In Education Week “Teacher”, a recent article reproduces the responses of four high school math teachers to the question: “What are some best practices for teaching high school mathematics?”
I got as far as the first teacher who starts off by telling us:
“Focus on the processes and connections between different processes rather than just finding the answer.”
Those who believe the big lie told at school board meetings and repeated ad nauseum by math reformers (in publications like Education Week and others of similar ilk) that “traditionally taught math failed thousands of students” likely nodded their heads at this with a big “Yup, that’s what’s wrong with how math is taught.” What goes unnoticed and unacknowledged is that a good textbook does everything claimed to be his, but in a more efficient way and one that focuses on individual success, not feel-good student reactions in class.
To be more specific, in elaborating on this idea about processes and connections, he starts off with: “Teach to big ideas rather than 180 disconnected lessons.”
I don’t know about you, but I haven’t heard of or witnessed any high school teacher who teaches 180 disconnected lessons. And though math books from previous eras like the 50’s and 60’s are held in disdain by today’s math reformers and others who claim the high ground in “best practices in math teaching”, the lessons contained therein were well sequenced, well connected and well scaffolded. Such textbooks taught to big ideas; in fact, my old algebra book was called “Algebra: Big Ideas and Basic Skills”.
But then he gets to the heart of what’s really bugging him:
The answer is important, but the processes that lead to answers are far more important to learn. The connections between processes and between different representations of that process are where the core mathematical ideas you want your children to learn reside. This means that while teaching processes, you should use these processes to teach students mathematical principles they can apply to solve problems, not how to solve problem types.
The connections he thinks are not happening have always been a part of traditional math. We learned big ideas from unit material. Given a governing equation, like distance equals rate times time, we solved straightforward problems with worked examples to get us started. Then, in working the homework problems, there were many problem variations. We learned how to rearrange the equation and solve for the unknown, to draw diagrams, label unknowns, and start writing down known equations to find an equal number of equations as unknowns. There was never one type of problem for any new unit of material.
There was no special process or set of ideas that guaranteed transference other than practice. From the unit material homework variations, we saw patterns of solutions that could be applied to other types of problems. The variations took us well past the initial worked example and required us to stretch our prior knowledge and apply it to new situations within the context of the original problem type. What if you had five people working on a job at different rates? What if they started at different times? If they could only work for 5 hours, how much might be left over for you to do? (For more on the topic of non-routine problems within the context of problem types, see this article. )
The teacher next exhorts us to:
Use instructional routines to support all of your students in having access to the mathematics. Instructional routines shift the cognitive load for students as they focus less on what their role is and what they are supposed to next, since these tasks are delegated to long-term memory, and are therefore more able to focus on learning mathematics.
He provides a link to his blog to provide an example of instructional routines. There, we see the typical “growing tile” type of problem presented as an instructional routine. He states that
[t]he high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.
traditional math 
These types of articles are tiresome. Parents, in particular, are fed up with this type of edu jargon only meant to confuse, demean and frustrate their kids and send them straight to Kumon.
You know what works? Teaching math the way it’s always been taught. And if you’re lucky through 12 years of schooling, consider yourself lucky if you had THAT ONE TEACHER who knew what he/she was talking about. It might just change your life.
Kids understand puhlenty. And one thing they’re particularly savvy about, is one they’re getting snowed over. Teach them properly, and leave the edubabble behind.
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I always laugh at that charge of “disconnected lessons”. Because it’s always spouted by someone who advocates current style lessons which almost always stand alone. It’s some lovely trick or activity someone dreamed up that only vaguely fits into whatever current grade-level work might be. Or if it isn’t no biggee, we’ll just adjust it a bit so it fits my Grade 4 …or 5… or 6 … or 7 … class. Whatever. Because it’s cool. It’s a tiling pattern, and if you look at the 1×1 then 2×2 then 3×3 up to 5×5 patterns you might guess the 6×6 and (for Grade 7) come up with a formula for Grade 7. And we’ll pair and share … or maybe get them to do an extended exercise in groups … and everyone tell their own method … then as an extension exercise they’ll make up their own patterns … Tomorrow we’re doing fractions, but with REAL pizza. They’ll love it. Then we’ll introduce alge-tiles …
In other words, the very definition of disconnected. Fuzzy math is the ULTIMATE in disconnected lessons.
Looking back through my high school math text (Dolciani, Berman & Wooten) from the 1970s I see the very epitome of CONNECTED. Each lesson follows from the previous one, building on ideas and methods previously learned, stretching toward those to come.
Yep. Disconnnected math stinks. Because it does not build a coherent body of knowledge. And that is precisely what’s wrong with proposed “new” approaches to teaching the subject. A glance at any primary teaching resources more than 40 years old makes it instantly clear where modern texts (and even worse those who eschew texts altogether) have gone wrong.
By all means let us talk about connected math. But I know this first teacher you cite; he’s no advocate for conventional instruction. And not really an advocate for the connectedness that comes from sequential learning along a well-defined continuum of knowledge-building. Today’s advocates of “connectedness” from idea to idea and technique to technique and fact to fact and between those three … appear to have no idea that this was once the de facto standard of math instruction, and still exists in conventional instructional resources available today.
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“…he’s no advocate for conventional instruction.”
Many of them now claim their ideas are pedagogically agnostic. Bwa Ha Ha! Dan Meyer now talks about the agnostic benefits of engagement and excitement. The implication is that these are not possible with traditionally-taught math. Never mind the fundamentally lower coverage and waste of time for their techniques. It’s the process that’s king and they proclaim the wonders of transference of their methods.
As Barry and we all ask: “Where is the proof?” There is none.
It’s the big lie.
You could make an argument that slower with more hands-on techniques is better for some students, but I don’t even see proof of that. They don’t separate the variables of slower and in-class teaching methods. They never talk about the math requirements of these students as they head for specific college degree programs or votech schools. They claim that process is king and their idea of transference works. If it doesn’t, then the students will be long gone by then and will probably blame themselves. Schools will point to kids like my son and never ask us what we had to do at home.
“[t]he high level goals of Contemplate then Calculate”
Bwa Ha Ha! In my algebra class in the 60’s, we had to write down the rule or identity that allowed us to do each step of our solution. We couldn’t combine steps for each line of change. We students hated it, but it worked. We didn’t do it in groups to make it more fun (with less individual thought) because we had far too many problem variations to do.
Why do we never hear about the size of their individual homework p-sets and how they ensure that mastery is achieved? Why don’t we hear about testing techniques to identify and enforce mastery? Why do they always assume that they have to do what’s best for whomever walks into their classrooms. Why is their viewpoint ONLY teacher-centric and never longitudinal curriculum centric? Why is it always about them as individual teachers? In high schools, AP classes enforce at least a level of absolute expectations in terms of content and skills, but these fuzzy pedagogues never talk about preparing students for that level. That doesn’t stop them from claiming some sort of higher ground of understanding.
They can’t have it both ways. The College Board wants it both ways with CCSS and AP classes. Their introduction (in the Fall of 2018) of Pre-AP in 9th grade is a farce that tries to provide cover for the low slope of CCSS for all in K-8. The Pre-AP people talk about social justice, but where is the social justice of us STEM parents helping out at home or with tutors, thereby increasing the academic gap. We are not helicopter parents. We are forced to do what the schools should be doing for all. In the old days, I got to calculus in high school with absolutely no help from my parents.
They lower expectations with full inclusion and CCSS, and then try to claim the higher ground with their magic hands-on transference techniques in classes that covers less material, thereby requiring Pre-AP classes in ninth grade that finally focus on content and skills. It’s too late. Pre-AP Algebra in 9th grade has no curriculum connection to AP calculus as a senior unless you double up in math one of the years. After 8 years of CCSS math, that is just not possible.
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I have been torturing myself lately reading Jo Boaler tweets, and this article reminds me of a video her team created and tweeted. In the video, two kids talk about how horrible school was because they had to do the “same thing” every day. Her fans loved the video, and said that it explained why “the textbooks stayed on the shelf in [their] classroom.”
For those kids, the “same thing” every day meant working out of the same textbook every day. I would not have considered that to be the same thing, because presumably the textbook would have been covering different topics.
A good textbook has connected lessons. Those disconnected lessons that are cropping up tend to come from teachers downloading all of their lessons off the internet.
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