“You Can Always Add. You Can’t Subtract.” Ctd.

Bryan Anderson and Joel Patterson simply subtracted elements from printed tasks, added them back in later, and watched their classrooms become more interesting places for students.

Anderson took a task from the Shell Centre and delayed all the calculation questions, making room for a lot of informal dialog first.


Patterson took a Discovering Geometry task and removed the part where the textbook specified that the solution space ran from zero to eight.


“It turns out that by shortening the question,” Joel Patterson said, “I opened the question up, and the kids surprised themselves and me!”

I believe EDC calls these “tail-less problems.” I call it being less helpful.

BTW. These are great task designers here. I spent the coldest winter of my life at the Shell Centre because I wanted to learn their craft. Discovering Geometry was written by friend-of-the-blog Michael Serra. This only demonstrates how unforgiving the print medium is to interesting math tasks, like asking Picasso to paint with a toilet plunger. You have to add everything at once.


I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.


  1. Dan – I actually think your claim at the end about print is incorrect. The problem is not print. The problem is unimaginative use of print (and teacher editions). As long as we assume that texts have to march through topics and provide simplistic low-level drill items and highly scaffolded prompts then we give up before we start. In fact, it is high time that smart people were enlisted to write more modern inquiry-based textbooks.

    In my work for Pearson BTW I pushed and pushed on this issue – to no avail. Why? Because the senior designers and editors said that there was no adequate market for such texts – Catch-22.

    Look at the good work done in math by the Georgia Dept of Ed – their problems are pretty good – your 3-acts are cited! – and they map out a potential text in most HS courses.

  2. One of the main problems with print is the idea of maximizing profit. That usually means squeezing in as much information on as few pages as possible.

    While we will forever be challenged to make starting at a textbook more interesting than watching math happen in real time, I think this idea of subtracting could really help publishers make their texts more engaging. A quick improvement might be taking the student text and call it the “teacher edition,” while subtracting the information overload and call that the “student edition.”

    I always found students to be much more engaged when we worked on tasks that were not part of a big, intimidating book the size of a concrete block. Scaffolding them along and gradually releasing information (including resources, such as the next section/chapter/etc.) could go a long way.

    Sounds like increasing costs for the publishers, so I don’t see something like this changing anytime soon!

  3. Graham Fletcher

    September 29, 2014 - 7:14 am -

    The simple idea of stripping away questions has tremendously impacted my students’ engagement in recent years. On the behalf of my students and personal growth, thanks Dan!
    Many times, the question posed within textbooks over-scaffolds the learning and prohibits students from the constructivist struggle. Whether intended or not is another matter. The conversations that happen in absence of questions, have been the gateway to students feeling comfortable about engaging in mathematical dialogue, which is something I really struggled to promote.
    I still have so much to learn in this process but I’ve come to realize that student inquiry far exceeds the questions asked within published material. The questions usually attached to problems are sub-steps to what students really care about…their questions.

  4. @Dan- I wholly agree, I use Shell Center all the time, they have great stuff, I have just switched my focus on how much to present and when. I have found the same as @Graham, that my student engagement has increased by simply stripping down the questions and presenting a simplified initial problem. I know some of the reasons behind how textbooks come to be, but I also agree with @Kyle that maybe they should strip down student content, keep it in the teacher’s edition, and encourage teachers to partake in the journey with their students. Structure would still be provided in Teacher Editions to ensure proper scaffolding for teachers who feel they can’t spend a ton of time planning (especially those elementary teachers who plan for all core subjects).

    I will continue to “Cut Out What I Don’t Need”, so far it has done an outstanding job engaging both my students and myself.

  5. Shell Centre’s website is a treasure trove of prime cuts of math. Their lesson on false-positives in medical tests is perfect–and if you read the Teacher Notes, they tell you that you MUST let kids talk out their ideas for a full day with NO teacher approval or disapproval, so they can sort out their difficulties themselves.

    As for all this “develop the question” and “edit the textbooks” stuff… didn’t IMP, the Interactive Math Project do that?

    I never taught IMP, but the one student in my Calculus class who took 4 years of IMP could solve problems with amazing mental flexibility, never restricting herself to a procedure, always considering new perspectives & connections.

  6. When I look at the tasks and problem sets of newer teachers my initial reaction is often, “Wow. That’s a lot of text.”

    You raise an interesting point about the limitations of print here, and how this could be an opportunity for technology to do some good. I think ideally we want students to build the scaffolding themselves, but delaying it (or adapting it?) could be a step in the process.

    I’m curious to hear more about what Grant has to say on the matter. As you point out, the Shell Centre stuff is great–what more can a textbook do to promote this without filling in the details?

  7. @Dan thank you for sharing this (and many other) threads. Less is more! We need to recapture student imagination – get back to “seeing the ripples” you wrote about recently. We’re aiming for the same with CueThink. That is the mindset behind much of our initial seed content. Here’s one example – “Shari says that all prime numbers are odd. Glenn says all odd numbers are prime. What do YOU say?” We’re seeing student engagement and teacher “ah ha’s” when they recognize problems presented in this manner are accessible to ALL students. We intend to continue building scaffolds to promote thinking, deeper learning and peer-to-peer conversation. #makemathsocial, if you will.

  8. @Grant, Kyle summarizes the problem well, I think:

    One of the main problems with print is the idea of maximizing profit. That usually means squeezing in as much information on as few pages as possible.

    It’s common to find part (c) of a problem providing the answer to part (a), or to find a given graph more suited for the last part of a task than the first. Good tasks are possible in print, but even the most progressive textbooks have to reckon with its cost and weight.

  9. Something else that came up in discussion today was not only the fact that the amount of text in a problem can kill great thinking and discussion before it begins, but it also introduces an extraneous variable that can get in the way of learning math. It is common that students who struggle the most, also struggle with literacy skills.

    When we layer problems with a huge blanket of text, their inability to start may have more to do with the words than the numbers. Using more visuals in class – including 3 act math tasks – has really helped me better identify student learning needs related to the “math” in my classroom.

  10. I agree with @Kyle regarding literacy. My school is slated to gain (net) 11 ELL students for each month for the remainder of the school year as per estimates from the ELL office. Many of these students have few (if any) English skills, so using problems with little or no reading components can actually help better assess the math skills that the students are coming with into the classroom. According to the National Center for Education Statistics, the percent of ELL students is growing in several states in the West, including Texas (http://nces.ed.gov/programs/coe/indicator_cgf.asp). Given Texas’ standing as one of the largest adopters of textbooks, perhaps the ever-increasing ELL population will (eventually) cause a textbook overhaul that includes less text in order to better support all learners.

  11. There are many problems with textbooks beyond the ‘scaffolding’ cited, but let’s not assume they are all bad. The question is ‘how do I cover a topic in a structured fashion in a limited amount of time?’ Student discussion is great, but, darn, there’s the bell. How many lectures have you attended where the lecturer said, “Well, we are out of time. If you have further questions, try to catch me in my office.” My own experience is to teach the higher level abstracts in class, and leave detailed exercises for afterwards.

  12. As much as possible, I try to set up the problem so students are problem solving without any math. Ie, I want them first to think about “what would make sense” before bringing in the numbers.

    As was suggested earlier, make the student editions stripped of all its meat. Most publishers offer disposable books, so why not print student editions where the question(s) and scenarios can be developed in a more natural way? This sounds like a fun experiment.