Step-by-step Math Learning – Our Socratic mission is to “make learning easier”. Our app lets you take a picture of a household question and we’ll teach you how to answer it – magic!
Millions of students use our app and website to learn, and math (especially algebra) is always at the top, and for good reason: everyone needs to learn math, they’ve been doing it for years, the concepts build on each other, and many discover that is hard. to understand.
Step-by-step Math Learning
To provide a great math learning experience, we wanted to help students solve math problems step by step. A good step-by-step solution to an algebra problem (eg “simplify
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”) should be detailed and include good explanations of what is happening on the road. These steps should also be intuitive—not just a step-by-step solution, but something a teacher would show their student.
We looked for existing solutions to integrate into our app, but the ones we found were closed, behind paywalls, or didn’t focus on step-by-step instructions, so we decided to build our own.
Today we’re excited to release mathsteps, the first open source project that teaches math step by step. We’d love for you to join us in making learning math easy and fun.
We’re using math operations to support the math experience in our latest update. Students can take a picture of a math question and we’ll teach you how to answer it.
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Our main goal for this project is to create a library of mathematical solvers focused on pedagogy (how best to teach). The math problems we are currently focusing on are pre-algebraic and algebraic problems that involve simplifying expressions, e.g.
. Our solution is a node module that creates a list of solution steps based on a mathematical chain. It is important that this step-by-step solution is similar to what a teacher would show a student.
As humans, we read and write math as a line of text. If you were to write a mathematical expression, it would probably look like this:
You can also look at this mathematical expression and use your intuition to prioritize where to start simplifying. However, the computer understands the expression best if it is stored in a tree. These trees can be surprisingly complex – even a short expression, for example
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There are many existing open source projects that parse math strings and build such trees. Some of these projects are also complete computer algebra systems (CAS) that can provide answers to math problems, although not with step-by-step explanations.
Steps to do so. SymPy, the well-known open source CAS, stood out as a great choice. However, as we dug deeper into the code, we realized that SymPy’s expression tree structure is optimized for finding answers, but not for learning. Its trees do not store division or subtraction, since these operations can be represented by multiplication, addition, and exponents.
Sympy introduces ambiguity; given a simpy tree, there are several user inputs that could have provided this and are mathematically equivalent, but not necessarily the same for the learner. When designing a CAS, it is useful to reduce the problem to make it easier to achieve the goal. But the goal of CAS, just getting the answer, is different from our goal, a step-by-step solution. And the step-by-step solution requires a different architecture.
Looking further, we found math.js, a powerful and extensive open source math library. Expression trees provide a lot of detail about the structure of a mathematical expression, which is great for building the necessary steps. Working with math.js was a great pleasure. His community is great and Joss was very responsive and supportive in building the math steps.
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It is important to note that when math.js creates an expression tree, it represents all operations as binary (ie, a node can have at most two children). This can be explained by the textbook definition of arithmetic operations. For example, + adds exactly
. This means math.js has to decide which two things to add. When building the tree, it implicitly adds parentheses to make operations binary.
But since + and * are commutative and associative binary operations, they intuitively appear to be non-binary but can take any number of arguments.
It seems to me to multiply 4 words, 3 of which are x and can be combined. This merging step turns out to be important for teaching, so we need to change the tree in math.js so that it is no longer binary.
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After using math.js to create a tree from a math string, we transform the tree using flattening operations. This flattening operation removes the grouping choices made by the math.js parser. By turning a binary tree into a tree that represents mathematics in a more humanly intuitive way, it is much easier to make incremental simplifications.
Finding similar terms requires iterating up and down the tree. These steps are much easier if we first transform the tree like this:
A transformed tree is much closer to how we all intuitively perceive addition. Then we can look at the (+) descendants, see that two of them are x and two of them are numbers, and collect the like words to get
Note that even if we change the tree, we still retain the user’s input and therefore the ability to teach what the student is asking. There is exactly one situation where the tree storage is slightly different from the one entered by the student: subtraction. When you see an expression
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It doesn’t really make sense and we assume the student will never get into it. So when we print the tree on the right, we replace
Once we have an expression tree that is tuned to best support incremental simplification, we reapply the simplification rules to the tree. Here are some examples of the main categories of simplification rules that we repeat at each step, used in the order in which the teacher presents them to their student:
Each of these simplification rules is a tree search that traverses the entire tree of mathematical expressions to see if we can make that simplification anywhere. For example, find a rule
During a tree search, the algorithm checks nodes one by one (shown in red in the gif) to see if they match the rule. For a
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Each tree search in math steps finds one place in the tree to apply a simplification and then returns from the search with that simplification. We keep looking for simplifications, always starting at the very top of the tree, until no more simplifications can be used. We gradually build a list of each simplification used, which then builds step-by-step to the final solution.
To provide the best learning experience possible, we sometimes add additional steps for detailed information. Ideally, there are as many details as possible so that we are less likely to leave the student confused.
In this example, the best solution is more steps and also requires more code. Pedagogical considerations add complexity to the math solver, but these extra details make for a more intuitive learning experience as we explain things more thoroughly.
Mathematics, the steps may cease to be mere simplifications, but this may lead to technical problems. For example, the best way to teach someone how to add fractions would be to first explain how to form the common denominator.
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Complicated! But if the search tree is changed immediately without context, this can happen:
To solve this, we need to remember that when we choose the next step, we add fractions. Our solution in mathematical operations is to group related simplifications (for example, all operations to add two fractions) in the same iteration of the tree search.
By grouping steps, we can also introduce substeps – extra information behind a step that isn’t shown at the top level. Here’s how it looks in our app:
The summarized substeps allow us to provide detailed steps without overwhelming students at first glance. At first, we can only reveal high-level changes and let the student explore the details of operations they don’t understand. The grouped steps are not just a technical simplification; they represent a real, intuitive, pedagogical concept.
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We tried to make the step descriptions as detailed and specific as possible; for example “count the readers” is better than just “count the numbers”. This brings us closer to the language that the teacher can of course use in a multi-step explanation. We also want to allow users of Math Steps to specify what specifically has changed in an expression, such as “adding 2 and 3 gives you 5”. We keep track of which part of the tree in a node has changed.
There are two types of tree searches in math steps: one that simplifies the children of a node first, and one that simplifies the parent first.
, which is harder to track down than simplifying parent before children (pre-order search) and going straight from
Arithmetic sequences, however, make more sense for sequential searches because we have learned to simplify everything that is most profound.
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