn it into an HTML5 page so anyone can use it.

  And now I have a video as well as a usable piece of content for learners to look and and use on their own at home. I am trying to model good teaching practice that I use at school.

And yet, the question of why do we make linear functions separate from other so it is harder to learn than it should be still comes to my mind. Why? I don’t have a clear answer, and I am not sure anyone else does either. That is sad.

Reflections after an AP workshop

 APStats, Personal, Success maybe  Comments Off on Reflections after an AP workshop
Apr 082013

Today is the first official day of Spring Break (Monday) and so far I have had an eventful weekend. I started by flying to Los Angeles and attending the 5th of the series of AP Workshops they have had. These are the one day workshop where teachers can attend and get some additional tips, hints, and prep for their AP classes.

They definitely inspired me!

I flew down on the districts dime. They offered to send myself and one other teacher to this workshop and paid for flights and hotel for us. This was very generous and even though the workshop was not offering Statistics I felt I could not refuse the offer to go and spend the day doing some calculus. I am glad I did. I spent the day talking to teachers from LA who work in very urban to not so urban schools about how to teach better. I also got a chance to speak with Don who works for the CollegeBoard and has some amazing insights into the AP process. More on that later.

First: This was the FIFTH in a series of these workshops for the LA Unified School District. Think about that. They have had 5 so far, and one more planned. If you teach AP, you are getting some great professional development. But it goes beyond “if you teach AP”. I was in the ‘little to no experience’ calculus room by choice, and in that room there were 7 others who DID NOT TEACH CALCULUS at all. One very eager teacher taught 10 years of geometry and 2 years of algebra 2, but wanted to make sure he was teaching correctly to help develop calculus learners. Another young teacher was in her 3rd year of teaching, and wanted to make sure she was still fresh and current on all levels of math.

Think about that. The depth of teachers LAUSD is developing by opening the sessions up to teachers who want to learn but are not currently teaching calculus is amazing. When Washoe County does their one workshop each year I doubt that it is this widely attended. I went once, but have not gone the last two years because of speech and debate meets being scheduled the same weekend. The one I attended was 100% only current AP teachers. What are we losing by limiting these?

Second: Some tips gleaned about what works for the “urban learner.” I won’t try to define what that means, but suffice it to say that LA has its fair share of them. Some of the tips here are pretty normal. A couple I never thought of. Worth the price of admission right there.

  1. Go over new material BEFORE questions about homework. <insert dopeslap here> DUH! What happens when you get lots of questions about the material from yesterday? You end up rushing the lesson for today. Which leads to more question tomorrow. Which leads to rushed lessons, and then you are always behind and the learners then learn to use that to their advantage and never let you get caught up.  Teach first, THEN go over homework if you need to. Teach first.
  2. Warmups in groups of 3 to 4. The urban learner has been trained NOT to be the smartest or the standout learner. The standout outside of the classroom is the mole that gets whacked (to use my own notes and putting it in different context.) Group work for the warmup allows them to talk about content without being “that kid”.
  3. Meet with parents at the beginning of the year in an AP class. Tell them what the expectations are. Let them know that there needs to be a homework time where NOONE watches TV in the house. Stress the importance of learning. Set the example. Have the parents form a support group of their own. Stress the importance of constant vigilance on homework and learning. Let the parents know there will be very stressful days for their learner. That is okay, support them.
  4. Encourage learners to build study groups by assigning problem sets from old AP exams that are due as group work. Start off by having 2 to 3 week deadlines. They need time to learn how to work as a group. Eventually shorten it to 3 to 5 days with more problems in the problems sets. It is about managing their workload and teaching them how to learn and talk about content.
  5. Praise often, and praise the right stuff. Look at a problem the learner did. Notice they did 85% of the problem right but made a consistent mistake. Point that out. Show them how much they know, and how little they really got wrong. The “100% is correct or nothing” mentality built by the “urban learner” is devastating to learning if they don’t see the progress.

These were the biggies. It is about creating culture of success in the classroom.

Finally, the next thing I learned was the importance of hammering the gatekeepers and using the AP Potential report. The AP Potential report is why the school district pays for every sophomore to take the PSAT. We get a report that itemizes for each learner what AP classes scores like theirs have been successful in AP classes. For 24 different AP classes.

I wonder if that report lists more than 60 learners for AP mathematics. Because that is how many learners are taking AP mathematics at my school out of 2100 learners. This means that at my school, only 2.9% of the learners take AP math of either calculus or statistics. Pretty sad, actually.

I am going to get a copy of our AP potential report so I can look at it and start pushing for AP math at my school. I am also going to share that with all the other AP teachers. I may ruffle some feathers but it is time to push, challenge, and ruffle some of the gatekeepers feathers.

AP Stats: how to remember all those conditions checks

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Feb 232013


Every year in AP stats for the last 4 years I have struggled with getting my learners to understand and use the all important conditions checks in Confidence Intervals and Hypothesis tests. This year I changed up how I taught it, and it has really made an impact. In fact, I can honestly say that all of my learners are using and completing the conditions checks with consistency.

Here is what I did.

First, there are two essential assumptions / conditions:

1. Randomization – is the data collected through some sort of random process (usually given)

2. Independence – what makes the data independent? Stop and think about the context.

Okay, those two generally are not the problem, because they are the same across all types. No mess, no fuss, stop, think and read and think again and you have those two. It is the next two that requires the real thought and memorization. I changed it up this year and put my own spin on them. Instead of naming them as the book, I realized that what they are trying to set is the maximum and minimum sample size that will give you an appropriate sample:

  Z CI or HT T CI or HT
3. Max Sample Size n < 10% of population of interest n < 10% of population of interest
4. Min Sample Size np & nq >= 10 3 parts to “nearly normal”


With this kind of setup, they are thinking about the context more. Why do they have to check the 10% condition? Because that gives you a ceiling on your sample size. Why the success / fail condition? It gives you the floor to the sample size.

My learners are much more focused and they are planning better this year on this topic. I am very pleased by how fluid and easily they are incorporating the CI’s and the HT’s into their language, whereas in prior years it was a struggle to get them to memorize what they were doing.

I think the fact they understand why is making a difference.

#Made4Math–homework research

 Personal, Research  Comments Off on #Made4Math–homework research
Feb 042013

Homework, to give, nor not? How much to give? How much is too much? What purpose does it serve? What is the purpose in assigning it?

I will be honest, I don’t have answers to these questions, but I do have some research, some documents downloaded that may help you shape your own answers to these questions.

by John Dunlosky, et al.

Let’s start off with some super new research (Jan 2013) that identifies 5 very useful study skills that make up homework and 5 that do not help. While this is related to homework it is not about homework. I feel we as teachers need to think about why we are assigning homework, and to make it completely useful we should follow some best practices. I read about this article online, and then followed the links to find the free download from the Association for Psychological Science. Thank you for providing the research for free! This article could shape the homework assignments given to make them more useful to the learner; and one hint, practice testing was found to be very useful!

by Joseph Murphy, et. al.

Okay, this article is a bit dated, but when I was researching homework for a paper, I didn’t find much that wasn’t shrill and emotional. It is relevant, because I think some teachers haven’t changed much of their homework planning from 1987 when this study was done. Again, I really think about why I am assigning this or that as homework, there are different uses for homework, and what use am I using.

by Etta Kralovec, John Buell and David Skinner

This short little 9 page section out of an older edition of a book entitled Taking Sides: Clashing views on educational issues by James Wm. Noll gives a pro and a con to the question. It is short, and gives both sides of the debate. In math, I think we have to give some homework. I am firmly on the Yes side of the question, but it does come down to the purpose of it.

by Nevada’s Northwestern Regional Professional Development Program for Educators

Okay, I have said “purpose” several times in this post, because that is something that resonated with me closely when I went through the training our RPDP did for our math department a couple of years ago. It was based around Cathy Vatterott’s book “Rethinking Homework” that I found to be very useful in shaping my own ideas. The Whole Homework guide above is a 138 page document created by our school district to give blackline masters, thinking guides, and tools to our teachers to help us think about homework in a more constructive manner. All in all, I recommend both the Guide and the Rethinking Homework book.

As always, I hope something here is useful to someone.

Living example of correlation between SES & acheivement

 General  Comments Off on Living example of correlation between SES & acheivement
Dec 282012

This month, Grant Wiggins wrote an article on the correlation between SES and academic achievement.  There is a strong correlation between SAT scores and the families income and there is not a single data point out of place in the table. Here is the full 2012 report.


Look at the scores climb as the family income climbs. Every educator will tell you this occurs, but as Grant points out, we have no real explanation for why. The number of lurking variables and confounding variables in this discussion is tremendous, and we don’t know how or why or what they are. We do know the correlation is strong, however. [I strongly encourage everyone to read Grant’s article. He has so many supporting links that are all very worthwhile and constructive.]

Which is why I am really annoyed at my local newspaper, the Reno Gazette Journal. They are running a series of articles on the “Smartest Seniors”. Guess what they are using to determine this. Yup, you guessed it, SAT scores.

So where do these seniors come from? 2 private (and very expensive) schools and 3 public schools that are all in the highest of income brackets in the county are the home schools of the 5 featured seniors. And don’t get me wrong, they all are very awesome kids who deserve the write up in the newspaper.

I am just frustrated because I don’t know how to push my learners to this level. What am I doing wrong that I don’t have any of my learners on the local lists? I don’t teach at a high SES school, in fact approximately 40% of my school is on free and reduced lunch. But correlation does not mean causation, and I should be able to get some of my learners in the top.

How? I just feel like I have way more questions than answers right now, and it is frustrating the heck out of me.

Can’t, Won’t, Failures and Recovery

 Failure, General  Comments Off on Can’t, Won’t, Failures and Recovery
Dec 162012

I have been thinking and struggling with these ideas for a week now. I read Dave’s post summarizing the study about repeating Algebra 1 and the lack of success in CA, and I really felt I needed to dive deeper in this topic.

So I read many link and downloaded almost every article that was linked in the following pages.

EdSource: Many math students are failing, repeating courses without success

Which leads to the Center for Teaching & Learning’s report: College Bound in Middle & High School.

As well as WestEd’s complete list of Reports (didn’t read all of these for this article) which features the above report. November 2012 is the date on it, so it doesn’t get more recent that that.

There is also this brief from EdSource on Math Readiness in CA.

Dave said something that caught my eye in my Google Reader, and started me down this road of thinking and stressing.

From my limited time in the classroom, too many students seem to have given up on their chance to go to college well before they even get to algebra I, much less algebra II, at least in terms of their effort towards improving their performance or achievement in mathematics.  Yet, if you ask these students, they nearly unanimously say they want to go to college.

It was as if he taught in my department at my school!

Let me backup and tell a story of my department and school.

For the last six years we have had essentially one red cell at my school, SPED Math. Sometimes we have had ELL Math in addition, and one time we had Math as a red cell across the board. We have an extended learning period that meets 4 days per week, and the Math Department has been on the Remediation Roller Coaster teaching proficiency classes 3 of the 4 days for the last 6 years.

No other department at my school teaches during this time, but the math department has stepped up and has voluntarily rode the coaster.

Finally, we said enough this year, and we jumped off that coaster (and have caught some huge flack for it from some in our admin) and focused on freshmen. Now we each have a freshman class of Alg 1 learners who are struggling, and we work with them 45 min per day on math support and skills.

And some of them are choosing to continue to fail, and some are failing because they don’t know how to do middle school math.

Some of them can’t add –11 to 5 to get –6.

The “negative times a negative” is confused with the “negative plus a negative” so some are saying –4 + –5 is + 9.

Yes, these learners are struggling in Alg 1. These learners are the “Can’ts” I mentioned above. They are trying, they are struggling, working, and learning and they will turn into “Cans” by the end of the school year because of this one on one support.

But will they earn credit? I don’t know. They have 2 weeks left in the semester and that time is ticking away quickly for them.

How do we take these learners and get them Algebra 1 Semester 1 credit? According to the report by WestEd it looks bleak. But I have confidence from working with my classes that if we continue to give these Cant’s the constant support they will be able to earn both semester of credits.

Then there is the other group in my support class, the Wont’s. I have 5 learners that just won’t try at all. I am there one on one, I have mentors who are sophomores working with them, and nothing works. They are completely shut down.

These learners have hopes, dreams; they all say they want to go to college and do something with their lives, but they won’t do anything to make those dreams come to pass. How do we remediate this group?

According to WestED, making them retake Algebra 1 will not work. My anecdotal evidence supports the research as well. The Wont’s have made a decision, whether consciously or not, that they will not try. And they will not go to college, let alone graduate from high school without the Alg 1 credit.

According to the WestED report, the reason why is they were pushed into mathematics at a higher level then they were probably ready for. Since they were working far higher then their cognitive skills allowed, they just gave up.

How do we get a learner who has given up to re-engage? This is a struggle I face daily in my support class and as a department chair. I need to come up with a plan to help them, but no research I have seen gives me any confidence in how to approach this.

All I know is I can’t just say “retake the class.” That is a path towards failure on top of failure. It is also what our district considers “Accepted Practice.” (see number 16).

If anyone has any ideas, research, articles, or any other thoughts, please send them along. I need them. Badly.

Aug 132012

This is a post that can not be written without a better understanding of how Exeter structures its school year. First off, they are on a 3 term schedule; 3 ten week (approx) terms per year. This is a FABULOUS schedule, and I can speak from experience. It is a similar schedule to what Knox College (where I attended college) uses .

The benefits of the 3 term schedule are many, but some of the best are the sense of urgency it places on the learner and the ability to be more flexible in scheduling. The sense of urgency is terrific. You have 10 weeks in your class, and you are counting down right from week 1; “9 weeks left, 8 weeks left … oh crap, finals are next week.” You are never allowed to kick back and think, “nah, I have time, I don’t need to worry about that project / assignment / test yet.”

The second benefit, flexibility, is the one thing that truly is what I want to discuss here. I asked our instructor how many new learners Exeter gets each year. I was thinking that they probably didn’t have many transfers in after the first year, but that was not true. The Freshman class starts around 250 each year. Then, each year, approximately 50 learners are added to the class until the Senior class graduates with around 400 learners. [Of course, these numbers are not set in stone, they were given as an example only.]

That means that Exeter has a huge problem each year. The Freshman class is coming in with a wide range of ability levels, and then every year after that they get more learners who should be at one level, but in reality may be way way below that level or even above that level. This then is their dilemma, how to place the learners in the correct course for their level of ability.

First off, they have a strong system of placement exams for the learners. Wow. Imagine that, a learner placed into the class they should be instead of the class that all Freshman or all Sophomores should be in. Of course, given Exeter’s commitment to sharing what works, they have put some old placement exams online for everyone to see and use. Sweet! [Turns out, they weren’t old placement exams, but an internal site that was made public on their end.]

Secondly, they teach all three of the Math1 and Math2 classes each term. If Math1 is broken into 3 pieces, which I will call 1a, 1b, and 1c, then during term 1 of the Freshman year, all three classes, Math1a, Math1b and Math1c AND Math2a, Math2b and Math2c are taught all 3 terms. Wow.

This means a learner might be placed into Math1c as a freshman during Term 1, not every learner is required to start with the first Math1a class. This is HUGE! Now you don’t have to worry about the bored learner sitting in class causing problems waiting for the material to catch up to them.

Okay, can this be used in the public schools? After all isn’t that the point of this? I am not sure.

This kind of system works for a private school (and Knox is a private school as well).  because the system is not set up to provide an education for farm laborers. You do know that is why we have long summers off, right? Because the kids in school were needed to work in the fields during the summer. That hasn’t been true for 30 years, but we still have that long summer break. Sue provided a link to the myth of the summer break below in the comments. It is worth reading and following up on.

It certainly could not be adopted because of the extremely underfunded nature of public schools. We are lucky to have 1 math teacher per 30 learners, while a private school can charge as much as they need to in order to cover the extra teachers to teach the same course every semester and hold the ratio down to 1:12.

So the short answer is no, we can’t do it like Exeter can. That is a cop out though. How can we do something similar?

How can we personalize the instruction so learners who are advanced can forge ahead?

How can we implement and use placement exams to tailor the instruction?

Kahn Academy is getting some traction on this, even though the videos contain mathematical errors and mistakes because it is a way to personalize instruction. Can we, as innovative, driven math teachers figure out a way to do it better?

I am willing to bet yes. Once we are willing to throw away the books and embrace the standards, then we can simply say, Johnny knows standard 1, 2, and 3, so an A on those, but he needs to work on standards 4, 5, and 6.

Next we need a way to assess well, and finally we need a way to locate those standards out all of the standards that are part of the problem sets.

Once we have these elements, then we can at least try to individualize.

Exeter’s problem solving method

 Lesson idea, Success YES!  Comments Off on Exeter’s problem solving method
Aug 112012

In this post I want to show Exeter’s problem solving strategy. This is important, because it is SO different from how a problem like this is typically approached.

First off, the problem I am going to model is M1:21:11 [Math 1, page 21, problem 11]

11. Alex was hired to unpack and clean 576 very small items of glassware, at five cents per piece successfully unpacked. For every item broken during the process, however, Alex had to pay $1.98. At the end of the job, Alex received $22.71. How many items did Alex break?

In a typical Algebra 1 class we would try to get the learner to see the equation is:

.05(576-x) + 1.98x = 22.71

In fact we try to get the learner to jump directly to the equation from the problem by deconstructing the sentences, and then solve the equation. x = 3, by the way.

Now, let’s see how Exeter expects and demands that ALL of the modeling problems are handled.

First off, we will be making a table. The headings in this table are mandatory and can not be short cut. The learners must label the table thoroughly so that it makes sense. Remember, this is the same problem as above. I am going to paste in my table all filled out, and then explain the essential elements.

Guess: # of broken bottles

# of unbroken

$ Paid for unbroken

$ subtracted for broken

Amount paid




























.05(576-B)-1.98B =




Okay, there we have. A decent example of what a modeling, problem solving solution would look like. At the beginning stages of Math 1, they would not demand the last row, the equation row. But quickly they would ask the learners to start generalizing their solution.

The guesses column are not set in stone. The guesses are going to be the learners guesses. They are going to guess whatever they want. I started with 0, because maybe he didn’t break any. Then I saw that was too high to my goal, so I figured Alex broke a few. Then I was too low, so I picked one in the middle.

Now, let’s examine what the columns mean. It is clear from the headings that each column has a very specific purpose and is clearly labeled. What are we guessing? We are guessing the number of glasses he broke. If he breaks 5, then he didn’t break 571. How do we get that, we subtract. Each column must have in it HOW they get the number, not just what the number is. And so on.

Notice that by the time the learner reaches the answer, they have worked several times the process, they know the multiplications, the subtractions, and they have the solution worked out. Where does the variable go? It goes into the spot where numbers change. What do we call the variable? Don’t care, use a letter that makes sense to the problem.

How do they start this process? The first problem that is a modeling / problem solving problem is M1:9:4. It looks like this:


Notice that they start by giving the table and even filling out the first row. The problem I worked above, didn’t have that level of detail. The learner had to provide it. That is the point.


Yea, I shouted that. We have this impression that Exeter is so fabulous, that they don’t have to ladder or work with learners. We think that the learners just will magically go *poof* and be able to do all these things that we struggle with.

Guess, what, they struggle with similar things there as we do in our schools. It might be easier because of smaller class sizes, but the root problems are the same.

Okay, off my soap box.

The Algebra 1 activities have some problem solving activities, and they even are sneaky by giving a blank table with fewer columns than the learners need! The learner is pushed to make the table for themselves.

Think about this type of problem solving for special ed, or EL Learners. They have the numbers set up, they can see where the Letter for the Unknown goes, because it is the only number that changes when they are doing the problems. Wouldn’t this method help them out so much?!

Think about your average learner who struggles with parsing the language of the problem. If they work 10 or 20 of these as starters, as homework, as in class activities, do you really think they are going to stress about a word problem?

Nope, they are going to say, “Mr. Waddell, these are easy, can we move on to something harder?” And you know they will.

Think about the really advanced learner. They are going to resent the table after a short time, but they will go to the generalization much faster because of it.

Can you think of any downside to this method of problem solving? I can’t. I have done Algebraic Thinking’s “SOLVE” method, and other methods. None of them are as straightforward and easy to put together as this method. We could spend THOUSANDS of dollars on professional development on problem solving, and none of that money would come close to the success of just creating a table, labeling, and working it out step by step.

Guess and Check. That is what Exeter calls it. I call it just downright successful for every level of learner.