To Recap: I made it a goal of mine this year to make better assessments. Enough has happened that I’ve started to process it all, which is one of the things I use this blog for: taking a snapshot of where my thoughts are right now, and documenting the things I want to keep and the things I want to change. This endeavor led to these monsters here, here, and here.

Also – thanks to everyone who’s given such positive feedback about my post on the Three Types of Standards. In looking at my curriculum and deciding what to assess, I realized I started to see three distinct types of skills/concepts/problems that I want my students to be responsible for. The post linked at the beginning of this paragraph talks primarily about Procedural and Conceptual skills and how I identify and assess them.

What I left out of the last post is what I mean by Synthesis Skills. These are the ones I’m still trying to get a handle on, both with identification and with assessing.

Some Background: My Assessments and Grading: Each page of my test contains a single skill that’s been isolated from other aspects of my curriculum. I do this so students know exactly what they’re being assessed on and how they can show mastery. Some skills inevitably build on previous ones, so some are less ‘isolated’ than others (ie: Geometric Probability necessarily relies on calculating area correctly and I can’t avoid that). However, the goal is to probe if my students truly understand a particular concept or skill, which means I want questions that target this skill or concept as much as possible. My goal isn’t to assess a wider variety of skills with a single question – it’s to assess a single skill at a deeper level. This helps me target students weak points and emphasize where their strengths are. It makes it easier to target remediation. It also makes it easier for me to teach concepts, since I’m more aware of when a problem has a potential pitfall that is ultimately unrelated to the skill or concept that I’m trying to teach (ie: If I were going to teach ‘solving 2-step equations’, my first example wouldn’t have a fraction in it).

Outside Influence: This post by Dan Meyer has an excellent explanation of what I try to accomplish with my assessments in terms of collecting data and targeting remediation. I’m going to reference this post in the next paragraph, so if you don’t read it now, here’s the gist: Dan was assessing the skill “I can find the surface area of a cone” and he debated including a problem that would have required his students to first apply the Pythagorean Theorem before finding surface area. Dan argues, and I agree, that this does not serve the goal of targeted assessment and remediation – if a student gets this question wrong, is it because the don’t know how to find the surface area? Or because they don’t know the Pythagorean Theorem? Also, if I were teaching how to find the surface area of a cone: I wouldn’t start with a problem that required them to use the Pythagorean Theorem. I would start with a problem that simply had them calculate the surface area so they could get familiar with the concept and procedure.

The Problem: Even if I don’t start a unit with this problem, and even if I don’t include it on an assessment, I still want my students to solve a problem like that. That requires an extra step or two or three. That requires them to apply a few separate skills in order to solve a larger problem. My fear is that if I ignore problems like this, students begin to see mathematics as isolated chunks of knowledge and skills that aren’t necessarily inter-related. That the types of problems they can solve are limited in depth and complexity. That students begin to see entire units within my curriculum as disconnected and unrelated – or, worse, that they see my whole curriculum as disconnected and unrelated.

This isn’t what I want – I want to feel like my units and curriculum are always building towards something. Something that, every once in a while, unifies what we’ve been doing for the past few weeks/months/year. That require those extra steps and combining skills together. Sometimes this place is within the units themselves, and sometimes they’re entire units of their own – a culminating concept or project whose whole purpose is to synthesis several other skills to solve a specific problem.

I call these Synthesis Skills – skills that remind my students that my curriculum and my unit are interconnected and help me fight this fear that students will see everything as isolated. Sometimes they’re a specific type of problem that requires my students to tie together everything from a unit – in this case, it’s probably better to call these Synthesis Problems. The problems are unit-specific and serve to cement everything we’ve talked about in a particular unit. But also looking at my curriculum, I can see examples of entire Synthesis Units – a collection of concepts and skills where the success of this unit is entirely dependent on how well my students have understood everything that came before it. These are the units where, if my students never mastered some previous skill, it becomes deadly as we work through the concepts and skills that make up this unit.

# Within a Unit: Synthesis Problems

When I go problem-hunting online or in textbooks, every once in a while I’ll see a problem where I think “Yah… if my students can do this, then I’m pretty sure they can do anything I could thrown at them. I want this to be the final problem of the unit”. They’re problems where each student may have solved it in a slightly different way. They’re problems with a small amount of information given, but whose execution requires several steps. They’re problems where when I think about an assessment, I think “I only need to give them one of these – if they can do this one, they’re in good shape”. They’re problems that I imagine would be free-response questions on an AP Exam. The statement of the skill is in its purest form – no caveats like “… using properties of triangles” or “… using the formulas for slope, distance, and midpoint”. The solution path is relatively open-ended compared to how questions on my Procedural and Conceptual pages. Here are some of mine:

Note: The problems on the second-page are sourced from various resources from the Discovering Geometry textbook published by Key Curriculum, including Patty Paper Geoemtry.

Skill: I Can Find Missing Angles

Synthesized Skills: Basic angle relationships (vertical angles, linear pair), Angle Relationships with triangles (sum to 180, isosceles triangles), Angle Relationships with Parallel Lines.

Analysis: I’m a huge fan of problems like this – give them a mess of lines and a few pieces of information, then tell them to figure the rest out. They’ll need to use parallel line relationships, polygon relationships, triangle relationships, etc. They have to choose which one to use and when. They have to know when they can’t use a certain relationship – otherwise it’ll mess up the rest of their calculations. These problems can keep appearing throughout the year as we learn more and more about angle relationships (like the problems on the second page).

Skill: I Can Describe a Transformation in the Coordinate Plane

Synthesized Skills: Describing Translations, Describing Reflections, Describing Rotations

Analysis: This was a gooooood assessment from my 1st Semester Honors class. There are many possible solutions. They require the student to use all the transformations we’ve learned so far, but there’s little direction on how. If they can do this, then I’m pretty confident they can problem-solve any transformation-type problem I could throw at them.

Assessing These Skills: When I put a problem like this on an assessment, there are a lot of different places where a student can go wrong. They could make a lot of procedural mistakes, in which case I know we need to work on our foundation first. Or, if it’s a small procedural mistake in the beginning, then the comparison to a free-response question on an AP Exam is especially valid: I follow their mistake through the problem and see if they truly understand what’s happening at a conceptual level. Or maybe they use vocabulary incorrectly or apply procedures incorrectly, indicating a deeper conceptual misunderstanding. Or maybe they get stuck in the problem-solving aspect of the problem – the open-ended nature of the problem causes students to get stuck and lose confidence, leading to an incorrect answer. Grading and remediation becomes a lot less straightforward with these problems.

All of this is contrary to the stated goals of my assessments way back at the top of this post: to target specific skills and collect data about where my students are at. But they still serve my overall philosophy of assessment: the things I assess are how I tell my students “Hey! This stuff is important and you are responsible for it!”, and these are all problems that I feel are important and that they should be responsible for. In fact, these are the skills that have the highest importance because they collect together everything – procedural, conceptual, and problem-solving. They are the ‘big idea’ of the unit in it’s purest form.

I’ve struggled this year trying to unify this tension between targeted SBG assessments that feel like isolated ‘checklists’ of skills, and my desire to teach and assess these bigger, broader problems that I still want my student to be held accountable for. And I think I’ve realized that these ideas are fundamentally opposed – that I can’t do quick SBG type assessments if I want to also assess how well my students can solve these complex problems which take time and whose remediation is complex. Which means I either need to change how I do my pen-and-paper assessments (which is what I’ve been doing), or I need to find another way to assess these synthesis skills (also something I’ve been trying via projects).

My goal for next year is to have something like this in mind for all of my units – some sort of complex problem that we’re building towards – and then find a way to incorporate this into a project or an assessment (or both).

# Synthesis Units Within the Curriculum

When I look at my curriculum, there are a few places where I find myself thinking “If I want to do this topic justice – to have my students really learn and appreciate it at a deep level and not just regurgitate for the test – then I either need to spend 2-3 weeks on it, or not mention it at all”. Since I teach Geometry, the place where I see this tension the most is with Centers of Triangles. I love the way Kate Nowak teaches them – hands on, incorporating coordinate geometry, incorporating the Pythagorean Theorem, incorporating area, and culminating with a scavenger hunt that has them construct these points – compass and straightedge and all. This is the right way to include Centers of Triangles in the curriculum – acknowledging that it will span several areas that we’ve already talked about and building up to a singular activity/task/assessment which has them apply all of their skills.

I haven’t found the time to fit this level of depth into my curriculum. If I were to teach Centers of Triangles, I would have to sandwich it between several other units that are already laid out. We wouldn’t be able to go to the depth that we need into order to really understandappreciate, and apply what we were learning. I would be too pressed for time to teach anything more than memorize the centers, their properties, and regurgitate for a test. Which isn’t really teaching. It’s ‘covering’ the material, which I try to avoid whenever possible.

In my mind, ‘Centers of Triangles’ is a Synthesis Unit – the success of this unit is mostly dependent on how well students have understood the material that builds up to it (constructions, coordinate geometry, triangle vocabulary) and it connects all of these things together. This unit is a place to show off a truly interesting problem that we, as developing mathematicians, now have enough tools to solve. And not just solve – but appreciate and understand why our solution works and how it involves all of this machinery we’ve learned throughout the year. This unit revolves around a singular problem/question – which point is equidistant from the other three? – and the ‘new knowledge’ serves the goal of finding an answer to this problem. You could fit this problem at the end of an existing unit – one on constructions or triangles maybe – but it feels awkward because the problem requires non-trivial connections between several different units and concepts. This means the real benefits of this problem revolve around communication and problem-solving, not as an excuse to teach brand-new material that we’ll use later. In fact, anything ‘new’ we discover in this unit probably won’t be used again anytime soon – which is fine, since it’s supposed to be the celebration of ideas and strategies across the curriculum all culminating with a single problem.

Here are some Synthesis Units I’ve tried over the last few years:

Skill: I Can Determine the type of Quadrilateral formed by 4 points in the Coordinate Plane

Synthesized Skills: Calculating Slope, Distance, and Midpoint; Understanding that congruent segments have the same length; Understanding that parallel lines have the same slope; Understanding that parallel lines have the same slope; Understanding that two segments with the same midpoint bisect each other; Using deductive reasoning to apply properties of quadrilaterals; Explaining your answer

Analysis: For the last few years, I’ve used this as the culminating problem in my unit on quadrilaterals. After having it go a bit poorly this year, I’m starting to think that it’s something that belongs independent of my quadrilateral unit. When we discuss these problems, the real emphasis is on deductive reasoning and how to explain your answer to a third party. These are the real reasons to talk about these problems – the deductive process and explaining your reasoning – not the focus on a single correct answer. The journey is the most meaningful part, not the final result. If I can’t spend the time on this discussion, then these problems become oversimplified and an exercise in procedures without connections.

Reflective Moment: The ‘failure’ of the unit above was the catalyst for this post about assessment and curriculum and ‘tricking’ my students into practicing old skills under new contexts. If I’ve done a poor job at teaching quadrilaterals or coordinate geometry, then this unit becomes an exercise in re-learning these skills and I can’t have the high-level deductive, logical discussions that I want to have. If I’ve done a great job teaching quadrilaterals and coordinate geometry, then the conversations happen at a higher cognitive level and have this argumentative and questioning quality. I think this is why we sometimes avoid these Synthesis concepts and units (especially Proofs and Constructions in Geometry – I’m guilty of this too), because it makes it apparent when our classes haven’t been mastering the concepts that are supposed to build up to these units, which means the units become a shadow of their potential for investigation and synthesis.

A Synthesis Unit in Calculus: Sam Shah’s Optimization Unit.

Assessing These Units: These are the problems where I’ve never been sure how to assess them on a pen-and-paper test. The problem itself is complex, and the important pieces seem like something bigger than just a pen-and-paper test.

And I think the answer I’ve decided on is: I can’t. I need a project. I need a presentation. I need writing and reflection. I need something more than just pen and paper for a student to show that their mastery of these problems. And even more than that – these should be units where we’re doing work that we should be proud of, so I should give my students opportunities to be proud of their work. So, this is something I want to try next year – having little ‘Synthesis Units’ every quarter or so that highlight how our curriculum is interconnected and then give students a meaningful project to work on that emphasizes this.

Outside Influence: A while ago, the Common Core Tools blog released one possible way to sequence the Common Core standards into high school units. Within each high-school course, there are places for ‘Modeling Unis’ and ‘Projects’. These Synthesis Units are the types of problems/projects/ideas that I think of when I imagine what would fit into those spaces (although they may not be what the Common Core has in mind).

Closing Thoughts: I’ve been writing this post over the last few days, and now that I’m at the end, I went back through and wondered “Why was I writing this in the first place?”. So here I am trying to answer that question:

I think I’ve realized that my idea of a traditional SBG assessment (several skills per assessment, given very frequently, targeted remediation and data collection) does not play well with these Synthesis Skills that I still want to assess. And I’ve decided that’s okay, which is why I’m changing my assessments so that I can address these Synthesis skills but still hold students accountable for their Procedural and Conceptual knowledge. I also think its important to realize when a certain problem or concept or strategy is too big for a pen and paper test or even too big for an individual unit – that it’s worthwhile to dedicate a decent period of time to it and some sort of project or presentation in order to assess it properly. And so now, when I find units like this, instead of looking for the right ‘pen and paper’ assessment, I need to look for the right project/presentation assessment.

Final Related Blog Post: One day, I hope to have some sort of all-encompassing-project on the same scale as David Cox’s Farming Project.